University Press of Florida
Theory of functions of a real variable
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Title: Theory of functions of a real variable
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Creator: Sternberg, Shlomo
Publication Date: 5/10/2005
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Subjects / Keywords: Math, Metric spaces, Stone Weierstrass theorem, Machado’s theorem, Hahn Banach theorem, Uniform Boundedness, Hilbert space, Scalar products, Cauchy Schwartz inequality, Pythagorean theorem, Fourier series, Heisenberg uncertainty principle, Sobolev Spaces, Lebesgue outer measure, Lebesgue inner measure, Riesz representation theorem, Wiener measure, Brownian …
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Abstract: This text is for a beginning graduate course in real variables and functional analysis. It assumes that the student has seen the basics of real variable theory and point set topology. Contents: 1) The topology of metric spaces. 2) Hilbert Spaces and Compact operators. 3) The Fourier Transform. 4) Measure theory. 5) The Lebesgue integral. 6) The Daniell integral. 7) Wiener measure, Brownian motion and white noise. 8) Haar measure. 9) Banach algebras and the spectral theorem. 10) The spectral theorem. 11) Stone’s theorem. 12) More about the spectral theorem. 13) Scattering theory.
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General Note: Shlomo Sternberg, Mathematics and Statistics, Harvard University
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General Note: MAS 105 - LINEAR ALGEBRA I (CALC. II PREREQU), MAA 306 - REAL VARIABLE THEORY I: GRADUATE, MAA 307 - REAL VARIABLE THEORY II: GRADUATE
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Theoryoffunctionsofarealvariable.ShlomoSternbergMay10,2005

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2Introduction.Ihavetaughtthebeginninggraduatecourseinrealvariablesandfunctionalanalysisthreetimesinthelastveyears,andthisbookistheresult.Thecourseassumesthatthestudenthasseenthebasicsofrealvariabletheoryandpointsettopology.TheelementsofthetopologyofmetricsspacesarepresentedinthenatureofarapidreviewinChapterI.Thecourseitselfconsistsoftwoparts:1measuretheoryandintegration,and2Hilbertspacetheory,especiallythespectraltheoremanditsapplications.InChapterIIIdothebasicsofHilbertspacetheory,i.e.whatIcandowithoutmeasuretheoryortheLebesgueintegral.TheherohereandperhapsforthersthalfofthecourseistheRieszrepresentationtheorem.Includedisthespectraltheoremforcompactself-adjointoperatorsandapplicationsofthistheoremtoellipticpartialdierentialequations.ThepdematerialfollowscloselythetreatmentbyBersandSchecterinPartialDierentialEquationsbyBers,JohnandSchecterAMSChapterIIIisarapidpresentationofthebasicsabouttheFouriertransform.ChapterIVisconcernedwithmeasuretheory.TherstpartfollowsCaratheodory'sclassicalpresentation.ThesecondpartdealingwithHausdormeasureanddi-mension,Hutchinson'stheoremandfractalsistakeninlargepartfromthebookbyEdgar,Measuretheory,Topology,andFractalGeometrySpringer.Thisbookcontainsmanymoredetailsandbeautifulexamplesandpictures.ChapterVisastandardtreatmentoftheLebesgueintegral.ChaptersVI,andVIIIdealwithabstractmeasuretheoryandintegration.ThesechaptersbasicallyfollowthetreatmentbyLoomisinhisAbstractHar-monicAnalysis.ChapterVIIdevelopsthetheoryofWienermeasureandBrownianmotionfollowingaclassicalpaperbyEdNelsonpublishedintheJournalofMathemat-icalPhysicsin1964.ThenwestudytheideaofageneralizedrandomprocessasintroducedbyGelfandandVilenkin,butfromapointofviewtaughttousbyDanStroock.Therestofthebookisdevotedtothespectraltheorem.Wepresentthreeproofsofthistheorem.Therst,whichiscurrentlythemostpopular,derivesthetheoremfromtheGelfandrepresentationtheoremforBanachalgebras.ThisispresentedinChapterIXforboundedoperators.InthischapterweagainfollowLoomisratherclosely.InChapterXweextendtheprooftounboundedoperators,followingLoomisandReedandSimonMethodsofModernMathematicalPhysics.ThenwegiveLorch'sproofofthespectraltheoremfromhisbookSpectralTheory.Thishastheavorofcomplexanalysis.ThethirdproofduetoDavies,presentedattheendofChapterXIIreplacescomplexanalysisbyalmostcomplexanalysis.Theremainingchapterscanbeconsideredasgivingmorespecializedin-formationaboutthespectraltheoremanditsapplications.ChapterXIisde-votedtooneparametersemi-groups,andespeciallytoStone'stheoremabouttheinnitesimalgeneratorofoneparametergroupsofunitarytransformations.ChapterXIIdiscussessometheoremswhichareofimportanceinapplicationsof

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3thespectraltheoremtoquantummechanicsandquantumchemistry.ChapterXIIIisabriefintroductiontotheLax-Phillipstheoryofscattering.

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Contents1Thetopologyofmetricspaces131.1Metricspaces.............................131.2Completenessandcompletion.....................161.3NormedvectorspacesandBanachspaces..............171.4Compactness..............................181.5TotalBoundedness...........................181.6Separability...............................191.7SecondCountability..........................201.8ConclusionoftheproofofTheorem1.5.1..............201.9Dini'slemma..............................211.10TheLebesgueoutermeasureofanintervalisitslength......211.11Zorn'slemmaandtheaxiomofchoice................231.12TheBairecategorytheorem.....................241.13Tychono'stheorem..........................241.14Urysohn'slemma............................251.15TheStone-Weierstrasstheorem....................271.16Machado'stheorem..........................301.17TheHahn-Banachtheorem......................321.18TheUniformBoundednessPrinciple.................352HilbertSpacesandCompactoperators.372.1Hilbertspace..............................372.1.1Scalarproducts........................372.1.2TheCauchy-Schwartzinequality...............382.1.3Thetriangleinequality....................392.1.4Hilbertandpre-Hilbertspaces................402.1.5ThePythagoreantheorem..................412.1.6ThetheoremofApollonius..................422.1.7ThetheoremofJordanandvonNeumann.........422.1.8Orthogonalprojection.....................452.1.9TheRieszrepresentationtheorem..............472.1.10WhatisL2T?........................482.1.11Projectionontoadirectsum.................492.1.12Projectionontoanitedimensionalsubspace........495

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6CONTENTS2.1.13Bessel'sinequality.......................492.1.14Parseval'sequation......................502.1.15Orthonormalbases......................502.2Self-adjointtransformations......................512.2.1Non-negativeself-adjointtransformations..........522.3Compactself-adjointtransformations................542.4Fourier'sFourierseries........................572.4.1Proofbyintegrationbyparts.................572.4.2Relationtotheoperatord dx..................602.4.3Garding'sinequality,specialcase...............622.5TheHeisenberguncertaintyprinciple................642.6TheSobolevSpaces..........................672.7Garding'sinequality..........................722.8ConsequencesofGarding'sinequality................762.9Extensionofthebasiclemmastomanifolds.............792.10Example:Hodgetheory........................802.11Theresolvent..............................833TheFourierTransform.853.1Conventions,especiallyabout2...................853.2Convolutiongoestomultiplication..................863.3Scaling.................................863.4FouriertransformofaGaussianisaGaussian...........863.5Themultiplicationformula......................883.6Theinversionformula.........................883.7Plancherel'stheorem.........................883.8ThePoissonsummationformula...................893.9TheShannonsamplingtheorem...................903.10TheHeisenbergUncertaintyPrinciple................913.11Tempereddistributions........................923.11.1ExamplesofFouriertransformsofelementsofS0......934Measuretheory.954.1Lebesgueoutermeasure........................954.2Lebesgueinnermeasure........................984.3Lebesgue'sdenitionofmeasurability................984.4Caratheodory'sdenitionofmeasurability..............1024.5Countableadditivity..........................1044.6-elds,measures,andoutermeasures................1084.7Constructingoutermeasures,MethodI...............1094.7.1Apathologicalexample....................1104.7.2Metricoutermeasures.....................1114.8Constructingoutermeasures,MethodII...............1134.8.1Anexample..........................1144.9Hausdormeasure...........................1164.10Hausdordimension..........................117

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CONTENTS74.11Pushforward..............................1194.12TheHausdordimensionoffractals................1194.12.1Similaritydimension......................1194.12.2Thestringmodel.......................1224.13TheHausdormetricandHutchinson'stheorem..........1244.14Aneexamples............................1264.14.1TheclassicalCantorset....................1264.14.2TheSierpinskiGasket....................1284.14.3Moran'stheorem.......................1295TheLebesgueintegral.1335.1Realvaluedmeasurablefunctions..................1345.2Theintegralofanon-negativefunction...............1345.3Fatou'slemma.............................1385.4Themonotoneconvergencetheorem.................1405.5ThespaceL1X;R..........................1405.6Thedominatedconvergencetheorem.................1435.7Riemannintegrability.........................1445.8TheBeppo-Levitheorem......................1455.9L1iscomplete.............................1465.10DensesubsetsofL1R;R......................1475.11TheRiemann-LebesgueLemma...................1485.11.1TheCantor-Lebesguetheorem................1505.12Fubini'stheorem............................1515.12.1Product-elds........................1515.12.2-systemsand-systems...................1525.12.3Themonotoneclasstheorem.................1535.12.4Fubinifornitemeasuresandboundedfunctions.....1545.12.5Extensionstounboundedfunctionsandto-nitemeasures.1566TheDaniellintegral.1576.1TheDaniellIntegral.........................1576.2Monotoneclasstheorems.......................1606.3Measure.................................1616.4Holder,Minkowski,LpandLq....................1636.5kk1istheessentialsupnorm....................1666.6TheRadon-NikodymTheorem....................1676.7ThedualspaceofLp.........................1706.7.1Thevariationsofaboundedfunctional...........1716.7.2DualityofLpandLqwhenS<1............1726.7.3ThecasewhereS=1..................1736.8IntegrationonlocallycompactHausdorspaces..........1756.8.1Rieszrepresentationtheorems................1756.8.2Fubini'stheorem........................1766.9TheRieszrepresentationtheoremredux...............1776.9.1Statementofthetheorem...................177

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8CONTENTS6.9.2Propositionsintopology...................1786.9.3ProofoftheuniquenessoftherestrictedtoBX....1806.10Existence................................1806.10.1Denition............................1806.10.2MeasurabilityoftheBorelsets................1826.10.3Compactsetshavenitemeasure..............1836.10.4Interiorregularity.......................1836.10.5Conclusionoftheproof....................1847Wienermeasure,Brownianmotionandwhitenoise.1877.1Wienermeasure............................1877.1.1TheBigPathSpace......................1877.1.2Theheatequation.......................1897.1.3Pathsarecontinuouswithprobabilityone.........1907.1.4EmbeddinginS0........................1947.2Stochasticprocessesandgeneralizedstochasticprocesses.....1957.3Gaussianmeasures...........................1967.3.1Generalitiesaboutexpectationandvariance........1967.3.2Gaussianmeasuresandtheirvariances...........1987.3.3ThevarianceofaGaussianwithdensity...........1997.3.4ThevarianceofBrownianmotion..............2007.4ThederivativeofBrownianmotioniswhitenoise..........2028Haarmeasure.2058.1Examples................................2068.1.1Rn...............................2068.1.2Discretegroups........................2068.1.3Liegroups...........................2068.2Topologicalfacts............................2118.3ConstructionoftheHaarintegral..................2128.4Uniqueness...............................2168.5G<1ifandonlyifGiscompact................2188.6Thegroupalgebra...........................2188.7Theinvolution.............................2208.7.1Themodularfunction.....................2208.7.2Denitionoftheinvolution..................2228.7.3Relationtoconvolution....................2238.7.4Banachalgebraswithinvolutions..............2238.8Thealgebraofnitemeasures....................2238.8.1Algebrasandcoalgebras....................2248.9Invariantandrelativelyinvariantmeasuresonhomogeneousspaces.225

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CONTENTS99Banachalgebrasandthespectraltheorem.2319.1Maximalideals.............................2329.1.1Existence............................2329.1.2Themaximalspectrumofaring...............2329.1.3Maximalidealsinacommutativealgebra..........2339.1.4Maximalidealsintheringofcontinuousfunctions.....2349.2Normedalgebras............................2359.3TheGelfandrepresentation......................2369.3.1InvertibleelementsinaBanachalgebraformanopenset.2389.3.2TheGelfandrepresentationforcommutativeBanachal-gebras..............................2419.3.3Thespectralradius......................2419.3.4ThegeneralizedWienertheorem...............2429.4Self-adjointalgebras..........................2449.4.1Animportantgeneralization.................2479.4.2Animportantapplication...................2489.5TheSpectralTheoremforBoundedNormalOperators,Func-tionalCalculusForm.........................2499.5.1Statementofthetheorem...................2509.5.2SpecBT=SpecAT:....................2519.5.3Adirectproofofthespectraltheorem............25310Thespectraltheorem.25510.1Resolutionsoftheidentity......................25610.2Thespectraltheoremforboundednormaloperators........26110.3Stone'sformula.............................26110.4Unboundedoperators.........................26210.5Operatorsandtheirdomains.....................26310.6Theadjoint...............................26410.7Self-adjointoperators.........................26510.8Theresolvent..............................26610.9Themultiplicationoperatorformofthespectraltheorem.....26810.9.1Cyclicvectors.........................26910.9.2Thegeneralcase........................27110.9.3Thespectraltheoremforunboundedself-adjointopera-tors,multiplicationoperatorform..............27110.9.4Thefunctionalcalculus....................27310.9.5Resolutionsoftheidentity..................27410.10TheRiesz-Dunfordcalculus......................27610.11Lorch'sproofofthespectraltheorem................27910.11.1Positiveoperators.......................27910.11.2Thepointspectrum......................28110.11.3Partitionintopuretypes...................28210.11.4Completionoftheproof....................28310.12Characterizingoperatorswithpurelycontinuousspectrum....28710.13Appendix.Theclosedgraphtheorem................288

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10CONTENTS11Stone'stheorem29111.1vonNeumann'sCayleytransform..................29211.1.1Anelementaryexample....................29711.2Equiboundedsemi-groupsonaFrechetspace............29911.2.1Theinnitesimalgenerator..................29911.3Thedierentialequation.......................30111.3.1Theresolvent..........................30311.3.2Examples............................30411.4Thepowerseriesexpansionoftheexponential...........30911.5TheHilleYosidatheorem.......................31011.6Contractionsemigroups........................31311.6.1Dissipationandcontraction..................31411.6.2Aspecialcase:exptB)]TJ/F11 9.963 Tf 9.963 0 Td[(IwithkBk1.........31611.7Convergenceofsemigroups......................31711.8TheTrotterproductformula.....................32011.8.1Lie'sformula..........................32011.8.2Cherno'stheorem......................32111.8.3Theproductformula.....................32211.8.4Commutators.........................32311.8.5TheKato-Rellichtheorem..................32311.8.6Feynmanpathintegrals....................32411.9TheFeynman-Kacformula......................32611.10ThefreeHamiltonianandtheYukawapotential..........32811.10.1TheYukawapotentialandtheresolvent...........32911.10.2ThetimeevolutionofthefreeHamiltonian.........33112Moreaboutthespectraltheorem33312.1Boundstatesandscatteringstates..................33312.1.1Schwartzschild'stheorem...................33312.1.2Themeanergodictheorem.................33512.1.3Generalconsiderations....................33612.1.4Usingthemeanergodictheorem...............33912.1.5TheAmrein-Georgescutheorem...............34012.1.6Katopotentials........................34112.1.7ApplyingtheKato-Rellichmethod..............34312.1.8Usingtheinequality.7..................34412.1.9Ruelle'stheorem........................34512.2Non-negativeoperatorsandquadraticforms............34512.2.1Fractionalpowersofanon-negativeself-adjointoperator.34512.2.2Quadraticforms........................34612.2.3Lowersemi-continuousfunctions...............34712.2.4Themaintheoremaboutquadraticforms..........34812.2.5Extensionsandcores.....................35012.2.6TheFriedrichsextension...................35012.3Dirichletboundaryconditions....................35112.3.1TheSobolevspacesW1;2andW1;20.........352

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CONTENTS1112.3.2Generalizingthedomainandthecoecients........35412.3.3ASobolevversionofRademacher'stheorem........35512.4Rayleigh-Ritzanditsapplications..................35712.4.1Thediscretespectrumandtheessentialspectrum.....35712.4.2Characterizingthediscretespectrum............35712.4.3Characterizingtheessentialspectrum...........35812.4.4Operatorswithemptyessentialspectrum..........35812.4.5Acharacterizationofcompactoperators..........36012.4.6Thevariationalmethod....................36012.4.7Variationsonthevariationalformula............36212.4.8Thesecularequation.....................36412.5TheDirichletproblemforboundeddomains............36512.6Valence.................................36612.6.1Twodimensionalexamples..................36712.6.2Huckeltheoryofhydrocarbons................36812.7Davies'sproofofthespectraltheorem...............36812.7.1Symbols............................36812.7.2Slowlydecreasingfunctions..................36912.7.3Stokes'formulaintheplane.................37012.7.4Almostholomorphicextensions................37112.7.5TheHeer-Sjostrandformula................37112.7.6Aformulafortheresolvent..................37312.7.7Thefunctionalcalculus....................37412.7.8Resolventinvariantsubspaces................37612.7.9Cyclicsubspaces........................37712.7.10Thespectralrepresentation..................38013Scatteringtheory.38313.1Examples................................38313.1.1Translation-truncation....................38313.1.2Incomingrepresentations...................38413.1.3Scatteringresidue.......................38613.2Breit-Wigner..............................38713.3Therepresentationtheoremforstronglycontractivesemi-groups.38813.4TheSinairepresentationtheorem..................39013.5TheStone-vonNeumanntheorem.................392

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12CONTENTS

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Chapter1Thetopologyofmetricspaces1.1MetricspacesAmetricforasetXisafunctiondfromXXtothenon-negativerealnumberswhichwedentebyR0,d:XX!R0suchthatforallx;y;z2X1.dx;y=dy;x2.dx;zdx;y+dy;z3.dx;x=04.Ifdx;y=0thenx=y.Theinequalityin2isknownasthetriangleinequalitysinceifXistheplaneanddtheusualnotionofdistance,itsaysthatthelengthofanedgeofatriangleisatmostthesumofthelengthsofthetwootheredges.Intheplane,theinequalityisstrictunlessthethreepointslieonaline.Condition4isinmanywaysinessential,anditisoftenconvenienttodropit,especiallyforthepurposesofsomeproofs.Forexample,wemightwanttoconsiderthedecimalexpansions:49999:::and:50000:::asdierent,butashavingzerodistancefromoneanother.Orwemightwanttoidentify"thesetwodecimalexpansionsasrepresentingthesamepoint.Afunctiondwhichsatisesonlyconditions1-3iscalledapseudo-metric.AmetricspaceisapairX;dwhereXisasetanddisametriconX.Almostalways,whendisunderstood,weengageintheabuseoflanguageandspeakofthemetricspaceX".13

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14CHAPTER1.THETOPOLOGYOFMETRICSPACESSimilarlyforthenotionofapseudo-metricspace.Inlikefashion,wecalldx;ythedistancebetweenxandy,thefunctiondbeingunderstood.Ifrisapositivenumberandx2X,theopenballofradiusraboutxisdenedtobethesetofpointsatdistancelessthanrfromxandisdenotedbyBrx.Insymbols,Brx:=fyjdx;yr+sbyvirtueofthetriangleinequality.Supposethatthisintersectionisnotemptyandthatw2BrxBsz:Ify2Xissuchthatdy;w0thereexistsa=x;>0suchthatfBxBy:Noticethatinthisdenitionisallowedtodependbothonxandon.Themapiscalleduniformlycontinuousifwecanchoosetheindependentlyofx.Anevenstrongerconditiononamapfromonepseudo-metricspacetoan-otheristheLipschitzcondition.Amapf:X!Yfromapseudo-metricspaceX;dXtoapseudo-metricspaceY;dYiscalledaLipschitzmapwithLipschitzconstantCifdYfx1;fx2CdXx1;x28x1;x22X:ClearlyaLipschitzmapisuniformlycontinuous.Forexample,supposethatAisaxedsubsetofapseudo-metricspaceX.DenethefunctiondA;fromXtoRbydA;x:=inffdx;w;w2Ag:

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1.1.METRICSPACES15Thetriangleinequalitysaysthatdx;wdx;y+dy;wforallw,inparticularforw2A,andhencetakinglowerboundsweconcludethatdA;xdx;y+dA;y:ordA;x)]TJ/F11 9.963 Tf 9.963 0 Td[(dA;ydx;y:ReversingtherolesofxandythengivesjdA;x)]TJ/F11 9.963 Tf 9.963 0 Td[(dA;yjdx;y:Usingthestandardmetricontherealnumberswherethedistancebetweenaandbisja)]TJ/F11 9.963 Tf 10.061 0 Td[(bjthislastinequalitysaysthatdA;isaLipschitzmapfromXtoRwithC=1.Aclosedsetisdenedtobeasetwhosecomplementisopen.Sincetheinverseimageofthecomplementofasetunderamapfisthecomplementoftheinverseimage,weconcludethattheinverseimageofaclosedsetunderacontinuousmapisagainclosed.Forexample,thesetconsistingofasinglepointinRisclosed.SincethemapdA;iscontinuous,weconcludethatthesetfxjdA;x=0gconsistingofallpointsatzerodistancefromAisaclosedset.ItclearlyisaclosedsetwhichcontainsA.SupposethatSissomeclosedsetcontainingA,andy62S.Thenthereissomer>0suchthatBryiscontainedinthecomplementofS,whichimpliesthatdy;wrforallw2S.ThusfxjdA;x=0gS.InshortfxjdA;x=0gisaclosedsetcontainingAwhichiscontainedinallclosedsetscontainingA.Thisisthedenitionoftheclosureofaset,whichisdenotedby A.Wehaveprovedthat A=fxjdA;x=0g:Inparticular,theclosureoftheonepointsetfxgconsistsofallpointsusuchthatdu;x=0.Nowtherelationdx;y=0isanequivalencerelation,callitR.Transitiv-itybeingaconsequenceofthetriangleinequality.ThisthendividesthespaceXintoequivalenceclasses,whereeachequivalenceclassisoftheform fxg,theclosureofaonepointset.Ifu2 fxgandv2 fygthendu;vdu;x+dx;y+dy;v=dx;y:sincex2 fugandy2 fvgweobtainthereverseinequality,andsodu;v=dx;y:

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16CHAPTER1.THETOPOLOGYOFMETRICSPACESInotherwords,wemaydenethedistancefunctiononthequotientspaceX=R,i.e.onthespaceofequivalenceclassesbyd fxg; fyg:=du;v;u2 fxg;v2 fygandthisdoesnotdependonthechoiceofuandv.Axioms1-3forametricspacecontinuetohold,butnowd fxg; fyg=0 fxg= fyg:Inotherwords,X=Risametricspace.Clearlytheprojectionmapx7! fxgisanisometryofXontoX=R.Anisometryisamapwhichpreservesdistances.Inparticularitiscontinuous.Itisalsoopen.Inshort,wehaveprovidedacanonicalwayofpassingviaanisometryfromapseudo-metricspacetoametricspacebyidentifyingpointswhichareatzerodistancefromoneanother.AsubsetAofapseudo-metricspaceXiscalleddenseifitsclosureisthewholespace.Fromtheaboveconstruction,theimageA=RofAinthequotientspaceX=Risagaindense.Wewillusethisfactinthenextsectioninthefollowingform:Iff:Y!XisanisometryofYsuchthatfYisadensesetofX,thenfdescendstoamapFofYontoadensesetinthemetricspaceX=R.1.2Completenessandcompletion.TheusualnotionofconvergenceandCauchysequencegooverunchangedtometricspacesorpseudo-metricspacesY.Asequencefyngissaidtoconvergetothepointyifforevery>0thereexistsanN=Nsuchthatdyn;y<8n>N:AsequencefyngissaidtobeCauchyifforany>0thereexistsanN=Nsuchthatdyn;ym<8m;n>N:ThetriangleinequalityimpliesthateveryconvergentsequenceisCauchy.ButnoteveryCauchysequenceisconvergent.Forexample,wecanhaveasequenceofrationalnumberswhichconvergetoanirrationalnumber,asintheapproxi-mationtothesquarerootof2.SoifwelookatthesetofrationalnumbersasametricspaceQinitsownright,noteveryCauchysequenceofrationalnumbersconvergesinQ.Wemustcomplete"therationalnumberstoobtainR,thesetofrealnumbers.Wewanttodiscussthisphenomenoningeneral.Sowesaythatapseudo-metricspaceiscompleteifeveryCauchysequenceconverges.Thekeyresultofthissectionisthatwecanalwayscomplete"ametricorpseudo-metricspace.Moreprecisely,weclaimthat

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1.3.NORMEDVECTORSPACESANDBANACHSPACES.17Anymetricorpseudo-metricspacecanbemappedbyaonetooneisometryontoadensesubsetofacompletemetricorpseudo-metricspace.Bytheitalicizedstatementoftheprecedingsection,itisenoughtoprovethisforapseudo-metricspacesX.LetXseqdenotethesetofCauchysequencesinX,anddenethedistancebetweentheCauchysequencesfxngandfyngtobedfxng;fyng:=limn!1dxn;yn:Itiseasytocheckthatddenesapseudo-metriconXseq.Letf:X!Xseqbethemapsendingxtothesequenceallofwhoseelementsarex;fx=x;x;x;x;:Itisclearthatfisonetooneandisanisometry.Theimageisdensesincebydenitionlimdfxn;fxng=0:NowsincefXisdenseinXseq,itsucestoshowthatanyCauchysequenceofpointsoftheformfxnconvergestoalimit.Butsuchasequenceconvergestotheelementfxng.QED1.3NormedvectorspacesandBanachspaces.Ofspecialinterestarevectorspaceswhichhaveametricwhichiscompatiblewiththevectorspacepropertiesandwhichiscomplete:LetVbeavectorspaceovertherealorcomplexnumbers.Anormisarealvaluedfunctionv7!kvkonVwhichsatises1.kvk0and>0ifv6=0,2.kcvk=jcjkvkforanyrealorcomplexnumberc,and3.kv+wkkvk+kwk8v;w2V.Thendv;w:=kv)]TJ/F11 9.963 Tf 8.252 0 Td[(wkisametriconV,whichsatisesdv+u;w+u=dv;wforallv;w;u2V.Theballofradiusrabouttheoriginisthenthesetofallvsuchthatkvk
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18CHAPTER1.THETOPOLOGYOFMETRICSPACES1.4Compactness.AtopologicalspaceXissaidtobecompactifithasoneandhencetheotherofthefollowingequivalentproperties:Everyopencoverhasanitesubcover.Inmoredetail:iffUgisacollectionofopensetswithX[Uthentherearenitelymany1;:::;nsuchthatXU1[[Un:IfFisafamilyofclosedsetssuchthatF2F=;thenaniteintersectionoftheF'sareempty:F1Fn=;:1.5TotalBoundedness.AmetricspaceXissaidtobetotallyboundedifforevery>0therearenitelymanyopenballsofradiuswhichcoverX.Theorem1.5.1Thefollowingassertionsareequivalentforametricspace:1.Xiscompact.2.EverysequenceinXhasaconvergentsubsequence.3.Xistotallyboundedandcomplete.Proofthat1.2.LetfyigbeasequenceofpointsinX.Werstshowthatthereisapointxwiththepropertyforevery>0,theopenballofradiuscenteredatxcontainsthepointsyiforinnitelymanyi.Supposenot.Thenforanyz2Xthereisan>0suchthattheballBzcontainsonlynitelymanyyi:Sincez2Bz,thesetofsuchballscoversX.Bycompactness,nitelymanyoftheseballscoverX,andhencethereareonlynitelymanyi,acontradiction.Nowchoosei1sothatyi1isintheballofradius1 2centeredatx.Thenchoosei2>i1sothatyi2isintheballofradius1 4centeredatxandkeepgoing.Wehaveconstructedasubsequencesothatthepointsyikconvergetox.Thuswehaveprovedthat1.implies2.

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1.6.SEPARABILITY.19Proofthat2.3.IffxjgisaCauchysequenceinX,ithasaconvergentsubsequencebyhypothesis,andthelimitofthissubsequenceisbythetriangleinequalitythelimitoftheoriginalsequence.HenceXiscomplete.Wemustshowthatitistotallybounded.Given>0,pickapointy12XandletBy1beopenballofradiusabouty1.IfBy1=Xthereisnothingfurthertoprove.Ifnot,pickapointy22X)]TJ/F11 9.963 Tf 9.038 0 Td[(By1andletBy2betheballofradiusabouty2.IfBy1[By2=Xthereisnothingtoprove.Ifnot,pickapointy32X)]TJ/F8 9.963 Tf 10.262 0 Td[(By1[By2etc.Thisprocedurecannotcontinueindenitely,forthenwewillhaveconstructedasequenceofpointswhichareallatamutualdistancefromoneanother,andthissequencehasnoCauchysubsequence.Proofthat3.2.LetfxjgbeasequenceofpointsinXwhichwerelabelasfx1;jg.LetB1;1 2;:::;Bn1;1 2beanitenumberofballsofradius1 2whichcoverX.Ourhypothesis3.assertsthatsuchanitecoverexists.Innitelymanyofthejmustbesuchthatthex1;jalllieinoneoftheseballs.Relabelthissubsequenceasfx2;jg.CoverXbynitelymanyballsofradius1 3.Theremustbeinnitelymanyjsuchthatallthex2;jlieinoneoftheballs.Relabelthissubsequenceasfx3;jg.Continue.Attheithstagewehaveasubsequencefxi;jgofouroriginalsequenceinfactoftheprecedingsubsequenceintheconstructionallofwhosepointslieinaballofradius1=i.Nowconsiderthediagonal"subsequencex1;1;x2;2;x3;3;::::Allthepointsfromxi;ionlieinaxedballofradius1=isothisisaCauchysequence.SinceXisassumedtobecomplete,thissubsequenceofouroriginalsequenceisconvergent.Wehaveshownthat2.and3.areequivalent.Thehardpartoftheproofconsistsinshowingthatthesetwoconditionsimply1.Forthisitisusefultointroducesometerminology:1.6Separability.AmetricspaceXiscalledseparableifithasacountablesubsetfxjgofpointswhicharedense.ForexampleRisseparablebecausetherationalsarecountableanddense.Similarly,Rnisseparablebecausethepointsallofwhosecoordinatesarerationalformacountabledensesubset.Proposition1.6.1AnysubsetYofaseparablemetricspaceXisseparableintheinducedmetric.Proof.LetfxjgbeacountabledensesequenceinX.Considerthesetofpairsj;nsuchthatB1=2nxjY6=;:Foreachsuchj;nletyj;nbeanypointinthisnon-emptyintersection.Weclaimthatthecountablesetofpointsyj;naredenseinY.Indeed,letybeanypointofY.Letnbeanypositiveinteger.Wecanndapointxjsuchthatdxj;y<1=2nsincethexjaredenseinX.Butthendy;yj;n<1=nbythetriangleinequality.QED

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20CHAPTER1.THETOPOLOGYOFMETRICSPACESProposition1.6.2AnytotallyboundedmetricspaceXisseparable.Proof.Foreachnletfx1;n;:::;xin;ngbethecentersofballsofradius1=nniteinnumberwhichcoverX.Putallofthesetogetherintoonesequencewhichisclearlydense.QEDAbasefortheopensetsinatopologyonaspaceXisacollectionBofopensetsuchthateveryopensetofXistheunionofsetsofBProposition1.6.3AfamilyBisabaseforthetopologyonXifandonlyifforeveryx2XandeveryopensetUcontainingxthereisaV2Bsuchthatx2VandVU.Proof.IfBisabase,thenUisaunionofmembersofBoneofwhichmustthereforecontainx.Conversely,letUbeanopensubsetofX.Foreachx2UthereisaVxUbelongingtoB.Theunionoftheseoverallx2UiscontainedinUandcontainsallthepointsofU,henceequalsU.SoBisabase.QED1.7SecondCountability.AtopologicalspaceXissaidtobesecondcountableortosatisfythesecondaxiomofcountabilityifithasabaseBwhichisniteorcountable.Proposition1.7.1AmetricspaceXissecondcountableifandonlyifitisseparable.Proof.SupposeXisseparablewithacountabledensesetfxig.Theopenballsofradius1=naboutthexiformacountablebase:Indeed,ifUisanopensetandx2UthentakensucientlylargesothatB2=nxU.Choosejsothatdxj;x<1=n.ThenV:=B1=nxjsatisesx2VUsobyProposition1.6.3thesetofballsB1=nxjformabaseandtheyconstituteacountableset.Conversely,letBbeacountablebase,andchooseapointxj2UjforeachUj2B.IfxisanypointofX,theballofradius>0aboutxincludessomeUjandhencecontainsxj.Sothexjformacountabledenseset.QEDProposition1.7.2Lindelof'stheorem.SupposethatthetopologicalspaceXissecondcountable.Theneveryopencoverhasacountablesubcover.LetUbeacover,notnecessarilycountable,andletBbeacountablebase.LetCBconsistofthoseopensetsVbelongingtoBwhicharesuchthatVUwhereU2U.ByProposition1.6.3theseformacountablecover.ForeachV2CchooseaUV2UsuchthatVUV.ThenthefUVgV2CformacountablesubsetofUwhichisacover.QED1.8ConclusionoftheproofofTheorem1.5.1.Supposethatcondition2.and3.ofthetheoremholdforthemetricspaceX.ByProposition1.6.2,Xisseparable,andhencebyProposition1.7.1,Xis

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1.9.DINI'SLEMMA.21secondcountable.HencebyProposition1.7.2,everycoverUhasacountablesubcover.SowemustprovethatifU1;U2;U3;:::isasequenceofopensetswhichcoverX,thenX=U1[U2[[Umforsomeniteintegerm.Supposenot.Foreachmchoosexm2Xwithxm62U1[[Um.Bycondition2.ofTheorem1.5.1,wemaychooseasubsequenceofthefxjgwhichconvergetosomepointx.SinceU1[[Umisopen,itscomplementisclosed,andsincexj62U1[[Umforj>mweconcludethatx62U1[[Umforanym.ThissaysthatthefUjgdonotcoverX,acontradiction.QEDPuttingthepiecestogether,weseethataclosedboundedsubsetofRmiscompact.ThisisthefamousHeine-Boreltheorem.SoTheorem1.5.1canbeconsideredasafarreachinggeneralizationoftheHeine-Boreltheorem.1.9Dini'slemma.LetXbeametricspaceandletLdenotethespaceofrealvaluedcontinuousfunctionsofcompactsupport.Sof2Lmeansthatfiscontinuous,andtheclosureofthesetofallxforwhichjfxj>0iscompact.ThusLisarealvectorspace,andf2Ljfj2L.Thusiff2Landg2Lthenf+g2Landalsomaxf;g=1 2f+g+jf)]TJ/F11 9.963 Tf 9.276 0 Td[(gj2Landminf;g=1 2f+g)-153(jf)]TJ/F11 9.963 Tf 9.276 0 Td[(gj2L.ForasequenceofelementsinLormoregenerallyinanyspaceofrealvaluedfunctionswewritefn#0tomeanthatthesequenceoffunctionsismonotonedecreasing,andateachxwehavefnx!0.Theorem1.9.1Dini'slemma.Iffn2Landfn#0thenkfnk1!0.Inotherwords,monotonedecreasingconvergenceto0impliesuniformconvergencetozeroforelementsofL.Proof.Given>0,letCn=fxjfnxg.ThentheCnarecompact,CnCn+1andTkCk=;.Henceaniteintersectionisalreadyempty,whichmeansthatCn=;forsomen.Thismeansthatkfnk1forsomen,andhence,sincethesequenceismonotonedecreasing,forallsubsequentn.QED1.10TheLebesgueoutermeasureofanintervalisitslength.ForanysubsetARwedeneitsLebesgueoutermeasurebymA:=infX`In:InareintervalswithA[In:.1Herethelength`IofanyintervalI=[a;b]isb)]TJ/F11 9.963 Tf 10.11 0 Td[(awiththesamedenitionforhalfopenintervalsa;b]or[a;b,oropenintervals.Ofcourseifa=andbisniteor+1,orifaisniteandb=+1thelengthisinnite.Sotheinmumin.1istakenoverallcoversofAbyintervals.Bytheusual=2ntrick,i.e.byreplacingeachIj=[aj;bj]byaj)]TJ/F11 9.963 Tf 10.249 0 Td[(=2j+1;bj+=2j+1wemay

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22CHAPTER1.THETOPOLOGYOFMETRICSPACESassumethattheinmumistakenoveropenintervals.Equallywell,wecouldusehalfopenintervalsoftheform[a;b,forexample..ItisclearthatifABthenmAmBsinceanycoverofBbyintervalsisacoverofA.Also,ifZisanysetofmeasurezero,thenmA[Z=mA.Inparticular,mZ=0ifZhasmeasurezero.Also,ifA=[a;b]isaninterval,thenwecancoveritbyitself,som[a;b]b)]TJ/F11 9.963 Tf 9.962 0 Td[(a;andhencethesameistruefora;b];[a;b,ora;b.Iftheintervalisinnite,itclearlycannotbecoveredbyasetofintervalswhosetotallengthisnite,sinceifwelinedthemupwithendpointstouchingtheycouldnotcoveraninniteinterval.WestillmustprovethatmI=`I.2ifIisaniteinterval.WemayassumethatI=[c;d]isaclosedintervalbywhatwehavealreadysaid,andthattheminimizationin1.1iswithrespecttoacoverbyopenintervals.Sowhatwemustshowisthatif[c;d][iai;bithend)]TJ/F11 9.963 Tf 9.963 0 Td[(cXibi)]TJ/F11 9.963 Tf 9.963 0 Td[(ai:WerstapplyHeine-Boreltoreplacethecountablecoverbyanitecover.Thisonlydecreasestherighthandsideofprecedinginequality.Soletnbethenumberofelementsinthecover.Wewanttoprovethatif[c;d]n[i=1ai;bithend)]TJ/F11 9.963 Tf 9.963 0 Td[(cnXi=1bi)]TJ/F11 9.963 Tf 9.962 0 Td[(ai:Weshalldothisbyinductiononn.Ifn=1thena1dsoclearlyb1)]TJ/F11 9.963 Tf 9.963 0 Td[(a1>d)]TJ/F11 9.963 Tf 9.963 0 Td[(c.Supposethatn2andweknowtheresultforallcoversofallintervals[c;d]withatmostn)]TJ/F8 9.963 Tf 10.746 0 Td[(1intervalsinthecover.Ifsomeintervalai;biisdisjointfrom[c;d]wemayeliminateitfromthecover,andthenweareinthecaseofn)]TJ/F8 9.963 Tf 10.111 0 Td[(1intervals.Soeveryai;bihasnon-emptyintersectionwith[c;d].Amongthetheintervalsai;bitherewillbeoneforwhichaitakesontheminimumpossiblevalue.Byrelabeling,wemayassumethatthisisa1;b1.Sinceciscovered,wemusthavea1dthena1;b1covers[c;d]andthereisnothingfurthertodo.Soassumeb1d.Wemusthaveb1>csincea1;b1[c;d]6=;.Sinceb12[c;d],atleastoneoftheintervalsai;bi;i>1containsthepointb1.Byrelabeling,wemayassumethatitisa2;b2.Butnowwehaveacoverof[c;d]byn)]TJ/F8 9.963 Tf 9.963 0 Td[(1intervals:[c;d]a1;b2[n[i=3ai;bi:

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1.11.ZORN'SLEMMAANDTHEAXIOMOFCHOICE.23Sobyinductiond)]TJ/F11 9.963 Tf 9.963 0 Td[(cb2)]TJ/F11 9.963 Tf 9.963 0 Td[(a1+nXi=3bi)]TJ/F11 9.963 Tf 9.963 0 Td[(ai:Butb2)]TJ/F11 9.963 Tf 9.963 0 Td[(a1b2)]TJ/F11 9.963 Tf 9.963 0 Td[(a2+b1)]TJ/F11 9.963 Tf 9.963 0 Td[(a1sincea2
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24CHAPTER1.THETOPOLOGYOFMETRICSPACESallofDwecouldaddasinglepointx0notinthedomainoffandy02Fx0contradictingthemaximalityoff.QED1.12TheBairecategorytheorem.Theorem1.12.1Inacompletemetricspaceanycountableintersectionofdenseopensetsisdense.Proof.LetXbethespace,letBbeanopenballinX,andletO1;O2:::beasequenceofdenseopensets.WemustshowthatBnOn!6=;:SinceO1isdense,BO16=;,andisopen.ThusBO1containstheclosure B1ofsomeopenballB1.WemaychooseB1smallerifnecessarysothatitsradiusis<1.SinceB1isopenandO2isdense,B1O2containstheclosure B2ofsomeopenballB2,ofradius<1 2,andsoon.SinceXiscomplete,theintersectionofthedecreasingsequenceofclosedballswehaveconstructedcontainssomepointxwhichbelongbothtoBandtotheintersectionofalltheOi.QEDABairespaceisdenedasatopologicalspaceinwhicheverycountableintersectionofdenseopensetsisdense.ThusBaire'stheoremassertsthateverycompletemetricspaceisaBairespace.AsetAinatopologicalspaceiscallednowheredenseifitsclosurecontainsnoopenset.Putanotherway,asetAisnowheredenseifitscomplementAccontainsanopendenseset.AsetSissaidtobeofrstcategoryifitisacountableunionofnowheredensesets.ThenBaire'scategorytheoremcanbereformulatedassayingthatthecomplementofanysetofrstcategoryinacompletemetricspaceorinanyBairespaceisdense.ApropertyPofpointsofaBairespaceissaidtoholdquasi-surelyorquasi-everywhereifitholdsonanintersectionofcountablymanydenseopensets.Inotherwords,ifthesetwherePdoesnotholdisofrstcategory.1.13Tychono'stheorem.LetIbeaset,servingasanindexset".Supposethatforeach2Iwearegivenanon-emptytopologicalspaceS.TheCartesianproductS:=Y2ISisdenedasthecollectionofallfunctionsxwhosedomaininIandsuchthatx2S.Thisspaceisnotemptybytheaxiomofchoice.Wefrequentlywritexinsteadofxandcalledxthecoordinateofx".Themapf:Y2IS!S;x7!x

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1.14.URYSOHN'SLEMMA.25iscalledtheprojectionofSontoS.WeputonStheweakesttopologysuchthatalloftheseprojectionsarecontinuous.SotheopensetsofSaregeneratedbythesetsoftheformf)]TJ/F7 6.974 Tf 6.226 0 Td[(1UwhereUSisopen:Theorem1.13.1[Tychono.]IfalltheSarecompact,thensoisS=Q2IS.Proof.LetFbeafamilyofclosedsubsetsofSwiththepropertythattheintersectionofanynitecollectionofsubsetsfromthisfamilyisnotempty.WemustshowthattheintersectionofalltheelementsofFisnotempty.UsingZorn,extendFtoamaximalfamilyF0ofnotnecessarilyclosedsubsetsofSwiththepropertythattheintersectionofanynitecollectionofelementsofF0isnotempty.Foreach,theprojectionfF0hasthepropertythatthereisapointx2SwhichisintheclosureofallthesetsbelongingtofF0.Letx2Sbethepointwhose-thcoordinateisx.WewillshowthatxisintheclosureofeveryelementofF0whichwillcompletetheproof.LetUbeanopensetcontainingx.Bythedenitionoftheproducttopology,therearenitelymanyiandopensubsetsUiSisuchthatx2ni=1f)]TJ/F7 6.974 Tf 6.226 0 Td[(1iUiU:Soforeachi=1;:::;n;xi2Ui.ThismeansthatUiintersectseverysetbelongingtofiF0.Sof)]TJ/F7 6.974 Tf 6.226 0 Td[(1iUiintersectseverysetbelongingtoF0andhencemustbelongtoF0bymaximality.Therefore,ni=1f)]TJ/F7 6.974 Tf 6.226 0 Td[(1iUi2F0;againbymaximality.ThissaysthatUintersectseverysetofF0.Inotherwords,anyneighborhoodofxintersectseverysetbelongingtoF0,whichisjustanotherwayofsayingxbelongstotheclosureofeverysetbelongingtoF0.QED1.14Urysohn'slemma.AtopologicalspaceSiscallednormalifitisHausdor,andifforanypairF1;F2ofclosedsetswithF1F2=;therearedisjointopensetsU1;U2withF1U1andF2U2.Forexample,supposethatSisHausdorandcompact.Foreachp2F1andq2F2thereareneighborhoodsOqofpandWqofqwithOqWq=;.ThisistheHausdoraxiom.AnitenumberoftheWqcoverF2sinceitiscompact.LettheintersectionofthecorrespondingOqbecalledUpandtheunionofthecorrespondingWqbecalledVp.Thusforeachp2F1wehavefoundaneighborhoodUpofpandanopensetVpcontainingF2with

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26CHAPTER1.THETOPOLOGYOFMETRICSPACESUpVp=;.Onceagain,nitelymanyoftheUpcoverF1.SotheunionUoftheseandtheintersectionVofthecorrespondingVpgivedisjointopensetsUcontainingF1andVcontainingF2.SoanycompactHausdorspaceisnormal.Theorem1.14.1[Urysohn'slemma.]IfF0andF1aredisjointclosedsetsinanormalspaceSthenthereisacontinuousrealvaluedfunctionf:S!Rsuchthat0f1,f=0onF0andf=1onF1.Proof.LetV1:=Fc1:WecanndanopensetV1 2containingF0andwhoseclosureiscontainedinV1,sincewecanchooseV1 2disjointfromanopensetcontainingF1.SowehaveF0V1 2; V1 2V1:ApplyingournormalityassumptiontothesetsF0andVc1 2wecanndanopensetV1 4withF0V1 4and V1 4V1 2.Similarly,wecanndanopensetV3 4with V1 2V3 4and V3 4V1:SowehaveF0V1 4; V1 4V1 2; V1 2V3 4; V3 4V1=Fc1:Continuinginthisway,foreach0ameansthatthereissomer>asuchthatx62 Vr.Thusf)]TJ/F7 6.974 Tf 6.226 0 Td[(1a;1]=[r>a Vrc;alsoaunionofopensets,henceopen.Sowehaveshownthatf)]TJ/F7 6.974 Tf 6.227 0 Td[(1[0;bandf)]TJ/F7 6.974 Tf 6.227 0 Td[(1a;1]areopen.Hencef)]TJ/F7 6.974 Tf 6.226 0 Td[(1a;bisopen.Sincetheintervals[0;b;a;1]anda;bformabasisfortheopensetsontheinterval[0;1],weseethattheinverseimageofanyopensetunderfisopen,whichsaysthatfiscontinuous.QEDWewillhaveseveraloccasionstousethisresult.

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1.15.THESTONE-WEIERSTRASSTHEOREM.271.15TheStone-Weierstrasstheorem.ThisisanimportantgeneralizationofWeierstrass'stheoremwhichassertedthatthepolynomialsaredenseinthespaceofcontinuousfunctionsonanycompactinterval,whenweusetheuniformtopology.Weshallhavemanyusesforthistheorem.AnalgebraAofrealvaluedfunctionsonasetSissaidtoseparatepointsifforanyp;q2S;p6=qthereisanf2Awithfp6=fq.Theorem1.15.1[Stone-Weierstrass.]LetSbeacompactspaceandAanalgebraofcontinuousrealvaluedfunctionsonSwhichseparatespoints.ThentheclosureofAintheuniformtopologyiseitherthealgebraofallcontinuousfunctionsonS,oristhealgebraofallcontinuousfunctionsonSwhichallvanishatasinglepoint,callitx1.Wewillgivetwodierentproofsofthisimportanttheorem.Forourrstproof,werststateandprovesomepreliminarylemmas:Lemma1.15.1AnalgebraAofboundedrealvaluedfunctionsonasetSwhichisclosedintheuniformtopologyisalsoclosedunderthelatticeoperations_and^.Proof.Sincef_g=1 2f+g+jf)]TJ/F11 9.963 Tf 10.261 0 Td[(gjandf^g=1 2f+g)-252(jf)]TJ/F11 9.963 Tf 10.26 0 Td[(gjwemustshowthatf2Ajfj2A:Replacingfbyf=kfk1wemayassumethatjfj1:TheTaylorseriesexpansionaboutthepoint1 2forthefunctiont7!t+21 2convergesuniformlyon[0;1].Sothereexists,forany>0thereisapolynomialPsuchthatjPx2)]TJ/F8 9.963 Tf 9.963 0 Td[(x2+21 2j
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28CHAPTER1.THETOPOLOGYOFMETRICSPACESAsQdoesnotcontainaconstantterm,andAisanalgebra,Qf22Aforanyf2A.Sinceweareassumingthatjfj1wehaveQf22A;andkQf2)-222(jfjk1<4:SinceweareassumingthatAisclosedunderkk1weconcludethatjfj2Acompletingtheproofofthelemma.Lemma1.15.2LetAbeasetofrealvaluedcontinuousfunctionsonacompactspaceSsuchthatf;g2Af^g2Aandf_g2A:ThentheclosureofAintheuniformtopologycontainseverycontinuousfunctiononSwhichcanbeapproximatedateverypairofpointsbyafunctionbelongingtoA.Proof.SupposethatfisacontinuousfunctiononSwhichcanbeapproxi-matedatanypairofpointsbyelementsofA.Soletp;q2Sand>0,andletfp;q;2Abesuchthatjfp)]TJ/F11 9.963 Tf 9.963 0 Td[(fp;q;pj<;jfq)]TJ/F11 9.963 Tf 9.963 0 Td[(fp;q;qj<:LetUp;q;:=fxjfp;q;xfx)]TJ/F11 9.963 Tf 9.962 0 Td[(g:Fixqand.ThesetsUp;q;coverSaspvaries.HenceanitenumbercoverSsinceweareassumingthatSiscompact.Wemaytaketheminimumfq;ofthecorrespondingnitecollectionoffp;q;.Thefunctionfq;hasthepropertythatfq;xfx)]TJ/F11 9.963 Tf 9.963 0 Td[(forx2pVp;q;wheretheintersectionisagainoverthesamenitesetofp's.Wehavenowfoundacollectionoffunctionsfq;suchthatfq;f)]TJ/F11 9.963 Tf 8.688 0 Td[(onsomeneighborhoodVq;ofq.WemaychooseanitenumberofqsothattheVq;coverallofS.Takingthemaximumofthecorrespondingfq;givesafunctionf2Awithf)]TJ/F11 9.963 Tf 9.962 0 Td[(
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1.15.THESTONE-WEIERSTRASSTHEOREM.29SinceweareassumingthatAisclosedintheuniformtopologyweconcludethatf2A,completingtheproofofthelemma.ProofoftheStone-Weierstrasstheorem.Supposerstthatforeveryx2Sthereisag2Awithgx6=0.Letx6=yandh2Awithhy6=0.Thenwemaychooserealnumberscanddsothatf=cg+dhissuchthat06=fx6=fy6=0:ThenforanyrealnumbersaandbwemayndconstantsAandBsuchthatAfx+Bf2x=aandAfy+Bf2y=b:Wecanthereforeapproximateinfacthitexactlyonthenoseanyfunctionatanytwodistinctpoints.WeknowthattheclosureofAisclosedunder_and^bytherstlemma.BythesecondlemmaweconcludethattheclosureofAisthealgebraofallrealvaluedcontinuousfunctions.Thesecondalternativeisthatthereisapoint,callitp1atwhichallf2Avanish.WewishtoshowthattheclosureofAcontainsallcontinuousfunctionsvanishingatp1.LetBbethealgebraobtainedfromAbyaddingtheconstants.ThenBsatisesthehypothesesoftheStone-Weierstrasstheoremandcontainsfunctionswhichdonotvanishatp1.sowecanapplytheprecedingresult.Ifgisacontinuousfunctionvanishingatp1wemay,forany>0ndanf2Aandaconstantcsothatkg)]TJ/F8 9.963 Tf 9.963 0 Td[(f+ck1< 2:Evaluatingatp1givesjcj<=2.Sokg)]TJ/F11 9.963 Tf 9.963 0 Td[(fk1<:QEDThereasonfortheapparentlystrangenotationp1hastodowiththenotionoftheonepointcompacticationofalocallycompactspace.AtopologicalspaceSiscalledlocallycompactifeverypointhasaclosedcompactneighborhood.WecanmakeScompactbyaddingasinglepoint.Indeed,letp1beapointnotbelongingtoSandsetS1:=S[p1:WeputatopologyonS1bytakingastheopensetsalltheopensetsofStogetherwithallsetsoftheformO[p1whereOisanopensetofSwhosecomplementiscompact.ThespaceS1iscompact,forifwehaveanopencoverofS1,atleastoneoftheopensetsinthiscovermustbeofthesecondtype,henceitscomplementiscompact,hencecoveredbynitelymanyoftheremainingsets.IfSitselfiscompact,thentheemptysethascompactcomplement,hencep1hasanopenneighborhood

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30CHAPTER1.THETOPOLOGYOFMETRICSPACESdisjointfromS,andallwehavedoneisaddadisconnectedpointtoS.ThespaceS1iscalledtheone-pointcompacticationofS.InapplicationsoftheStone-Weierstrasstheorem,weshallfrequentlyhavetodowithanalgebraoffunctionsonalocallycompactspaceconsistingoffunctionswhichvanishatinnity"inthesensethatforany>0thereisacompactsetCsuchthatjfj
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1.16.MACHADO'STHEOREM.31Soeverychainhasanupperbound,andhencebyZorn'slemma,thereexistsamaximum,i.e.thereexistsanon-emptyclosedsubsetEsatisfying.3whichhasthepropertythatnopropersubsetofEsatises1.3.Weshallcallsuchasubsetf-minimal.Theorem1.16.1[Machado.]SupposethatACRMisasubalgebrawhichcontainstheconstantsandwhichisclosedintheuniformtopology.Thenforeveryf2CRMthereexistsanAlevelsetsatisfying.3.Infact,everyf-minimalsetisanAlevelset.Proof.LetEbeanf-minimalset.SupposeitisnotanAlevelset.Thismeansthatthereissomeh2AwhichisnotconstantonA.Replacinghbyah+cwhereaandcareconstant,wemayarrangethatminx2Eh=0andmaxx2Eh=1:LetE0:=fx2Ej0hx2 3gandE1:fx2Ej1 3x1g:Thesearenon-emptyclosedpropersubsetsofE,andhencetheminimalityofEimpliesthatthereexistg0;g12Asuchthatkf)]TJ/F11 9.963 Tf 9.963 0 Td[(g0kE0dfM)]TJ/F11 9.963 Tf 9.962 0 Td[(andkf)]TJ/F11 9.963 Tf 9.963 0 Td[(g1kE1dfM)]TJ/F11 9.963 Tf 9.963 0 Td[(forsome>0.Denehn:=)]TJ/F11 9.963 Tf 9.963 0 Td[(hn2nandkn:=hng0+)]TJ/F11 9.963 Tf 9.963 0 Td[(hng1:BothhnandknbelongtoAand0hn1onE,withstrictinequalityonE0E1.Ateachx2E0E1iwehavejfx)]TJ/F11 9.963 Tf 9.963 0 Td[(knxj=jhnxfx)]TJ/F11 9.963 Tf 9.963 0 Td[(hnxg0x+)]TJ/F11 9.963 Tf 9.963 0 Td[(hnxfx)]TJ/F8 9.963 Tf 9.962 0 Td[()]TJ/F11 9.963 Tf 9.962 0 Td[(hnxg1xjhnxkf)]TJ/F11 9.963 Tf 9.963 0 Td[(g0jkE0E1+)]TJ/F11 9.963 Tf 9.963 0 Td[(hnkf)]TJ/F11 9.963 Tf 9.963 0 Td[(g1jkE0E1hnxkf)]TJ/F11 9.963 Tf 9.963 0 Td[(g0kE0+)]TJ/F11 9.963 Tf 9.963 0 Td[(hnxkf)]TJ/F11 9.963 Tf 9.963 0 Td[(g1jkE1dfM)]TJ/F11 9.963 Tf 9.962 0 Td[(:Wewillnowshowthathn!1onE0nE1andhn!0onE1nE0.Indeed,onE0nE1wehavehn<1 3nsohn=1)]TJ/F11 9.963 Tf 9.962 0 Td[(hn2n1)]TJ/F8 9.963 Tf 9.963 0 Td[(2nhn1)]TJ/F1 9.963 Tf 9.963 14.047 Td[(2 3n!1sincethebinomialformulagivesanalternatingsumwithdecreasingterms.Ontheotherhand,hn+hn2n=1)]TJ/F11 9.963 Tf 9.963 0 Td[(h22n1orhn1 +hn2n:

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32CHAPTER1.THETOPOLOGYOFMETRICSPACESNowthebinomialformulaimpliesthatforanyintegerkandanypositivenumberawehaveka+akor+a)]TJ/F10 6.974 Tf 6.227 0 Td[(k1=ka.Sowehavehn1 2nhn:OnE0nE1wehavehn)]TJ/F7 6.974 Tf 5.762 -4.147 Td[(2 3nsotherewehavehn3 4n!0:Thuskn!g0uniformlyonE0nE1andkn!g1uniformlyonE1nE0.Weconcludethatfornlargeenoughkf)]TJ/F11 9.963 Tf 9.962 0 Td[(knkE
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1.17.THEHAHN-BANACHTHEOREM.33Wewanttochooseavalue=FysothatifwethendeneFx+y:=Fx+Fy=Fx+;8x2M;2RwedonotincreasethenormofF.IfF=0wetake=0.IfF6=0,wemayreplaceFbyF=kFk,extendandthenmultiplybykFksowithoutlossofgeneralitywemayassumethatkFk=1.Wewanttochoosetheextensiontohavenorm1,whichmeansthatwewantjFx+jkx+yk8x2M;2R:If=0thisistruebyhypothesis.If6=0dividethisinequalitybyandreplace1=xbyx.WewantjFx+jkx+yk8x2M:Wecanwritethisastwoseparateconditions:Fx2+kx2+yk8x22Mand)]TJ/F11 9.963 Tf 9.962 0 Td[(Fx1)]TJ/F11 9.963 Tf 9.963 0 Td[(kx1+yk8x12M:Rewritingthesecondinequalitythisbecomes)]TJ/F11 9.963 Tf 7.749 0 Td[(Fx1)-222(kx1+yk)]TJ/F11 9.963 Tf 18.265 0 Td[(Fx2+kx2+yk:Thequestioniswhethersuchachoiceispossible.Inotherwords,isthesupre-mumofthelefthandsideoverallx12Mlessthanorequaltotheinmumoftherighthandsideoverallx22M?Iftheanswertothisquestionisyes,wemaychoosetobeanyvaluebetweenthesupoftheleftandtheinfoftherighthandsidesoftheprecedinginequality.Soourquestionis:IsFx2)]TJ/F11 9.963 Tf 9.963 0 Td[(Fx1jkx2+yk+kx1+yk8x1;x22M?Butx1)]TJ/F11 9.963 Tf 10.064 0 Td[(x2=x1+y)]TJ/F8 9.963 Tf 10.065 0 Td[(x2+yandsousingthefactthatkFk=1andthetriangleinequalitygivesjFx2)]TJ/F11 9.963 Tf 9.962 0 Td[(Fx1jkx2)]TJ/F11 9.963 Tf 9.963 0 Td[(x1kkx2+yk+kx1+yk:Thiscompletestheproofoftheproposition,andhenceoftheHahn-Banachtheoremovertherealnumbers.Wenowdealwiththecomplexcase.IfBisacomplexnormedvectorspace,thenitisalsoarealvectorspace,andtherealandimaginarypartsofacomplexlinearfunctionarereallinearfunctions.Inotherwords,wecanwriteanycomplexlinearfunctionFasFx=Gx+iHx

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34CHAPTER1.THETOPOLOGYOFMETRICSPACESwhereGandHarereallinearfunctions.ThefactthatFiscomplexlinearsaysthatFix=iFxorGix=)]TJ/F11 9.963 Tf 7.748 0 Td[(HxorHx=)]TJ/F11 9.963 Tf 7.749 0 Td[(GixorFx=Gx)]TJ/F11 9.963 Tf 9.962 0 Td[(iGix:ThefactthatkFk=1impliesthatkGk1.SowecanadjointherealonedimensionalspacespannedbyytoMandextendthereallinearfunctiontoit,keepingthenorm1.NextadjointherealonedimensionalspacespannedbyiyandextendGtoit.WenowhaveGextendedtoMCywithnoincreaseinnorm.TrytodeneFz:=Gz)]TJ/F11 9.963 Tf 9.962 0 Td[(iGizonMCy.ThismapofMCy!CisR-linear,andcoincideswithFonM.Wemustcheckthatitiscomplexlinearandthatitsnormis1:TocheckthatitiscomplexlinearitisenoughtoobservethatFiz=Giz)]TJ/F11 9.963 Tf 9.963 0 Td[(iG)]TJ/F11 9.963 Tf 7.749 0 Td[(z=i[Gz)]TJ/F11 9.963 Tf 9.963 0 Td[(iGiz]=iFz:Tocheckthenorm,wemay,foranyz,choosesothateiFzisrealandisnon-negative.ThenjFzj=jeiFzj=jFeizj=Geizkeizk=kzksokFk1.QEDSupposethatMisaclosedsubspaceofBandthaty62M.LetddenotethedistanceofytoM,sothatd:=infx2Mky)]TJ/F11 9.963 Tf 9.962 0 Td[(xk:SupposewestartwiththezerofunctiononM,andextenditrsttoMybyFy)]TJ/F11 9.963 Tf 9.963 0 Td[(x=d:ThisisalinearfunctiononM+fyganditsnormis1.IndeedkFk=sup;xjdj ky)]TJ/F11 9.963 Tf 9.962 0 Td[(xk=supx02Md ky)]TJ/F11 9.963 Tf 9.963 0 Td[(x0k=d d=1:LetM0bethesetofallcontinuouslinearfunctionsonBwhichvanishonM.Then,usingtheHahn-BanachtheoremwegetProposition1.17.2Ify2Bandy62MwhereMisaclosedlinearsubspaceofB,thenthereisanelementF2M0withkFk1andFy6=0.InfactwecanarrangethatFy=dwheredisthedistancefromytoM.

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1.18.THEUNIFORMBOUNDEDNESSPRINCIPLE.35WehaveanembeddingB!Bx7!xwherexF:=Fx:TherstpartoftheprecedingpropositioncanbeformulatedasM00=MifMisaclosedsubspaceofB.Themapx7!xisclearlylinearandjxFj=jFxjkFkkxk:TakingthesupofjxFj=kFkshowsthatkxkkxkwherethenormontheleftisthenormonthespaceB.Ontheotherhand,ifwetakeM=f0gintheprecedingproposition,wecanndanF2BwithkFk=1andFx=kxk.ForthisFwehavejxFj=kxk.Sokxkkxk:WehaveprovedTheorem1.17.2ThemapB!Bgivenaboveisanormpreservinginjec-tion.1.18TheUniformBoundednessPrinciple.Theorem1.18.1LetBbeaBanachspaceandfFngbeasequenceofelementsinBsuchthatforeveryxedx2BthesequenceofnumbersfjFnxjgisbounded.ThenthesequenceofnormsfkFnkgisbounded.Proof.TheproofwillbebyaBairecategorystyleargument.WewillproveProposition1.18.1ThereexistssomeballB=By;r;r>0aboutapointywithkyk1andaconstantKsuchthatjFnzjKforallz2B.Proofthatthepropositionimpliesthetheorem.Foranyzwithkzk<1wehavez)]TJ/F11 9.963 Tf 9.962 0 Td[(y=2 rr 2z)]TJ/F11 9.963 Tf 9.963 0 Td[(yandkr 2z)]TJ/F11 9.963 Tf 9.962 0 Td[(ykrsincekz)]TJ/F11 9.963 Tf 9.963 0 Td[(yk2.jFnzjjFnz)]TJ/F11 9.963 Tf 9.963 0 Td[(yj+jFnyj2K r+K:

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36CHAPTER1.THETOPOLOGYOFMETRICSPACESSokFnk2K r+Kforallnprovingthetheoremfromtheproposition.Proofoftheproposition.Ifthepropositionisfalse,wecanndn1suchthatjFn1xj>1atsomex2B;1andhenceinsomeballofradius<1 2aboutx.Thenwecanndann2withjFn2zj>2insomenon-emptyclosedballofradius<1 3lyinginsidetherstball.Continuinginductively,wechooseasubsequencenmandafamilyofnestednon-emptyballsBmwithjFnmzj>mthroughoutBmandtheradiioftheballstendingtozero.SinceBiscomplete,thereisapointxcommontoalltheseballs,andfjFnxjgisunbounded,contrarytohypothesis.QEDWewillhaveoccasiontousethistheoreminareversedform".RecallthatwehavethenormpreservinginjectionB!Bsendingx7!xwherexF=Fx.SinceBisaBanachspaceevenifBisincompletewehaveCorollary1.18.1IffxngisasequenceofelementsinanormedlinearspacesuchthatthenumericalsequencefjFxnjgisboundedforeachxedF2Bthenthesequenceofnormsfkxnkgisbounded.

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Chapter2HilbertSpacesandCompactoperators.2.1Hilbertspace.2.1.1Scalarproducts.Visacomplexvectorspace.Aruleassigningtoeverypairofvectorsf;g2Vacomplexnumberf;giscalledasemi-scalarproductif1.f;gislinearinfwhengisheldxed.2.g;f= f;g.Thisimpliesthatf;gisanti-linearingwhenfisheldxed.Inotherwords.f;ag+bh= af;g+ bf;h.Italsoimpliesthatf;fisreal.3.f;f0forallf2V.If3.isreplacedbythestrongercondition4.f;f>0forallnon-zerof2Vthenwesaythat;isascalarproduct.Examples.V=Cn,soanelementzofVisacolumnvectorofcomplexnumbers:z=0B@z1...zn1CAandz;wisgivenbyz;w:=nX1zi wi:37

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38CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.Vconsistsofallcontinuouscomplexvaluedfunctionsonthereallinewhichareperiodicofperiod2andf;g:=1 2Z)]TJ/F10 6.974 Tf 6.227 0 Td[(fx gxdx:WewilldenotethisspacebyCT.HeretheletterTstandsfortheonedimensionaltorus,i.e.thecircle.Weareidentifyingfunctionswhichareperiodicwithperiod2withfunctionswhicharedenedonthecircleR=2Z.Vconsistsofalldoublyinnitesequencesofcomplexnumbersa=:::;a)]TJ/F7 6.974 Tf 6.226 0 Td[(2;a)]TJ/F7 6.974 Tf 6.226 0 Td[(1;a0;a1;a2;:::whichsatisfyXjaij2<1:Herea;b:=Xai bi:Allthreeareexamplesofscalarproducts.2.1.2TheCauchy-Schwartzinequality.Thissaysthatif;isasemi-scalarproductthenjf;gjf;f1 2g;g1 2:.1Proof.Foranyrealnumbertcondition3.abovesaysthatf)]TJ/F11 9.963 Tf 9.424 0 Td[(tg;f)]TJ/F11 9.963 Tf 9.424 0 Td[(tg0.Expandingoutgives0f)]TJ/F11 9.963 Tf 9.963 0 Td[(tg;f)]TJ/F11 9.963 Tf 9.963 0 Td[(tg=f;f)]TJ/F11 9.963 Tf 9.963 0 Td[(t[f;g+g;f]+t2g;g:Sinceg;f= f;g,thecoecientoftintheaboveexpressionistwicetherealpartoff;g.SotherealquadraticformQt:=f;f)]TJ/F8 9.963 Tf 9.963 0 Td[(2Ref;gt+t2g:gisnowherenegative.Soitcannothavedistinctrealroots,andhencebytheb2)]TJ/F8 9.963 Tf 9.963 0 Td[(4acruleweget4Ref;g2)]TJ/F8 9.963 Tf 9.963 0 Td[(4f;fg;g0orRef;g2f;fg;g:.2Thisisusefulandalmostbutnotquitewhatwewant.Butwemayapplythisinequalitytoh=eigforany.Thenh;h=g;g.Choosesothatf;g=rei

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2.1.HILBERTSPACE.39wherer=jf;gj.Thenf;h=f;eig=e)]TJ/F10 6.974 Tf 6.227 0 Td[(if;g=jf;gjandtheprecedinginequalitywithgreplacedbyhgivesjf;gj2f;fg;gandtakingsquarerootsgives.1.2.1.3ThetriangleinequalityForanysemiscalarproductdenekfk:=f;f1 2sowecanwritetheCauchy-Schwartzinequalityasjf;gjkfkkgk:Thetriangleinequalitysaysthatkf+gkkfk+kgk:.3Proof.kf+gk2=f+g;f+g=f;f+2Ref;g+g;gf;f+2kfkkgk+g;gby.2=kfj2+2kfkkgk+kgk2=kfk+kgk2:Takingsquarerootsgivesthetriangleinequality2.3.Noticethatkcfk=jcjkfk.4sincecf;cf=c cf;f=jcj2kfk2.SupposewetrytodenethedistancebetweentwoelementsofVbydf;g:=kf)]TJ/F11 9.963 Tf 9.963 0 Td[(gk:Noticethatthendf;f=0,df;g=dg;fandforanythreeelementsdf;hdf;g+dg;hbyvirtueofthetriangleinequality.Theonlytroublewiththisdenitionisthatwemighthavetwodistinctelementsatzerodistance,i.e.0=df;g=kf)]TJ/F11 9.963 Tf 9.146 0 Td[(gk.Butthiscannothappenif;isascalarproduct,i.e.satisescondition4.

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40CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.AcomplexvectorspaceVendowedwithascalarproductiscalledapre-Hilbertspace.LetVbeacomplexvectorspaceandletkkbeamapwhichassignstoanyf2Vanon-negativerealkfknumbersuchthatkfk>0forallnon-zerof.Ifkksatisesthetriangleinequality.3andequation.4itiscalledanorm.Avectorspaceendowedwithanormiscalledanormedspace.Thepre-Hilbertspacescanbecharacterizedamongallnormedspacesbytheparallelogramlawaswewilldiscussbelow.Lateron,wewillhavetoweakencondition.4inourgeneralstudy.Butitistoocomplicatedtogivethegeneraldenitionrightnow.2.1.4Hilbertandpre-Hilbertspaces.Thereasonfortheprexpre"isthefollowing:Thedistanceddenedabovehasallthedesiredpropertieswemightexpectofadistance.Inparticular,wecandenethenotionsoflimit"andofaCauchysequence"asisdonefortherealnumbers:IffnisasequenceofelementsofV,andf2Vwesaythatfisthelimitofthefnandwritelimn!1fn=f;orfn!fif,foranypositivenumberthereisanN=Nsuchthatdfn;f0thereisanK=Ksuchthatdfm;fn<8;m;nK:Ifthesequencefnhasalimit,thenitisCauchy-justchooseK=N1 2andusethetriangleinequality.ButitisquitepossiblethataCauchysequencehasnolimit.Asanexampleofthistypeofphenomenon,thinkoftherationalnumberswithjr)]TJ/F11 9.963 Tf 10.28 0 Td[(sjasthedistance.ThewholepointofintroducingtherealnumbersistoguaranteethateveryCauchysequencehasalimit.Sowesaythatapre-HilbertspaceisaHilbertspaceifitiscomplete"intheabovesense-ifeveryCauchysequencehasalimit.Sincethecomplexnumbersarecompletebecausetherealnumbersare,itfollowsthatCniscomplete,i.e.isaHilbertspace.Indeed,wecansaythatanynitedimensionalpre-HilbertspaceisaHilbertspacebecauseitisisomorphicasapre-HilbertspacetoCnforsomen.Seebelowwhenwediscussorthonormalbases.Thetroubleisintheinnitedimensionalcase,suchasthespaceofcontinuousperiodicfunctions.Thisspaceisnotcomplete.Forexample,letfnbethefunctionwhichisequaltooneon)]TJ/F11 9.963 Tf 7.749 0 Td[(+1 n;)]TJ/F7 6.974 Tf 9.421 3.922 Td[(1 n,equaltozeroon1 n;)]TJ/F7 6.974 Tf 12.109 3.922 Td[(1 n

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2.1.HILBERTSPACE.41andextendedlinearly)]TJ/F7 6.974 Tf 9.421 3.923 Td[(1 nto1 nandfrom)]TJ/F7 6.974 Tf 11.245 3.923 Td[(1 nto+1 nsoastobecontinuousandthenextendedsoastobeperiodic.Thusontheinterval)]TJ/F7 6.974 Tf 11.129 3.923 Td[(1 n;+1 nthefunctionisgivenbyfnx=n 2x)]TJ/F8 9.963 Tf 9.386 0 Td[()]TJ/F7 6.974 Tf 11.059 3.923 Td[(1 n.Ifmn,thefunctionsfmandfnagreeoutsidetwointervalsoflength2 mandontheseintervalsjfmx)]TJ/F11 9.963 Tf 9.091 0 Td[(fnxj1.Sokfm)]TJ/F11 9.963 Tf 10.221 0 Td[(fnk21 22=mshowingthatthesequenceffngisCauchy.Butthelimitwouldhavetoequaloneon)]TJ/F11 9.963 Tf 7.748 0 Td[(;0andequalzeroon0;andsobediscontinuousattheoriginandat.ThusthespaceofcontinuousperiodicfunctionsisnotaHilbertspace,onlyapre-Hilbertspace.Butjustaswecompletetherationalnumbersbythrowinginideal"el-ementstogettherealnumbers,wemaysimilarlycompleteanypre-HilbertspacetogetauniqueHilbertspace.SeethesectionCompletioninthechapteronmetricspacesforageneraldiscussionofhowtocompleteanymetricspace.Inparticular,thecompletionofanynormedvectorspaceisacompletenormedvectorspace.AcompletenormedspaceiscalledaBanachspace.ThegeneralconstructionimpliesthatanynormedvectorspacecanbecompletedtoaBa-nachspace.Fromtheparallelogramlawdiscussedbelow,itwillfollowthatthecompletionofapre-HilbertspaceisaHilbertspace.ThecompletionofthespaceofcontinuousperiodicfunctionswillbedenotedbyL2T.2.1.5ThePythagoreantheorem.LetVbeapre-Hilbertspace.Wehavekf+gk2=kfk2+2Ref;g+kgk2:Sokf+gk2=kfk2+kgk2,Ref;g=0:.5Wemakethedenitionf?g,f;g=0andsaythatfisperpendiculartogorthatfisorthogonaltog.NoticethatthisisastrongerconditionthantheconditionforthePythagoreantheorem,therighthandconditionin.5.Forexamplekf+ifk2=2kfk2butf;if=)]TJ/F11 9.963 Tf 7.749 0 Td[(ikfk26=0ifkfk6=0.Ifuiissomenitecollectionofmutuallyorthogonalvectors,thensoareziuiwheretheziareanycomplexnumbers.Soifu=XiziuithenbythePythagoreantheoremkuk2=Xijzij2kuik2:

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42CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.Inparticular,iftheui6=0,thenu=0zi=0foralli.Thisshowsthatanysetofmutuallyorthogonalnon-zerovectorsislinearlyindependent.Noticethatthesetoffunctionseinisanorthonormalsetinthespaceofcontinuousperiodicfunctionsinthatnotonlyaretheymutuallyorthogonal,buteachhasnormone.2.1.6ThetheoremofApollonius.Addingtheequationskf+gk2=kfk2+2Ref;g+kgk2.6kf)]TJ/F11 9.963 Tf 9.963 0 Td[(gk2=kfk2)]TJ/F8 9.963 Tf 9.962 0 Td[(2Ref;g+kgk2.7giveskf+gk2+kf)]TJ/F11 9.963 Tf 9.962 0 Td[(gk2=2)]TJ/F14 9.963 Tf 4.566 -8.07 Td[(kfk2+kgk2:.8Thisisknownastheparallelogramlaw.ItisthealgebraicexpressionofthetheoremofApolloniuswhichassertsthatthesumoftheareasofthesquaresonthesidesofaparallelogramequalsthesumoftheareasofthesquaresonthediagonals.Ifwesubtract.7from.6wegetRef;g=1 4)]TJ/F14 9.963 Tf 4.566 -8.069 Td[(kf+gk2)-222(kf)]TJ/F11 9.963 Tf 9.963 0 Td[(gk2:.9Nowif;g=if;gandRefif;gg=)]TJ/F8 9.963 Tf 7.749 0 Td[(Imf;gsoImf;g=)]TJ/F8 9.963 Tf 7.748 0 Td[(Reif;g=Ref;igsof;g=1 4)]TJ/F14 9.963 Tf 4.567 -8.07 Td[(kf+gk2)-222(kf)]TJ/F11 9.963 Tf 9.963 0 Td[(gk2+ikf+igk2)]TJ/F11 9.963 Tf 9.963 0 Td[(ikf)]TJ/F11 9.963 Tf 9.963 0 Td[(igk2:.10Ifwenowcompleteapre-Hilbertspace,therighthandsideofthisequationisdenedonthecompletion,andisacontinuousfunctionthere.Itthereforefollowsthatthescalarproductextendstothecompletion,and,bycontinuity,satisesalltheaxiomsforascalarproduct,plusthecompletenessconditionfortheassociatednorm.Inotherwords,thecompletionofapre-HilbertspaceisaHilbertspace.2.1.7ThetheoremofJordanandvonNeumann.ThisisessentiallyaconversetothetheoremofApollonius.ItsaysthatifkkisanormonacomplexvectorspaceVwhichsatises.8,thenVisinfactapre-Hilbertspacewithkfk2=f;f.Ifthetheoremistrue,thenthescalarproductmustbegivenby.10.Sowemustprovethatifwetake.10asthe

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2.1.HILBERTSPACE.43denition,thenalltheaxiomsonascalarproducthold.Theeasiestaxiomtoverifyisg;f= f;g:Indeed,therealpartoftherighthandsideof.10isunchangedundertheinterchangeoffandgsinceg)]TJ/F11 9.963 Tf 8.972 0 Td[(f=)]TJ/F8 9.963 Tf 7.749 0 Td[(f)]TJ/F11 9.963 Tf 8.973 0 Td[(gandk)]TJ/F11 9.963 Tf 15.178 0 Td[(hk=khkforanyhisoneofthepropertiesofanorm.Alsog+if=if)]TJ/F11 9.963 Tf 9.502 0 Td[(igandkihk=khksothelasttwotermsontherightof.10getinterchanged,provingthatg;f= f;g.Itisjustaseasytoprovethatif;g=if;g:Indeedreplacingfbyifsendskf+igk2intokif+igk2=kf+gk2andsendskf+gk2intokif+gk2=kif)]TJ/F11 9.963 Tf 9.768 0 Td[(igk2=kf)]TJ/F11 9.963 Tf 9.768 0 Td[(igk2=i)]TJ/F11 9.963 Tf 7.748 0 Td[(ikf)]TJ/F11 9.963 Tf 9.769 0 Td[(igk2sohastheeectofmultiplyingthesumoftherstandfourthtermsbyi,andsimilarlyforthesumofthesecondandthirdtermsontherighthandsideof.10.Now.10implies.9.Supposewereplacef;gin.8byf1+g;f2andbyf1)]TJ/F11 9.963 Tf 9.963 0 Td[(g;f2andsubtractthesecondequationfromtherst.Wegetkf1+f2+gk2)-222(kf1+f2)]TJ/F11 9.963 Tf 9.963 0 Td[(gk2+kf1)]TJ/F11 9.963 Tf 9.963 0 Td[(f2+gk2)-222(kf1)]TJ/F11 9.963 Tf 9.962 0 Td[(f2)]TJ/F11 9.963 Tf 9.963 0 Td[(gk2=2)]TJ/F14 9.963 Tf 4.566 -8.07 Td[(kf1+gk2)-222(kf1)]TJ/F11 9.963 Tf 9.963 0 Td[(gk2:Inviewof.9wecanwritethisasRef1+f2;g+Ref1)]TJ/F11 9.963 Tf 9.963 0 Td[(f2;g=2Ref1;g:.11Nowtherighthandsideof.9vanisheswhenf=0sincekgk=k)]TJ/F11 9.963 Tf 17.168 0 Td[(gk.Soifwetakef1=f2=fin2.11wegetRef;g=2Ref;g:Wecanthuswrite.11asRef1+f2;g+Ref1)]TJ/F11 9.963 Tf 9.963 0 Td[(f2;g=Ref1;g:Inthisequationmakethesubstitutionsf17!1 2f1+f2;f27!1 2f1)]TJ/F11 9.963 Tf 9.963 0 Td[(f2:ThisyieldsRef1+f2;g=Ref1;g+Ref2;g:Sinceitfollowsfrom.10and.9thatf;g=Ref;g)]TJ/F11 9.963 Tf 9.963 0 Td[(iReif;gweconcludethatf1+f2;g=f1;g+f2;g:

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44CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.Takingf1=)]TJ/F11 9.963 Tf 7.748 0 Td[(f2showsthat)]TJ/F11 9.963 Tf 7.749 0 Td[(f;g=)]TJ/F8 9.963 Tf 7.749 0 Td[(f;g:ConsiderthecollectionCofcomplexnumberswhichsatisfyf;g=f;gforallf;g.Weknowfromf1+f2;g=f1;g+f2;gthat;2C+2C:SoCcontainsallintegers.If06=2Cthenf;g==f;g==f;gso)]TJ/F7 6.974 Tf 6.227 0 Td[(12C.ThusCcontainsallcomplexrationalnumbers.Thetheoremwillbeprovedifwecanprovethatf;giscontinuousin.Butthetriangleinequalitykf+gkkfk+kgkappliedtof=f2;g=f1)]TJ/F11 9.963 Tf 9.963 0 Td[(f2impliesthatkf1kkf1)]TJ/F11 9.963 Tf 9.963 0 Td[(f2k+kf2korkf1k)-222(kf2kkf1)]TJ/F11 9.963 Tf 9.962 0 Td[(f2k:Interchangingtheroleoff1andf2givesjkf1k)-222(kf2kjkf1)]TJ/F11 9.963 Tf 9.963 0 Td[(f2k:Thereforejkfgk)-222(kfgkjk)]TJ/F11 9.963 Tf 9.962 0 Td[(fk:Sincek)]TJ/F11 9.963 Tf 10.273 0 Td[(fk!0as!thisshowsthattherighthandsideof.10whenappliedtofandgisacontinuousfunctionof.ThusC=C.WehaveprovedTheorem2.1.1[P.JordanandJ.vonNeumann]IfVisanormedspacewhosenormsatises.8thenVisapre-Hilbertspace.Noticethatthecondition.8involvesonlytwovectorsatatime.SoweconcludeasanimmediateconsequenceofthistheoremthatCorollary2.1.1Anormedvectorspaceispre-HilbertspaceifandonlyifeverytwodimensionalsubspaceisaHilbertspaceintheinducednorm.Actually,aweakerversionofthiscorollary,withtworeplacedbythreehadbeenprovedbyFrechet,AnnalsofMathematics,July1935,whoraisedtheproblemofgivinganabstractcharacterizationofthosenormsonvectorspaceswhichcomefromscalarproducts.IntheimmediatelyfollowingpaperJordanandvonNeumannprovedthetheoremaboveleadingtothestrongercorollarythattwodimensionssuce.

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2.1.HILBERTSPACE.452.1.8Orthogonalprojection.WecontinuewiththeassumptionthatVispre-Hilbertspace.IfAandBaretwosubsetsofV,wewriteA?Bifu2Aandv2Bu?v,inotherwordsifeveryelementofAisperpendiculartoeveryelementofB.Similarly,wewillwritev?AiftheelementvisperpendiculartoallelementsofA.Finally,wewillwriteA?forthesetofallvwhichsatisfyv?A.NoticethatA?isalwaysalinearsubspaceofV,foranyA.NowletMbealinearsubspaceofV.LetvbesomeelementofV,notnecessarilybelongingtoM.Wewanttoinvestigatetheproblemofndingaw2Msuchthatv)]TJ/F11 9.963 Tf 10.065 0 Td[(w?M.Ofcourse,ifv2Mthentheonlychoiceistotakew=v.Sotheinterestingproblemiswhenv62M.Supposethatsuchawexists,andletxbeanyotherpointofM.ThenbythePythagoreantheorem,kv)]TJ/F11 9.963 Tf 9.962 0 Td[(xk2=kv)]TJ/F11 9.963 Tf 9.962 0 Td[(w+w)]TJ/F11 9.963 Tf 9.963 0 Td[(xk2=kv)]TJ/F11 9.963 Tf 9.963 0 Td[(wk2+kw)]TJ/F11 9.963 Tf 9.963 0 Td[(xk2sincev)]TJ/F11 9.963 Tf 9.963 0 Td[(w?Mandw)]TJ/F11 9.963 Tf 9.963 0 Td[(x2M.Sokv)]TJ/F11 9.963 Tf 9.963 0 Td[(wkkv)]TJ/F11 9.963 Tf 9.962 0 Td[(xkandthisinequalityisstrictifx6=w.Inwords:ifwecanndaw2Msuchthatv)]TJ/F11 9.963 Tf 10.186 0 Td[(w?MthenwistheuniquesolutionoftheproblemofndingthepointinMwhichisclosesttov.Conversely,supposewefoundaw2Mwhichhasthisminimizationproperty,andletxbeanyelementofM.Thenforanyrealnumbertwehavekv)]TJ/F11 9.963 Tf 9.963 0 Td[(wk2kv)]TJ/F11 9.963 Tf 9.963 0 Td[(w+txk2=kv)]TJ/F11 9.963 Tf 9.962 0 Td[(wk2+2tRev)]TJ/F11 9.963 Tf 9.963 0 Td[(w;x+t2kxk2:Sincetheminimumofthisquadraticpolynomialintoccurringontherightisachievedatt=0,weconcludebydierentiatingwithrespecttotandsettingt=0,forexamplethatRev)]TJ/F11 9.963 Tf 9.962 0 Td[(w;x=0:Byourusualtrickofreplacingxbyeixweconcludethatv)]TJ/F11 9.963 Tf 9.963 0 Td[(w;x=0:Sincethisholdsforallx2M,weconcludethatv)]TJ/F11 9.963 Tf 9.898 0 Td[(w?M.Sotondwwesearchfortheminimumofkv)]TJ/F11 9.963 Tf 9.962 0 Td[(xk;x2M.Nowkv)]TJ/F11 9.963 Tf 9.609 0 Td[(xk0andissomenitenumberforanyx2M.Sotherewillbesomerealnumbermsuchthatmkv)]TJ/F11 9.963 Tf 8.837 0 Td[(xkforx2M,andsuchthatnostrictlylargerrealnumberwillhavethisproperty.misknownasthegreatestlowerbound"ofthevalueskv)]TJ/F11 9.963 Tf 10.343 0 Td[(xk;x2M.Sowecanndasequenceofvectorsxn2Msuchthatkv)]TJ/F11 9.963 Tf 9.963 0 Td[(xnk!m:WeclaimthatthexnformaCauchysequence.Indeed,kxi)]TJ/F11 9.963 Tf 9.963 0 Td[(xjk2=kv)]TJ/F11 9.963 Tf 9.963 0 Td[(xj)]TJ/F8 9.963 Tf 9.963 0 Td[(v)]TJ/F11 9.963 Tf 9.962 0 Td[(xik2

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46CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.andbytheparallelogramlawthisequals2)]TJ/F14 9.963 Tf 4.566 -8.07 Td[(kv)]TJ/F11 9.963 Tf 9.963 0 Td[(xik2+kv)]TJ/F11 9.963 Tf 9.963 0 Td[(xjk2)-222(k2v)]TJ/F8 9.963 Tf 9.963 0 Td[(xi+xjk2:Nowtheexpressioninparenthesisconvergesto2m2.Thelasttermontherightisk2v)]TJ/F8 9.963 Tf 11.158 6.74 Td[(1 2xi+xjk2:Since1 2xi+xj2M,weconcludethatk2v)]TJ/F8 9.963 Tf 9.963 0 Td[(xi+xjk24m2sokxi)]TJ/F11 9.963 Tf 9.963 0 Td[(xjk24m+2)]TJ/F8 9.963 Tf 9.963 0 Td[(4m2foriandjlargeenoughthatkv)]TJ/F11 9.963 Tf 10.37 0 Td[(xikm+andkv)]TJ/F11 9.963 Tf 10.37 0 Td[(xjkm+.ThisprovesthatthesequencexnisCauchy.Hereisthecruxofthematter:IfMiscomplete,thenwecanconcludethatthexnconvergetoalimitwwhichisthentheuniqueelementinMsuchthatv)]TJ/F11 9.963 Tf 8.962 0 Td[(w?M.Itisatthispointthatcompletenessplayssuchanimportantrole.Putanotherway,wecansaythatifMisasubspaceofVwhichiscompleteunderthescalarproduct;restrictedtoMthenwehavetheorthogonaldirectsumdecompositionV=MM?;whichsaysthateveryelementofVcanbeuniquelydecomposedintothesumofanelementofMandavectorperpendiculartoM.Forexample,ifMistheonedimensionalsubspaceconsistingofallcomplexmultiplesofanon-zerovectory,thenMiscomplete,sinceCiscomplete.Sowexists.SinceallelementsofMareoftheformay,wecanwritew=ayforsomecomplexnumbera.Thenv)]TJ/F11 9.963 Tf 9.962 0 Td[(ay;y=0orv;y=akyk2soa=v;y kyk2:WecallatheFouriercoecientofvwithrespecttoy.Particularlyusefulisthecasewherekyk=1andwecanwritea=v;y:.12Gettingbacktothegeneralcase,ifV=MM?holds,sothattoeveryvtherecorrespondsauniquew2Msatisfyingv)]TJ/F11 9.963 Tf 10.023 0 Td[(w2M?themapv7!wiscalledorthogonalprojectionofVontoMandwillbedenotedbyM.

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2.1.HILBERTSPACE.472.1.9TheRieszrepresentationtheorem.LetVandWbetwocomplexvectorspaces.AmapT:V!WiscalledlinearifTx+y=Tx+TY8x;y2V;;2Candiscalledanti-linearifTx+y= Tx+ TY8x;y2V;2C:If`:V!Cisalinearmap,alsoknownasalinearfunctionthenker`:=fx2Vj`x=0ghascodimensiononeunless`0.Indeed,if`y6=0then`x=1wherex=1 `yyandforanyz2V,z)]TJ/F11 9.963 Tf 9.963 0 Td[(`zx2ker`:IfVisanormedspaceand`iscontinuous,thenker`isaclosedsubspace.ThespaceofcontinuouslinearfunctionsisdenotedbyV.Ithasitsownnormdenedbyk`k:=supx2V;kxk6=0j`xj=kxk:SupposethatHisapre-hilbertspace.Thenwehaveanantilinearmap:H!H;gf:=f;g:TheCauchy-Schwartzinequalityimpliesthatkgkkgkandinfactg;g=kgk2showsthatkgk=kgk:Inparticularthemapisinjective.TheRieszrepresentationtheoremsaysthatifHisaHilbertspace,thenthismapissurjective:

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48CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.Theorem2.1.2EverycontinuouslinearfunctiononHisgivenbyscalarprod-uctbysomeelementofH.TheproofisaconsequenceofthetheoremaboutprojectionsappliedtoN:=ker`:If`=0thereisnothingtoprove.If`6=0thenNisaclosedsubspaceofcodimensionone.Choosev62N.Thenthereisanx2Nwithv)]TJ/F11 9.963 Tf 9.218 0 Td[(x?N:Lety:=1 kv)]TJ/F11 9.963 Tf 9.962 0 Td[(xkv)]TJ/F11 9.963 Tf 9.963 0 Td[(x:Theny?Nandkyk=1:Foranyf2H,[f)]TJ/F8 9.963 Tf 9.962 0 Td[(f;yy]?ysof)]TJ/F8 9.963 Tf 9.962 0 Td[(f;yy2Nor`f=f;y`y;soifwesetg:= `yythenf;g=`fforallf2H.QED2.1.10WhatisL2T?WehavedenedthespaceL2TtobethecompletionofthespaceCTundertheL2normkfk2=f;f1 2.Inparticular,everylinearfunctiononCTwhichiscontinuouswithrespecttothethisL2normextendstoauniquecontinuouslinearfunctiononL2T.BytheRieszrepresentationtheoremweknowthateverysuchcontinuouslinearfunctionisgivenbyscalarproductbyanelementofL2T.ThuswemaythinkoftheelementsofL2TasbeingthelinearfunctionsonCTwhicharecontinuouswithrespecttotheL2norm.AnelementofL2Tshouldnotbethoughtofasafunction,butratherasalinearfunctiononthespaceofcontinuousfunctionsrelativetoaspecialnorm-theL2norm.

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2.1.HILBERTSPACE.492.1.11Projectionontoadirectsum.SupposethattheclosedsubspaceMofapre-HilbertspaceistheorthogonaldirectsumofanitenumberofsubspacesM=MiMimeaningthattheMiaremutuallyperpendicularandeveryelementxofMcanbewrittenasx=Xxi;xi2Mi:Theorthogonalityguaranteesthatsuchadecompositionisunique.SupposefurtherthateachMiissuchthattheprojectionMiexists.ThenMexistsandMv=XMiv:.13Proof.ClearlytherighthandsidebelongstoM.Wemustshowv)]TJ/F1 9.963 Tf 8.923 7.472 Td[(PiMivisorthogonaltoeveryelementofM.ForthisitisenoughtoshowthatitisorthogonaltoeachMjsinceeveryelementofMisasumofelementsoftheMj.Sosupposexj2Mj.ButMiv;xj=0ifi6=j.Sov)]TJ/F1 9.963 Tf 9.963 9.464 Td[(XMiv;xj=v)]TJ/F11 9.963 Tf 9.963 0 Td[(Mjv;xj=0bythedeningpropertyofMj.2.1.12Projectionontoanitedimensionalsubspace.Wenowwillputtheequations.12and.13together:SupposethatMisanitedimensionalsubspacewithanorthonormalbasisi.ThisimpliesthatMisanorthogonaldirectsumoftheonedimensionalspacesspannedbytheiandhenceMexistsandisgivenbyMv=Xaiiwhereai=v;i:.142.1.13Bessel'sinequality.Wenowlookattheinnitedimensionalsituationandsupposethatwearegivenanorthonormalsequencefig11.Anyv2VhasitsFouriercoecientsai=v;irelativetothemembersofthissequence.Bessel'sinequalityassertsthat1X1jaij2kvk2;.15inparticularthesumontheleftconverges.

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50CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.Proof.Letvn:=nXi=1aii;sothatvnistheprojectionofvontothesubspacespannedbytherstnofthei.Inanyevent,v)]TJ/F11 9.963 Tf 9.962 0 Td[(vn?vnsobythePythagoreanTheoremkvk2=kv)]TJ/F11 9.963 Tf 9.962 0 Td[(vnk2+kvnk2=kv)]TJ/F11 9.963 Tf 9.962 0 Td[(vnk2+nXi=1jaij2:ThisimpliesthatnXi=1jaij2kvk2andlettingn!1showsthattheseriesontheleftofBessel'sinequalityconvergesandthatBessel'sinequalityholds.2.1.14Parseval'sequation.Continuingtheaboveargument,observethatkv)]TJ/F11 9.963 Tf 9.962 0 Td[(vnk2!0,Xjaij2=kvk2:Butkv)]TJ/F11 9.963 Tf 9.54 0 Td[(vnk2!0ifandonlyifkv)]TJ/F11 9.963 Tf 9.54 0 Td[(vnk!0whichisthesameassayingthatvn!v.Butvnisthen-thpartialsumoftheseriesPaii,andinthelanguageofseries,wesaythataseriesconvergestoalimitvandwritePaii=vifandonlyifthepartialsumsapproachv.SoXaii=v,Xijaij2=kvk2:.16Ingeneral,wewillcalltheseriesPiaiitheFourierseriesofvrelativetothegivenorthonormalsequencewhetherornotitconvergestov.ThusParseval'sequalitysaysthattheFourierseriesofvconvergestovifandonlyifPjaij2=kvk2.2.1.15Orthonormalbases.WestillsupposethatVismerelyapre-Hilbertspace.Wesaythatanorthonor-malsequencefigisabasisofVifeveryelementofVisthesumofitsFourierseries.Forexample,oneofourtaskswillbetoshowthattheexponentialsfeinxg1n=formabasisofCT.Iftheorthonormalsequenceiisabasis,thenanyvcanbeapproximatedascloselyaswelikebynitelinearcombinationsofthei,infactbythepartialsumsofitsFourierseries.WesaythatthenitelinearcombinationsoftheiaredenseinV.Conversely,supposethatthenitelinearcombinationsofthe

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2.2.SELF-ADJOINTTRANSFORMATIONS.51iaredenseinV.Thismeansthatforanyvandany>0wecanndannandasetofncomplexnumbersbisuchthatkv)]TJ/F1 9.963 Tf 9.963 9.465 Td[(Xbiik:Butweknowthatvnistheclosestvectortovamongallthelinearcombinationsoftherstnofthei.sowemusthavekv)]TJ/F11 9.963 Tf 9.963 0 Td[(vnk:ButthissaysthattheFourierseriesofvconvergestov,i.e.thattheiformabasis.Forexample,weknowfromFejer'stheoremthattheexponentialseikxaredenseinCT.Henceweknowthattheyformabasisofthepre-HilbertspaceCT.Wewillgivesomealternativeproofsofthisfactbelow.InthecasethatVisactuallyaHilbertspace,andnotmerelyapre-Hilbertspace,thereisanalternativeandveryusefulcriterionforanorthonormalse-quencetobeabasis:LetMbethesetofalllimitsofnitelinearcombinationsofthei.AnyCauchysequenceinMconvergesinVsinceVisaHilbertspace,andthislimitbelongstoMsinceitisitselfalimitofnitelinearcombinationsoftheibythediagonalargumentforexample.ThusV=MM?,andtheiformabasisofM.SotheiformabasisofVifandonlyifM?=f0g.Butthisisthesameassayingthatnonon-zerovectorisorthogonaltoallthei.SowehaveprovedProposition2.1.1InaHilbertspace,theorthonormalsetfigisabasisifandonlyifnonon-zerovectorisorthogonaltoallthei.2.2Self-adjointtransformations.WecontinuetoletVdenoteapre-Hilbertspace.LetTbealineartransfor-mationofVintoitself.Thismeansthatforeveryv2VthevectorTv2VisdenedandthatTvdependslinearlyonv:Tav+bw=aTv+bTwforanytwovectorsvandwandanytwocomplexnumbersaandb.Werecallfromlinearalgebrathatanon-zerovectorviscalledaneigenvectorofTifTvisascalartimesv,inotherwordsifTv=vwherethenumberiscalledthecorrespondingeigenvalue.AlineartransformationTonViscalledsymmetricifforanypairofelementsvandwofVwehaveTv;w=v;Tw:NoticethatifvisaneigenvectorofasymmetrictransformationTwitheigenvalue,thenv;v=v;v=Tv;v=v;Tv=v;v= v;v;so= .Inotherwords,alleigenvaluesofasymmetrictransformationarereal.

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52CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.WewillletS=SVdenotetheunitsphere"ofV,i.e.Sdenotesthesetofall2Vsuchthatkk=1.AlineartransformationTiscalledboundedifkTkisboundedasrangesoverallofS.IfTisbounded,weletkTk:=max2SkTk:ThenkTvkkTkkvkforallv2V.Alineartransformationonanitedimensionalspaceisautomat-icallybounded,butnotsoforaninnitedimensionalspace.Also,foranylineartransformationT,wewillletNTdenotethekernelofT,soNT=fv2VjTv=0gandRTdenotetherangeofT,soRT:=fvjv=Twforsomew2Vg:BothNTandRTarelinearsubspacesofV.Forboundedtransformations,thephraseself-adjoint"issynonymouswithsymmetric".Lateronwewillneedtostudynon-boundednoteverywheredenedsymmetrictransformations,andthenarathersubtleandimportantdistinctionwillbemadebetweenself-adjointtransformationsandthosewhicharemerelysymmetric.Butfortherestofthissectionwewillonlybecon-sideringboundedlineartransformations,andsowewillfreelyusethephraseself-adjoint",andusuallydroptheadjectivebounded"sinceallourtrans-formationswillbeassumedtobebounded.Wedenotethesetofallboundedself-adjointtransformationsbyA,orbyAVifweneedtomakeVexplicit.2.2.1Non-negativeself-adjointtransformations.IfTisaself-adjointtransformation,thenTv;v=v;Tv= Tv;vsoTv;visalwaysarealnumber.Moregenerally,foranypairofelementsvandw,Tv;w= Tw;v:SinceTv;wdependslinearlyonvforxedw,weseethattherulewhichassignstoeverypairofelementsvandwthenumberTv;wsatisesthersttwoconditionsinourdenitionofasemi-scalarproduct.SinceTv;vmightbenegative,condition3.ofthedenitionneednotbesatised.Thisleadstothefollowingdenition:Aself-adjointtransformationTiscallednon-negativeifTv;v08v2V:

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2.2.SELF-ADJOINTTRANSFORMATIONS.53SoifTisanon-negativeself-adjointtransformation,thentherulewhichassignstoeverypairofelementsvandwthenumberTv;wisasemi-scalarproducttowhichwemayapplytheCauchy-SchwartzinequalityandconcludethatjTv;wjTv;v1 2Tw;w1 2:NowletusassumeinadditionthatTisboundedwithnormkTk.Letustakew=Tvintheprecedinginequality.WegetkTvk2=jTv;TvjTv;v1 2TTv;Tv1 2:NowapplytheCauchy-Schwartzinequalityfortheoriginalscalarproducttothelastfactorontheright:TTv;Tv1 2kTTvk1 2kTvk1 2kTk1 2kTvk1 2kTvk1 2=kTk1 2kTvk;wherewehaveusedthedeningpropertyofkTkintheformkTTvkkTkkTvk.SubstitutingthisintothepreviousinequalitywegetkTvk2Tv;v1 2kTk1 2kTvk:IfkTvk6=0wemaydividethisinequalitybykTvktoobtainkTvkkTk1 2Tv;v1 2:.17ThisinequalityisclearlytrueifkTvk=0andsoholdsinallcases.Wewillmakemuchuseofthisinequality.Forexample,itfollowsfrom.17thatTv;v=0Tv=0:.18Italsofollowsfrom.17thatifwehaveasequencefvngofvectorswithTvn;vn!0thenkTvvk!0andsoTvn;vn!0Tvn!0:.19NoticethatifTisaboundedselfadjointtransformation,notnecessarilynon-negative,thenrI)]TJ/F11 9.963 Tf 10.109 0 Td[(Tisanon-negativeself-adjointtransformationifrkTk:Indeed,rI)]TJ/F11 9.963 Tf 9.963 0 Td[(Tv;v=rv;v)]TJ/F8 9.963 Tf 9.963 0 Td[(Tv;vr)-222(kTkv;v0since,byCauchy-Schwartz,Tv;vjTv;vjkTvkkvkkTkkvk2=kTkv;v:SowemayapplytheprecedingresultstorI)]TJ/F11 9.963 Tf 9.963 0 Td[(T.

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54CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.2.3Compactself-adjointtransformations.Wesaythattheself-adjointtransformationTiscompactifithasthefollowingproperty:Givenanysequenceofelementsun2S,wecanchooseasubsequenceunisuchthatthesequenceTuniconvergestoalimitinV.Someremarksaboutthiscomplicatedlookingdenition:IncaseVisnitedimensional,everylineartransformationisbounded,hencethesequenceTunliesinaboundedregionofournitedimensionalspace,andhencebythecom-pletenesspropertyoftherealandhencecomplexnumbers,wecanalwaysndsuchaconvergentsubsequence.SoinnitedimensionseveryTiscompact.Moregenerally,thesameargumentshowsthatifRTisnitedimensionalandTisboundedthenTiscompact.SothedenitionisofinterestessentiallyinthecasewhenRTisinnitedimensional.AlsonoticethatifTiscompact,thenTisbounded.OtherwisewecouldndasequenceunofelementsofSsuchthatkTunknandsonosubsequenceTunicanconverge.Wenowcometothekeyresultwhichwewilluseoverandoveragain:Theorem2.3.1LetTbeacompactself-adjointoperator.ThenRThasanorthonormalbasisfigconsistingofeigenvectorsofTandifRTisinnitedimensionalthenthecorrespondingsequencefrngofeigenvaluesconvergesto0.Proof.WeknowthatTisbounded.IfT=0thereisnothingtoprove.SoassumethatT6=0andletm1:=kTk>0:BythedenitionofkTkwecanndasequenceofvectorsun2SsuchthatkTunk!kTk.BythedenitionofcompactnesswecanndasubsequenceofthissequencesothatTuni!wforsomew2V.Ontheotherhand,thetransformationT2isself-adjointandboundedbykTk2.HencekTk2I)]TJ/F11 9.963 Tf 10.29 0 Td[(T2isnon-negative,andkTk2I)]TJ/F11 9.963 Tf 9.962 0 Td[(T2un;un=kTk2)-222(kTunk2!0:Soweknowfrom.19thatkTk2un)]TJ/F11 9.963 Tf 9.963 0 Td[(T2un!0:PassingtothesubsequencewehaveT2uni=TTuni!TwandsokTk2uni!Tworuni!1 m21Tw:ApplyingTtothiswegetTuni!1 m21T2w

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2.3.COMPACTSELF-ADJOINTTRANSFORMATIONS.55orT2w=m21w:Alsokwk=kTk=m16=0.Sow6=0.SowisaneigenvectorofT2witheigenvaluem21.Wehave0=T2)]TJ/F11 9.963 Tf 9.963 0 Td[(m21w=T+m1T)]TJ/F11 9.963 Tf 9.963 0 Td[(m1w:IfT)]TJ/F11 9.963 Tf 10.453 0 Td[(m1w=0,thenwisaneigenvectorofTwitheigenvaluem1andwenormalizebysetting1:=1 kwkw:Thenk1k=1andT1=m11:IfT)]TJ/F11 9.963 Tf 10.131 0 Td[(m1w6=0theny:=T)]TJ/F11 9.963 Tf 10.131 0 Td[(m1wisaneigenvectorofTwitheigenvalue)]TJ/F11 9.963 Tf 7.749 0 Td[(m1andagainwenormalizebysetting1:=1 kyky:Sowehavefoundaunitvector12RTwhichisaneigenvectorofTwitheigenvaluer1=m1.NowletV2:=?1:Ifw;1=0,thenTw;1=w;T1=r1w;1=0:Inotherwords,TV2V2andwecanconsiderthelineartransformationTrestrictedtoV2whichisagaincompact.Ifweletm2denotethenormofthelineartransformationTwhenrestrictedtoV2thenm2m1andwecanapplytheprecedingproceduretondauniteigenvector2witheigenvaluem2.Weproceedinductively,lettingVn:=f1;:::;n)]TJ/F7 6.974 Tf 6.226 0 Td[(1g?andndaneigenvectornofTrestrictedtoVnwitheigenvaluemn6=0iftherestrictionofTtoVnisnotzero.Sotherearetwoalternatives:aftersomenitestagetherestrictionofTtoVniszero.InthiscaseRTisnitedimensionalwithorthonormalbasis1;:::;n)]TJ/F7 6.974 Tf 6.227 0 Td[(1.OrTheprocesscontinuesindenitelysothatateachstagetherestrictionofTtoVnisnotzeroandwegetaninnitesequenceofeigenvectorsandeigenvaluesriwithjrijjri+1j.

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56CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.Therstcaseisoneofthealternativesinthetheorem,soweneedtolookatthesecondalternative.Werstprovethatjrnj!0.Ifnot,thereissomec>0suchthatjrnjcforallnsincethejrnjaredecreasing.Ifi6=j,thenbythePythagoreantheoremwehavekTi)]TJ/F11 9.963 Tf 9.963 0 Td[(Tjk2=krii)]TJ/F11 9.963 Tf 9.963 0 Td[(rjjk2=r2ikik2+r2jkjk2:Sincekik=kjj=1thisgiveskTi)]TJ/F11 9.963 Tf 9.963 0 Td[(Tjk2=r2i+r2j2c2:HencenosubsequenceoftheTicanconverge,sinceallthesevectorsareatleastadistancecp 2apart.ThiscontradictsthecompactnessofT.TocompletetheproofofthetheoremwemustshowthattheiformabasisofRT.Soifw=TvwemustshowthattheFourierseriesofwwithrespecttotheiconvergestow.WebeginwiththeFouriercoecientsofvrelativetotheiwhicharegivenbyan=v;n:ThentheFouriercoecientsofwaregivenbybi=w;i=Tv;i=v;Ti=v;rii=riai:Sow)]TJ/F10 6.974 Tf 14.695 12.454 Td[(nXi=1bii=Tv)]TJ/F10 6.974 Tf 14.696 12.454 Td[(nXi=1airii=Tv)]TJ/F10 6.974 Tf 14.696 12.454 Td[(nXi=1aii:Nowv)]TJ/F1 9.963 Tf 9.962 7.472 Td[(Pni=1aiiisorthogonalto1;:::;nandhencebelongstoVn+1.SokTv)]TJ/F10 6.974 Tf 14.696 12.454 Td[(nXi=1aiikjrn+1jkv)]TJ/F10 6.974 Tf 14.695 12.454 Td[(nXi=1aiik:BythePythagoreantheorem,kv)]TJ/F10 6.974 Tf 14.696 12.453 Td[(nXi=1aiikkvk:Puttingthetwopreviousinequalitiestogetherwegetkw)]TJ/F10 6.974 Tf 14.696 12.453 Td[(nXi=1biik=kTv)]TJ/F10 6.974 Tf 14.695 12.453 Td[(nXi=1aiikjrn+1jkvk!0:ThisprovesthattheFourierseriesofwconvergestowconcludingtheproofofthetheorem.Theconverse"oftheaboveresultiseasy.Hereisaversion:SupposethatHisaHilbertspacewithanorthonormalbasisfigconsistingofeigenvectors

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2.4.FOURIER'SFOURIERSERIES.57ofanoperatorT,soTi=ii,andsupposethati!0asi!1.ThenTiscompact.Indeed,foreachjwecanndanN=Njsuchthatjrj<1 j8r>Nj:WecanthenletHjdenotetheclosedsubspacespannedbyalltheeigenvectorsr;r>Nj,sothatH=H?jHjisanorthogonaldecompositionandH?jisnitedimensional,infactisspannedtherstNjeigenvectorsofT.Nowletfuigbeasequenceofvectorswithkuik1say.Wedecomposeeachelementasui=u0iu00i;u0i2H?1;u00i2Hj:Wecanchooseasubsequencesothatu0ikconverges,becausetheyallbelongtoanitedimensionalspace,andhencesodoesTuiksinceTisbounded.WecandecomposeeveryelementofthissubsequenceintoitsH?2andH2components,andchooseasubsequencesothattherstcomponentconverges.Proceedinginthisway,andthenusingtheCantordiagonaltrickofchoosingthek-thtermofthek-thselectedsubsequence,wehavefoundasubsequencesuchthatforanyxedj,thenowrelabeledsubsequence,theH?jcomponentofTujconverges.ButtheHjcomponentofTujhasnormlessthan1=j,andsothesequenceconvergesbythetriangleinequality.2.4Fourier'sFourierseries.Wewanttoapplythetheoremaboutcompactself-adjointoperatorsthatweprovedintheprecedingsectiontoconcludethatthefunctionseinxformanorthonormalbasisofthespaceCT.Infact,adirectproofofthisfactiselementary,usingintegrationbyparts.Sowewillpausetogiventhisdirectproof.Thenwewillgobackandgiveamorecomplicatedproofofthesamefactusingourtheoremoncompactoperators.Thereasonforgivingthemorecomplicatedproofisthatitextendstofarmoregeneralsituations.2.4.1Proofbyintegrationbyparts.WehaveletCTdenotethespaceofcontinuousfunctionsonthereallinewhichareperiodicwithperiod2.WewillletC1TdenotethespaceofperiodicfunctionswhichhaveacontinuousrstderivativenecessarilyperiodicandbyC2Tthespaceofperiodicfunctionswithtwocontinuousderivatives.IffandgbothbelongtoC1Tthenintegrationbypartsgives1 2Z)]TJ/F10 6.974 Tf 6.226 0 Td[(f0 gdx=)]TJ/F8 9.963 Tf 11.963 6.74 Td[(1 2Z)]TJ/F10 6.974 Tf 6.227 0 Td[(f g0dx

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58CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.sincetheboundaryterms,whichnormallyariseintheintegrationbypartsformula,cancel,duetotheperiodicityoffandg.Ifwetakeg=einx=in;n6=0theintegralontherighthandsideofthisequationistheFouriercoecient:cn=1 2Z)]TJ/F10 6.974 Tf 6.226 0 Td[(fxe)]TJ/F10 6.974 Tf 6.227 0 Td[(inxdx:Wethusobtaincn=1 in1 2Z)]TJ/F10 6.974 Tf 6.227 0 Td[(f0xe)]TJ/F10 6.974 Tf 6.227 0 Td[(inxdxso,forn6=0,jcnjA nwhereA:=1 2Z)]TJ/F10 6.974 Tf 6.226 0 Td[(jf0xjdxisaconstantindependentofnbutdependingonf.Iff2C2Twecantakegx=)]TJ/F11 9.963 Tf 7.749 0 Td[(einx=n2andintegratebypartstwice.Weconcludethatforn6=0jcnjB n2whereB:=1 2Z)]TJ/F10 6.974 Tf 6.227 0 Td[(jf00xj2isagainindependentofn.ButthisprovesthattheFourierseriesoff,Xcneinxconvergesuniformlyandabsolutelyforandf2C2T.Thelimitofthisseriesisthereforesomecontinuousperiodicfunction.Wemustprovethatthislimitequalsf.SowemustprovethatateachpointfXcneiny!fy:Replacingfxbyfx)]TJ/F11 9.963 Tf 9.057 0 Td[(yitisenoughtoprovethisformulaforthecasey=0.Sowemustprovethatforanyf2C2TwehavelimN;M!1MX)]TJ/F10 6.974 Tf 6.226 0 Td[(Ncn!f:Writefx=fx)]TJ/F11 9.963 Tf 10.622 0 Td[(f+f.TheFouriercoecientsofanyconstantfunctioncallvanishexceptforthec0termwhichequalsc.Sotheabovelimitistriviallytruewhenfisaconstant.Hence,inprovingtheaboveformula,itisenoughtoproveitundertheadditionalassumptionthatf=0,andweneedtoprovethatinthiscaselimN;M!1c)]TJ/F10 6.974 Tf 6.226 0 Td[(N+c)]TJ/F10 6.974 Tf 6.227 0 Td[(N+1++cM!0:Theexpressioninparenthesisis1 2Z)]TJ/F10 6.974 Tf 6.227 0 Td[(fx gN;Mxdx

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2.4.FOURIER'SFOURIERSERIES.59wheregN;Mx=e)]TJ/F10 6.974 Tf 6.226 0 Td[(iNx+e)]TJ/F10 6.974 Tf 6.226 0 Td[(iN)]TJ/F7 6.974 Tf 6.226 0 Td[(1x++eiMx=e)]TJ/F10 6.974 Tf 6.226 0 Td[(iNx1+eix++eiM+Nx=e)]TJ/F10 6.974 Tf 6.227 0 Td[(iNx1)]TJ/F11 9.963 Tf 9.963 0 Td[(eiM+N+1x 1)]TJ/F11 9.963 Tf 9.963 0 Td[(eix=e)]TJ/F10 6.974 Tf 6.227 0 Td[(iNx)]TJ/F11 9.963 Tf 9.962 0 Td[(eiM+1x 1)]TJ/F11 9.963 Tf 9.963 0 Td[(eix;x6=0wherewehaveusedtheformulaforageometricsum.Byl'H^opital'srule,thisextendscontinuouslytothevalueM+N+1forx=0.Nowf=0,andsincefhastwocontinuousderivatives,thefunctionhx:=fx 1)]TJ/F11 9.963 Tf 9.962 0 Td[(e)]TJ/F10 6.974 Tf 6.227 0 Td[(ixdenedforx6=0oranymultipleof2extends,byl'H^opital'srule,toafunc-tiondenedatallvalues,andwhichiscontinuouslydierentiableandperiodic.Hencethelimitwearecomputingis1 2Z)]TJ/F10 6.974 Tf 6.226 0 Td[(hxeiNxdx)]TJ/F8 9.963 Tf 14.176 6.74 Td[(1 2Z)]TJ/F10 6.974 Tf 6.227 0 Td[(hxe)]TJ/F10 6.974 Tf 6.227 0 Td[(iM+1xdxandweknowthateachofthesetermstendstozero.WehavethusprovedthattheFourierseriesofanytwicedierentiableperi-odicfunctionconvergesuniformlyandabsolutelytothatfunction.IfweconsiderthespaceC2Twithourusualscalarproductf;g=1 2Z)]TJ/F10 6.974 Tf 6.227 0 Td[(f gdxthenthefunctionseinxaredenseinthisspace,sinceuniformconvergenceimpliesconvergenceinthekknormassociatedto;.So,ongeneralprinciples,Bessel'sinequalityandParseval'sequationhold.ItisnottrueingeneralthattheFourierseriesofacontinuousfunctionconvergesuniformlytothatfunctionorconvergesatallinthesenseofuniformconvergence.HoweveritistruethatwedohaveconvergenceintheL2norm,i.e.theHilbertspacekknormonCT.Toprovethis,weneedonlyprovethattheexponentialfunctionseinxaredense,andsincetheyaredenseinC2T,itisenoughtoprovethatC2TisdenseinCT.Forthis,letbeafunctiondenedonthelinewithatleasttwocontinuousboundedderivativeswith=1andoftotalintegralequaltooneandwhichvanishesrapidlyatinnity.AfavoriteistheGaussnormalfunctionx:=1 p 2e)]TJ/F10 6.974 Tf 6.227 0 Td[(x2=2Equallywell,wecouldtaketobeafunctionwhichactuallyvanishesoutsideofsomeneighborhoodoftheorigin.Lettx:=1 tx t:

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60CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.Ast!0thefunctiontbecomesmoreandmoreconcentratedabouttheorigin,butstillhastotalintegralone.Hence,foranyboundedcontinuousfunctionf,thefunctiont?fdenedbyt?fx:=Z1fx)]TJ/F11 9.963 Tf 9.962 0 Td[(ytydy=Z1futx)]TJ/F11 9.963 Tf 9.963 0 Td[(udu:satisest?f!funiformlyonanyniteinterval.Fromtherightmostexpres-sionfort?faboveweseethatt?fhastwocontinuousderivatives.Fromtherstexpressionweseethatt?fisperiodiciffis.ThisprovesthatC2TisdenseinCT.WehavethusprovedconvergenceintheL2norm.2.4.2Relationtotheoperatord dx.EachofthefunctionseinxisaneigenvectoroftheoperatorD=d dxinthatD)]TJ/F11 9.963 Tf 4.566 -8.069 Td[(einx=ineinx:SotheyarealsoeigenvaluesoftheoperatorD2witheigenvalues)]TJ/F11 9.963 Tf 7.749 0 Td[(n2.Also,onthespaceoftwicedierentiableperiodicfunctionstheoperatorD2satisesD2f;g=1 2Z)]TJ/F10 6.974 Tf 6.227 0 Td[(f00x gxdx=f0x gx)]TJ/F10 6.974 Tf 6.226 0 Td[()]TJ/F8 9.963 Tf 14.177 6.74 Td[(1 2Z)]TJ/F10 6.974 Tf 6.227 0 Td[(f0x g0xdxbyintegrationbyparts.Sincef0andgareassumedtobeperiodic,theendpointtermscancel,andintegrationbypartsoncemoreshowsthatD2f;g=f;D2g=)]TJ/F8 9.963 Tf 7.749 0 Td[(f0;g0:ButofcourseDandcertainlyD2isnotdenedonCTsincesomeofthefunc-tionsbelongingtothisspacearenotdierentiable.Furthermore,theeigenvaluesofD2aretendingtoinnityratherthantozero.SosomehowtheoperatorD2mustbereplacedwithsomethinglikeitsinverse.Infact,wewillworkwiththeinverseofD2)]TJ/F8 9.963 Tf 9.963 0 Td[(1,butrstsomepreliminaries.WewillletC2[)]TJ/F11 9.963 Tf 7.749 0 Td[(;]denotethefunctionsdenedon[)]TJ/F11 9.963 Tf 7.749 0 Td[(:]andtwicedif-ferentiablethere,withcontinuoussecondderivativesuptotheboundary.WedenotebyC[)]TJ/F11 9.963 Tf 7.749 0 Td[(;]thespaceoffunctionsdenedon[)]TJ/F11 9.963 Tf 7.749 0 Td[(;]whicharecontinu-ousuptotheboundary.WecanregardCTasthesubspaceofC[)]TJ/F11 9.963 Tf 7.749 0 Td[(;]con-sistingofthosefunctionswhichsatisfytheboundaryconditionsf=f)]TJ/F11 9.963 Tf 7.749 0 Td[(andthenextendedtothewholelinebyperiodicity.WeregardC[)]TJ/F11 9.963 Tf 7.749 0 Td[(;]asapre-Hilbertspacewiththesamescalarproductthatwehavebeenusing:f;g=1 2Z)]TJ/F10 6.974 Tf 6.227 0 Td[(fx gxdx:

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2.4.FOURIER'SFOURIERSERIES.61IfwecanshowthateveryelementofC[)]TJ/F11 9.963 Tf 7.748 0 Td[(;]isasumofitsFourierseriesinthepre-HilbertspacesensethenthesamewillbetrueforCT.SowewillworkwithC[)]TJ/F11 9.963 Tf 7.749 0 Td[(;].WecanconsidertheoperatorD2)]TJ/F8 9.963 Tf 9.963 0 Td[(1asalinearmapD2)]TJ/F8 9.963 Tf 9.963 0 Td[(1:C2[)]TJ/F11 9.963 Tf 7.749 0 Td[(;]!C[)]TJ/F11 9.963 Tf 7.749 0 Td[(;]:Thismapissurjective,meaningthatgivenanycontinuousfunctiongwecanndatwicedierentiablefunctionfsatisfyingthedierentialequationf00)]TJ/F11 9.963 Tf 9.963 0 Td[(f=g:Infactwecanndawholetwodimensionalfamilyofsolutionsbecausewecanaddanysolutionofthehomogeneousequationh00)]TJ/F11 9.963 Tf 9.963 0 Td[(h=0tofandstillobtainasolution.Wecouldwritedownanexplicitsolutionfortheequationf00)]TJ/F11 9.963 Tf 9.994 0 Td[(f=g,butwewillnotneedto.Itisenoughforustoknowthatthesolutionexists,whichfollowsfromthegeneraltheoryofordinarydierentialequations.Thegeneralsolutionofthehomogeneousequationisgivenbyhx=aex+be)]TJ/F10 6.974 Tf 6.226 0 Td[(x:LetMC2[)]TJ/F11 9.963 Tf 7.749 0 Td[(;]bethesubspaceconsistingofthosefunctionswhichsatisfytheperiodicbound-aryconditions"f=f)]TJ/F11 9.963 Tf 7.749 0 Td[(;f0=f0)]TJ/F11 9.963 Tf 7.749 0 Td[(:Givenanyfwecanalwaysndasolutionofthehomogeneousequationsuchthatf)]TJ/F11 9.963 Tf 9.969 0 Td[(h2M.Indeed,weneedtochoosethecomplexnumbersaandbsuchthatifhisasgivenabove,thenh)]TJ/F11 9.963 Tf 9.962 0 Td[(h)]TJ/F11 9.963 Tf 7.748 0 Td[(=f)]TJ/F11 9.963 Tf 9.962 0 Td[(f)]TJ/F11 9.963 Tf 7.748 0 Td[(;andh0)]TJ/F11 9.963 Tf 9.962 0 Td[(h0)]TJ/F11 9.963 Tf 7.749 0 Td[(=f0)]TJ/F11 9.963 Tf 9.962 0 Td[(f0)]TJ/F11 9.963 Tf 7.749 0 Td[(:Collectingcoecientsanddenotingtherighthandsideoftheseequationsbycanddwegetthelinearequationse)]TJ/F11 9.963 Tf 9.963 0 Td[(e)]TJ/F10 6.974 Tf 6.226 0 Td[(a)]TJ/F11 9.963 Tf 9.963 0 Td[(b=c;e)]TJ/F11 9.963 Tf 9.963 0 Td[(e)]TJ/F10 6.974 Tf 6.227 0 Td[(a+b=dwhichhasauniquesolution.SothereexistsauniqueoperatorT:C[)]TJ/F11 9.963 Tf 7.749 0 Td[(;]!MwiththepropertythatD2)]TJ/F11 9.963 Tf 9.962 0 Td[(IT=I:

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62CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.WewillprovethatTisselfadjointandcompact..20Oncewewillhaveprovedthisfact,thenweknoweveryelementofMcanbeexpandedintermsofaseriesconsistingofeigenvectorsofTwithnon-zeroeigenvalues.ButifTw=wthenD2w=D2)]TJ/F11 9.963 Tf 9.963 0 Td[(Iw+w=1 [D2)]TJ/F11 9.963 Tf 9.962 0 Td[(IT]w+w=1 +1w:SowmustbeaneigenvectorofD2;itmustsatisfyw00=w:Soif=0thenw=aconstantisasolution.If=r2>0thenwisalinearcombinationoferxande)]TJ/F10 6.974 Tf 6.227 0 Td[(rxandasweshowedabove,nonon-zerosuchcombinationcanbelongtoM.If=)]TJ/F11 9.963 Tf 7.749 0 Td[(r2thenthesolutionisalinearcombinationofeirxande)]TJ/F10 6.974 Tf 6.226 0 Td[(irxandtheaboveargumentshowsthatrmustbesuchthateir=e)]TJ/F10 6.974 Tf 6.227 0 Td[(irsor=nisaninteger.Thus.20willshowthattheeinxareabasisofM,andalittlemoreworkthatwewilldoattheendwillshowthattheyareinfactalsoabasisofC[)]TJ/F11 9.963 Tf 7.749 0 Td[(;].Butrstletusworkon.20.ItiseasytoseethatTisselfadjoint.Indeed,letf=Tuandg=TvsothatfandgareinMandu;Tv=[D2)]TJ/F8 9.963 Tf 9.962 0 Td[(1]f;g=)]TJ/F8 9.963 Tf 7.748 0 Td[(f0;g0)]TJ/F8 9.963 Tf 9.963 0 Td[(f;g=f;[D2)]TJ/F8 9.963 Tf 9.962 0 Td[(1]g=Tu;vwherewehaveusedintegrationbypartsandtheboundaryconditionsdeningMforthetwomiddleequalities.2.4.3Garding'sinequality,specialcase.Wenowturntothecompactness.Wehavealreadyveriedthatforanyf2Mwehave[D2)]TJ/F8 9.963 Tf 9.962 0 Td[(1]f;f=)]TJ/F8 9.963 Tf 7.749 0 Td[(f0;f0)]TJ/F8 9.963 Tf 9.963 0 Td[(f;f:Takingabsolutevalueswegetkf0k2+kfk2j[D2)]TJ/F8 9.963 Tf 9.963 0 Td[(1]f;fj:.21Weactuallygetequalityhere,themoregeneralversionofthisthatwewilldeveloplaterwillbeaninequality.Letu=[D2)]TJ/F8 9.963 Tf 9.962 0 Td[(1]fandusetheCauchy-Schwartzinequalityj[D2)]TJ/F8 9.963 Tf 9.962 0 Td[(1]f;fj=ju;fjkukkfk

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2.4.FOURIER'SFOURIERSERIES.63ontherighthandsideof.21toconcludethatkfk2kukkfkorkfkkuk:Use.21againtoconcludethatkf0k2kukkfkkuk2bytheprecedinginequality.Wehavef=Tu,andletusnowsupposethatuliesontheunitspherei.e.thatkuk=1.Thenwehaveprovedthatkfk1;andkf0k1:.22Wewishtoshowthatfromanysequenceoffunctionssatisfyingthesetwocondi-tionswecanextractasubsequencewhichconverges.Hereconvergencemeans,ofcourse,withrespecttothenormgivenbykfk2=1 2Z)]TJ/F10 6.974 Tf 6.226 0 Td[(jfxj2dx:Infact,wewillprovesomethingstronger:thatgivenanysequenceoffunctionssatisfying.22wecanndasubsequencewhichconvergesintheuniformnormkfk1:=maxx2[)]TJ/F10 6.974 Tf 6.227 0 Td[(;]jfxj:Noticethatkfk=1 2Z)]TJ/F10 6.974 Tf 6.227 0 Td[(jfxj2dx1 21 2Z)]TJ/F10 6.974 Tf 6.226 0 Td[(kfk12dx1 2=kfk1soconvergenceintheuniformnormimpliesconvergenceinthenormwehavebeenusing.Toproveourresult,noticethatforanya
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64CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.Weconcludethatjfb)]TJ/F11 9.963 Tf 9.963 0 Td[(faj1 2jb)]TJ/F11 9.963 Tf 9.962 0 Td[(aj1 2:.23Inthisinequality,letustakebtobeapointwherejfjtakesonitsmaximumvalue,sothatjfbj=kfk1.Letabeapointwherejfjtakesonitsminimumvalue.Ifnecessaryinterchangetheroleofaandbtoarrangethata
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2.5.THEHEISENBERGUNCERTAINTYPRINCIPLE.65LetVbeapre-Hilbertspace,andSdenotetheunitsphereinV.Ifandaretwounitvectorsi.e.elementsofStheirscalarproduct;isacomplexnumberwith0j;j21.Inquantummechanics,thisnumberistakenasaprobability.Althoughintherealworld"Visusuallyinnitedimensional,wewillwarmupbyconsideringthecasewhereVisnitedimensional.Givena2Sandanorthonormalbasis1;:::;nofV,wehave1=kk2=j;1j2++j;nj2:Thesaysthatthevariousprobabilitiesj;ij2adduptoone.Werecallsomelanguagefromelementaryprobabilitytheory:Supposewehaveanexperimentresultinginanitenumberofmeasurednumericaloutcomesi,eachwithprob-abilitypiofoccurring.Thenthemeanhiisdenedbyhi:=1p1++npnanditsvariance22:=1)-222(hi2p1++n)-222(hi2pnandanimmediatecomputationshowsthat2=h2i)-222(hi2:Thesquarerootofthevarianceiscalledthestandarddeviation.Thevari-anceorthestandarddeviationmeasuresthespread"ofthepossiblevaluesof.TounderstanditsmeaningwehaveChebychev'sinequalitywhichestimatestheprobabilitythatkcandeviatefromhibyasmuchasrforanyposi-tivenumberr.Chebychev'sinequalitysaysthatthisprobabilityis1=r2.InsymbolsProbjk)-222(hijr1 r2:Indeed,theprobabilityontheleftisthesumofallthepksuchthatjk)-133(hijr.DenotingthissumbyPrwehaveXrpkXrpk)-222(hi2 r22Xallkpk)-222(hi2 r22=1 r22Xallk)-222(hi2pk=1 r2:Replacingibyi+cdoesnotchangethevariance.NowsupposethatAisaself-adjointoperatoronV,thattheiaretheeigenvaluesofAwitheigenvectorsiconstitutinganorthonormalbasis,andthatthepi=j;ij2asabove.1.Showthathi=A;andthat2=A2;)]TJ/F8 9.963 Tf 9.963 0 Td[(A;2:

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66CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.WewillwritetheexpressionA;ashAi,Inquantummechanicsaunitvectoriscalledastateandaself-adjointoperatoriscalledanobservableandtheexpressionhAiiscalledtheexpectationoftheobservableAinthestate.Similarlywedenote)]TJ/F8 9.963 Tf 4.566 -8.069 Td[(A2;)]TJ/F8 9.963 Tf 9.963 0 Td[(A;21=2byA.ItiscalledtheuncertaintyoftheobservableAinthestate.NoticethatA2=hA)-222(hAiI2iwhereIdenotestheidentityoperator.IndeedA)]TJ/F8 9.963 Tf 9.962 0 Td[(A;I2=A2)]TJ/F8 9.963 Tf 9.962 0 Td[(2A;A+A;2IsohA)-222(hAi2i=A2;)]TJ/F8 9.963 Tf 9.963 0 Td[(2hAi2+hAi2=hA2i)-222(hAi2:Whenthestateisxedinthecourseofdiscussion,wewilldropthesub-scriptandwritehAiandAinsteadofhAiandA.Forexample,wewouldwritethepreviousresultasA=hA)-222(hAiI2i:IfAandBareoperatorswelet[A;B]denotethecommutator:[A;B]:=AB)]TJ/F11 9.963 Tf 9.962 0 Td[(BA:Noticethat[A;B]=)]TJ/F8 9.963 Tf 7.749 0 Td[([B;A]and[I;B]=0foranyB.SoifAandBareselfadjoint,soisi[A;B]andreplacingAbyA)-194(hAiIandBbyB)-194(hBiIdoesnotchangeA,Bori[A;B].TheuncertaintyprinciplesaysthatforanytwoobservablesAandBwehaveAB1 2jhi[A;B]ijq:Proof.SetA1:=A)-222(hAi;B1:=B)-222(hBisothat[A1;B1]=[A;B]:Let:=A1+ixB1:Then;=A2)]TJ/F11 9.963 Tf 9.963 0 Td[(xhi[A;B]i+B2:Since;0forallxthisimpliesthatb24acthathi[A;B]i24A2B2;andtakingsquarerootsgivestheresult.ThepurposeofthenextfewsectionsistoprovideavastgeneralizationoftheresultsweobtainedfortheoperatorD2.Wewillprovethecorrespondingresultsforanyelliptic"dierentialoperatordenitionsbelow.

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2.6.THESOBOLEVSPACES.67Iplantostudydierentialoperatorsactingonvectorbundlesovermanifolds.Butitrequiressomeeorttosetthingsup,andIwanttogettothekeyanalyticideaswhichareessentiallyrepeatedapplicationsofintegrationbyparts.SoIwillstartwithellipticoperatorsLactingonfunctionsonthetorusT=Tn,wheretherearenoboundarytermswhenweintegratebyparts.Thenanimmediateextensiongivestheresultforellipticoperatorsonfunctionsonmanifolds,andalsoforboundaryvalueproblemssuchastheDirichletproblem.ThetreatmenthereratherslavishlyfollowsthetreatmentbyBersandSchechterinPartialDierentialEquationsbyBers,JohnandSchechterAMS.2.6TheSobolevSpaces.RecallthatTnowstandsforthen-dimensionaltorus.LetP=PTdenotethespaceoftrigonometricpolynomials.Thesearefunctionsonthetorusoftheformux=Xa`ei`xwhere`=`1;:::;`nisann-tupletofintegersandthesumisnite.Foreachintegertpositive,zeroornegativeweintroducethescalarproductu;vt:=X`+``ta` b`:.24Fort=0thisisthescalarproductu;v0=1 nZTux vxdx:Thisdiersbyafactorof)]TJ/F10 6.974 Tf 6.227 0 Td[(nfromthescalarproductthatisusedbyBersandSchecter.Wewilldenotethenormcorrespondingtothescalarproduct;sbykks.If:=)]TJ/F1 9.963 Tf 9.409 14.047 Td[(@2 @x12++@2 @xn2theoperator+satises+u=X+``a`ei`xandso+tu;vs=u;+tvs=u;vs+tandk+tuks=kuks+2t:.25

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68CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.WethengetthegeneralizedCauchy-Schwartzinequality"ju;vsjkuks+tkvks)]TJ/F10 6.974 Tf 6.227 0 Td[(t.26foranyt,asaconsequenceoftheusualCauchy-Schwartzinequality.Indeed,X`+``sa` b`=X`+``s+t 2a`+``s)]TJ/F9 4.981 Tf 5.397 0 Td[(t 2 b`=1+s+t 2u;+s)]TJ/F9 4.981 Tf 5.396 0 Td[(t 2v0k+s+t 2uk0k+s)]TJ/F9 4.981 Tf 5.396 0 Td[(t 2vk0=kuks+tkvks)]TJ/F10 6.974 Tf 6.227 0 Td[(t:ThegeneralizedCauchy-SchwartzinequalityreducestotheusualCauchy-Schwartzinequalitywhent=0.Clearlywehavekukskuktifst:IfDpdenotesapartialderivative,Dp=@jpj @x1p1@xnpmthenDpu=Xi`pa`ei`x:Intheseequationsweareusingthefollowingnotations:Ifp=p1;:::;pnisavectorwithnon-negativeintegerentrieswesetjpj:=p1++pn:If=1;:::;nisarowvectorwesetp:=p11p22pnnItisthenclearthatkDpuktkukt+jpj.27andsimilarlykuktconstantdependingontXjpjtkDpuk0ift0:.28Inparticular,

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2.6.THESOBOLEVSPACES.69Proposition2.6.1Thenormsu7!kuktt0andu7!XjpjtkDpuk0areequivalent.WeletHtdenotethecompletionofthespacePwithrespecttothenormkkt.EachHtisaHilbertspace,andwehavenaturalembeddingsHt,!Hsifs
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70CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.thenv2Hsfors<)]TJ/F10 6.974 Tf 8.945 3.923 Td[(n 2.Ifuisgivenbyux=P`a`ei`xisanytrigonometricpolynomial,thenu;v0=Xa`=u:Sothenaturalpairing.29allowsustoextendthelinearfunctionsendingu7!utoallofHtift>n 2.Wecannowgivevitstruename":itistheDiracdeltafunction"onthetoruswhereu;0=u:So2H)]TJ/F10 6.974 Tf 6.227 0 Td[(tfort>n 2,andtheprecedingequationisusuallywrittensymbolicallyas1 nZTuxxdx=u;butthetruemathematicalinterpretationisasgivenabove.WesetH1:=Ht;H:=[Ht:ThespaceH0isjustL2T,andwecanthinkofthespaceHt;t>0asconsistingofthosefunctionshavinggeneralizedL2derivativesuptoordert".CertainlyafunctionofclassCtbelongstoHt.Withalossofdegreeofdierentiabilitytheconverseistrue:Lemma2.6.1[Sobolev.]Ifu2Htandthn 2i+k+1thenu2CkTandsupx2TjDpuxjconst.kuktforjpjk:.31ByapplyingthelemmatoDpuitisenoughtoprovethelemmafork=0.Soweassumethatu2Htwitht[n=2]+1.ThenXja`j2X+``tja`j2X+``)]TJ/F10 6.974 Tf 6.227 0 Td[(t<1;sincetheseriesP+``)]TJ/F10 6.974 Tf 6.227 0 Td[(tconvergesfort[n=2]+1.Soforthisrangeoft,theFourierseriesforuconvergesabsolutelyanduniformly.Therighthandsideoftheaboveinequalitygivesthedesiredbound.QEDAdistributiononTnisalinearfunctionTonC1TnwiththecontinuityconditionthathT;ki!0wheneverDpk!0

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2.6.THESOBOLEVSPACES.71uniformlyforeachxedp.Ifu2H)]TJ/F10 6.974 Tf 6.226 0 Td[(twemaydenehu;i:=; u0andsinceC1TisdenseinHtwemayconcludeLemma2.6.2H)]TJ/F10 6.974 Tf 6.226 0 Td[(tisthespaceofthosedistributionsTwhicharecontinuousinthekktnorm,i.e.whichsatisfykkkt!0hT;ki!0:WethenobtainTheorem2.6.1[LaurentSchwartz.]Histhespaceofalldistributions.Inotherwords,anydistributionbelongstoH)]TJ/F10 6.974 Tf 6.227 0 Td[(tforsomet.Proof.SupposethatTisadistributionthatdoesnotbelongtoanyH)]TJ/F10 6.974 Tf 6.227 0 Td[(t.Thismeansthatforanyk>0wecanndaC1functionkwithkkkk<1 kandjhT;kij1:ButbyLemma2.6.1weknowthatkkkk<1 kimpliesthatDpk!0uniformlyforanyxedpcontradictingthecontinuitypropertyofT.QEDSupposethatisaC1functiononT.MultiplicationbyisclearlyaboundedoperatoronH0=L2T,andsoitisalsoaboundedoperatoronHt;t>0sincewecanexpandDpubyapplicationsofLeibnitz'srule.Fort=)]TJ/F11 9.963 Tf 7.749 0 Td[(s<0weknowbythegeneralizedCauchySchwartzinequalitythatkukt=supjv;u0j=kvks=supju; vj=kvkskuktk vks=kvks:Soinallcaseswehavekuktconst.dependingonandtkukt:.32LetL=XjpjmpxDpbeadierentialoperatorofdegreemwithC1coecients.ThenitfollowsfromtheabovethatkLukt)]TJ/F10 6.974 Tf 6.227 0 Td[(mconstantkukt.33wheretheconstantdependsonLandt.Lemma2.6.3[Rellich'slemma.]Ifs
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72CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.Proof.WemustshowthattheimageoftheunitballBofHtinHscanbecoveredbynitelymanyballsofradius.ChooseNsolargethat+``s)]TJ/F10 6.974 Tf 6.227 0 Td[(t=2< 2when``>N.LetZtbethesubspaceofHtconsistingofallusuchthata`=0when``N.Thisisaspaceofnitecodimension,andhencetheunitballofZ?tHtcanbecoveredbynitelymanyballsofradius 2.ThespaceZ?tconsistsofallusuchthata`=0when``>N.TheimageofZ?tinHsistheorthogonalcomplementoftheimageofZt.Ontheotherhand,foru2BZwehavekuk2s+Ns)]TJ/F10 6.974 Tf 6.227 0 Td[(tkuk2t 22:SotheimageofBZiscontainedinaballofradius 2.EveryelementoftheimageofBcanbewrittenasanorthogonalsumofanelementintheimageofBZ?tandanelementofBZtandsotheimageofBiscoveredbynitelymanyballsofradius.QED2.7Garding'sinequality.Letx;a;andbbepositivenumbers.Thenxa+x)]TJ/F10 6.974 Tf 6.227 0 Td[(b1becauseifx1therstsummandis1andifx1thesecondsummandis1.Settingx=1=aAgives1Aa+)]TJ/F10 6.974 Tf 6.227 0 Td[(b=aA)]TJ/F10 6.974 Tf 6.227 0 Td[(bifandAarepositive.Supposethatt1>s>t2andweseta=t1)]TJ/F11 9.963 Tf 8.492 0 Td[(s;b=s)]TJ/F11 9.963 Tf 8.493 0 Td[(t2andA=1+``.Thenweget+``s+``t1+)]TJ/F7 6.974 Tf 6.227 0 Td[(s)]TJ/F10 6.974 Tf 6.227 0 Td[(t2=t1)]TJ/F10 6.974 Tf 6.227 0 Td[(s+``t2andthereforekukskukt1+)]TJ/F7 6.974 Tf 6.226 0 Td[(s)]TJ/F10 6.974 Tf 6.226 0 Td[(t2=t1)]TJ/F10 6.974 Tf 6.226 0 Td[(skukt2ift1>s>t2;>0.34forallu2Ht1.Thiselementaryinequalitywillbethekeytoseveralargumentsinthissectionwherewewillcombine.34withintegrationbyparts.AdierentialoperatorL=PjpjmpxDpwithrealcoecientsandmeveniscalledellipticifthereisaconstantc>0suchthat)]TJ/F8 9.963 Tf 7.749 0 Td[(1m=2Xjpj=mapxpcm=2:.35Inthisinequality,thevectorisadummyvariable".Itstrueinvariantsignif-icanceisthatitisacovector,i.e.anelementofthecotangentspaceatx.The

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2.7.GARDING'SINEQUALITY.73expressionontheleftofthisinequalityiscalledthesymboloftheoperatorL.Itisahomogeneouspolynomialofdegreeminthevariablewhosecoecientsarefunctionsofx.ThesymbolofLissometimeswrittenasLorLx;.Anotherwayofexpressingcondition.35istosaythatthereisapositiveconstantcsuchthatLx;cforallxandsuchthat=1:WewillassumeuntilfurthernoticethattheoperatorLisellipticandthatmisapositiveeveninteger.Theorem2.7.1[Garding'sinequality.]Foreveryu2C1Twehaveu;Lu0c1kuk2m=2)]TJ/F11 9.963 Tf 9.963 0 Td[(c2kuk20.36wherec1andc2areconstantsdependingonL.Remark.Ifu2Hm=2,thenbothsidesoftheinequalitymakesense,andwecanapproximateuinthekkm=2normbyC1functions.Soonceweprovethetheorem,weconcludethatitisalsotrueforallelementsofHm=2.Wewillprovethetheoreminstages:1.WhenLisconstantcoecientandhomogeneous.2.WhenLishomogeneousandapproximatelyconstant.3.WhentheLcanhavelowerordertermsbutthehomogeneouspartofLisapproximatelyconstant.4.Thegeneralcase.Stage1.L=Pjpj=mpDpwherethepareconstants.Thenu;Lu0=0@Xa`ei`x;X`0@Xjpj=mpi`p1Aa`ei`x1A0cX```m=2ja`j2by.35=cX[1+``m=2]ja`j2)]TJ/F11 9.963 Tf 9.962 0 Td[(ckuk20cCkuk2m=2)]TJ/F11 9.963 Tf 9.962 0 Td[(ckuk0whereC=supr01+rm=2 +rm=2:Thistakescareofstage1.

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74CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.Stage2.L=L0+L1whereL0isasinstage1andL1=Pjpj=mpxDpandmaxp;xjpxj<;wheresucientlysmall.Howsmallwillbedeterminedverysooninthecourseofthediscussion.Wehaveu;L0u0c0kuk2m=2)]TJ/F11 9.963 Tf 9.963 0 Td[(ckuk20fromstage1.Weintegrateu;L1u0bypartsm=2times.Therearenoboundarytermssinceweareonthetorus.Inintegratingbypartssomeofthederivativeswillhitthecoecients.LetuscollectallthethesetermsasI2.TheothertermswecollectasI1,soI1=XZbp0+p00Dp0u Dp00udxwherejp0j=jp00j=m=2andbr=r.WecanestimatethissumbyjI1jconst.kuk2m=2andsowillrequirethatconst.0,kukm 2)]TJ/F7 6.974 Tf 6.227 0 Td[(1kukm 2+)]TJ/F10 6.974 Tf 6.227 0 Td[(m=2kuk0:SubstitutingthisintotheaboveestimateforI2givesjI2jconst.kuk2m=2+)]TJ/F10 6.974 Tf 6.227 0 Td[(m=2const.kukm=2kuk0:Foranypositivenumbersa;bandtheinequalitya)]TJ/F11 9.963 Tf 9.03 0 Td[()]TJ/F7 6.974 Tf 6.226 0 Td[(1b20impliesthat2ab2a2+)]TJ/F7 6.974 Tf 6.227 0 Td[(2b2.Taking2=m 2+1wecanreplacethesecondtermontherightintheprecedingestimateforjI2jby)]TJ/F10 6.974 Tf 6.226 0 Td[(m)]TJ/F7 6.974 Tf 6.227 0 Td[(1const.kuk20atthecostofenlargingtheconstantinfrontofkuk2m 2.WehavethusestablishedthatjI1jconst.1kuk2m=2

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2.7.GARDING'SINEQUALITY.75wheretheconstantdependsonlyonm,andjI2jconst.2kuk2m=2+)]TJ/F10 6.974 Tf 6.227 0 Td[(m)]TJ/F7 6.974 Tf 6.227 0 Td[(1const.kuk20wheretheconstantsdependonL1butisatourdisposal.Soifconst.1
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76CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.whichisGarding'sinequality.QEDForthetimebeingwewillcontinuetostudythecaseofthetorus.Butalookaheadisinorder.Inthislaststepoftheargument,whereweappliedthepartitionofunityargument,wehavereallyfreedourselvesoftherestrictionofbeingonthetorus.Oncewemaketheappropriatedenitions,wewillthengetGarding'sinequalityforellipticoperatorsonmanifolds.Furthermore,theconsequencesweareabouttodrawfromGarding'sinequalitywillbeequallyvalidinthemoregeneralsetting.2.8ConsequencesofGarding'sinequality.Proposition2.8.1Foreveryintegertthereisaconstantct=ct;Landapositivenumber=t;LsuchthatkuktctkLu+ukt)]TJ/F10 6.974 Tf 6.226 0 Td[(m.38when>forallsmoothu,andhenceforallu2Ht:Proof.Letsbesomenon-negativeinteger.Wewillrstprove.38fort=s+m 2.WehavekuktkLu+ukt)]TJ/F10 6.974 Tf 6.227 0 Td[(m=kuktkLu+uks)]TJ/F9 4.981 Tf 7.422 2.677 Td[(m 2=kuktk+sLu++suk)]TJ/F10 6.974 Tf 6.226 0 Td[(s)]TJ/F9 4.981 Tf 7.422 2.677 Td[(m 2u;+sLu++su0bythegeneralizedCauchy-Schwartzinequality2.26.Theoperator+sLisellipticoforderm+2sso.25andGarding'sinequalitygivesu;+sLu++su0c1kuk2s+m 2)]TJ/F11 9.963 Tf 9.962 0 Td[(c2kuk20+kuk2s:Sincekukskuk0wecancombinethetwopreviousinequalitiestogetkuktkLu+ukt)]TJ/F10 6.974 Tf 6.227 0 Td[(mc1kuk2t+)]TJ/F11 9.963 Tf 9.963 0 Td[(c2kuk20:If>c2wecandropthesecondtermanddividebykukttoobtain.38.Wenowprovethepropositionforthecaset=m 2)]TJ/F11 9.963 Tf 10.357 0 Td[(sbythesamesortofargument:WehavekuktkLu+uk)]TJ/F10 6.974 Tf 6.227 0 Td[(s)]TJ/F9 4.981 Tf 7.422 2.677 Td[(m 2=k+)]TJ/F10 6.974 Tf 6.227 0 Td[(suks+m 2kLu+uk)]TJ/F10 6.974 Tf 6.227 0 Td[(s)]TJ/F9 4.981 Tf 7.422 2.677 Td[(m 2+)]TJ/F10 6.974 Tf 6.227 0 Td[(su;L+s+)]TJ/F10 6.974 Tf 6.227 0 Td[(su+u0:

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2.8.CONSEQUENCESOFGARDING'SINEQUALITY.77NowusethefactthatL+sisellipticandGarding'sinequalitytocontinuetheaboveinequalitiesasc1k+)]TJ/F10 6.974 Tf 6.226 0 Td[(suk2s+m 2)]TJ/F11 9.963 Tf 9.962 0 Td[(c2k+)]TJ/F10 6.974 Tf 6.227 0 Td[(suk20+kuk2)]TJ/F10 6.974 Tf 6.227 0 Td[(s=c1kuk2t)]TJ/F11 9.963 Tf 9.962 0 Td[(c2kuk)]TJ/F7 6.974 Tf 6.227 0 Td[(2s+kuk2)]TJ/F10 6.974 Tf 6.227 0 Td[(sc1kuk2tif>c2.Againwemaythendividebykukttogettheresult.QEDTheoperatorL+IisaboundedoperatorfromHttoHt)]TJ/F10 6.974 Tf 6.227 0 Td[(mforanyt.Supposewextandchoosesolargethat.38holds.Then.38saysthatL+Iisinvertibleonitsimage,andboundedtherewithaboundindependentof>,andthisimageisaclosedsubspaceofHt)]TJ/F10 6.974 Tf 6.227 0 Td[(m.LetusshowthatthisimageisallofHt)]TJ/F10 6.974 Tf 6.227 0 Td[(mforlargeenough.Supposenot,whichmeansthatthereissomew2Ht)]TJ/F10 6.974 Tf 6.226 0 Td[(mwithw;Lu+ut)]TJ/F10 6.974 Tf 6.226 0 Td[(m=0forallu2Ht.Wecanwritethislastequationas+t)]TJ/F10 6.974 Tf 6.226 0 Td[(mw;Lu+u0=0:IntegrationbypartsgivestheadjointdierentialoperatorLcharacterizedby;L0=L;0forallsmoothfunctionsand,andbypassingtothelimitthisholdsforallelementsofHrforrm.TheoperatorLhasthesameleadingtermasLandhenceiselliptic.Soletuschoosesucientlylargethat2.38holdsforLaswellasforL.Now0=)]TJ/F8 9.963 Tf 4.566 -8.07 Td[(+t)]TJ/F10 6.974 Tf 6.226 0 Td[(mw;Lu+u0=)]TJ/F11 9.963 Tf 4.566 -8.07 Td[(L+t)]TJ/F10 6.974 Tf 6.226 0 Td[(mw++t)]TJ/F10 6.974 Tf 6.227 0 Td[(mw;u0forallu2HtwhichisdenseinH0soL+t)]TJ/F10 6.974 Tf 6.226 0 Td[(mw++t)]TJ/F10 6.974 Tf 6.227 0 Td[(mw=0andhenceby.38+t)]TJ/F10 6.974 Tf 6.227 0 Td[(mw=0sow=0.WehaveprovedProposition2.8.2ForeverytandforlargeenoughdependingonttheoperatorL+ImapsHtbijectivelyontoHt)]TJ/F10 6.974 Tf 6.227 0 Td[(mandL+I)]TJ/F7 6.974 Tf 6.226 0 Td[(1isboundedindependentlyof.AsanimmediateapplicationwegettheimportantTheorem2.8.1IfuisadistributionandLu2Hsthenu2Hs+m.Proof.Writef=Lu.BySchwartz'stheorem,weknowthatu2Hkforsomek.Sof+u2Hmink;sforany.Choosinglargeenough,weconcludethatu=L+I)]TJ/F7 6.974 Tf 6.226 0 Td[(1f+u2Hmink+m;s+m.Ifk+m
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78CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.theargumenttoconcludethatu2Hmink+2m;s+m.wecankeepgoinguntilweconcludethatu2Hs+m.QEDNoticeasanimmediatecorollarythatanysolutionofthehomogeneousequa-tionLu=0isC1.WenowobtainasecondimportantconsequenceofProposition2.8.2.ChoosesolargethattheoperatorsL+I)]TJ/F7 6.974 Tf 6.227 0 Td[(1andL+I)]TJ/F7 6.974 Tf 6.226 0 Td[(1existasoperatorsfromH0!Hm.Followtheseoperatorswiththeinjectionm:Hm!H0andsetM:=mL+I)]TJ/F7 6.974 Tf 6.227 0 Td[(1;M:=mL+I)]TJ/F7 6.974 Tf 6.227 0 Td[(1:SincemiscompactRellich'slemmaandthecompositeofacompactoperatorwithaboundedoperatoriscompact,weconcludeTheorem2.8.2TheoperatorsMandMarecompact.SupposethatL=L.ThisisusuallyexpressedbysayingthatLisfor-mallyself-adjoint".Moreonthisterminologywillcomelater.ThisimpliesthatM=M.Inotherwords,Misacompactselfadjointoperator,andwecanapplyTheorem2.3.1toconcludethateigenvectorsofMformabasisofRMandthatthecorrespondingeigenvaluestendtozero.Prop2.8.2saysthatRMisthesameasmHmwhichisdenseinH0=L2T.Wecon-cludethattheeigenvectorsofMformabasisofL2T.IfMu=ruthenu=L+IMu=rLu+rusouisaneigenvectorofLwitheigenvalue1)]TJ/F11 9.963 Tf 9.962 0 Td[(r r:WeconcludethattheeigenvectorsofLareabasisofH0.WeclaimthatonlynitelymanyoftheseeigenvaluesofLcanbenegative.Indeed,sinceweknowthattheeigenvaluesrnofMapproachzero,thenumeratorintheaboveexpressionispositive,forlargeenoughn,andhenceiftherewereinnitelymanynegativeeigenvaluesk,theywouldhavetocorrespondtonegativerkandsothesek!.Buttakingsk=)]TJ/F11 9.963 Tf 7.748 0 Td[(kasthein.38inProp.2.8.1weconcludethatu=0,ifLu=kuifkislargeenough,contradictingthedenitionofaneigenvector.Soallbutanitenumberofthernarepositive,andthesetendtozero.Tosummarize:Theorem2.8.3TheeigenvectorsofLareC1functionswhichformabasisofH0.OnlynitelymanyoftheeigenvalueskofLarenegativeandn!1asn!1.Itiseasytoextendtheresultsobtainedaboveforthetorusintwodirections.Oneistoconsiderfunctionsdenedinadomain=boundedopensetGofRnandtheotheristoconsiderfunctionsdenedonacompactmanifold.Inbothcasesafewelementarytricksallowustoreducetothetoruscase.Wesketchwhatisinvolvedforthemanifoldcase.

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2.9.EXTENSIONOFTHEBASICLEMMASTOMANIFOLDS.792.9Extensionofthebasiclemmastomanifolds.LetE!Mbeavectorbundleoveramanifold.WeassumethatMisequippedwithadensitywhichweshalldenotebyjdxjandthatEisequippedwithapositivedenitesmoothlyvaryingscalarproduct,sothatwecandenetheL2normofasmoothsectionsofEofcompactsupport:ksk20:=ZMjsj2xjdxj:SupposefortherestofthissectionthatMiscompact.LetfUigbeanitecoverofMbycoordinateneighborhoodsoverwhichEhasagiventrivialization,andiapartitionofunitysubordinatetothiscover.LetibeadieomorphismorUiwithanopensubsetofTnwherenisthedimensionofM.ThenifsisasmoothsectionofE,wecanthinkofis)]TJ/F7 6.974 Tf 6.227 0 Td[(1iasanRmorCmvaluedfunctiononTn,andconsiderthesumofthekkknormsappliedtoeachcomponent.Weshallcontinuetodenotethissumbykif)]TJ/F7 6.974 Tf 6.227 0 Td[(1ikkandthendenekfkk:=Xikif)]TJ/F7 6.974 Tf 6.227 0 Td[(1ikkwherethenormsontherightareinthenormsonthetorus.Thesenormsdependonthetrivializationsandonthepartitionsofunity.Butanytwonormsareequivalent,andthekk0normisequivalenttotheintrinsic"L2normdenedabove.WedenetheSobolevspacesWktobethecompletionofthespaceofsmoothsectionsofErelativetothenormkkkfork0,andthesespacesarewelldenedastopologicalvectorspacesindependentlyofthechoices.SinceSobolev'slemmaholdslocally,itgoesthroughunchanged.SimilarlyRellich'slemma:ifsnisasequenceofelementsofW`whichisboundedinthekk`normfor`>k,theneachoftheelementsisn)]TJ/F7 6.974 Tf 6.227 0 Td[(1ibelongtoH`onthetorus,andareboundedinthekk`norm,hencewecanselectasubsequenceof1sn)]TJ/F7 6.974 Tf 6.227 0 Td[(11whichconvergesinHk,thenasubsubsequencesuchthatisn)]TJ/F7 6.974 Tf 6.227 0 Td[(1ifori=1;2convergeetc.arrivingatasubsequenceofsnwhichconvergesinWk.AdierentialoperatorLmappingsectionsofEintosectionsofEisanoperatorwhoselocalexpressionintermsofatrivializationandacoordinatecharthastheformLs=XjpjmpxDpsHeretheaparelinearmapsormatricesifourtrivializationsareintermsofRm.Underchangesofcoordinatesandtrivializationsthechangeinthecoecientsarerathercomplicated,butthesymbolofthedierentialoperatorL:=Xjpj=mapxp2TMxiswelldened.

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80CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.IfweputaRiemannmetriconthemanifold,wecantalkaboutthelengthjjofanycotangentvector.IfLisadierentialoperatorfromEtoitselfi.e.F=EweshallcallLevenellipticifmisevenandthereexistssomeconstantCsuchthathv;LviCjjmjvj2forallx2M;v2Ex;2TMxandh;idenotesthescalarproductonEx.Garding'sinequalityholds.Indeed,locally,thisisjustarestatementofthevectorvaluedversionofGarding'sinequalitythatwehavealreadyprovedforthetorus.ButStage4intheproofextendsunchangedotherthanthereplacementofscalarvaluedfunctionsbyvectorvaluedfunctionstothemoregeneralcase.2.10Example:Hodgetheory.Weassumeknowledgeofthebasicfactsaboutdierentiablemanifolds,inpar-ticulartheexistenceofanoperatord:k!k+1withitsusualproperties,wherekdenotesthespaceofexteriork-forms.Also,ifMisorientableandcarriesaRiemannmetricthentheRiemannmetricinducesascalarproductontheexteriorpowersofTMandalsopicksoutavolumeform.Sothereisaninducedscalarproduct;=;konkandaformaladjointofd:k!k)]TJ/F7 6.974 Tf 6.227 0 Td[(1andsatisesd;=;whereisak+1-formandisak-form.Then:=d+disasecondorderdierentialoperatoronkandsatises;=kdk2+kk2wherekkj2=;istheintrinsicL2normsokk=kk0intermsofthenotationoftheprecedingsection.Furthermore,if=XIIdxIisalocalexpressionforthedierentialform,wheredxI=dxi1^^dxikI=i1;:::;ikthenalocalexpressionforis=)]TJ/F1 9.963 Tf 9.409 9.465 Td[(Xgij@I @xi@xj+

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2.10.EXAMPLE:HODGETHEORY.81wheregij=hdxi;dxjiandthearelowerorderderivatives.Inparticulariselliptic.Let2kandsupposethatd=0:LetC,thecohomologyclassofbethesetofall2kwhichsatisfy)]TJ/F11 9.963 Tf 9.963 0 Td[(=d;2k)]TJ/F7 6.974 Tf 6.227 0 Td[(1andlet CdenotetheclosureofCintheL2norm.ItisaclosedsubspaceoftheHilbertspaceobtainedbycompletingkrelativetoitsL2norm.LetusdenotethisspacebyLk2,so CisaclosedsubspaceofLk2.Proposition2.10.1If2kandd=0,thereexistsaunique2 Csuchthatkkkk82C:Furthermore,issmooth,andd=0and=0:IfchooseaminimizingsequenceforkkinC.IfwechooseaminimizingsequenceforkkinCweknowitisCauchy,cf.theproofoftheexistenceoforthogonalprojectionsinaHilbertspace.Soweknowthatexistsandisunique.Forany2k+1wehave;=lim;=limd;=0asrangesoveraminimizingsequence.Theequation;=0forall2k+1saysthatisaweaksolutionoftheequationd=0.Weclaimthat;d=082k)]TJ/F7 6.974 Tf 6.227 0 Td[(1whichsaysthatisaweaksolutionof=0.Indeed,foranyt2R,kk2k+tdk2=kk2+t2kdk2+2t;dso)]TJ/F8 9.963 Tf 7.748 0 Td[(2t;dt2kdk2:If;d6=0,wecanchooset=)]TJ/F11 9.963 Tf 7.749 0 Td[(;d j;dj;>0

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82CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.soj;djjdj2:Asisarbitrary,thisimpliesthat;d=0.So;=;[d+d]=0forany2k.Henceisaweaksolutionof=0andsoissmooth.ThespaceHkofweak,andhencesmoothsolutionsof=0isnitedimensionalbythegeneraltheory.ItiscalledthespaceofHarmonicforms.Wehaveseenthatthereisauniqueharmonicforminthecohomologyclassofanyclosedform,sthecohomologygroupsarenitedimensional.Infact,thegeneraltheorytellsusthatLk2MEkHilbertspacedirectsumwhereEkistheeigenspacewitheigenvalueof.EachEisnitedimensionalandconsistsofsmoothforms,andthe!1.TheeigenspaceEk0isjustHk,thespaceofharmonicforms.Also,since;=kdk2+kk2weknowthatalltheeigenvaluesarenon-negative.Sinced=dd+d=dd=d,weseethatd:Ek!Ek+1andsimilarly:Ek!Ek)]TJ/F7 6.974 Tf 6.226 0 Td[(1:For6=0,if2Ekandd=0,then==dso=d=sodrestrictedtotheEisexact,andsimilarlysois.Furthermore,onLkEkwehaveI==d+2sowehaveEk=dEk)]TJ/F7 6.974 Tf 6.227 0 Td[(1Ek+1andthisdecompositionisorthogonalsinced;=d2;=0.AsarstconsequenceweseethatLk2=Hk dk)]TJ/F7 6.974 Tf 6.226 0 Td[(1 k)]TJ/F7 6.974 Tf 6.227 0 Td[(1theHodgedecomposition.IfHdenotesprojectionontotherstcomponent,thenisinvertibleontheimageofI)]TJ/F11 9.963 Tf 10.613 0 Td[(Hwithaaninversetherewhichiscompact.SoifweletNdenotethisinverseonimI)]TJ/F11 9.963 Tf 9.968 0 Td[(HandsetN=0onHkwegetN=I)]TJ/F11 9.963 Tf 9.962 0 Td[(HNd=dNN=NN=NNH=0

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2.11.THERESOLVENT.83whicharethefundamentalassertionsofHodgetheory,togetherwiththeasser-tionprovedabovethatHistheuniqueminimizingelementinitscohomologyclass.Wehaveseenthatd+:MkE2k!MkE2k+1isanisomorphismfor6=0.39whichofcourseimpliesthatXk)]TJ/F8 9.963 Tf 7.749 0 Td[(1kdimEk=0Thisshowsthattheindexoftheoperatord+actingonLLk2istheEulercharacteristicofthemanifold.Theindexofanyoperatoristhedierencebetweenthedimensionsofthekernelandcokernel.LetPk;denotetheprojectionofLk2ontoEk.Soe)]TJ/F10 6.974 Tf 6.227 0 Td[(t=Xe)]TJ/F10 6.974 Tf 6.226 0 Td[(tPk;isthesolutionoftheheatequationonLk2.Ast!1thisapproachestheoperatorHprojectingLk2ontoHk.LettingkdenotetheoperatoronLk2weseethattre)]TJ/F10 6.974 Tf 6.227 0 Td[(tk=Xe)]TJ/F10 6.974 Tf 6.226 0 Td[(kwherethesumisoveralleigenvalueskofkcountedwithmultiplicity.Itfollowsfrom.39thatthealternatingsumoverkofthecorrespondingsumovernon-zeroeigenvaluesvanishes.HenceX)]TJ/F8 9.963 Tf 7.749 0 Td[(1ktre)]TJ/F10 6.974 Tf 6.227 0 Td[(tk=Misindependentoft.Theindextheoremcomputesthistraceforsmallvaluesoftintermsoflocalgeometricinvariants.Theoperatord+isanexampleofaDiracoperatorwhosegeneralde-nitionwewillnotgivehere.ThecorrespondingassertionandlocalevaluationisthecontentofthecelebratedAtiyah-Singerindextheorem,oneofthemostimportanttheoremsdiscoveredinthetwentiethcentury.2.11Theresolvent.Inordertoconnectwhatwehavedoneherenotationthatwillcomelater,itisconvenienttoletA=)]TJ/F11 9.963 Tf 7.749 0 Td[(LsothatnowtheoperatorzI)]TJ/F11 9.963 Tf 9.963 0 Td[(A)]TJ/F7 6.974 Tf 6.227 0 Td[(1iscompactasanoperatoronH0forzsucientlynegative.Ihavedroppedthemwhichshouldcomeinfrontofthisexpression.TheoperatorAnowhasonly

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84CHAPTER2.HILBERTSPACESANDCOMPACTOPERATORS.nitelymanypositiveeigenvalues,withthecorrespondingspacesofeigenvectorsbeingnitedimensional.Infact,theeigenvectorsn=nAcountedwithmultiplicityapproachasn!1andtheoperatorzI)]TJ/F11 9.963 Tf 10.125 0 Td[(A)]TJ/F7 6.974 Tf 6.227 0 Td[(1existsandisaboundedinfactcompactoperatorsolongasz6=nforanyn.Indeed,wecanwriteanyu2H0asu=XnannwherenisaneigenvectorofAwitheigenvaluenandtheformanorthonor-malbasisofH0.ThenzI)]TJ/F11 9.963 Tf 9.962 0 Td[(A)]TJ/F7 6.974 Tf 6.226 0 Td[(1u=X1 z)]TJ/F11 9.963 Tf 9.962 0 Td[(nann:TheoperatorzI)]TJ/F11 9.963 Tf 8.655 0 Td[(A)]TJ/F7 6.974 Tf 6.227 0 Td[(1iscalledtheresolventofAatthepointzanddenotedbyRz;AorsimplybyRzifAisxed.SoRz;A:=zI)]TJ/F11 9.963 Tf 9.963 0 Td[(A)]TJ/F7 6.974 Tf 6.227 0 Td[(1forthosevaluesofz2Cforwhichtherighthandsideisdened.IfzandarecomplexnumberswithRez>Rea,thentheintegralZ10e)]TJ/F10 6.974 Tf 6.227 0 Td[(zteatdtconverges,andwecanevaluateitas1 z)]TJ/F11 9.963 Tf 9.962 0 Td[(a=Z10e)]TJ/F10 6.974 Tf 6.227 0 Td[(zteatdt:IfRezisgreaterthanthelargestoftheeigenvaluesofAwecanwriteRz;A=Z10e)]TJ/F10 6.974 Tf 6.226 0 Td[(ztetAdtwherewemayinterpretthisequationasashorthandfordoingtheintegralforthecoecientofeacheigenvector,asabove,orasanactualoperatorvaluedintegral.Wewillspendalotoftimelateroninthiscoursegeneralizingthisformulaandderivingmanyconsequencesfromit.

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Chapter3TheFourierTransform.3.1Conventions,especiallyabout2.ThespaceSconsistsofallfunctionsonmathbbRwhichareinnitelydier-entiableandvanishatinnityrapidlywithalltheirderivativesinthesensethatkfkm;n:=supfjxmfnxjg<1:Thekkm;ngiveafamilyofsemi-normsonSmakingSintoaFrechetspace-thatis,avectorspacespacewhosetopologyisdeterminedbyacountablefamilyofsemi-norms.Moreaboutthislaterinthecourse.Weusethemeasure1 p 2dxonRandsodenetheFouriertransformofanelementofSby^f:=1 p 2ZRfxe)]TJ/F10 6.974 Tf 6.227 0 Td[(ixdxandtheconvolutionoftwoelementsofSbyf?gx:=1 p 2ZRfx)]TJ/F11 9.963 Tf 9.963 0 Td[(tgtdt:TheFouriertransformiswelldenedonSandd dxm)]TJ/F11 9.963 Tf 7.749 0 Td[(ixnf^=imd dn^f;asfollowsbydierentiationundertheintegralsignandbyintegrationbyparts.ThisshowsthattheFouriertransformmapsStoS.85

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86CHAPTER3.THEFOURIERTRANSFORM.3.2Convolutiongoestomultiplication.f?g^=1 2ZZfx)]TJ/F11 9.963 Tf 9.962 0 Td[(tgtdxe)]TJ/F10 6.974 Tf 6.227 0 Td[(ixdx=1 2ZZfugte)]TJ/F10 6.974 Tf 6.227 0 Td[(iu+tdudt=1 p 2ZRfue)]TJ/F10 6.974 Tf 6.227 0 Td[(iudu1 p 2ZRgte)]TJ/F10 6.974 Tf 6.227 0 Td[(itdtsof?g^=^f^g:3.3Scaling.Foranyf2Sanda>0deneSafbySafx:=fax.Thensettingu=axsodx=1=aduwehaveSaf^=1 p 2ZRfaxe)]TJ/F10 6.974 Tf 6.227 0 Td[(ixdx=1 p 2ZR=afue)]TJ/F10 6.974 Tf 6.227 0 Td[(iu=adusoSaf^==aS1=a^f:3.4FouriertransformofaGaussianisaGaus-sian.Thepolarcoordinatetrickevaluates1 p 2ZRe)]TJ/F10 6.974 Tf 6.227 0 Td[(x2=2dx=1:Theintegral1 p 2ZRe)]TJ/F10 6.974 Tf 6.226 0 Td[(x2=2)]TJ/F10 6.974 Tf 6.227 0 Td[(xdxconvergesforallcomplexvaluesof,uniformlyinanycompactregion.Henceitdenesananalyticfunctionofthatcanbeevaluatedbytakingtoberealandthenusinganalyticcontinuation.Forrealwecompletethesquareandmakeachangeofvariables:1 p 2ZRe)]TJ/F10 6.974 Tf 6.227 0 Td[(x2=2)]TJ/F10 6.974 Tf 6.226 0 Td[(xdx=1 p 2ZRe)]TJ/F7 6.974 Tf 6.226 0 Td[(x+2=2+2=2dx=e2=21 p 2ZRe)]TJ/F7 6.974 Tf 6.227 0 Td[(x+2=2dx=e2=2:

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3.4.FOURIERTRANSFORMOFAGAUSSIANISAGAUSSIAN.87Setting=igives^n=nifnx:=e)]TJ/F10 6.974 Tf 6.227 0 Td[(x2=2:Ifweseta=inourscalingequationanddene:=Snsox=e)]TJ/F10 6.974 Tf 6.227 0 Td[(2x2=2;then^x=1 e)]TJ/F10 6.974 Tf 6.227 0 Td[(x2=22:Noticethatforanyg2SwehaveZR=aS1=agd=ZRgdsosettinga=weconcludethat1 p 2ZR^d=1forall.Let:=1:=1^and:=^:Then=1 so?g)]TJ/F11 9.963 Tf 9.963 0 Td[(g=1 p 2ZR[g)]TJ/F11 9.963 Tf 9.962 0 Td[()]TJ/F11 9.963 Tf 9.962 0 Td[(g]1 d==1 p 2ZR[g)]TJ/F11 9.963 Tf 9.963 0 Td[()]TJ/F11 9.963 Tf 9.962 0 Td[(g]d:Sinceg2SitisuniformlycontinuousonR,sothatforany>0wecannd0sothattheaboveintegralislessthaninabsolutevalueforall0<<0.Inshort,k?g)]TJ/F11 9.963 Tf 9.962 0 Td[(gk1!0;as!0:

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88CHAPTER3.THEFOURIERTRANSFORM.3.5Themultiplicationformula.ThissaysthatZR^fxgxdx=ZRfx^gxdxforanyf;g2S.Indeedthelefthandsideequals1 p 2ZRZRfye)]TJ/F10 6.974 Tf 6.227 0 Td[(ixydygxdx:Wecanwritethisintegralasadoubleintegralandtheninterchangetheorderofintegrationwhichgivestherighthandside.3.6Theinversionformula.Thissaysthatforanyf2Sfx=1 p 2ZR^feixd:Toprovethis,werstobservethatforanyh2StheFouriertransformofx7!eixhxisjust7!^h)]TJ/F11 9.963 Tf 9.963 0 Td[(asfollowsdirectlyfromthedenition.Takinggx=eitxe)]TJ/F10 6.974 Tf 6.226 0 Td[(2x2=2inthemultiplicationformulagives1 p 2ZR^fteitxe)]TJ/F10 6.974 Tf 6.226 0 Td[(2t2=2dt=1 p 2ZRftt)]TJ/F11 9.963 Tf 9.963 0 Td[(xdt=f?x:Weknowthattherighthandsideapproachesfxas!0.Also,e)]TJ/F10 6.974 Tf 6.227 0 Td[(2t2=2!1foreachxedt,andinfactuniformlyonanyboundedtinterval.Furthermore,0
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3.8.THEPOISSONSUMMATIONFORMULA.89Theinversionformulaappliedtof?~fandevaluatedat0givesf?~f=1 p 2ZRj^fj2dx:Thelefthandsideofthisequationis1 p 2ZRfx~f)]TJ/F11 9.963 Tf 9.963 0 Td[(xdx=1 p 2ZRjfxj2dx:ThuswehaveprovedPlancherel'sformula1 p 2ZRjfxj2dx=1 p 2ZRj^fxj2dx:DeneL2RtobethecompletionofSwithrespecttotheL2normgivenbythelefthandsideoftheaboveequation.SinceSisdenseinL2RweconcludethattheFouriertransformextendstounitaryisomorphismofL2Rontoitself.3.8ThePoissonsummationformula.Thissaysthatforanyg2SwehaveXkgk=1 p 2Xm^gm:Toprovethislethx:=Xkgx+2ksohisasmoothfunction,periodicofperiod2andh=Xkgk:WemayexpandhintoaFourierserieshx=Xmameimxwheream=1 2Z20hxe)]TJ/F10 6.974 Tf 6.227 0 Td[(imxdx=1 2ZRgxe)]TJ/F10 6.974 Tf 6.226 0 Td[(imxdx=1 p 2^gm:Settingx=0intheFourierexpansionhx=1 p 2X^gmeimxgivesh=1 p 2Xm^gm:

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90CHAPTER3.THEFOURIERTRANSFORM.3.9TheShannonsamplingtheorem.Letf2SbesuchthatitsFouriertransformissupportedintheinterval[)]TJ/F11 9.963 Tf 7.749 0 Td[(;].Thenaknowledgeoffnforalln2Zdeterminesf.Moreexplicitly,ft=1 1Xn=fnsinn)]TJ/F11 9.963 Tf 9.962 0 Td[(t n)]TJ/F11 9.963 Tf 9.963 0 Td[(t:.1Proof.Letgbetheperiodicfunctionofperiod2whichextends^f,theFouriertransformoff.Sog=^f;2[)]TJ/F11 9.963 Tf 7.749 0 Td[(;]andisperiodic.ExpandgintoaFourierseries:g=Xn2Zcnein;wherecn=1 2Z)]TJ/F10 6.974 Tf 6.227 0 Td[(ge)]TJ/F10 6.974 Tf 6.227 0 Td[(ind=1 2Z1^fe)]TJ/F10 6.974 Tf 6.226 0 Td[(ind;orcn=1 1 2f)]TJ/F11 9.963 Tf 7.749 0 Td[(n:Butft=1 1 2Z1^feitd=1 1 2Z)]TJ/F10 6.974 Tf 6.226 0 Td[(geitd=1 1 2Z)]TJ/F10 6.974 Tf 6.226 0 Td[(X1 1 2f)]TJ/F11 9.963 Tf 7.749 0 Td[(nein+td:Replacingnby)]TJ/F11 9.963 Tf 7.749 0 Td[(ninthesum,andinterchangingsummationandintegration,whichislegitimatesincethefndecreaseveryfast,thisbecomesft=1 2XnfnZ)]TJ/F10 6.974 Tf 6.226 0 Td[(eit)]TJ/F10 6.974 Tf 6.226 0 Td[(nd:ButZ)]TJ/F10 6.974 Tf 6.227 0 Td[(eit)]TJ/F10 6.974 Tf 6.227 0 Td[(nd=eit)]TJ/F10 6.974 Tf 6.227 0 Td[(n it)]TJ/F11 9.963 Tf 9.963 0 Td[(n)]TJ/F10 6.974 Tf 6.227 0 Td[(=eit)]TJ/F10 6.974 Tf 6.227 0 Td[(n)]TJ/F11 9.963 Tf 9.963 0 Td[(eit)]TJ/F10 6.974 Tf 6.226 0 Td[(n it)]TJ/F11 9.963 Tf 9.963 0 Td[(n=2sinn)]TJ/F11 9.963 Tf 9.963 0 Td[(t n)]TJ/F11 9.963 Tf 9.962 0 Td[(t:QEDItisusefultoreformulatethisviarescalingsothattheinterval[)]TJ/F11 9.963 Tf 7.748 0 Td[(;]isreplacedbyanarbitraryintervalsymmetricabouttheorigin:Intheengineeringliteraturethefrequencyisdenedby=2:

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3.10.THEHEISENBERGUNCERTAINTYPRINCIPLE.91Supposewewanttoapply.1tog=Saf.WeknowthattheFouriertransformofgis1=aS1=a^fandsuppS1=a^f=asupp^f:Soifsupp^f[)]TJ/F8 9.963 Tf 7.749 0 Td[(2c;2c]wewanttochooseasothata2cora1 2c:.2Forainthisrange.1saysthatfax=1 Xfnasinx)]TJ/F11 9.963 Tf 9.962 0 Td[(n x)]TJ/F11 9.963 Tf 9.962 0 Td[(n;orsettingt=ax,ft=1Xn=fnasin at)]TJ/F11 9.963 Tf 9.963 0 Td[(na at)]TJ/F11 9.963 Tf 9.963 0 Td[(na:.3ThisholdsinL2undertheassumptionthatfsatisessupp^f[)]TJ/F8 9.963 Tf 7.749 0 Td[(2c;2c].Wesaythatfhasnitebandwidthorisbandlimitedwithbandlimitc.Thecriticalvalueac=1=2cisknownastheNyquistsamplingintervaland=a=2cisknownastheNyquistsamplingrate.ThustheShannonsamplingtheoremsaysthataband-limitedsignalcanberecoveredcompletelyfromasetofsamplestakenataratetheNyquistsamplingrate.3.10TheHeisenbergUncertaintyPrinciple.Letf2SRwithZjfxj2dx=1:Wecanthinkofx7!jfxj2asaprobabilitydensityontheline.Themeanofthisprobabilitydensityisxm:=Zxjfxj2dx:IfwetaketheFouriertransform,thenPlancherelsaysthatZj^fj2d=1aswell,soitdenesaprobabilitydensitywithmeanm:=Zj^fj2d:

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92CHAPTER3.THEFOURIERTRANSFORM.Supposeforthemomentthatthesemeansbothvanish.TheHeisenbergUn-certaintyPrinciplesaysthatZjxfxj2dxZj^fj2d1 4:Proof.Write)]TJ/F11 9.963 Tf 7.748 0 Td[(ifastheFouriertransformoff0andusePlanchereltowritethesecondintegralasRjf0xj2dx.ThentheCauchy-SchwarzinequalitysaysthatthelefthandsideisthesquareofZjxfxf0xjdxZRexfx f0xdx=1 2Zxfx f0x+ fxf0xdx=1 2Zxd dxjfj2dx=1 2Zjfj2dx=1 2:QEDIffhasnormonebutthemeanoftheprobabilitydensityjfj2isnotnecessar-ilyofzeroandsimilarlyforforitsFouriertransformtheHeisenberguncertaintyprinciplesaysthatZjx)]TJ/F11 9.963 Tf 9.963 0 Td[(xmfxj2dxZj)]TJ/F11 9.963 Tf 9.963 0 Td[(m^fj2d1 4:Thegeneralcaseisreducedtothespecialcasebyreplacingfxbyfx+xmeimx:3.11Tempereddistributions.ThespaceSwasdenedtobethecollectionofallsmoothfunctionsonRsuchthatkfkm;n:=supxfjxmfnxjg<1:ThecollectionofthesenormsdeneatopologyonSwhichismuchnerthattheL2topology:Wedeclarethatasequenceoffunctionsffkgapproachesg2Sifandonlyifkfk)]TJ/F11 9.963 Tf 9.963 0 Td[(gkm;n!0foreverymandn.AlinearfunctiononSwhichiscontinuouswithrespecttothistopologyiscalledatempereddistribution.ThespaceoftempereddistributionsisdenotedbyS0.Forexample,everyelementf2SdenesalinearfunctiononSby7!h;fi=1 p 2ZRx fxdx:

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3.11.TEMPEREDDISTRIBUTIONS.93Butthislastexpressionmakessenseforanyelementf2L2R,orforanypiecewisecontinuousfunctionfwhichgrowsatinnitynofasterthananypoly-nomial.Forexample,iff1,thelinearfunctionassociatedtofassignstothevalue1 p 2ZRxdx:ThisisclearlycontinuouswithrespecttothetopologyofSbutthisfunctionofdoesnotmakesenseforageneralelementofL2R.AnotherexampleofanelementofS0istheDirac-functionwhichassignsto2Sitsvalueat0.ThisisanelementofS0butmakesnosensewhenevaluatedonageneralelementofL2R.Iff2S,thenthePlancherelformulaformulaimpliesthatitsFouriertrans-formFf=^fsatises;f=F;Ff:ButwecannowusethisequationtodenetheFouriertransformofanarbitraryelementofS0:If`2S0wedeneF`tobethelinearfunctionF`:=`F)]TJ/F7 6.974 Tf 6.226 0 Td[(1:3.11.1ExamplesofFouriertransformsofelementsofS0.If`correspondstothefunctionf1,thenF`=1 p 2ZRF)]TJ/F7 6.974 Tf 6.226 0 Td[(1d=F)]TJ/F14 9.963 Tf 4.566 -8.07 Td[(F)]TJ/F7 6.974 Tf 6.227 0 Td[(1=:SotheFouriertransformofthefunctionwhichisidenticallyoneistheDirac-function.IfdenotestheDirac-function,thenF=F)]TJ/F7 6.974 Tf 6.227 0 Td[(1=)]TJ/F8 9.963 Tf 4.566 -8.07 Td[(F)]TJ/F7 6.974 Tf 6.227 0 Td[(1=1 p 2ZRxdx:SotheFouriertransformoftheDiracfunctionisthefunctionwhichisidenticallyone.Infact,thislastexamplefollowsfromtheprecedingone:Ifm=F`thenFm=mF)]TJ/F7 6.974 Tf 6.226 0 Td[(1=`F)]TJ/F7 6.974 Tf 6.226 0 Td[(1F)]TJ/F7 6.974 Tf 6.227 0 Td[(1:ButF)]TJ/F7 6.974 Tf 6.227 0 Td[(2x=)]TJ/F11 9.963 Tf 7.749 0 Td[(x:Soifm=F`thenFm=`where`:=`\017:

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94CHAPTER3.THEFOURIERTRANSFORM.TheFouriertransformofthefunctionx:Thisassignstoevery2Sthevalue1 p 2Zeixxddx=1 p 2Z1 id d)]TJ/F11 9.963 Tf 4.566 -8.07 Td[(eixddx=i1 p 2Zd dxeixddx=iFF)]TJ/F7 6.974 Tf 6.226 0 Td[(1d dx=id dx:NowforanelementofSwehaveZd dx fdx=)]TJ/F8 9.963 Tf 16.113 6.74 Td[(1 p 2Z df dxdx:Sowedenethederivativeofan`2S0byd` dx=`)]TJ/F11 9.963 Tf 8.944 6.739 Td[(d dx:SotheFouriertransformofxis)]TJ/F11 9.963 Tf 7.748 0 Td[(id dx.

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Chapter4Measuretheory.4.1Lebesgueoutermeasure.Werecallsomeresultsfromthechapteronmetricspaces:ForanysubsetARwedeneditsLebesgueoutermeasurebymA:=infX`In:InareintervalswithA[In:.1Herethelength`IofanyintervalI=[a;b]isb)]TJ/F11 9.963 Tf 10.11 0 Td[(awiththesamedenitionforhalfopenintervalsa;b]or[a;b,oropenintervals.Ofcourseifa=andbisniteor+1,orifaisniteandb=+1thelengthisinnite.Sotheinmumin.1istakenoverallcoversofAbyintervals.Bytheusual=2ntrick,i.e.byreplacingeachIj=[aj;bj]byaj)]TJ/F11 9.963 Tf 10.249 0 Td[(=2j+1;bj+=2j+1wemayassumethattheinmumistakenoveropenintervals.Equallywell,wecouldusehalfopenintervalsoftheform[a;b,forexample..ItisclearthatifABthenmAmBsinceanycoverofBbyintervalsisacoverofA.Also,ifZisanysetofmeasurezero,thenmA[Z=mA.Inparticular,mZ=0ifZhasmeasurezero.Also,ifA=[a;b]isaninterval,thenwecancoveritbyitself,som[a;b]b)]TJ/F11 9.963 Tf 9.963 0 Td[(a;andhencethesameistruefora;b];[a;b,ora;b.Iftheintervalisinnite,itclearlycannotbecoveredbyasetofintervalswhosetotallengthisnite,sinceifwelinedthemupwithendpointstouchingtheycouldnotcoveraninniteinterval.WerecalltheproofthatmI=`I.2ifIisaniteinterval:WemayassumethatI=[c;d]isaclosedintervalbywhatwehavealreadysaid,andthattheminimizationin.1iswithrespecttoacoverbyopenintervals.Sowhatwemustshowisthatif[c;d][iai;bi95

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96CHAPTER4.MEASURETHEORY.thend)]TJ/F11 9.963 Tf 9.962 0 Td[(cXibi)]TJ/F11 9.963 Tf 9.963 0 Td[(ai:WerstappliedHeine-Boreltoreplacethecountablecoverbyanitecover.Thisonlydecreasestherighthandsideofprecedinginequality.Soletnbethenumberofelementsinthecover.Weneededtoprovethatif[c;d]n[i=1ai;bithend)]TJ/F11 9.963 Tf 9.963 0 Td[(cnXi=1bi)]TJ/F11 9.963 Tf 9.962 0 Td[(ai;andwedidthisthisbyinductiononn.Ifn=1thena1dsoclearlyb1)]TJ/F11 9.963 Tf 9.963 0 Td[(a1>d)]TJ/F11 9.963 Tf 9.963 0 Td[(c.Supposethatn2andweknowtheresultforallcoversofallintervals[c;d]withatmostn)]TJ/F8 9.963 Tf 10.745 0 Td[(1intervalsinthecover.Ifsomeintervalai;biisdisjointfrom[c;d]wemayeliminateitfromthecover,andthenweareinthecaseofn)]TJ/F8 9.963 Tf 10.111 0 Td[(1intervals.Soeveryai;bihasnon-emptyintersectionwith[c;d].Amongthetheintervalsai;bitherewillbeoneforwhichaitakesontheminimumpossiblevalue.Byrelabeling,wemayassumethatthisisa1;b1.Sinceciscovered,wemusthavea1dthena1;b1covers[c;d]andthereisnothingfurthertodo.Soassumeb1d.Wemusthaveb1>csincea1;b1[c;d]6=;.Sinceb12[c;d],atleastoneoftheintervalsai;bi;i>1containsthepointb1.Byrelabeling,wemayassumethatitisa2;b2.Butnowwehaveacoverof[c;d]byn)]TJ/F8 9.963 Tf 9.963 0 Td[(1intervals:[c;d]a1;b2[n[i=3ai;bi:Sobyinductiond)]TJ/F11 9.963 Tf 9.963 0 Td[(cb2)]TJ/F11 9.963 Tf 9.963 0 Td[(a1+nXi=3bi)]TJ/F11 9.963 Tf 9.963 0 Td[(ai:Butb2)]TJ/F11 9.963 Tf 9.963 0 Td[(a1b2)]TJ/F11 9.963 Tf 9.962 0 Td[(a2+b1)]TJ/F11 9.963 Tf 9.963 0 Td[(a1sincea2
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4.1.LEBESGUEOUTERMEASURE.972.ABmAmB:3.m[iAiXimAi:4.IfdistA;B>0thenmA[B=mA+mB:5.mA=inffmU:UA;Uopeng:6.ForanintervalmI=`I:Theonlyitemsthatwehavenotdonealreadyareitems4and5.Buttheseareimmediate:for4wemaychoosetheintervalsin.1alltohavelength0ifwechooseacloseenoughapproximationtotheinmum.IshouldalsoaddthatalltheaboveworksforRninsteadofRifwereplacethewordinterval"byrectangle",meaningarectangularparallelepiped,i.easetwhichisaproductofonedimensionalintervals.Wealsoreplacelengthbyvolumeorareaintwodimensions.WhatisneededisthefollowingLemma4.1.1LetCbeanitenon-overlappingcollectionofclosedrectanglesallcontainedintheclosedrectangleJ.ThenvolJXI2CvolI:IfCisanynitecollectionofrectanglessuchthatJ[I2CIthenvolJXI2CvolI:Thislemmaoccursonpage1ofStrook,Aconciseintroductiontothetheoryofintegrationtogetherwithitsproof.Iwilltakethisforgranted.InthenextfewparagraphsIwilltalkasifweareinR,buteverythinggoesthroughunchangedifRisreplacedbyRn.

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98CHAPTER4.MEASURETHEORY.4.2Lebesgueinnermeasure.Item5.intheprecedingparagraphsaysthattheLebesgueoutermeasureofanysetisobtainedbyapproximatingitfromtheoutsidebyopensets.TheLebesgueinnermeasureisdenedasmA=supfmK:KA;Kcompactg:.4ClearlymAmAsincemKmAforanyKA.WealsohaveProposition4.2.1ForanyintervalIwehavemI=`I:.5Proof.If`I=1theresultisobvious.SowemayassumethatIisaniteintervalwhichwemayassumetobeopen,I=a;b.IfKIiscompact,thenIisacoverofKandhencefromthedenitionofoutermeasuremK`I.SomI`I.Ontheotherhand,forany>0;<1 2b)]TJ/F11 9.963 Tf 10.184 0 Td[(atheinterval[a+;b)]TJ/F11 9.963 Tf 10.288 0 Td[(]iscompactandm[a)]TJ/F11 9.963 Tf 10.289 0 Td[(;a+]=b)]TJ/F11 9.963 Tf 10.289 0 Td[(a)]TJ/F8 9.963 Tf 10.288 0 Td[(2mI.Letting!0provestheproposition.QED4.3Lebesgue'sdenitionofmeasurability.AsetAwithmA<1issaidtomeasurableinthesenseofLebesgueifmA=mA:.6IfAismeasurableinthesenseofLebesgue,wewritemA=mA=mA:.7IfKisacompactset,thenmK=mKsinceKisacompactsetcon-tainedinitself.HenceallcompactsetsaremeasurableinthesenseofLebesgue.IfIisaboundedinterval,thenIismeasurableinthesenseofLebesguebyProposition4.2.1.IfmA=1,wesaythatAismeasurableinthesenseofLebesgueifallofthesetsA[)]TJ/F11 9.963 Tf 7.749 0 Td[(n;n]aremeasurable.Proposition4.3.1IfA=SAiisaniteorcountabledisjointunionofsetswhicharemeasurableinthesenseofLebesgue,thenAismeasurableinthesenseofLebesgueandmA=XimAi:

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4.3.LEBESGUE'SDEFINITIONOFMEASURABILITY.99Proof.WemayassumethatmA<1-otherwiseapplytheresulttoA[)]TJ/F11 9.963 Tf 7.748 0 Td[(n;n]andAi[)]TJ/F11 9.963 Tf 7.749 0 Td[(n;n]foreachn.WehavemAXnmAn=XnmAn:Let>0,andforeachnchoosecompactKnAnwithmKnmAn)]TJ/F11 9.963 Tf 14.338 6.74 Td[( 2n=mAn)]TJ/F11 9.963 Tf 14.339 6.74 Td[( 2nsinceAnismeasurableinthesenseofLebesgue.ThesetsKnarepairwisedisjoint,hence,beingcompact,atpositivedistancesfromoneanother.HencemK1[[Kn=mK1++mKnandK1[[KniscompactandcontainedinA.HencemAmK1++mKn;andsincethisistrueforallnwehavemAXnmAn)]TJ/F11 9.963 Tf 9.963 0 Td[(:Sincethisistrueforall>0wegetmAXmAn:ButthenmAmAandsotheyareequal,soAismeasurableinthesenseofLebesgue,andmA=PmAi.QEDProposition4.3.2OpensetsandclosedsetsaremeasurableinthesenseofLebesgue.Proof.AnyopensetOcanbewrittenasthecountableunionofopenintervalsIi,andJn:=Innn)]TJ/F7 6.974 Tf 6.226 0 Td[(1[i=1Iiisadisjointunionofintervalssomeopen,someclosed,somehalfopenandOisthedisjontunionoftheJn.SoeveryopensetisadisjointunionofintervalshencemeasurableinthesenseofLebesgue.IfFisclosed,andmF=1,thenF[)]TJ/F11 9.963 Tf 7.748 0 Td[(n;n]iscompact,andsoFismeasurableinthesenseofLebesgue.SupposethatmF<1:

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100CHAPTER4.MEASURETHEORY.Forany>0considerthesetsG1;:=[)]TJ/F8 9.963 Tf 7.749 0 Td[(1+ 22;1)]TJ/F11 9.963 Tf 13.861 6.739 Td[( 22]FG2;:=[)]TJ/F8 9.963 Tf 7.748 0 Td[(2+ 23;)]TJ/F8 9.963 Tf 7.748 0 Td[(1]F[[1;2)]TJ/F11 9.963 Tf 13.862 6.74 Td[( 23]FG3;:=[)]TJ/F8 9.963 Tf 7.748 0 Td[(3+ 24;)]TJ/F8 9.963 Tf 7.748 0 Td[(2]F[[2;3)]TJ/F11 9.963 Tf 13.862 6.74 Td[( 24]F...andsetG:=[iGi;:TheGi;areallcompact,andhencemeasurableinthesenseofLebesgue,andtheunioninthedenitionofGisdisjoint,soismeasurableinthesenseofLebesgue.Furthermore,thesumofthelengthsofthegaps"betweentheintervalsthatwentintothedenitionoftheGi;is.SomG+=mG+mFmG=mG=XimGi;:Inparticular,thesumontherightconverges,andhencebyconsideringanitenumberofterms,wewillhaveanitesumwhosevalueisatleastmG)]TJ/F11 9.963 Tf 10.24 0 Td[(.ThecorrespondingunionofsetswillbeacompactsetKcontainedinFwithmKmF)]TJ/F8 9.963 Tf 9.963 0 Td[(2:HenceallclosedsetsaremeasurableinthesenseofLebesgue.QEDTheorem4.3.1AismeasurableinthesenseofLebesgueifandonlyifforevery>0thereisanopensetUAandaclosedsetFAsuchthatmUnF<:Proof.SupposethatAismeasurableinthesenseofLebesguewithmA<1.ThenthereisanopensetUAwithmU
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4.3.LEBESGUE'SDEFINITIONOFMEASURABILITY.101mayenlargeeachFnifnecessarysothatFn[)]TJ/F11 9.963 Tf 7.749 0 Td[(n+1;n)]TJ/F8 9.963 Tf 9.804 0 Td[(1]Fn)]TJ/F7 6.974 Tf 6.226 0 Td[(1.WemayalsodecreasetheUnifnecessarysothatUn)]TJ/F11 9.963 Tf 7.749 0 Td[(n+1)]TJ/F11 9.963 Tf 9.963 0 Td[(n;n)]TJ/F8 9.963 Tf 9.963 0 Td[(1+nUn)]TJ/F7 6.974 Tf 6.227 0 Td[(1:Indeed,ifwesetCn:=[)]TJ/F11 9.963 Tf 7.748 0 Td[(n+1)]TJ/F11 9.963 Tf 9.779 0 Td[(n;n)]TJ/F8 9.963 Tf 9.779 0 Td[(1+n]Ucn)]TJ/F7 6.974 Tf 6.227 0 Td[(1thenCnisaclosedsetwithCnA=;.ThenUnCcnisstillanopensetcontaining[)]TJ/F11 9.963 Tf 7.749 0 Td[(n)]TJ/F8 9.963 Tf 9.448 0 Td[(2n+1;n+2n+1]AandUnCcn)]TJ/F11 9.963 Tf 7.749 0 Td[(n+1)]TJ/F11 9.963 Tf 9.653 0 Td[(n;n)]TJ/F8 9.963 Tf 9.652 0 Td[(1+nCcn)]TJ/F11 9.963 Tf 7.749 0 Td[(n+1)]TJ/F11 9.963 Tf 9.653 0 Td[(n;n)]TJ/F8 9.963 Tf 9.652 0 Td[(1+nUn)]TJ/F7 6.974 Tf 6.227 0 Td[(1:TakeU:=SUnsoUisopen.TakeF:=[Fn[)]TJ/F11 9.963 Tf 7.748 0 Td[(n;)]TJ/F11 9.963 Tf 7.749 0 Td[(n+1][[n)]TJ/F8 9.963 Tf 9.962 0 Td[(1;n]:ThenFisclosed,UAFandUnF[Un=Fn[)]TJ/F11 9.963 Tf 7.749 0 Td[(n;)]TJ/F11 9.963 Tf 7.749 0 Td[(n+1][[n)]TJ/F8 9.963 Tf 9.963 0 Td[(1;n][UnnFnIntheotherdirection,supposethatforeach,thereexistUAFwithmUnF<.SupposethatmA<1.ThenmF<1andmUmUnF+mF<+mF<1.ThenmAmU0weconcludethatmAmAsotheyareequalandAismeasurableinthesenseofLebesgue.IfmA=1,wehaveU)]TJ/F11 9.963 Tf 7.749 0 Td[(n)]TJ/F11 9.963 Tf 9.746 0 Td[(;n+A[)]TJ/F11 9.963 Tf 7.749 0 Td[(n;n]F[)]TJ/F11 9.963 Tf 7.749 0 Td[(n;n]andmU)]TJ/F11 9.963 Tf 7.749 0 Td[(n)]TJ/F11 9.963 Tf 9.962 0 Td[(;n+nF[)]TJ/F11 9.963 Tf 7.749 0 Td[(n;n]<2+=3sowecanproceedasbeforetoconcludethatmA[)]TJ/F11 9.963 Tf 7.749 0 Td[(n;n]=mA[)]TJ/F11 9.963 Tf 7.749 0 Td[(n;n].QEDSeveralfactsemergeimmediatelyfromthistheorem:Proposition4.3.3IfAismeasurableinthesenseofLebesgue,soisitscom-plementAc=RnA.Indeed,ifFAUwithFclosedandUopen,thenFcAcUcwithFcopenandUcclosed.Furthermore,FcnUc=UnFsoifAsatisestheconditionofthetheoremsodoesAc.Proposition4.3.4IfAandBaremeasurableinthesenseofLebesguesoisAB.For>0chooseUAAFAandUBBFBwithmUAnFA<=2andmUBnFB<=2.ThenUAUBABFAFBandUAUBnFAFBUAnFA[UBnFB.QEDPuttingtheprevioustwopropositionstogethergives

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102CHAPTER4.MEASURETHEORY.Proposition4.3.5IfAandBaremeasurableinthesenseofLebesguethensoisA[B.Indeed,A[B=AcBcc.SinceAnB=ABcwealsogetProposition4.3.6IfAandBaremeasurableinthesenseofLebesguethensoisAnB.4.4Caratheodory'sdenitionofmeasurability.AsetERissaidtobemeasurableaccordingtoCaratheodoryifforanysetARwehavemA=mAE+mAEc.8wherewerecallthatEcdenotesthecomplementofE.Inotherwords,AEc=AnE.Thisdenitionhasmanyadvantages,asweshallsee.OurrsttaskistoshowthatitisequivalenttoLebesgue's:Theorem4.4.1AsetEismeasurableinthesenseofCaratheodoryifandonlyifitismeasurableinthesenseofLebesgue.Proof.WealwayshavemAmAE+mAnEsocondition4.8isequivalenttomAE+mAnEmA.9forallA.SupposeEismeasurableinthesenseofLebesgue.Let>0.ChooseUEFwithUopen,FclosedandmU=F
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4.4.CARATHEODORY'SDEFINITIONOFMEASURABILITY.103Intheotherdirection,supposethatEismeasurableinthesenseofCaratheodory.FirstsupposethatmE<1:Thenforany>0thereexistsanopensetUEwithmUmU)]TJ/F11 9.963 Tf 9.963 0 Td[(:SothereisaclosedsetFUnVwithmF>mU)]TJ/F11 9.963 Tf 9.053 0 Td[(.ButsinceVUnE,wehaveUnVE.SoFE.SoFEUandmUnF=mU)]TJ/F11 9.963 Tf 9.962 0 Td[(mF<:HenceEismeasurableinthesenseofLebesgue.IfmE=1,wemustshowthatE[)]TJ/F11 9.963 Tf 7.749 0 Td[(n;n]ismeasurableinthesenseofCaratheodory,forthenitismeasurableinthesenseofLebesguefromwhatwealreadyknow.Weknowthattheinterval[)]TJ/F11 9.963 Tf 7.749 0 Td[(n;n]itself,beingmeasurableinthesenseofLebesgue,ismeasurableinthesenseofCaratheodory.SowewillhavecompletedtheproofofthetheoremifweshowthattheintersectionofEwith[)]TJ/F11 9.963 Tf 7.748 0 Td[(n;n]ismeasurableinthesenseofCaratheodory.Moregenerally,wewillshowthattheunionorintersectionoftwosetswhicharemeasurableinthesenseofCaratheodoryisagainmeasurableinthesenseofCaratheodory.Noticethatthedenition.8issymmetricinEandEcsoifEismeasurableinthesenseofCaratheodorysoisEc.Soitsucestoprovethenextlemmatocompletetheproof.Lemma4.4.1IfE1andE2aremeasurableinthesenseofCaratheodorysoisE1[E2.ForanysetAwehavemA=mAE1+mAEc1by.8appliedtoE1.Applying.8toAEc1andE2givesmAEc1=mAEc1E2+mAEc1Ec2:

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104CHAPTER4.MEASURETHEORY.SubstitutingthisbackintotheprecedingequationgivesmA=mAE1+mAEc1E2+mAEc1Ec2:.10SinceEc1Ec2=E1[E2cwecanwritethisasmA=mAE1+mAEc1E2+mAE1[E2c:NowAE1[E2=AE1[AEc1E2somAE1+mAEc1E2mAE1[E2:Substitutingthisforthetwotermsontherightofthepreviousdisplayedequa-tiongivesmAmAE1[E2+mAE1[E2cwhichisjust.9forthesetE1[E2.Thisprovesthelemmaandthetheorem.WeletMdenotetheclassofmeasurablesubsetsofR-measurability"inthesenseofLebesgueorCaratheodorythesebeingequivalent.Noticebyinductionstartingwithtwotermsasinthelemma,thatanyniteunionofsetsinMisagaininM4.5Countableadditivity.TherstmaintheoreminthesubjectisthefollowingdescriptionofMandthefunctionmonit:Theorem4.5.1Mandthefunctionm:M!Rhavethefollowingproperties:R2M.E2MEc2M.IfEn2Mforn=1;2;3;:::thenSnEn2M.IfFn2MandtheFnarepairwisedisjoint,thenF:=SnFn2MandmF=1Xn=1mFn:Proof.Wealreadyknowthersttwoitemsonthelist,andweknowthataniteunionofsetsinMisagaininM.WealsoknowthelastassertionwhichisProposition4.3.1.ButitwillbeinstructiveandusefulforustohaveaproofstartingdirectlyfromCaratheodory'sdenitionofmeasurablity:IfF12M;F22MandF1F2=;thentakingA=F1[F2;E1=F1;E2=F2

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4.5.COUNTABLEADDITIVITY.105in.10givesmF1[F2=mF1+mF2:InductionthenshowsthatifF1;:::;FnarepairwisedisjointelementsofMthentheirunionbelongstoMandmF1[F2[[Fn=mF1+mF2++mFn:Moregenerally,ifweletAbearbitraryandtakeE1=F1;E2=F2in.10wegetmA=mAF1+mAF2+mAF1[F2c:IfF32MisdisjointfromF1andF2wemayapply.8withAreplacedbyAF1[F2candEbyF3togetmAF1[F2c=mAF3+mAF1[F2[F3c;sinceF1[F2cFc3=Fc1Fc2Fc3=F1[F2[F3c:SubstitutingthisbackintotheprecedingequationgivesmA=mAF1+mAF2+mAF3+mAF1[F2[F3c:Proceedinginductively,weconcludethatifF1;:::;Fnarepairwisedisjointele-mentsofMthenmA=nX1mAFi+mAF1[[Fnc:.11NowsupposethatwehaveacountablefamilyfFigofpairwisedisjointsetsbelongingtoM.Sincen[i=1Fi!c1[i=1Fi!cweconcludefrom.11thatmAnX1mAFi+mA1[i=1Fi!c!andhencepassingtothelimitmA1X1mAFi+mA1[i=1Fi!c!:NowgivenanycollectionofsetsBkwecanndintervalsfIk;jgwithBk[jIk;j

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106CHAPTER4.MEASURETHEORY.andmBkXj`Ik;j+ 2k:So[kBk[k;jIk;jandhencem[BkXmBk;theinequalitybeingtriviallytrueifthesumontherightisinnite.So1Xi=1mAFkmA1[i=1Fi!!:ThusmA1X1mAFi+mA1[i=1Fi!c!mA1[i=1Fi!!+mA1[i=1Fi!c!:Theextremerightofthisinequalityisthelefthandsideof4.9appliedtoE=[iFi;andsoE2Mandtheprecedingstringofinequalitiesmustbeequalitiessincethemiddleistrappedbetweenbothsideswhichmustbeequal.HencewehaveprovedthatifFnisadisjointcountablefamilyofsetsbelongingtoMthentheirunionbelongstoMandmA=XimAFi+mA1[i=1Fi!c!:.12IfwetakeA=SFiweconcludethatmF=1Xn=1mFn.13iftheFjaredisjointandF=[Fj:SowehavereprovedthelastassertionofthetheoremusingCaratheodory'sdenition.Forthethirdassertion,weneedonlyobservethatacountableunionofsetsinMcanbealwayswrittenasacountabledisjointunionofsetsinM.Indeed,setF1:=E1;F2:=E2nE1=E1Ec2

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4.5.COUNTABLEADDITIVITY.107F3:=E3nE1[E2etc.TherighthandsidesallbelongtoMsinceMisclosedundertakingcomplementsandniteunionsandhenceintersections,and[jFj=[Ej:Wehavecompletedtheproofofthetheorem.Anumberofeasyconsequencesfollow:Thesymmetricdierencebetweentwosetsisthesetofpointsbelongingtooneortheotherbutnotboth:AB:=AnB[BnA:Proposition4.5.1IfA2MandmAB=0thenB2MandmA=mB.Proof.ByassumptionAnBhasmeasurezeroandhenceismeasurablesinceitiscontainedinthesetABwhichisassumedtohavemeasurezero.SimilarlyforBnA.AlsoAB2MsinceAB=AnAnB:ThusB=AB[BnA2M:SinceBnAandABaredisjoint,wehavemB=mAB+mBnA=mAB=mAB+mAnB=mA:QEDProposition4.5.2SupposethatAn2MandAnAn+1forn=1;2;:::.Thenm[An=limn!1mAn:Indeed,settingBn:=AnnAn)]TJ/F7 6.974 Tf 6.226 0 Td[(1withB1=A1theBiarepairwisedisjointandhavethesameunionastheAisom[An=1Xi=1mBi=limn!1nXi=1mBn=limn!1mn[i=1Bi!=limn!1mAn:QEDProposition4.5.3IfCnCn+1isadecreasingfamilyofsetsinMandmC1<1thenmCn=limn!1mCn:

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108CHAPTER4.MEASURETHEORY.Indeed,setA1:=;;A2:=C1nC2;A3:=C1nC3etc.TheA'sareincreasingsom[C1nCi=limn!1mC1nCn=mC1)]TJ/F8 9.963 Tf 13.449 0 Td[(limn!1mCnbytheprecedingproposition.SincemC1<1wehavemC1nCn=mC1)]TJ/F11 9.963 Tf 9.963 0 Td[(mCn:Also[nC1nCn=C1nnCn!:Som[nC1nCn!=mC1)]TJ/F11 9.963 Tf 9.962 0 Td[(mCn=mC1)]TJ/F8 9.963 Tf 13.449 0 Td[(limn!1mCn:SubtractingmC1frombothsidesofthelastequationgivestheequalityintheproposition.QED4.6-elds,measures,andoutermeasures.WewillnowtaketheitemsinTheorem4.5.1asaxioms:LetXbeaset.UsuallyXwillbeatopologicalspaceorevenametricspace.AcollectionFofsubsetsofXiscalledaeldif:X2F,IfE2FthenEc=XnE2F,andIffEngisasequenceofelementsinFthenSnEn2F,Theintersectionofanyfamilyof-eldsisagaina-eld,andhencegivenanycollectionCofsubsetsofX,thereisasmallest-eldFwhichcontainsit.ThenFiscalledthe-eldgeneratedbyC.IfXisametricspace,the-eldgeneratedbythecollectionofopensetsiscalledtheBorel-eld,usuallydenotedbyBorBXandasetbelongingtoBiscalledaBorelset.Givena-eldFanon-negativemeasureisafunctionm:F![0;1]suchthatm;=0andCountableadditivity:IfFnisadisjointcollectionofsetsinFthenm[nFn!=XnmFn:

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4.7.CONSTRUCTINGOUTERMEASURES,METHODI.109Inthecountableadditivityconditionitisunderstoodthatbothsidesmightbeinnite.AnoutermeasureonasetXisamapmto[0;1]denedonthecollectionofallsubsetsofXwhichsatisesm;=0,Monotonicity:IfABthenmAmB,andCountablesubadditivity:mSnAnPnmAn.Givenanoutermeasure,m,wedenedasetEtobemeasurablerelativetomifmA=mAE+mAEcforallsetsA.ThenCaratheodory'stheoremthatweprovedintheprecedingsectionassertsthatthecollectionofmeasurablesetsisa-eld,andthere-strictionofmtothecollectionofmeasurablesetsisameasurewhichweshallusuallydenotebym.Thereisanunfortunatedisagreementinterminology,inthatmanyoftheprofessionals,especiallyingeometricmeasuretheory,usethetermmeasure"forwhatwehavebeencallingoutermeasure".Howeverwewillfollowtheaboveconventionswhichusedtobetheoldfashionedstandard.Anobvioustask,givenCaratheodory'stheorem,istolookforwaysofcon-structingoutermeasures.4.7Constructingoutermeasures,MethodI.LetCbeacollectionofsetswhichcoverX.ForanysubsetAofXletcccAdenotethesetofniteorcountablecoversofAbysetsbelongingtoC.Inotherwords,anelementofcccAisaniteorcountablecollectionofelementsofCwhoseunioncontainsA.Supposewearegivenafunction`:C![0;1]:Theorem4.7.1ThereexistsauniqueoutermeasuremonXsuchthatmA`AforallA2CandIfnisanyoutermeasuresatisfyingtheprecedingconditionthennAmAforallsubsetsAofX.

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110CHAPTER4.MEASURETHEORY.ThisuniqueoutermeasureisgivenbymA=infD2cccAXD2D`D:.14Inotherwords,foreachcountablecoverofAbyelementsofCwecomputethesumabove,andthenminimizeoverallsuchcoversofA.Ifwehadtwooutermeasuressatisfyingbothconditionstheneachwouldhavetobetheother,sotheuniquenessisobvious.Tocheckthatthemdenedby.14isanoutermeasure,observethatfortheemptysetwemaytaketheemptycover,andtheconventionaboutanemptysumisthatitiszero,som;=0.IfABthenanycoverofBisacoverofA,sothatmAmB.Tocheckcountablesubadditivityweusetheusual=2ntrick:IfmAn=1foranyAnthesubadditivityconditionisobviouslysatised.Otherwise,wecanndaDn2cccAnwithXD2Dn`DmAn+ 2n:ThenwecancollectalltheDtogetherintoacountablecoverofAsomAXnmAn+;andsincethisistrueforall>0weconcludethatmiscountablysubadditive.Sowehaveveriedthatmdenedby.14isanoutermeasure.Wemustcheckthatitsatisesthetwoconditionsinthetheorem.IfA2CthenthesingleelementcollectionfAg2cccA,somA`A,sotherstconditionisobvious.Astothesecondcondition,supposenisanoutermeasurewithnD`DforallD2C.ThenforanysetAandanycountablecoverDofAbyelementsofCwehaveXD2D`DXD2DnDn[D2DD!nA;whereinthesecondinequalityweusedthecountablesubadditivityofnandinthelastinequalityweusedthemonotonicityofn.MinimizingoverallD2cccAshowsthatmAnA.QEDThisargumentisbasicallyarepeatperformanceoftheconstructionofLebesguemeasurewedidabove.Howeverthereissometrouble:4.7.1Apathologicalexample.SupposewetakeX=R,andletCconsistofallhalfopenintervalsoftheform[a;b.However,insteadoftaking`tobethelengthoftheinterval,wetakeittobethesquarerootofthelength:`[a;b:=b)]TJ/F11 9.963 Tf 9.963 0 Td[(a1 2:

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4.7.CONSTRUCTINGOUTERMEASURES,METHODI.111Iclaimthatanyhalfopenintervalsay[0;1oflengthonehasm[a;b=1.Since`istranslationinvariant,itdoesnotmatterwhichintervalwechoose.Indeed,m[0;11bytherstconditioninthetheorem,since`[0;1=1.Ontheotherhand,if[0;1[i[ai;bithenweknowfromtheHeine-BorelargumentthatXbi)]TJ/F11 9.963 Tf 9.963 0 Td[(ai1;sosquaringgivesXbi)]TJ/F11 9.963 Tf 9.963 0 Td[(ai1 22=Xibi)]TJ/F11 9.963 Tf 9.962 0 Td[(ai+Xi6=jbi)]TJ/F11 9.963 Tf 9.963 0 Td[(ai1 2bj)]TJ/F11 9.963 Tf 9.963 0 Td[(aj1 21:Som[0;1=1.Ontheotherhand,consideraninterval[a;boflength2.Sinceitcoversitself,m[a;bp 2.ConsidertheclosedintervalI=[0;1].ThenI[)]TJ/F8 9.963 Tf 7.749 0 Td[(1;1=[0;1andIc[)]TJ/F8 9.963 Tf 7.749 0 Td[(1;1=[)]TJ/F8 9.963 Tf 7.749 0 Td[(1;0somI[)]TJ/F8 9.963 Tf 7.748 0 Td[(1;1+mIc[)]TJ/F8 9.963 Tf 7.749 0 Td[(1;1=2>p 2m[)]TJ/F8 9.963 Tf 7.749 0 Td[(1;1:Inotherwords,theclosedunitintervalisnotmeasurablerelativetotheoutermeasuremdeterminedbythetheorem.WewouldlikeBorelsetstobemeasur-able,andtheabovecomputationshowsthatthemeasureproducedbyMethodIasabovedoesnothavethisdesirableproperty.Infact,ifweconsidertwohalfopenintervalsI1andI2oflengthoneseparatedbyasmalldistanceofsize,say,thentheirunionI1[I2iscoveredbyanintervaloflength2+,andhencemI1[I2p 2+0:.15TheconditiondA;B>0meansthatthereisan>0dependingonAandBsothatdx:y>forallx2A;y2B.ThemainresulthereisduetoCaratheodory:

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112CHAPTER4.MEASURETHEORY.Theorem4.7.2IfmisametricoutermeasureonametricspaceX,thenallBorelsetsofXaremmeasurable.Proof.Sincethe-eldofBorelsetsisgeneratedbytheclosedsets,itisenoughtoprovethateveryclosedsetFismeasurableinthesenseofCaratheodory,i.e.thatforanysetAmAmAF+mAnF:LetAj:=fx2Ajdx;F1 jg:WehavedAj;AF1=jso,sincemisametricoutermeasure,wehavemAF+mAj=mAF[AjmA.16sinceAF[AjA.NowAnF=[AjsinceFisclosed,andhenceeverypointofAnotbelongingtoFmustbeatapositivedistancefromF.Wewouldliketobeabletopasstothelimitin.16.Ifthelimitontheleftisinnite,thereisnothingtoprove.Sowemayassumeitisnite.Nowifx2AnF[Aj+1thereisaz2Fwithdx;z<1=j+1whileify2Ajwehavedy;z1=jsodx;ydy;z)]TJ/F11 9.963 Tf 9.963 0 Td[(dx;z1 j)]TJ/F8 9.963 Tf 19.582 6.739 Td[(1 j+1>0:LetB1:=A1andB2:=A2nA1;B3=A3nA2etc.Thusifij+2;thenBjAjandBiAnF[Ai)]TJ/F7 6.974 Tf 6.227 0 Td[(1AnF[Aj+1andsodBi;Bj>0.SomisadditiveonniteunionsofevenoroddB's:mn[k=1B2k)]TJ/F7 6.974 Tf 6.227 0 Td[(1!=nXk=1mB2k)]TJ/F7 6.974 Tf 6.226 0 Td[(1;mn[k=1B2k!=nXk=1mB2k:BothofthesearemA2nsincetheunionofthesetsinvolvedarecontainedinA2n.SincemA2nisincreasing,andassumedbounded,bothoftheabove

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4.8.CONSTRUCTINGOUTERMEASURES,METHODII.113seriesconverge.ThusmA=F=m[Ai=m0@Aj[[kj+1Bj1AmAj+1Xk=j+1mBjlimn!1mAn+1Xk=j+1mBj:Butthesumontherightcanbemadeassmallaspossiblebychoosingjlarge,sincetheseriesconverges.HencemA=Flimn!1mAnQED.4.8Constructingoutermeasures,MethodII.LetCEbetwocovers,andsupposethat`isdenedonE,andhence,byrestriction,onC.Inthedenition.14oftheoutermeasurem`;Cassociatedto`andC,weareminimizingoverasmallercollectionofcoversthanincomputingthemetricoutermeasurem`;EusingallthesetsofE.Hencem`;CAm`;EAforanysetA.WewanttoapplythisremarktothecasewhereXisametricspace,andwehaveacoverCwiththepropertythatforeveryx2Xandevery>0thereisaC2Cwithx2CanddiamC<.Inotherwords,weareassumingthattheC:=fC2CjdiamC0.ThenforeverysetAthem`;CAareincreasing,sowecanconsiderthefunctiononsetsgivenbymIIA:=sup!0m`;CA:Theaxiomsforanoutermeasurearepreservedbythislimitoperation,somIIisanoutermeasure.IfAandBaresuchthatdA;B>2,thenanysetofCwhichintersectsAdoesnotintersectBandviceversa,sothrowingawayextraneoussetsinacoverofA[Bwhichdoesnotintersecteither,weseethatmIIA[B=mIIA+mIIB.ThemethodIIconstructionalwaysyieldsametricoutermeasure.

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114CHAPTER4.MEASURETHEORY.4.8.1Anexample.LetXbethesetofallonesidedinnitesequencesof0'sand1's.SoapointofXisanexpressionoftheforma1a2a3whereeachaiis0or1.Foranynitesequenceof0'sor1's,let[]denotethesetofallsequenceswhichbeginwith.Wealsoletjjdenotethelengthof,thatis,thenumberofbitsin.Foreach00,wemustnda>0suchthatifdrx;y
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4.8.CONSTRUCTINGOUTERMEASURES,METHODII.115Thereissomethingspecialaboutthevaluer=1 2:LetCbethecollectionofallsetsoftheform[]andlet`bedenedonCby`[]=1 2jj:WecanconstructthemethodIIoutermeasureassociatedwiththisfunction,whichwillsatisfymII[]mI[]wheremIdenotesthemethodIoutermeasureassociatedwith`.Whatisspecialaboutthevalue1 2isthatifk=jjthen`[]=1 2k=1 2k+1+1 2k+1=`[0]+`[1]:Soifwealsousethemetricd1 2,wesee,byrepeatingtheabove,thatevery[]canbewrittenasthedisjointunionC1[[CnofsetsinCwith`[]=P`Ci.Thusm`;C[]`andsom`;C[]AmIAormII=mI.Italsofollowsfromtheabovecomputationthatm[]=`[]:Thereisalsosomethingspecialaboutthevalues=1 3:RecallthatoneofthedenitionsoftheCantorsetCisthatitconsistsofallpointsx2[0;1]whichhaveabase3expansioninvolvingonlythesymbols0and2.Leth:X!Cwherehsendsthebit1intothesymbol2,e.g.h:::=:022002::::Inotherwords,foranysequencezhz=hz 3;hz=hz+2 3:.18Iclaimthat:1 3d1 3x;yjhx)]TJ/F11 9.963 Tf 9.962 0 Td[(hyjd1 3x;y.19Proof.Ifxandystartwithdierentbits,sayx=0x0andy=1y0thend1 3x;y=1whilehxliesintheinterval[0;1 3]andhyliesintheinterval[2 3;1]ontherealline.Sohxandhyareatleastadistance1 3andatmostadistance1apart,whichiswhat.19says.Soweproceedbyinduction.Supposeweknowthat.19istruewhenx=x0andy=y0withx0;y0startingwithdierentdigits,andjjn.Theabovecasewaswherejj=0.

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116CHAPTER4.MEASURETHEORY.Soifjj=n+1theneitherand=0or=1andtheargumentforeithercaseissimilar:Weknowthat.19holdsforx0andy0andd1 3x;y=1 3d1 3x0;y0whilejhx)]TJ/F11 9.963 Tf 11.034 0 Td[(hyj=1 3jhx0)]TJ/F11 9.963 Tf 11.033 0 Td[(hy0jby.18.Hence.19holdsbyinduction.QEDInotherwords,themaphisaLipschitzmapwithLipschitzinversefromX;d1 3totheCantorsetC.Inashortwhile,aftermakingtheappropriatedenitions,thesetwocom-putations,onewiththemeasureassociatedto`[]=)]TJ/F7 6.974 Tf 5.762 -4.147 Td[(1 2jjandtheotherassociatedwithd1 3willshowthattheHausdordimension"oftheCantorsetislog2=log3.4.9Hausdormeasure.LetXbeametricspace.RecallthatifAisanysubsetofX,thediameterofAisdenedasdiamA=supx;y2Adx;y:TakeCtobethecollectionofallsubsetsofX,andforanypositiverealnumbersdene`sA=diamAswith0s=0.TakeCtoconsistofallsubsetsofX.ThemethodIIoutermeasureiscalledthes-dimensionalHausdoroutermeasure,anditsre-strictiontotheassociated-eldofCaratheodorymeasurablesetsiscalledthes-dimensionalHausdormeasure.Wewillletms;denotethemethodIoutermeasureassociatedto`sand,andletHsdenotetheHausdoroutermeasureofdimensions,sothatHsA=lim!0ms;A:Forexample,weclaimthatforX=R,H1isexactlyLebesgueoutermea-sure,whichwewilldenoteherebyL.Indeed,ifAhasdiameterr,thenAiscontainedinaclosedintervaloflengthr.HenceLAr.TheMethodIconstructiontheoremsaysthatm1;isthelargestoutermeasuresatisfyingmAdiamAforsetsofdiameterlessthan.Hencem1;ALAforallsetsAandall,andsoH1L:Ontheotherhand,anyboundedhalfopeninterval[a;bcanbebrokenupintoaniteunionofhalfopenintervalsoflength<,whosesumofdiametersisb)]TJ/F11 9.963 Tf 9.962 0 Td[(a.Som1;[a;bb)]TJ/F11 9.963 Tf 9.961 0 Td[(a.ButthemethodIconstructiontheoremsaysthatListhelargestoutermeasuresatisfyingm[a;bb)]TJ/F11 9.963 Tf 9.963 0 Td[(a:

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4.10.HAUSDORFFDIMENSION.117HenceH1LSotheyareequal.Intwoormoredimensions,theHausdormeasureHkonRkdiersfromLebesguemeasurebyaconstant.Thisisessentiallybecausetheyassigndierentvaluestotheballofdiameterone.Intwodimensionsforexample,theHausdormeasureH2assignsthevalueonetothediskofdiameterone,whileitsLebesguemeasureis=4.Forthisreason,someauthorsprefertoputthiscorrectionfactor"intothedenitionoftheHausdormeasure,whichwouldinvolvetheGammafunctionfornon-integrals.Iamfollowingtheconventionthatndsitsimplertodropthisfactor.Theorem4.9.1LetFXbeaBorelset.Let00HsF=1:Indeed,ifdiamA,thenmt;AdiamAtt)]TJ/F10 6.974 Tf 6.227 0 Td[(sdiamAssobythemethodIconstructiontheoremwehavemt;Bt)]TJ/F10 6.974 Tf 6.227 0 Td[(sms;BforallB.IfwetakeB=Finthisequality,thentheassumptionHsF<1impliesthatthelimitoftherighthandsidetendsto0as!0,soHtF=0.Thesecondassertioninthetheoremisthecontrapositiveoftherst.4.10Hausdordimension.ThislasttheoremimpliesthatforanyBorelsetF,thereisauniquevalues0whichmightbe0or1suchthatHtF=1forallts0.ThisvalueiscalledtheHausdordimensionofF.Itisoneofmanycompetingandnon-equivalentdenitionsofdimension.Noticethatitisametricinvariant,andinfactisthesamefortwospacesdierentbyaLipschitzhomeomorphismwithLipschitzinverse.Butitisnotatopologicalinvariant.Infact,weshallshowthatthespaceXofallsequencesofzerosandonestudiedabovehasHausdordimension1relativetothemetricd1 2whileithasHausdordimensionlog2=log3ifweusethemetricd1 3.SincewehaveshownthatX;d1 3isLipschitzequivalenttotheCantorsetC,thiswillalsoprovethatChasHausdordimensionlog2=log3.Werstdiscussthed1 2caseandusethefollowinglemmaLemma4.10.1IfdiamA>0,thenthereisansuchthatA[]anddiam[]=diamA.

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118CHAPTER4.MEASURETHEORY.Proof.GivenanysetA,ithasalongestcommonprex".Indeed,considerthesetoflengthsofcommonprexesofelementsofA.Thisisnitesetofnon-negativeintegerssinceAhasatleasttwodistinctelements.Letnbethelargestofthese,andletbeacommonprexofthislength.ThenitisclearlythelongestcommonprexofA.HenceA[]anddiam[]=diamA.QEDLetCdenotethecollectionofallsetsoftheform[]andlet`bethefunctiononCgivenby`[]=1 2jj;andlet`betheassociatedmethodIoutermeasure,andmtheassociatedmeasure;alltheseasweintroducedabove.Wehave`A`[]=diam[]=diamA:BythemethodIconstructiontheorem,m1;isthelargestoutermeasurewiththepropertythatnAdiamAforsetsofdiameter<.Hence`m1;,andsincethisistrueforall>0,weconcludethat`H1:Ontheotherhand,foranyandany>0,thereisannsuchthat2)]TJ/F10 6.974 Tf 6.227 0 Td[(n
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4.11.PUSHFORWARD.119Hences=log2=log3istheHausdordimensionofX.ThematerialabovewithsomeslightchangesinnotationwastakenfromthebookMeasure,Topology,andFractalGeometrybyGeraldEdgar,whereathoroughanddelightfullycleardiscussioncanbefoundofthesubjectslistedinthetitle.4.11Pushforward.Theabovediscussionisasamplingofintroductorymaterialtowhatisknownasgeometricmeasuretheory".Howevertheconstructionofmeasuresthatwewillbemainlyworkingwithwillbeanabstractionofthesimulation"approachthatwehavebeendevelopingintheproblemsets.Thesetupisasfollows:LetX;F;mbeasetwitha-eldandameasureonit,andletY;Gbesomeothersetwitha-eldonit.Amapf:X!Yiscalledmeasurableiff)]TJ/F7 6.974 Tf 6.226 0 Td[(1B2F8B2G:WemaythendeneameasurefmonY;GbyfmB=mf)]TJ/F7 6.974 Tf 6.227 0 Td[(1B:Forexample,ifYisthePoissonrandomvariablefromtheexercises,anduistheuniformmeasuretherestrictionofLebesguemeasuretoon[0;1],thenfuisthemeasureonthenon-negativeintegersgivenbyfufkg=e)]TJ/F10 6.974 Tf 6.227 0 Td[(k k!:Itwillbethisconstructionofmeasuresandvariantsonitwhichwilloccupyusoverthenextfewweeks.4.12TheHausdordimensionoffractals4.12.1Similaritydimension.Contractingratiolists.Anitecollectionofrealnumbersr1;:::;rniscalledacontractingratiolistif0
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120CHAPTER4.MEASURETHEORY.Proposition4.12.1Letr1;:::;rnbeacontractingratiolist.Thereexistsauniquenon-negativerealnumberssuchthatnXi=1rsi=1:.20Thenumbersis0ifandonlyifn=1.Proof.Ifn=1thens=0worksandisclearlytheonlysolution.Ifn>1,denethefunctionfon[0;1byft:=nXi=1rti:Wehavef=nandlimt!1ft=0<1:Sincefiscontinuous,thereissomepostivesolutionto.20.Toshowthatthissolutionisunique,itisenoughtoshowthatfismonotonedecreasing.ThisfollowsfromthefactthatitsderivativeisnXi=1rtilogri<0:QEDDenition4.12.1Thenumbersin.20iscalledthesimilaritydimensionoftheratiolistr1;:::;rn.Iteratedfunctionsystemsandfractals.Amapf:X!YbetweentwometricspacesiscalledasimilaritywithsimilarityratiorifdYfx1;fx2=rdXx1;x28x1;x22X:RecallthatamapiscalledLipschitzwithLipschitzconstantrifweonlyhadaninequality,,insteadofanequalityintheabove.LetXbeacompletemetricspace,andletr1;:::;rnbeacontractingratiolist.Acollectionf1;:::;fn;fi:X!Xiscalledaniteratedfunctionsystemwhichrealizesthecontractingratiolistiffi:X!X;i=1;:::;nisasimilaritywithratiori.Wealsosaythatf1;:::;fnisarealizationoftheratiolistr1;:::;rn.ItisaconsequenceofHutchinson'stheorem,seebelow,that

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4.12.THEHAUSDORFFDIMENSIONOFFRACTALS121Proposition4.12.2Iff1;:::;fnisarealizationofthecontractingratiolistr1;:::;rnonacompletemetricspace,X,thenthereexistsauniquenon-emptycompactsubsetKXsuchthatK=f1K[[fnK:Infact,Hutchinson'stheoremassertsthecorrespondingresultwherethefiaremerelyassumedtobeLipschitzmapswithLipschitzconstantsr1;:::;rn.ThesetKissometimescalledthefractalassociatedwiththerealizationf1;:::;fnofthecontractingratiolistr1;:::;rn.Thefactswewanttoestablishare:First,dimKs.21wheredimdenotesHausdordimension,andsisthesimilaritydimensionofr1;:::;rn.Ingeneral,wecanonlyassertaninequalityhere,forthethesetKdoesnotxr1;:::;rnoritsrealization.Forexample,wecanrepeatsomeoftheriandthecorrespondingfi.Thiswillgiveusalongerlist,andhencealargers,butwillnotchangeK.Butwecandemandaratherstrongformofnon-redundancyknownasMoran'scondition:ThereexistsanopensetOsuchthatOfiO8iandfiOfjO=;8i6=j:.22ThenTheorem4.12.1Iff1;:::;fnisarealizationofr1;:::;rnonRdandifMoran'sconditionholdsthendimK=s:Themethodofproofof.21willbetoconstructamodel"completemetricspaceEwitharealizationg1;:::;gnofr1;:::;rnonit,whichisuniversal"inthesensethatEisitselfthefractalassociatedtog1;:::;gn.TheHausdordimensionofEiss.Iff1;:::;fnisarealizationofr1;:::;rnonacompletemetricspaceXthenthereexistsauniquecontinuousmaph:E!Xsuchthathgi=fih:.23TheimagehEofhisK.ThemaphisLipschitz.Thisisclearlyenoughtoprove.21.AlittlemoreworkwillthenproveMoran'stheorem.

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122CHAPTER4.MEASURETHEORY.4.12.2Thestringmodel.Constructionofthemodel.Letr1;:::;rnbeacontractingratiolist,andletAdenotethealphabetcon-sistingofthelettersf1;:::;ng.LetEdenotethespaceofonesidedinnitestringsoflettersfromthealphabetA.IfdenotesanitestringwordoflettersfromA,weletwdenotetheproductoverallioccurringinoftheri.Thusw;=1where;istheemptystring,and,inductively,we=wwe;e2A:Ifx6=yaretwoelementsofE,theywillhavealongestcommoninitialstring,andwethendenedx;y:=w:ThismakesEintoacompleteultrameticspace.Denethemapsgi:E!Ebygix=ix:Thatis,gishiftstheinnitestringoneunittotherightandinsertstheletteriintheinitialposition.Intermsofourmetric,clearlyg1;:::;gnisarealizationofr1;:::;rnandthespaceEitselfisthecorrespondingfractalset.Welet[]denotethesetofallstringsbeginningwith,i.e.whoserstwordoflengthequaltothelengthofis.Thediameterofthissetisw.TheHausdordimensionofEiss.Webeginwithalemma:Lemma4.12.1LetAEhavepositivediameter.ThenthereexistsawordsuchthatA[]anddiamA=diam[]=w:Proof.SinceAhasatleasttwoelements,therewillbeawhichisaprexofoneandnottheother.SotherewillbeanintegernpossiblyzerowhichisthelengthofthelongestcommonprexofallelementsofA.TheneveryelementofAwillbeginwiththiscommonprexwhichthussatisestheconditionsofthelemma.QEDThelemmaimpliesthatincomputingtheHausdormeasureordimension,weneedonlyconsidercoversbysetsoftheform[].Nowifwechoosestobethesolutionof.20,thendiam[]s=nXi=1diam[i]s=diam[]snXi=1rsi:

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4.12.THEHAUSDORFFDIMENSIONOFFRACTALS123ThismeansthatthemethodIIoutermeasureassosicatedtothefunctionA7!diamAscoincideswiththemethodIoutermeasureandassignstoeachset[]themeasurews.InparticularthemeasureofEisone,andsotheHausdordimensionofEiss.TheuniversalityofE.Letf1;:::;fnarealizationofr1;:::;rnonacompletemetricspaceX.Chooseapointa2Xanddeneh0:E!Xbyh0z:a:Inductivelydenethemapshpbydeninghp+1oneachoftheopensets[fig]byhp+1iz:=fihpz:ThesequenceofmapsfhpgisCauchyintheuniformnorm.Indeed,ify2[fig]soy=gizforsomez2EthendXhp+1y;hpy=dXfihpz;fihp)]TJ/F7 6.974 Tf 6.227 0 Td[(1z=ridXhpz;hp)]TJ/F7 6.974 Tf 6.226 0 Td[(1z:Soifweletc:=maxirisothat0
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124CHAPTER4.MEASURETHEORY.4.13TheHausdormetricandHutchinson'sthe-orem.LetXbeacompletemetricspace.LetHXdenotethespaceofnon-emptycompactsubsetsofX.ForanyA2HXandanypositivenumber,letA=fx2Xjdx;y;forsomey2Ag:WecallAthe-collarofA.Recallthatwedeneddx;A=infy2Adx;ytobethedistancefromanyx2XtoA,thenwecanwritethedenitionofthe-collarasA=fxjdx;Ag:Noticethattheinmuminthedenitionofdx;Aisactuallyachieved,thatis,thereissomepointy2Asuchthatdx;A=dx;y:ThisisbecauseAiscompact.Forapairofnon-emptycompactsets,AandB,denedA;B=maxx2Adx;B:SodA;B,AB:NoticethatthisconditionisnotsymmetricinAandB.SoHausdorintroducedhA;B=maxfdA;B;dB;Ag.24=inffjABandBAg:.25asadistanceonHX.HeprovedProposition4.13.1ThefunctionhonHXHXsatsiestheaxiomsforametricandmakesHXintoacompletemetricspace.Furthermore,ifA;B;C;D2HXthenhA[B;C[DmaxfhA;C;hB;Dg:.26Proof.Webeginwith.26.IfissuchthatACandBDthenclearlyA[BC[D=C[D.RepeatingthisargumentwiththerolesofA;CandB;Dinterchangedproves.26.Weprovethathisametric:hissymmetric,bydenition.Also,hA;A=0;andifhA;B=0,theneverypointofAiswithinzerodistanceofB,and

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4.13.THEHAUSDORFFMETRICANDHUTCHINSON'STHEOREM.125hencemustbelongtoBsinceBiscompact,soABandsimilarlyBA.SohA;B=0impliesthatA=B.Wemustprovethetriangleinequality.ForthisitisenoughtoprovethatdA;BdA;C+dC;B;becauseinterchangingtheroleofAandBgivesthedesiredresult.Nowforanya2Awehaveda;B=minb2Bda;bminb2Bda;c+dc;b8c2C=da;c+minb2Bdc;b8c2C=da;c+dc;B8c2Cda;c+dC;B8c2C:Thesecondterminthelastexpressiondoesnotdependonc,sominimizingovercgivesda;Bda;C+dC;B:Maximizingoveraontherightgivesda;BdA;C+dC;B:MaximizingontheleftgivesthedesireddA;BdA;C+dC;A:Wesketchtheproofofcompleteness.LetAnbeasequenceofcompactnon-emptysubsetsofXwhichisCauchyintheHausdormetric.DenethesetAtobethesetofallx2Xwiththepropertythatthereexistsasequenceofpointsxn2Anwithxn!x.ItisstraighforwardtoprovethatAiscompactandnon-emptyandisthelimitoftheAnintheHausdormetric.Supposethat:X!Xisacontraction.ThendenesatransformationonthespaceofsubsetsofXwhichwecontinuetodenoteby:A=fxjx2Ag:Sinceiscontinuous,itcarriesHXintoitself.LetcbetheLipschitzconstantof.ThendA;B=maxa2A[minb2Bda;b]maxa2A[minb2Bcda;b]=cdA;B:Similarly,dB;AcdB;AandhencehA;BchA;B:.27

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126CHAPTER4.MEASURETHEORY.Inotherwords,acontractiononXinducesacontractiononHX.ThepreviousremarktogetherwiththefollowingobservationisthekeytoHutchinson'sremarkableconstructionoffractals:Proposition4.13.2LetT1;:::;TnbeacollectionofcontractionsonHXwithLipschitzconstantsc1;:::;cn,andletc=maxci.Denethetransforma-tionTonHXbyTA=T1A[T2A[[TnA:ThenTisacontractionwithLipschitzconstantc.Proof.Byinduction,itisenoughtoprovethisforthecasen=2.By.26hTA;TB=hT1A[T2A;T1B[T2BmaxfhT1A;hT1B;hT2A;T2Bgmaxfc1hA;B;c2hA;Bg=hA;Bmaxfc1;c2g=chA;B.PuttingthepreviousfactstogetherwegetHutchinson'stheorem;Theorem4.13.1LetT1;:::;TnbecontractionsonacompletemetricspaceandletcbethemaximumoftheirLipschitzcontants.DenetheHutchinosonoperatorTonHXbyTA:=T1A[[TnA:ThenTisacontractionwithLipschtzconstantc.4.14AneexamplesWedescribeseveralexamplesinwhichXisasubsetofavectorspaceandeachoftheTiinHutchinson'stheoremareanetransformationsoftheformTi:x7!Aix+biwherebi2XandAiisalineartransformation.4.14.1TheclassicalCantorset.TakeX=[0;1],theunitinterval.TakeT1:x7!x 3;T2:x7!x 3+2 3:Thesearebothcontractions,sobyHutchinson'stheoremthereexistsauniqueclosedxedsetC.ThisistheCantorset.

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4.14.AFFINEEXAMPLES127TorelateittoCantor'soriginalconstruction,letusgobacktotheproofofthecontractionxedpointtheoremappliedtoTactingonHX.Itsaysthatifwestartwithanynon-emptycompactsubsetA0andkeepapplyingTtoit,i.e.setAn=TnA0thenAn!CintheHausdormetric,h.SupposewetaketheintervalIitselfasourA0.ThenA1=TI=[0;1 3][[2 3;1]:inotherwords,applyingtheHutchinsonoperatorTtotheinterval[0;1]hastheeectofdeletingthemiddlethird"openinterval1 3;2 3.ApplyingToncemoregivesA2=T2[0;1]=[0;1 9][[2 9;1 3][[2 3;7 9][[8 9;1]:Inotherwords,A2isobtainedfromA1bydeletingthemiddlethirdsofeachofthetwointervalsofA1andsoon.ThiswasCantor'soriginalconstruction.SinceAn+1Anforthischoiceofinitialset,theHausdorlimitcoincideswiththeintersection.ButofcourseHutchinson'stheoremandtheproofofthecontractionsxedpointtheoremsaysthatwecanstartwithanynon-emptyclosedsetasourinitialseed"andthenkeepapplyingT.Forexample,supposewestartwiththeonepointsetB0=f0g.ThenB1=TB0isthetwopointsetB1=f0;2 3g;B2consistsofthefourpointsetB2=f0;2 9;2 3;8 9gandsoon.WethenmusttaketheHausdorlimitofthisincreasingcollectionofsets.Todescribethelimitingsetcfromthispointofview,itisusefultousetriadicexpansionsofpointsin[0;1].Thus0=:00000002=3=:20000002=9=:02000008=9=:2200000andsoon.ThusthesetBnwillconsistofpointswhosetriadicexpansionhasonlyzerosortwosintherstnpositionsfollowedbyastringofallzeros.ThusapointwilllieinCbethelimitofsuchpointsifandonlyifithasatriadicexpansionconsistingentirelyofzerosortwos.Thisincludesthepossibilityofaninnitestringofalltwosatthetailoftheexpansion.forexample,thepoint1whichbelongstotheCantorsethasatriadicexpansion1=:222222.Similarlythepoint2 3hasthetriadicexpansion2 3=:0222222andsoisin

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128CHAPTER4.MEASURETHEORY.thelimitofthesetsBn.Butapointsuchas:101isnotinthelimitoftheBnandhencenotinC.ThisdescriptionofCisalsoduetoCantor.Noticethatforanypointawithtriadicexpansiona=:a1a2a2T1a=:0a1a2a3;whileT2a=:2a1a2a3:Thusifalltheentriesintheexpansionofaareeitherzeroortwo,thiswillalsobetrueforT1aandT2a.ThisshowsthattheCgivenbythissecondCantordescriptionsatisesTCC.Ontheotherhand,T1:a2a3=:0a2a3;T2:a2a3=:2a2a3whichshowsthat:a1a2a3isintheimageofT1ifa1=0orintheimageofT2ifa1=2.ThisshowsthatTC=C.SinceCaccordingtoCantor'sseconddescriptionisclosed,theuniquenesspartofthexedpointtheoremguaranteesthattheseconddescriptioncoincideswiththerst.ThestatementthatTC=CimpliesthatCisself-similar".4.14.2TheSierpinskiGasketConsiderthethreeanetransformationsoftheplane:T1:xy7!1 2xy;T2:xy7!1 2xy+1 210;T3:xy7!1 2xy+1 201:ThexedpointoftheHutchinsonoperatorforthischoiceofT1;T2;T3iscalledtheSierpinskigasket,S.IfwetakeourinitialsetA0tobetherighttrianglewithverticesat00;10;and01theneachoftheTiA0isasimilarrighttrianglewhoselineardimensionsareone-halfaslarge,andwhichsharesonecommonvertexwiththeoriginaltriangle.Inotherwords,A1=TA0isobtainedfromouroriginaltrianglebedeletingtheinteriorofthereversedrighttrianglewhoseverticesarethemidpointsofourorigninaltriangle.JustasinthecaseoftheCantorset,successiveapplicationsofTtothischoiceoforiginalsetamountstosuccessivedeletionsofthemiddle"andtheHausdorlimitistheintersectionofallofthem:S=TAi.WecanalsostartwiththeoneelementsetB000

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4.14.AFFINEEXAMPLES129Usingabinaryexpansionforthexandycoordinates,applicationofTtoB0givesthethreeelementset00;:10;0:1:ThesetB2=TB1willcontainninepoints,whosebinaryexpansionsareob-tainedfromtheabovethreebyshiftingthexandyexapnsionsoneunittotherightandeitherinsertinga0beforebothexpansionstheeectofT1,inserta1beforetheexpansionofxandazerobeforetheyorviceversa.Procedinginthisfashion,weseethatBnconsistsof3npointswhichhaveall0inthebinaryexpansionofthexandycoordinates,pastthen-thposition,andwhicharefurtherconstrainedbytheconditionthatatnoearlerpointdowehavebothxi=1andyi=1.PassingtothelimitshowsthatSconsistsofallpointsforwhichwecanndpossibleininitebinaryexpansionsofthexandycoordi-natessothatxi=1=yineveroccurs.Forexamplex=1 2;y=1 2belongstoSbecausewecanwritex=:10000;y=:011111:::.Again,fromthisseconddescriptionofSintermsofbinaryexpansionsitisclearthatTS=S.4.14.3Moran'stheoremIfAisanysetsuchthatf[A]A,thenclearlyfp[A]Abyinduction.IfAisnon-emptyandclosed,thenforanya2A,andanyx2E,thelimitofthefabelongstoKasrangesovertherstwordsofsizepofx,andsobelongstoKandalsotoA.SincethesepointsconsitituteallofK,weseethatKAandhencefKfA.28foranyword.NowsupposethatMoran'sopensetconditionissatised,andletuswriteO:=fO:ThenOO=;ifisnotaprexoforisnotaprexof.Furthermore, fO=f Osowecanusethesymbol Ounambiguouslytodenotethesetwoequalsets.Byvirtueof.28wehaveK O

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130CHAPTER4.MEASURETHEORY.whereweuseKtodenotefK.Supposethatisnotaprexoforviceversa.ThenKO=;since OO=;.LetmdenotethemeasureonthestringmodelEthatweconstructedabove,sothatmE=1andmoregenerallym[]=ws.ThenwewillhaveprovedthattheHausdordimensionofKiss,andhence=sifwecanprovethatthereexistsaconstantbsuchthatforeveryBorelsetBKmh)]TJ/F7 6.974 Tf 6.227 0 Td[(1BbdiamBs;.29whereh:E!KisthemapweconstructedabovefromthestringmodeltoK.Letusintroducethefollowingnotation:Foranynitenon-emptystring,let)]TJ/F8 9.963 Tf 10.453 -3.615 Td[(denotethestringofcardinalityonelessobtainedbyremovingthelastletterin.Lemma4.14.1ThereexistsanintegerNsuchthatforanysubsetBKthesetQBofallnitestringssuchthat OB6=;anddiamO
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4.14.AFFINEEXAMPLES131orm2dDd Vrd:SoanyintegergreaterthattherighthandsideofthisinequalitywhichisindependentofBwilldo.Nowweturntotheproofof.29whichwillthencompletetheproofofMoran'stheorem.LetBbeaBorelsubsetofK.ThenB[2QB Osoh)]TJ/F7 6.974 Tf 6.227 0 Td[(1B[2QB[]:Now[]=diam[]s=1 DdiamOs1 DsdiamBsandsomh)]TJ/F7 6.974 Tf 6.226 0 Td[(1BX2QBmN1 DsdiamBsandhencewemaytakeb=N1 DsdiamBsandthen.29willhold.

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132CHAPTER4.MEASURETHEORY.

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Chapter5TheLebesgueintegral.Inwhatfollows,X;F;misaspacewitha-eldofsets,andmameasureonF.ThepurposeofthischapteristodevelopthetheoryoftheLebesgueintegralforfunctionsdenedonX.Thetheorystartswithsimplefunctions,thatisfunctionswhichtakeononlynitelymanynon-zerovalues,sayfa1;:::;angandwhereAi:=f)]TJ/F7 6.974 Tf 6.226 0 Td[(1ai2F:Inotherwords,westartwithfunctionsoftheformx=nXi=1ai1AiAi2F:.1Then,foranyE2FwewouldliketodenetheintegralofasimplefunctionoverEasZEdm=nXi=1aimAiE.2andextendthisdenitionbysomesortoflimitingprocesstoabroaderclassoffunctions.Ihaven'tyetspeciedwhattherangeofthefunctionsshouldbe.Certainly,eventogetstarted,wehavetoallowourfunctionstotakevaluesinavectorspaceoverR,inorderthattheexpressionontherightof.2makesense.Infact,IwilleventuallyallowftotakevaluesinaBanachspace.Howeverthetheoryisabitsimplerforrealvaluedfunctions,wherethelinearorderoftherealsmakessomeargumentseasier.Ofcourseitwouldthenbenoproblemtopasstoanynitedimensionalspaceoverthereals.ButwewillonoccasionneedintegralsininnitedimensionalBanachspaces,andthatwillrequirealittlereworkingofthetheory.133

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134CHAPTER5.THELEBESGUEINTEGRAL.5.1Realvaluedmeasurablefunctions.RecallthatifX;FandY;Garespaceswith-elds,thenf:X!Yiscalledmeasurableiff)]TJ/F7 6.974 Tf 6.226 0 Td[(1E2F8E2G:.3NoticethatthecollectionofsubsetsofYforwhich.3holdsisa-eld,andhenceifitholdsforsomecollectionC,itholdsforthe-eldgeneratedbyC.ForthenextfewsectionswewilltakeY=RandG=B,theBoreleld.SincethecollectionofopenintervalsonthelinegeneratetheBoreleld,arealvaluedfunctionf:X!Rismeasurableifandonlyiff)]TJ/F7 6.974 Tf 6.226 0 Td[(1I2FforallopenintervalsI:Equallywell,itisenoughtocheckthisforintervalsoftheform;aforallrealnumbersa.Proposition5.1.1IfF:R2!Risacontinuousfunctionandf;garetwomeasurablerealvaluedfunctionsonX,thenFf;gismeasurable.Proof.ThesetF)]TJ/F7 6.974 Tf 6.227 0 Td[(1;aisanopensubsetoftheplane,andhencecanbewrittenasthecountableunionofproductsofopenintervalsIJ.Soifweseth=Ff;gthenh)]TJ/F7 6.974 Tf 6.226 0 Td[(1;aisthecountableunionofthesetsf)]TJ/F7 6.974 Tf 6.227 0 Td[(1Ig)]TJ/F7 6.974 Tf 6.227 0 Td[(1JandhencebelongstoF.QEDFromthiselementarypropositionweconcludethatiffandgaremeasurablerealvaluedfunctionsthenf+gismeasurablesincex;y7!x+yiscontinuous,fgismeasurablesincex;y7!xyiscontinuous,hencef1AismeasurableforanyA2Fhencef+ismeasurablesincef)]TJ/F7 6.974 Tf 6.226 0 Td[(1[0;1]2Fandsimilarlyforf)]TJ/F8 9.963 Tf 10.046 -3.615 Td[(sojfjismeasurableandsoisjf)]TJ/F11 9.963 Tf 9.962 0 Td[(gj.Hencef^gandf_garemeasurableandsoon.5.2Theintegralofanon-negativefunction.Wearegoingtoallowforthepossibilitythatafunctionvalueoranintegralmightbeinnite.Weadopttheconventionthat01=0:

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5.2.THEINTEGRALOFANON-NEGATIVEFUNCTION.135Recallthatissimpleiftakesonanitenumberofdistinctnon-negativenitevalues,a1;:::;an,andthateachofthesetsAi=)]TJ/F7 6.974 Tf 6.227 0 Td[(1aiismeasurable.ThesesetspartitionX:X=A1[[An:Ofcoursesincethevaluesaredistinct,AiAj=;fori6=j:Withthisdenition,asimplefunctioncanbewrittenasin.1andthisex-pressionisunique.Sowemaytake.2asthedenitionoftheintegralofasimplefunction.Wenowextendthedenitiontoanarbitrary[0;1]valuedfunctionfbyZEfdm:=supIE;f.4whereIE;f=ZEdm:0f;simple:.5Inotherwords,wetakeallintegralsofexpressionsofsimplefunctionssuchthatxfxatallx.Wethendenetheintegraloffasthesupremumofthesevalues.NoticethatifA:=f)]TJ/F7 6.974 Tf 6.226 0 Td[(11haspositivemeasure,thenthesimplefunctionsn1AareallfandsoRXfdm=1:Proposition5.2.1Forsimplefunctions,thedenition.4coincideswithdenition.2.Proof.Sinceisitself,therighthandsideof.2belongstoIE;andhenceisREdmasgivenby.5.Wemustshowthereverseinequality:Supposethat=Pbj1Bj:Wecanwritetherighthandsideof.2asXbjmEBj=Xi;jbjmEAiBjsinceEBjisthedisjointunionofthesetsEAiBjbecausetheAipartitionX,andmisadditiveondisjointniteevencountableunions.OneachofthesetsAiBjwemusthavebjai.HenceXi;jbjmEAiBjXi;jaimEAiBj=XaimEAisincetheBjpartitionX.QEDInthecourseoftheproofoftheabovepropositionwehavealsoestablishedforsimplefunctionsimpliesZEdmZEdm:.6

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136CHAPTER5.THELEBESGUEINTEGRAL.SupposethatEandFaredisjointmeasurablesets.ThenmAiE[F=mAiE+mAiFsoeachtermontherightof.2breaksupintoasumoftwotermsandweconcludethatIfissimpleandEF=;;thenZE[Fdm=ZEdm+ZFdm:.7Also,itisimmediatefrom.2thatifa0thenIfissimplethenZEadm=aZEdm:.8Itisnowimmediatethattheseresultsextendtoallnon-negativemeasurablefunctions.Welisttheresultsandthenprovethem.Inwhatfollowsfandgarenon-negativemeasurablefunctions,a0isarealnumberandEandFaremeasurablesets:fgZEfdmZEgdm:.9ZEfdm=ZX1Efdm.10EFZEfdmZFfdm:.11ZEafdm=aZEfdm:.12mE=0ZEfdm=0:.13EF=;ZE[Ffdm=ZEfdm+ZFfdm:.14f=0a.e.,ZXfdm=0:.15fga.e.ZXfdmZXgdm:.16Proofs..9:IE;fIE;g..10:Ifisasimplefunctionwithf,thenmultiplyingby1Egivesafunctionwhichisstillfandisstillasimplefunction.ThesetIE;fisunchangedbyconsideringonlysimplefunctionsoftheform1Eandtheseconstituteallsimplefunctions1Ef..11:Wehave1Ef1Ffandwecanapply.9and.10..12:IE;af=aIE;f..13:Inthedenition.2allthetermsontherightvanishsincemEAi=0:SoIE;fconsistsofthesingleelement0.

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5.2.THEINTEGRALOFANON-NEGATIVEFUNCTION.137.14:Thisistrueforsimplefunctions,soIE[F;f=IE;f+IF;fmeaningthateveryelementofIE[F;fisasumofanelementofIE;fandanelementofIF;f.Thusthesupontheleftisthesumofthesupsontheright,provingthatthelefthandsideof.14isitsrighthandside.Toprovethereverseinequality,chooseasimplefunction1Efandasimplefunction1Ff.Then+1E[FfsinceEF=;.So+isasimplefunctionfandhenceZEdm+ZFdmZE[Ffdm:IfwenowmaximizethetwosummandsseparatelywegetZEfdm+ZFfdmZE[Ffdmwhichiswhatwewant..15:Iff=0almosteverywhere,andfthen=0a.e.since0.Thismeansthatallsetswhichenterintotherighthandsideof.2withai6=0havemeasurezero,sotherighthandsidevanishes.SoIX;fconsistsofthesingleelement0.Thisprovesin.15.Wewishtoprovethereverseimplication.LetA=fxjfx>0g.WewishtoshowthatmA=0.NowA=[AnwhereAn:=fxjfx>1 ng:ThesetsAnareincreasing,soweknowthatmA=limn!1mAn.SoitisenoughtoprovethatmAn=0foralln.But1 n1Anfandisasimplefunction.So1 nZX1Andm=1 nmAnZXfdm=0implyingthatmAn=0..16:LetE=fxjfxgxg.ThenEismeasurableandEcisofmeasurezero.Bydenition,1Ef1Egeverywhere,henceby.11ZX1EfdmZX1Egdm:ButZX1Efdm+ZX1Ecfdm=ZEfdm+ZEcfdm=ZXfdmwherewehaveused.14and.13.Similarlyforg.QED

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138CHAPTER5.THELEBESGUEINTEGRAL.5.3Fatou'slemma.Thissays:Theorem5.3.1Ifffngisasequenceofnon-negativefunctions,thenlimn!1infknZfkdmZlimn!1infknfkdm:.17Recallthatthelimitinferiorofasequenceofnumbersfangisdenedasfollows:Setbn:=infknaksothatthesequencefbngisnon-decreasing,andhencehasalimitpossiblyinnitewhichisdenedastheliminf.Forasequenceoffunctions,liminffnisobtainedbytakingliminffnxforeveryx.Considerthesequenceofsimplefunctionsf1[n;n+1]g.Ateachpointxtheliminfis0,infact1[n;n+1]xbecomesandstays0assoonasn>x.Thustherighthandsideof.17iszero.Thenumberswhichenterintothelefthandsideareall1,sothelefthandsideis1.Similarly,ifwetakefn=n1;1=n],thelefthandsideis1andtherighthandsideis0.Sowithoutfurtherassumptions,wegenerallyexpecttogetstrictinequalityinFatou'slemma.Proof:Setgn:=infknfksothatgngn+1andsetf:=limn!1infknfn=limn!1gn:Letfbeasimplefunction.WemustshowthatZdmlimn!1infknZfkdm:.18Therearetwocasestoconsider:amfx:x>0g=1.InthiscaseRdm=1andhenceRfdm=1sincef.WemustshowthatliminfRfndm=1.LetD:=fx:x>0gsomD=1:Choosesomepositivenumberb
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5.3.FATOU'SLEMMA.139LetDn:=fxjgnx>bg:TheDn%Dsinceb0g<1.Choose>0sothatitislessthantheminimumofthepositivevaluestakenonbyandsetx=x)]TJ/F11 9.963 Tf 9.962 0 Td[(ifx>00ifx=0:LetCn:=fxjgnxgandC=fx:fxg:ThenCn%C.WehaveZCndmZCngndmZCnfkdmknZCfkdmknZfkdmkn:SoZCndmliminfZfkdm:Wewillnextletn!1:Letcibethenon-zerovaluesofso=Xci1BiforsomemeasurablesetsBiC.ThenZCndm=XcimBiCn!XcimBi=Zdm

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140CHAPTER5.THELEBESGUEINTEGRAL.sinceBiCn%BiC=Bi.SoZdmliminfZfkdm:NowZdm=Zdm)]TJ/F11 9.963 Tf 9.962 0 Td[(mfxjx>0g:Sinceweareassumingthatmfxjx>0g<1,wecanlet!0andconcludethatRdmliminfRfkdm.QED5.4Themonotoneconvergencetheorem.Weassumethatffngisasequenceofnon-negativemeasurablefunctions,andthatfnxisanincreasingsequenceforeachx.Denefxtobethelimitpossibly+1ofthissequence.Wedescribethissituationbyfn%f.Themonotoneconvergencetheoremassertsthat:fn0;fn%flimn!1Zfndm=Zfdm:.19ThefnareincreasingandallfsotheRfndmaremonotoneincreasingandallRfdm.SothelimitexistsandisRfdm.Ontheotherhand,Fatou'slemmagivesZfdmliminfZfndm=limZfndm:QEDInthemonotoneconvergencetheoremweneedonlyknowthatfn%fa.e:Indeed,letCbethesetwhereconvergenceholds,somCc=0.Letgn=1Cfnandg=1Cf.Thengn%geverywhere,sowemayapply.19tognandg.ButRgndm=RfndmandRgdm=Rfdmsothetheoremholdsforfnandfaswell.5.5ThespaceL1X;R.WewillsayanRvaluedmeasurablefunctionisintegrableifbothRf+dm<1andRf)]TJ/F11 9.963 Tf 6.725 -3.616 Td[(dm<1.Ifthishappens,wesetZfdm:=Zf+dm)]TJ/F1 9.963 Tf 9.962 13.56 Td[(Zf)]TJ/F11 9.963 Tf 6.725 -4.113 Td[(dm:.20Sincebothnumbersontherightarenite,thisdierencemakessense.Someauthorsprefertoallowoneortheothernumbersbutnotbothtobeinnite,

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5.5.THESPACEL1X;R.141inwhichcasetherighthandsideof.20mightbe=1or.Wewillstickwiththeaboveconvention.WewilldenotethesetofallrealvaluedintegrablefunctionsbyL1orL1XorL1X;Rdependingonhowprecisewewanttobe.Noticethatiffgthenf+g+andf)]TJ/F14 9.963 Tf 9.492 -3.615 Td[(g)]TJ/F8 9.963 Tf 9.869 -3.615 Td[(allofthesefunctionsbeingnon-negative.SoZf+dmZg+dm;Zf)]TJ/F11 9.963 Tf 6.725 -4.113 Td[(dmZg)]TJ/F11 9.963 Tf 6.724 -4.113 Td[(dmhenceZf+dm)]TJ/F1 9.963 Tf 9.963 13.56 Td[(Zf)]TJ/F11 9.963 Tf 6.725 -4.114 Td[(dmZg+dm)]TJ/F1 9.963 Tf 9.963 13.56 Td[(Zg)]TJ/F11 9.963 Tf 6.725 -4.114 Td[(dmorfgZfdmZgdm:.21Ifaisanon-negativenumber,thenaf=af.Ifa<0thenaf=)]TJ/F11 9.963 Tf 7.749 0 Td[(afsoinallcaseswehaveZafdm=aZfdm:.22Wenowwishtoestablishf;g2L1f+g2L1andZf+gdm=Zfdm+Zgdm:.23Proof.Weprovethisinstages:Firstassumef=Pai1Ai;g=Pbi1Biarenon-negativesimplefunc-tions,wheretheAipartitionXasdotheBj.Thenwecandecomposeandrecombinethesetstoyield:Zf+gdm=Xi;jai+bjmAiBj=XiXjaimAiBj+XjXibjmAiBj=XiaimAi+XjbjmBj=Zfdm+ZgdmwherewehaveusedthefactthatmisadditiveandtheAiBjaredisjointsetswhoseunionoverjisAiandwhoseunionoveriisBj.

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142CHAPTER5.THELEBESGUEINTEGRAL.Nextsupposethatfandgarenon-negativemeasurablefunctionswithniteintegrals.Setfn:=22nXk=0k 2n1f)]TJ/F6 4.981 Tf 5.396 0 Td[(1[k 2n;k+1 2n]:Eachfnisasimplefunctionf,andpassingfromfntofn+1involvessplittingeachofthesetsf)]TJ/F7 6.974 Tf 6.227 0 Td[(1[k 2n;k+1 2n]inthesumintotwo,andchoosingalargervalueonthesecondportion.Sothefnareincreasing.Also,iffx<1,thenfx<2mforsomem,andforanyn>mfnxdiersfromfxbyatmost2)]TJ/F10 6.974 Tf 6.226 0 Td[(n.Hencefn%fa.e.,sincefisnitea.ebecauseitsintegralisnite.Similarlywecanconstructgn%g.Alsofn+gn%f+ga.e.Bythea.e.monotoneconvergencetheoremZf+gdm=limZfn+gndm=limZfndm+limZgndm=Zfdm+Zgdm;wherewehaveused.23forsimplefunctions.ThisargumentshowsthatRf+gdm<1ifbothintegralsRfdmandRgdmarenite.Foranyf2L1weconcludefromtheprecedingthatZjfjdm=Zf++f)]TJ/F8 9.963 Tf 6.725 -4.114 Td[(dm<1:Similarlyforg.Sincejf+gjjfj+jgjweconcludethatbothf+g+andf+g)]TJ/F8 9.963 Tf 10.045 -3.615 Td[(haveniteintegrals.Nowf+g+)]TJ/F8 9.963 Tf 9.962 0 Td[(f+g)]TJ/F8 9.963 Tf 9.492 -4.113 Td[(=f+g=f+)]TJ/F11 9.963 Tf 9.962 0 Td[(f)]TJ/F8 9.963 Tf 6.725 -4.113 Td[(+g+)]TJ/F11 9.963 Tf 9.963 0 Td[(g)]TJ/F8 9.963 Tf 6.725 -4.113 Td[(orf+g++f)]TJ/F8 9.963 Tf 8.939 -4.113 Td[(+g)]TJ/F8 9.963 Tf 9.492 -4.113 Td[(=f++g++f+g)]TJ/F11 9.963 Tf 6.724 -4.113 Td[(:Allexpressionsarenon-negativeandintegrable.Sointegratebothsidestoget.23.QEDWehavethusestablishedTheorem5.5.1ThespaceL1X;Risarealvectorspaceandf7!RfdmisalinearfunctiononL1X;R.WealsohaveProposition5.5.1Ifh2L1andRAhdm0forallA2Fthenh0a.e.Proof:LetAn:fxjhx)]TJ/F7 6.974 Tf 19.937 3.922 Td[(1 ng.ThenZAnhdmZAn)]TJ/F8 9.963 Tf 7.749 0 Td[(1 ndm=)]TJ/F8 9.963 Tf 9.443 6.74 Td[(1 nmAn

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5.6.THEDOMINATEDCONVERGENCETHEOREM.143somAn=0.ButifweletA:=fxjhx<0gthenAn%AandhencemA=0.QEDWehavedenedtheintegralofanyfunctionfasRfdm=Rf+dm)]TJ/F1 9.963 Tf -335.962 -3.929 Td[(Rf)]TJ/F11 9.963 Tf 6.725 -3.616 Td[(dm,andRjfjdm=Rf+dm+Rf)]TJ/F11 9.963 Tf 6.725 -3.616 Td[(dm.Sinceforanytwonon-negativerealnumbersa)]TJ/F11 9.963 Tf 9.963 0 Td[(ba+bweconcludethatZfdmZjfjdm:.24Ifwedenekfk1:=Zjfjdmwehaveveriedthatkf+gk1kfk1+kgk1;andhavealsoveriedthatkcfk1=jcjkfk1:Inotherwords,kk1isasemi-normonL1.Fromtheprecedingpropositionweknowthatkfk1=0ifandonlyiff=0a.e.Thequestionofwhetherwewanttopasstothequotientandidentifytwofunctionswhichdieronasetofmeasurezeroisamatteroftaste.5.6Thedominatedconvergencetheorem.ThissaysthatTheorem5.6.1Letfnbeasequenceofmeasurablefunctionssuchthatjfnjga.e.;g2L1:Thenfn!fa.e.f2L1andZfndm!Zfdm:Proof.Thefunctionsfnareallintegrable,sincetheirpositiveandnegativepartsaredominatedbyg.Assumeforthemomentthatfn0.ThenFatou'slemmasaysthatZfdmliminfZfndm:Fatou'slemmaappliedtog)]TJ/F11 9.963 Tf 9.963 0 Td[(fnsaysthatZg)]TJ/F11 9.963 Tf 9.962 0 Td[(fdmliminfZg)]TJ/F11 9.963 Tf 9.963 0 Td[(fndm=liminfZgdm)]TJ/F1 9.963 Tf 9.963 13.56 Td[(Zfndm=Zgdm)]TJ/F8 9.963 Tf 9.963 0 Td[(limsupZfndm:

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144CHAPTER5.THELEBESGUEINTEGRAL.SubtractingRgdmgiveslimsupZfndmZfdm:SolimsupZfndmZfdmliminfZfndmwhichcanonlyhappenifallthreeareequal.Wehaveprovedtheresultfornon-negativefn.Forgeneralfnwecanwriteourhypothesisas)]TJ/F11 9.963 Tf 7.749 0 Td[(gfnga.e.:Addinggtobothsidesgives0fn+g2ga.e.:Wenowapplytheresultfornon-negativesequencestog+fnandthensubtractoRgdm.5.7Riemannintegrability.SupposethatX=[a;b]isaninterval.WhatistherelationbetweentheLebesgueintegralandtheRiemannintegral?Letussupposethat[a;b]isboundedandthatfisaboundedfunction,sayjfjM.EachpartitionP:a=a0
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5.8.THEBEPPO-LEVITHEOREM.145Supposethatfismeasurable.AllthefunctionsintheaboveinequalityareLebesgueintegrable,sodominatedconvergenceimpliesthatlimUn=limZbaundx=Zbaudxwhereu=limunwithasimilarequationforthelowerbounds.TheRiemannintegralisdenedasthecommonvalueoflimLnandlimUnwhenevertheselimitsareequal.Proposition5.7.1fisRiemannintegrableifandonlyiffiscontinuousal-mosteverywhere.Proof.Noticethatifxisnotanendpointofanyintervalinthepartitions,thenfiscontinuousatxifandonlyifux=`x:Riemann'sconditionforintegrabilitysaysthatRu)]TJ/F11 9.963 Tf 8.667 0 Td[(`dm=0whichimpliesthatfiscontinuousalmosteverywhere.Conversely,iffiscontinuousa.e.thenu=f=`a.e..Sinceuismeasurablesoisf,andsinceweareassumingthatfisbounded,weconcludethatfLebesgueintegrable.As`=f=ua.e.theirLebesgueintegralscoincide.ButthestatementthattheLebesgueintegralofuisthesameasthatof`ispreciselythestatementofRiemannintegrability.QEDNoticethatinthecourseoftheproofwehavealsoshownthattheLebesgueandRiemannintegralscoincidewhenbothexist.5.8TheBeppo-Levitheorem.Webeginwithalemma:Lemma5.8.1Letfgngbeasequenceofnon-negativemeasurablefunctions.ThenZ1Xn=1gndm=1Xn=1Zgndm:Proof.WehaveZnXk=1gkdm=nXk=1Zgkdmfornitenbythelinearityoftheintegral.Sincethegk0,thesumsundertheintegralsignareincreasing,andbydenitionconvergetoP1k=1gk.Themonotoneconvergencetheoremimpliesthelemma.QEDButbothsidesoftheequationinthelemmamightbeinnite.Theorem5.8.1Beppo-Levi.Letfn2L1andsupposethat1Xk=1Zjfkjdm<1:

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146CHAPTER5.THELEBESGUEINTEGRAL.ThenPfkxconvergestoanitelimitforalmostallx,thesumisintegrable,andZ1Xk=1fkdm=1Xk=1Zfkdm:Proof.Takegn:=jfnjinthelemma.Ifwesetg=P1n=1gn=P1n=1jfnjthenthelemmasaysthatZgdm=1Xn=1Zjfnjdm;andweareassumingthatthissumisnite.Sogisintegrable,inparticularthesetofxforwhichgx=1musthavemeasurezero.Inotherwords,Xn=1jfnxj<1a.e.:Ifaseriesisabsolutelyconvergent,thenitisconvergent,sowecansaythatPfnxconvergesalmosteverywhere.Letfx=1Xn=1fnxatallpointswheretheseriesconverges,andsetfx=0atallotherpoints.Now1Xn=0fnxgxatallpoints,andhencebythedominatedconvergencetheorem,f2L1andZfdm=Zlimn!1nXk=1fkdm=limn!1XZfkdm=1Xk=1ZfkdmQED5.9L1iscomplete.ThisisanimmediatecorollaryoftheBeppo-LevitheoremandFatou'slemma.Indeed,supposethatfhngisaCauchysequenceinL1.Choosen1sothatkhn)]TJ/F11 9.963 Tf 9.963 0 Td[(hn1k1 28nn1:Thenchoosen2>n1sothatkhn)]TJ/F11 9.963 Tf 9.962 0 Td[(hn2k1 228nn2:

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5.10.DENSESUBSETSOFL1R;R.147Continuingthisway,wehaveproducedasubsequencehnjsuchthatkhnj+1)]TJ/F11 9.963 Tf 9.962 0 Td[(hnjk1 2j:Letfj:=hnj+1)]TJ/F11 9.963 Tf 9.962 0 Td[(hnj:ThenZjfjjdm<1 2jsothehypothesesoftheBeppo-Levytheoremaresatised,andPfjconvergesalmosteverywheretosomelimitf2L1.Buthn1+kXj=1fj=hnk+1:Sothesubsequencehnkconvergesalmosteverywheretosomeh2L1.Wemustshowthatthishisthelimitofthehninthekk1norm.ForthiswewilluseFatou'slemma.Foragiven>0,chooseNsothatkhn)]TJ/F11 9.963 Tf 10.496 0 Td[(hmkN.Sinceh=limhnjwehave,fork>N,kh)]TJ/F11 9.963 Tf 9.963 0 Td[(hkk1=Zjh)]TJ/F11 9.963 Tf 9.963 0 Td[(hkjdm=Zlimj!1jhnj)]TJ/F11 9.963 Tf 9.963 0 Td[(hkjdmliminfZjhnj)]TJ/F11 9.963 Tf 9.962 0 Td[(hkjdm=liminfkhnj)]TJ/F11 9.963 Tf 9.963 0 Td[(hkk<:QED5.10DensesubsetsofL1R;R.UpuntilnowwehavebeenstudyingintegrationonanarbitrarymeasurespaceX;F;m.Inthissectionandthenext,wewilltakeX=R;Ftobethe-eldofLebesguemeasurablesets,andmtobeLebesguemeasure,inordertosimplifysomeoftheformulationsandarguments.SupposethatfisaLebesgueintegrablenon-negativefunctiononR.Weknowthatforany>0thereisasimplefunctionsuchthatfandZfdm)]TJ/F1 9.963 Tf 9.962 13.56 Td[(Zdm=Zf)]TJ/F11 9.963 Tf 9.962 0 Td[(dm=kf)]TJ/F11 9.963 Tf 9.963 0 Td[(k1<:Tosaythatissimpleimpliesthat=Xai1Ai

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148CHAPTER5.THELEBESGUEINTEGRAL.nitesumwhereeachoftheai>0andsinceRdm<1eachAihasnitemeasure.SincemAi[)]TJ/F11 9.963 Tf 7.749 0 Td[(n;n]!mAiasn!1,wemaychoosensucientlylargesothatkf)]TJ/F11 9.963 Tf 9.963 0 Td[(k1<2where=Xai1Ai[)]TJ/F10 6.974 Tf 6.226 0 Td[(n;n]:ForeachofthesetsAi[)]TJ/F11 9.963 Tf 7.749 0 Td[(n;n]wecanndaboundedopensetUiwhichcontainsit,andsuchthatmUi=Aiisassmallasweplease.SowecanndnitelymanyboundedopensetsUisuchthatkf)]TJ/F1 9.963 Tf 9.963 9.464 Td[(Xai1Uik1<3:EachUiisacountableunionofdisjointopenintervals,Ui=SjIi;j,andsincemUi=PjmIi;j,wecanndnitelymanyIi;j,jrangingoveranitesetofintegers,JisuchthatmSj2JiisascloseasweliketomUi.SoletuscallastepfunctionafunctionoftheformPbi1IiwheretheIiareboundedintervals.Wehaveshownthatwecanndastepfunctionwithpositivecoecientswhichisascloseaswelikeinthekk1normtof.Iffisnotnecessarilynon-negative,weknowbydenition!thatf+andf)]TJ/F8 9.963 Tf 9.76 -3.615 Td[(areinL1,andsowecanapproximateeachbyastepfunction.thetriangleinequalitythengivesProposition5.10.1ThestepfunctionsaredenseinL1R;R.If[a;b];a
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5.11.THERIEMANN-LEBESGUELEMMA.149Forexample,ifht=cost;6=0,thentheexpressionunderthelimitsignintheaveragingconditionis1 csintwhichtendstozeroasjcj!1.Heretheoscillationsinharewhatgiverisetotheaveragingcondition.Asanotherexample,letht=1jtjt1=jtjjtj1:Thenthelefthandsideof.25is1 jcj+logjcj;jcj1:Heretheaveragingconditionissatisedbecausetheintegralin5.25growsmoreslowlythatjcj.Theorem5.11.1[GeneralizedRiemann-LebesgueLemma].Letf2L1[c;d];R;c
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150CHAPTER5.THELEBESGUEINTEGRAL.Wecanmakethesecondterm< 2bychoosingrlargeenough.QED5.11.1TheCantor-Lebesguetheorem.Thissays:Theorem5.11.2Ifatrigonometricseriesa0 2+Xndncosnt)]TJ/F11 9.963 Tf 9.963 0 Td[(ndn2RconvergesonasetEofpositiveLebesguemeasurethendn!0:Ihavewrittenthegeneralformofarealtrigonometricseriesasacosineserieswithphasessincewearetalkingaboutonlyrealvaluedfunctionsatthepresent.Ofcourse,appliedtotherealandimaginaryparts,thetheoremassertsthatifPaneinxconvergesonasetofpositivemeasure,thenthean!0.Also,thenotationsuggests-andthisismyintention-thatthen'sareintegers.Butintheproofbelowallthatwewillneedisthatthen'sareanysequenceofrealnumberstendingto1.Proof.Theproofisaniceapplicationofthedominatedconvergencetheorem,whichwasinventedbyLebesgueinpartpreciselytoprovethistheorem.WemayassumebypassingtoasubsetifnecessarythatEiscontainedinsomeniteinterval[a;b].Ifdn6!0thenthereisan>0andasubsequencejdnkj>forallk.Iftheseriesconverges,allitstermsgoto0,sothismeansthatcosnkt)]TJ/F11 9.963 Tf 9.963 0 Td[(k!08t2E:Socos2nkt)]TJ/F11 9.963 Tf 9.963 0 Td[(k!08t2E:NowmE<1andcos2nkt)]TJ/F11 9.963 Tf 10.489 0 Td[(k1andtheconstant1isintegrableon[a;b].Sowemaytakethelimitundertheintegralsignusingthedominatedconvergencetheoremtoconcludethatlimk!1ZEcos2nkt)]TJ/F11 9.963 Tf 9.963 0 Td[(kdt=ZElimk!1cos2nkt)]TJ/F11 9.963 Tf 9.963 0 Td[(kdt=0:Butcos2nkt)]TJ/F11 9.963 Tf 9.963 0 Td[(k=1 2[1+cos2nkt)]TJ/F11 9.963 Tf 9.962 0 Td[(k]soZEcos2nkt)]TJ/F11 9.963 Tf 9.963 0 Td[(kdt=1 2ZE[1+cos2nkt)]TJ/F11 9.963 Tf 9.962 0 Td[(k]dt=1 2mE+ZEcos2nkt)]TJ/F11 9.963 Tf 9.963 0 Td[(k=1 2mE+1 2ZR1Ecos2nkt)]TJ/F11 9.963 Tf 9.962 0 Td[(kdt:

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5.12.FUBINI'STHEOREM.151But1E2L1R;Rsothesecondtermonthelastlinegoesto0bytheRiemannLebesgueLemma.Sothelimitis1 2mEinsteadof0,acontradiction.QED5.12Fubini'stheorem.Thisfamoustheoremassertsthatundersuitableconditions,adoubleintegralisequaltoaniteratedintegral.Wewillproveitforrealandhencenitedimen-sionalvaluedfunctionsonarbitrarymeasurespaces.TheproofforBanachspacevaluedfunctionsisabitmoretricky,andweshallomititaswewillnotneedit.Thisisoneofthereasonswhywehavedevelopedtherealvaluedtheoryrst.Webeginwithsomefactsaboutproduct-elds.5.12.1Product-elds.LetX;FandY;Gbespaceswith-elds.OnXYwecanconsiderthecollectionPofallsetsoftheformAB;A2F;B2G:The-eldgeneratedbyPwill,byabuseoflanguage,bedenotedbyFG:IfEisanysubsetofXY,byanevenmoreseriousabuseoflanguagewewillletEx:=fyjx;y2EgandcontradictorilywewillletEy:=fxjx;y2Eg:ThesetExwillbecalledthex-sectionofEandthesetEywillbecalledthey-sectionofE.FinallywewillletCPdenotethecollectionofcylindersets,thatissetsoftheformAYA2ForXB;B2G:Inotherwords,anelementofPisacylindersetwhenoneofthefactorsisthewholespace.Theorem5.12.1.FGisgeneratedbythecollectionofcylindersetsC.

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152CHAPTER5.THELEBESGUEINTEGRAL.FGisthesmallest-eldonXYsuchthattheprojectionsprX:XY!XprXx;y=xprY:XY!YprYx;y=yaremeasurablemaps.ForeachE2FGandallx2Xthex-sectionExofEbelongstoGandforally2Ythey-sectionEyofEbelongstoF.Proof.AB=AYXBsoany-eldcontainingCmustalsocontainP.Thisprovestherstitem.Sincepr)]TJ/F7 6.974 Tf 6.227 0 Td[(1XA=AY,themapprXismeasurable,andsimilarlyforY.Butalso,any-eldcontainingallAYandXBmustcontainPbywhatwejustproved.Thisprovestheseconditem.Astothethirditem,anysetEoftheformABhasthedesiredsectionproperties,sinceitsxsectionisBifx2Aortheemptysetifx62A.Similarlyforitsysections.SoletHdenotethecollectionofsubsetsEwhichhavethepropertythatallEx2GandallEy2F.IfweshowthatHisa-eldwearedone.NowEcx=Excandsimilarlyfory,soGisclosedundertakingcomple-ments.Similarlyforcountableunions:[nEn!x=[nEnx:QED5.12.2-systemsand-systems.Recallthatthe-eldCgeneratedbyacollectionCofsubsetsofXistheintersectionofallthe-eldscontainingC.SometimesthecollectionCisclosedunderniteintersection.Inthatcase,wecallCa-system.Examples:Xisatopologicalspace,andCisthecollectionofopensetsinX.X=R,andCconsistsofallhalfinniteintervalsoftheform;a].WewilldenotethissystembyR.AcollectionHofsubsetsofXwillbecalleda-systemif1.X2H,2.A;B2HwithAB=;A[B2H,3.A;B2HandBAAnB2H,and4.fAng11HandAn%AA2H.

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5.12.FUBINI'STHEOREM.153Fromitems1and3weseethata-systemisclosedundercomplementa-tion,andsince;=Xcitcontainstheemptyset.IfBisbotha-systemandasystem,itisclosedunderanyniteunion,sinceA[B=A[B=ABwhichisadisjointunion.AnycountableunioncanbewrittenintheformA=%AnwheretheAnarenitedisjointunionsaswehavealreadyargued.SowehaveprovedProposition5.12.1IfHisbotha-systemanda-systemthenitisa-eld.Also,wehaveProposition5.12.2[Dynkin'slemma.]IfCisa-system,thenthe-eldgeneratedbyCisthesmallest-systemcontainingC.LetMbethe-eldgeneratedbyC,andHthesmallest-systemcontainingC.SoMH.Bytheprecedingproposition,allweneedtodoisshowthatHisa-system.LetH1:=fAjAC2H8C2Cg:ClearlyH1isa-systemcontainingC,soHH1whichmeansthatAC2HforallA2HandC2C.LetH2:=fAjAH2H8H2Hg:H2isagaina-system,anditcontainsCbywhatwehavejustproved.SoH2H,whichmeansthattheintersectionoftwoelementsofHisagaininH,i.e.Hisa-system.QED5.12.3Themonotoneclasstheorem.Theorem5.12.2LetBbeaclassofboundedrealvaluedfunctionsonaspaceZsatisfying1.BisavectorspaceoverR.2.Theconstantfunction1belongstoB.3.Bcontainstheindicatorfunctions1AforallAbelongingtoa-systemI.4.Ifffngisasequenceofnon-negativefunctionsinBandfn%fwherefisaboundedfunctiononZ,thenf2B.ThenBcontainseveryboundedMmeasurablefunction,whereMisthe-eldgeneratedbyI.Proof.LetHdenotetheclassofsubsetsofZwhoseindicatorfunctionsbelongtoB.ThenZ2Hbyitem2.IfBAarebothinH,then1AnB=1A)]TJ/F34 9.963 Tf 7.972 0 Td[(1BandsoAnBbelongstoHbyitem1.Similarly,ifAB=;then1A[B=1A+1B

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154CHAPTER5.THELEBESGUEINTEGRAL.andsoifAandBbelongtoHsodoesA[BwhenAB=;.Finally,condition4inthetheoremimpliescondition4inthedenitionofa-system.SowehaveprovedthatthatHisa-systemcontainingI.SobyDynkin'slemma,itcontainsM.Nowsupposethat0fKisaboundedMmeasurablefunction,wherewemaytakeKtobeaninteger.Foreachintegern0dividetheinterval[0;K]upintosubintervalsofsize2)]TJ/F10 6.974 Tf 6.227 0 Td[(n,andletAn;i:=fzji2)]TJ/F10 6.974 Tf 6.226 0 Td[(nfz
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5.12.FUBINI'STHEOREM.155isboundedandGmeasurable.Henceithasanintegralwithrespecttothemeasuren,whichwewilldenotebyZYfx;yndy:ThisisaboundedfunctionofxwhichwewillprovetobeFmeasurableinjustamoment.SimilarlywecanformZXfx;ymdxwhichisafunctionofy.Proposition5.12.4LetBdenotethespaceofboundedFGmeasurablefunc-tionssuchthatRYfx;yndyisaFmeasurablefunctiononX,RXfx;ymdxisaGmeasurablefunctiononYandZXZYfx;yndymdx=ZYZXfx;ymdxndy:.27ThenBconsistsofallboundedFGmeasurablefunctions.Proof.Wehaveveriedthatthersttwoitemsholdfor1AB.Bothsidesof.27equalmAnBasisclearfromtheproofofProposition5.12.3.Soconditions1-3ofthemonotoneclasstheoremareclearlysatised,andcondition4isaconsequenceoftwodoubleapplicationsofthemonotoneconvergencetheorem.QEDNowforanyC2FGwedenemnC:=ZXZY1Cx;yndymdx=ZYZX1Cx;ymdxndy;.28bothsidesbeingequalonaccountoftheprecedingproposition.ThismeasureassignsthevaluemAnBtoanysetAB2P,andsincePgeneratesFGasasigmaeld,anytwomeasureswhichagreeonPmustagreeonFG.HencemnistheuniquemeasurewhichassignsthevaluemAnBtosetsofP.Furthermore,weknowthatZXYfx;ymn=ZXZYfx;yndymdx=ZYZXfx;ymdxndy.29istrueforfunctionsoftheform1ABandhencebythemonotoneclasstheoremitistrueforallboundedfunctionswhicharemeasurablerelativetoFG.TheaboveassertionsarethecontentofFubini'stheoremforboundedmea-suresandfunctions.Wesummarize:

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156CHAPTER5.THELEBESGUEINTEGRAL.Theorem5.12.3LetX;F;mandY;G;nbemeasurespaceswithmX<1andnY<1.ThereexistsauniquemeasureonFGwiththepropertythatmnAB=mAnB8AB2P:ForanyboundedFGmeasurablefunction,thedoubleintegralisequaltotheiteratedintegralinthesensethat.29holds.5.12.5Extensionstounboundedfunctionsandto-nitemeasures.SupposethatwetemporarilykeeptheconditionthatmX<1andnY<1.Letfbeanynon-negativeFG-measurablefunction.Weknowthat.29holdsforallboundedmeasurablefunctions,inparticularforallsimplefunctions.Weknowthatwecanndasequenceofsimplefunctionssnsuchthatsn%f.Hencebyseveralapplicationsofthemonotoneconvergencetheorem,weknowthat.29istrueforallnon-negativeFG-measurablefunctionsinthesensethatallthreetermsareinnitetogether,ornitetogetherandequal.NowwehaveagreedtocallaFG-measurablefunctionfintegrableifandonlyiff+andf)]TJ/F8 9.963 Tf 10.046 -3.615 Td[(haveniteintegrals.Inthiscase.29holds.AmeasurespaceX;F;miscalled-niteifX=SnXnwheremXn<1.Inotherwords,Xis-niteifitisacountableunionofnitemeasurespaces.Asusual,wecanthenwriteXasacountableunionofdisjointnitemeasurespaces.SoifXandYare-nite,wecanwritethevariousintegralsthatoccurin.29assumsofintegralswhichoccurovernitemeasurespaces.AbitofstandardargumentationshowsthatFubinicontinuestoholdinthiscase.IfXorYisnot-nite,or,eveninthenitecase,iffisnotnon-negativeormnintegrable,thenFubinineednothold.Ihopetopresentthestandardcounter-examplesintheproblemset.

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Chapter6TheDaniellintegral.Daniell'sideawastotaketheaxiomaticpropertiesoftheintegralasthestart-ingpointanddevelopintegrationforbroaderandbroaderclassesoffunctions.Thenderivemeasuretheoryasaconsequence.MuchofthepresentationhereistakenfromthebookAbstractHarmonicAnalysisbyLynnLoomis.Someofthelemmas,propositionsandtheoremsindicatethecorrespondingsectionsinLoomis'sbook.6.1TheDaniellIntegralLetLbeavectorspaceofboundedrealvaluedfunctionsonasetSclosedunder^and_.Forexample,Smightbeacompletemetricspace,andLmightbethespaceofcontinuousfunctionsofcompactsupportonS.AmapI:L!RiscalledanIntegralif1.Iislinear:Iaf+bg=aIf+bIg2.Iisnon-negative:f0If0orequivalentlyfgIfIg.3.fn&0Ifn&0:Forexample,wemighttakeS=Rn;L=thespaceofcontinuousfunctionsofcompactsupportonRn,andItobetheRiemannintegral.Thersttwoitemsontheabovelistareclearlysatised.Astothethird,werecallDini'slemmafromthenotesonmetricspaces,whichsaysthatasequenceofcontinuousfunctionsofcompactsupportffngonametricspacewhichsatisesfn&0actuallyconvergesuniformlyto0.Furthermorethesupportsofthefnareallcontainedinaxedcompactset-forexamplethesupportoff1.Thisestablishesthethirditem.157

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158CHAPTER6.THEDANIELLINTEGRAL.Theplanisnowtosuccessivelyincreasetheclassoffunctionsonwhichtheintegralisdened.DeneU:=flimitsofmonotonenon-decreasingsequencesofelementsofLg:Wewillusethewordincreasing"assynonymouswithmonotonenon-decreasing"soastosimplifythelanguage.Lemma6.1.1IffnisanincreasingsequenceofelementsofLandifk2LsatisesklimfnthenlimIfnIk.Proof.Ifk2Landlimfnk,thenfn^kkandfnfn^ksoIfnIfn^kwhile[k)]TJ/F8 9.963 Tf 9.963 0 Td[(fn^k]&0soI[k)]TJ/F11 9.963 Tf 9.963 0 Td[(fn^k]&0by3orIfn^k%Ik:HencelimIfnlimIfn^k=Ik.QEDLemma6.1.2[12C]IfffngandfgngareincreasingsequencesofelementsofLandlimgnlimfnthenlimIgnlimIfn.Proof.Fixmandtakek=gminthepreviouslemma.ThenIgmlimIfn.Nowletm!1.QEDThusfn%fandgn%flimIfn=limIgnsowemayextendItoUbysettingIf:=limIfnforfn%f:Iff2L,thiscoincideswithouroriginalI,sincewecantakegn=fforallnintheprecedinglemma.WehavenowextendedIfromLtoU.ThenextlemmashowsthatifwenowstartwithIonUandapplythesameprocedureagain,wedonotgetanyfurther.Lemma6.1.3[12D]Iffn2Uandfn%fthenf2UandIfn%If.

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6.1.THEDANIELLINTEGRAL159Proof.Foreachxednchoosegmn%mfn.Sethn:=gn1__gnnsohn2Landhnisincreasingwithgnihnfnforin:Letn!1.Thenfilimhnf:Nowleti!1.Wegetflimhnf:SowehavewrittenfasalimitofanincreasingsequenceofelementsofL,Sof2U.AlsoIgniIhnIfsolettingn!1wegetIfiIflimIfnsopassingtothelimitsgivesIf=limIfn.QEDWehaveIf+g=If+Igforf;g2U:Dene)]TJ/F11 9.963 Tf 7.749 0 Td[(U:=f)]TJ/F11 9.963 Tf 12.731 0 Td[(fjf2UgandIf:=)]TJ/F11 9.963 Tf 7.748 0 Td[(I)]TJ/F11 9.963 Tf 7.749 0 Td[(ff2)]TJ/F11 9.963 Tf 17.158 0 Td[(U:Iff2Uand)]TJ/F11 9.963 Tf 7.748 0 Td[(f2UthenIf+I)]TJ/F11 9.963 Tf 7.749 0 Td[(f=If)]TJ/F11 9.963 Tf 8.077 0 Td[(f=I=0soI)]TJ/F11 9.963 Tf 7.749 0 Td[(f=)]TJ/F11 9.963 Tf 7.748 0 Td[(Ifinthiscase.Sothedenitionisconsistent.)]TJ/F11 9.963 Tf 7.749 0 Td[(Uisclosedundermonotonedecreasinglimits.etc.Ifg2)]TJ/F11 9.963 Tf 17.439 0 Td[(Uandh2Uwithghthen)]TJ/F11 9.963 Tf 7.748 0 Td[(g2Usoh)]TJ/F11 9.963 Tf 10.074 0 Td[(g2Uandh)]TJ/F11 9.963 Tf 10.075 0 Td[(g0soIh)]TJ/F11 9.963 Tf 9.963 0 Td[(Ig=Ih+)]TJ/F11 9.963 Tf 7.748 0 Td[(g=Ih)]TJ/F11 9.963 Tf 9.963 0 Td[(g0.AfunctionfiscalledI-summableifforevery>0;9g2)]TJ/F11 9.963 Tf 18.204 0 Td[(U;h2Uwithgfh;jIgj<1;jIhj<1andIh)]TJ/F11 9.963 Tf 9.963 0 Td[(g:ForsuchfdeneIf=glbIh=lubIg:Iff2Utakeh=fandfn2Lwithfn%f.Then)]TJ/F11 9.963 Tf 7.749 0 Td[(fn2LUsofn2)]TJ/F11 9.963 Tf 17.172 0 Td[(U.IfIf<1thenwecanchoosensucientlylargesothatIf)]TJ/F11 9.963 Tf 10.292 0 Td[(Ifn<.Thespaceofsummablefunctionsisdenotedby L1.Itisclearlyavectorspace,andIsatisesconditions1and2above,i.e.islinearandnon-negative.

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160CHAPTER6.THEDANIELLINTEGRAL.Theorem6.1.1[12G]Monotoneconvergencetheorem.fn2 L1,fn%fandlimIfn<1f2 L1andIf=limIfn.Proof.Replacingfnbyfn)]TJ/F11 9.963 Tf 9.963 0 Td[(f0wemayassumethatf0=0.Choosehn2U;suchthatfn)]TJ/F11 9.963 Tf 9.963 0 Td[(fn)]TJ/F7 6.974 Tf 6.227 0 Td[(1hnandIhnIfn)]TJ/F11 9.963 Tf 9.963 0 Td[(fn)]TJ/F7 6.974 Tf 6.227 0 Td[(1+ 2n:ThenfnnX1hiandnXi=1IhiIfn+:SinceUisclosedundermonotoneincreasinglimits,h:=1Xi=1hi2U;fhandIhlimIfn+:Sincefm2 L1wecanndagm2)]TJ/F11 9.963 Tf 17.158 0 Td[(UwithIfm)]TJ/F11 9.963 Tf 9.556 0 Td[(Igm
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6.3.MEASURE.161Lemma6.2.2Iff2Bthereexistsag2Usuchthatfg.Proof.ThelimitofamonotoneincreasingsequenceoffunctionsinUbelongstoU.HencethesetoffforwhichthelemmaistrueisamonotoneclasswhichcontainsL.henceitcontainsB.QEDAfunctionfisL-boundedifthereexistsag2L+withjfjg.AclassFoffunctionsissaidtobeL-monotoneifFisclosedundermonotonelimitsofL-boundedfunctions.Theorem6.2.2ThesmallestL-monotoneclassincludingL+isB+.Proof.CallthissmallestfamilyF.Ifg2L+,thesetofallf2B+suchthatf^g2FformamonotoneclasscontainingL+,hencecontainingB+henceequaltoB+.Iff2B+andfgthenf^g=f2F.SoFcontainsallLboundedfunctionsbelongingtoB+.Letf2B+.Bythelemma,chooseg2Usuchthatfg,andchoosegn2L+withgn%g.Thenf^gngnandsoisLbounded,sof^gn2F.Sincef^gn!fweseethatf2F.SoB+F:WeknowthatB+isamonotoneclass,inparticularanL-monotoneclass.HenceF=B+.QEDDeneL1:= L1B:Since L1andBarebothclosedunderthelatticeoperations,f2L1f2L1jfj2L1:Theorem6.2.3Iff2Bthenf2L1,9g2L1withjfjg.Wehaveproved:simplytakeg=jfj.Fortheconversewemayassumethatf0byapplyingtheresulttof+andf)]TJ/F8 9.963 Tf 6.725 -3.615 Td[(.Thefamilyofallh2B+suchthath^g2L1ismonotoneandincludesL+soincludesB+.Sof=f^g2L1.QEDExtendItoallofB+besettingit=1onfunctionswhichdonotbelongtoL1.6.3Measure.LoomiscallsasetAintegrableif1A2B.ThemonotoneclasspropertiesofBimplythattheintegrablesetsforma-eld.ThendeneA:=Z1Aandthemonotoneconvergencetheoremguaranteesthatisameasure.

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162CHAPTER6.THEDANIELLINTEGRAL.AddStone'saxiomf2Lf^12L:ThenthemonotoneclasspropertyimpliesthatthisistruewithLreplacedbyB.Theorem6.3.1f2Banda>0thenAa:=fpjfp>agisanintegrableset.Iff2L1thenAa<1:Proof.Letfn:=[nf)]TJ/F11 9.963 Tf 9.963 0 Td[(f^a]^12B:Thenfnx=8<:1iffxa+1 n0iffxanfx)]TJ/F11 9.963 Tf 9.963 0 Td[(aifa0thenf2B.Proof.For>1deneAm:=fxjm
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6.4.HOLDER,MINKOWSKI,LPANDLQ.163andZfdZfdZfd:SoifeitherofIforRfdisnitetheybothareandIf)]TJ/F1 9.963 Tf 9.963 13.56 Td[(Zfd)]TJ/F8 9.963 Tf 9.962 0 Td[(1If)]TJ/F8 9.963 Tf 9.963 0 Td[(1If:SoZfd=If:Iff2B+anda>0thenfxjfxa>bg=fxjfx>b1 ag:Sof2B+fa2B+andhencetheproductoftwoelementsofB+belongstoB+becausefg=1 4f+g2)]TJ/F8 9.963 Tf 9.963 0 Td[(f)]TJ/F11 9.963 Tf 9.963 0 Td[(g2:6.4Holder,Minkowski,LpandLq.Thenumbersp;q>1arecalledconjugateif1 p+1 q=1:Thisisthesameaspq=p+qorp)]TJ/F8 9.963 Tf 9.963 0 Td[(1q)]TJ/F8 9.963 Tf 9.962 0 Td[(1=1:Thislastequationsaysthatify=xp)]TJ/F7 6.974 Tf 6.227 0 Td[(1thenx=yq)]TJ/F7 6.974 Tf 6.226 0 Td[(1:Theareaunderthecurvey=xp)]TJ/F7 6.974 Tf 6.226 0 Td[(1from0toaisA=ap pwhiletheareabetweenthesamecurveandthey-axisuptoy=bB=bq q:

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164CHAPTER6.THEDANIELLINTEGRAL.Supposeb1jf+gjp[2maxjfj;jgj]p2p[jfjp+jgjp]impliesthatf+g2Lp.Writekf+gkppIjf+gjp)]TJ/F7 6.974 Tf 6.226 0 Td[(1jfj+Ijf+gjp)]TJ/F7 6.974 Tf 6.227 0 Td[(1jgj:Nowqp)]TJ/F8 9.963 Tf 9.962 0 Td[(1=qp)]TJ/F11 9.963 Tf 9.963 0 Td[(q=p

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6.4.HOLDER,MINKOWSKI,LPANDLQ.165sojf+gjp)]TJ/F7 6.974 Tf 6.227 0 Td[(12LqanditskkqnormisIjf+gjp1 q=Ijf+qjp1)]TJ/F6 4.981 Tf 7.57 2.677 Td[(1 p=Ijf+gjpp)]TJ/F6 4.981 Tf 5.397 0 Td[(1 p=kf+gkp)]TJ/F7 6.974 Tf 6.227 0 Td[(1p:Sowecanwritetheprecedinginequalityaskf+gkppjfj;jf+gjp)]TJ/F7 6.974 Tf 6.227 0 Td[(1+jgj;jf+gjp)]TJ/F7 6.974 Tf 6.226 0 Td[(1andapplyHolder'sinequalitytoconcludethatkf+gkpkf+gkp)]TJ/F7 6.974 Tf 6.227 0 Td[(1kfkp+kgkp:Wemaydividebykf+gkp)]TJ/F7 6.974 Tf 6.227 0 Td[(1ptogetMinkowski'sinequalityunlesskf+gkp=0inwhichcaseitisobvious.QEDTheorem6.4.1Lpiscomplete.Proof.Supposefn0,fn2Lp,andPkfnkp<1Thenkn:=nX1fj2LpbyMinkowskiandsincekn%fwehavejknjp%fpandhencebythemonotoneconvergencetheoremf:=P1j=1fn2Lpandkfkp=limkknkpPkfjkp:NowletffngbeanyCauchysequenceinLp.Bypassingtoasubsequencewemayassumethatkfn+1)]TJ/F11 9.963 Tf 9.963 0 Td[(fnkp<1 2n:SoP1njfi+1)]TJ/F11 9.963 Tf 9.963 0 Td[(fij2Lpandhencegn:=fn)]TJ/F13 6.974 Tf 13.187 12.454 Td[(1Xnjfi+1)]TJ/F11 9.963 Tf 9.962 0 Td[(fij2Lpandhn:=fn+1Xnjfi+1)]TJ/F11 9.963 Tf 9.963 0 Td[(fij2Lp:Wehavegn+1)]TJ/F11 9.963 Tf 9.963 0 Td[(gn=fn+1)]TJ/F11 9.963 Tf 9.963 0 Td[(fn+jfn+1)]TJ/F11 9.963 Tf 9.962 0 Td[(fnj0sognisincreasingandsimilarlyhnisdecreasing.Hencef:=limgn2Lpandkf)]TJ/F11 9.963 Tf 10.25 0 Td[(fnkpkhn)]TJ/F11 9.963 Tf 10.25 0 Td[(gnkp2)]TJ/F10 6.974 Tf 6.227 0 Td[(n+2!0.Sothesubsequencehasalimitwhichthenmustbethelimitoftheoriginalsequence.QEDProposition6.4.2LisdenseinLpforany1p<1.Proof.Forp=1thiswasadeningpropertyofL1.Moregenerally,supposethatf2Lpandthatf0.LetAn:=fx:1 n
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166CHAPTER6.THEDANIELLINTEGRAL.andletgn:=f1An:Thenf)]TJ/F11 9.963 Tf 7.89 0 Td[(gn&0asn!1.Choosensucientlylargesothatkf)]TJ/F11 9.963 Tf 7.89 0 Td[(gnkp<=2.Since0gnn1AnandAn0thenkfk1=limq!1kfkq:.2

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6.6.THERADON-NIKODYMTHEOREM.167Remark.Inthestatementofthetheorem,bothsidesof.2areallowedtobe1.Proof.Ifkfk1=0,thenkfkq=0forallq>0sotheresultistrivialinthiscase.Soletusassumethatkfk1>0andletabeanypositivenumbersmallerthatkfk1.Inotherwords,0ag:Thissethaspositivemeasurebythechoiceofa,anditsmeasureisnitesincef2Lp.AlsokfkqZAajfjq1=qaAa1=q:Lettingq!1givesliminfq!1kfkqaandsinceacanbeanynumberpwehavejfjqjfjpkfk1q)]TJ/F10 6.974 Tf 6.226 0 Td[(palmosteverywhere.Integratingandtakingtheq-throotgiveskfkqkfkpp qkfk11)]TJ/F9 4.981 Tf 7.422 3.221 Td[(p q:Lettingq!1givesthedesiredresult.QED6.6TheRadon-NikodymTheorem.Supposewearegiventwointegrals,IandJonthesamespaceL.Thatis,bothIandJsatisfythethreeconditionsoflinearity,positivity,andthemonotonelimitpropertythatwentintoourdenitionofthetermintegral".WesaythatJisabsolutelycontinuouswithrespecttoIifeverysetwhichisInulli.e.hasmeasurezerowithrespecttothemeasureassociatedtoIisJnull.TheintegralIissaidtobeboundedifI1<1;or,whatamountstothesamething,thatIS<1

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168CHAPTER6.THEDANIELLINTEGRAL.whereIisthemeasureassociatedtoI.WewillrstformulatetheRadon-Nikodymtheoremforthecaseofboundedintegrals,wherethereisaverycleverproofduetovon-NeummanwhichreducesittotheRieszrepresentationtheoreminHilbertspacetheory.Theorem6.6.1[Radon-Nikodym]LetIandJbeboundedintegrals,andsupposethatJisabsolutelycontinuouswithrespecttoI.Thenthereexistsanelementf02L1IsuchthatJf=Iff08f2L1J:.3Theelementf0isuniqueuptoequalityalmosteverywherewithrespecttoI.Proof.Aftervon-Neumann.ConsiderthelinearfunctionK:=I+JonL.ThenKsatisesallthreeconditionsinourdenitionofanintegral,andinadditionisbounded.Weknowfromthecasep=2ofTheorem6.4.1thatL2KisarealHilbertspace.AssumeforthisargumentthatwehavepassedtothequotientspacesoanelementofL2Kisanequivalenceclassofoffunctions.ThefactthatKisbounded,saysthat1:=1S2L2K.Iff2L2KthentheCauchy-SchwartzinequalitysaysthatKjfj=Kjfj1=jfj;12;Kkfk2;Kk1k2;K<1sojfjandhencefareelementsofL1K.Furthermore,jJfjJjfjKjfjkfk2;Kk1k2;Kforallf2L.SinceweknowthatLisdenseinL2KbyProposition6.4.2,JextendstoauniquecontinuouslinearfunctionalonL2K.WeconcludefromtherealversionoftheRieszrepresentationtheorem,thatthereexistsauniqueg2L2KsuchthatJf=f;g2;K=Kfg:IfAisanysubsetofSofpositivemeasure,thenJ1A=K1Agsogisnon-negative.Moreprecisely,gisequivalentalmosteverywheretoafunctionwhichisnon-negative.WeobtaininductivelyJf=Kfg=Ifg+Jfg=Ifg+Ifg2+Jfg2=...=IfnXi=1gi!+Jfgn:LetNbethesetofallxwheregx1.Takingf=1NintheprecedingstringofequalitiesshowsthatJ1NnI1N:Sincenisarbitrary,wehaveproved

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6.6.THERADON-NIKODYMTHEOREM.169Lemma6.6.1Thesetwhereg1hasImeasurezero.WehavenotyetusedtheassumptionthatJisabsolutelycontinuouswithrespecttoI.LetusnowusethisassumptiontoconcludethatNisalsoJ-null.Thismeansthatiff0andf2L1Jthenfgn&0almosteverywhereJ,andhencebythedominatedconvergencetheoremJfgn&0:PluggingthisbackintotheabovestringofequalitiesshowsbythemonotoneconvergencetheoremforIthatf1Xi=1gnconvergesintheL1InormtoJf.Inparticular,sinceJ1<1,wemaytakef=1andconcludethatP1i=1giconvergesinL1I.Sosetf0:=1Xi=1gi2L1I:Wehavef0=1 1)]TJ/F11 9.963 Tf 9.963 0 Td[(galmosteverywheresog=f0)]TJ/F8 9.963 Tf 9.963 0 Td[(1 f0almosteverywhereandJf=Iff0forf0;f2L1J.Bybreakinganyf2L1Jintothedierenceofitspositiveandnegativeparts,weconcludethat.3holdsforallf2L1J.Theuniquenessoff0almosteverywhereIfollowsfromtheuniquenessofginL2K.QEDTheRadonNikodymtheoremcanbeextendedintwodirections.Firstofall,letuscontinuewithourassumptionthatIandJarebounded,butdroptheabsolutecontinuityrequirement.LetussaythatanintegralHisabsolutelysingularwithrespecttoIifthereisasetNofI-measurezerosuchthatJh=0foranyhvanishingonN.LetusnowgobacktoLemma6.6.1.DeneJsingbyJsingf=J1Nf:ThenJsingissingularwithrespecttoI,andwecanwriteJ=Jcont+JsingwhereJcont=J)]TJ/F11 9.963 Tf 9.962 0 Td[(Jsing=J1Nc:

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170CHAPTER6.THEDANIELLINTEGRAL.ThenwecanapplytherestoftheproofoftheRadonNikodymtheoremtoJconttoconcludethatJcontf=Iff0wheref0=P1i=11NcgiisanelementofL1Iasbefore.Inparticular,JcontisabsolutelycontinuouswithrespecttoI.AsecondextensionistocertainsituationswhereSisnotofnitemeasure.WesaythatafunctionfislocallyL1iff1A2L1foreverysetAwithA<1.WesaythatSis-nitewithrespecttoifSisacountableunionofsetsofnitemeasure.Thisisthesameassayingthat1=1S2B.IfSis-nitethenitcanbewrittenasadisjointunionofsetsofnitemeasure.IfSis-nitewithrespecttobothIandJitcanbewrittenasthedisjointunionofcountablymanysetswhicharebothIandJnite.SoifJisabsolutelycontinuouswithrespectI,wecanapplytheRadon-Nikodymtheoremtoeachofthesesetsofnitemeasure,andconcludethatthereisanf0whichislocallyL1withrespecttoI,suchthatJf=Iff0forallf2L1J,andf0isuniqueuptoalmosteverywhereequality.6.7ThedualspaceofLp.RecallthatHolder'sinequality.1saysthatZfgdkfkpkgkqiff2Lpandg2Lqwhere1 p+1 q=1:Fortherestofthissectionwewillassumewithoutfurthermentionthatthisrelationbetweenpandqholds.Holder'sinequalityimpliesthatwehaveamapfromLq!Lpsendingg2LqtothecontinuouslinearfunctiononLpwhichsendsf7!Ifg=Zfgd:Furthermore,Holder'sinequalitysaysthatthenormofthismapfromLq!Lpis1.Inparticular,thismapisinjective.ThetheoremwewanttoproveisthatundersuitableconditionsonSandIwhicharemoregeneraleventhat-nitenessthismapissurjectivefor1p<1.WewillrstprovethetheoreminthecasewhereS<1,thatiswhenIisaboundedintegral.Forthiswewillwillneedalemma:

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6.7.THEDUALSPACEOFLP.1716.7.1Thevariationsofaboundedfunctional.SupposewestartwithanarbitraryLandI.Foreach1p1wehavethenormkkponLwhichmakesLintoarealnormedlinearspace.LetFbealinearfunctiononLwhichisboundedwithrespecttothisnorm,sothatjFfjCkfkpforallf2LwhereCissomenon-negativeconstant.TheleastupperboundofthesetofCwhichworkiscalledkFkpasusual.Iff02L,deneF+f:=lubfFg:0gf;g2Lg:ThenF+f0andF+fkFkpkfkpsinceFgjFgjkFkpkgkpkFkpkfkpforall0gf,g2L,since0gfimpliesjgjpjfjpfor1p<1andalsoimplieskgk1kfk1.AlsoF+cf=cF+f8c0asfollowsdirectlyfromthedenition.Supposethatf1andf2arebothnon-negativeelementsofL.Ifg1;g22Lwith0g1f1and0g2f2thenF+f1+f2lubFg1+g1=lubFg1+lubFg2=F+f1+F+f2:Ontheotherhand,ifg2Lsatises0gf1+f2then0g^f1f1,andg^f12L.Alsog)]TJ/F11 9.963 Tf 8.593 0 Td[(g^f12Landvanishesatpointsxwheregxf1xwhileatpointswheregx>f1xwehavegx)]TJ/F11 9.963 Tf 10.025 0 Td[(g^f1x=gx)]TJ/F11 9.963 Tf 10.024 0 Td[(f1xf2x.Sog)]TJ/F11 9.963 Tf 9.963 0 Td[(g^f1f2andsoF+f1+f2=lubFglubFg^f1+lubFg)]TJ/F11 9.963 Tf 9.476 0 Td[(g^f1F+f1+F+f2:SoF+f1+f2=F+f1+F+f2ifbothf1andf2arenon-negativeelementsofL.Nowwriteanyf2Lasf=f1)]TJ/F11 9.963 Tf 10.626 0 Td[(g1wheref1andg1arenon-negative.Forexamplewecouldtakef1=f+andg1=f)]TJ/F8 9.963 Tf 6.724 -3.615 Td[(.DeneF+f=F+f1)]TJ/F11 9.963 Tf 9.963 0 Td[(F+g1:

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172CHAPTER6.THEDANIELLINTEGRAL.Thisiswelldened,forifwealsohadf=f2)]TJ/F11 9.963 Tf 9.963 0 Td[(g2thenf1+g2=f2+g1soF+f1+F+g2=F+f1+g2=F+f2+g1=F+f2+F+g1soF+f1)]TJ/F11 9.963 Tf 9.962 0 Td[(F+g1=F+f2)]TJ/F11 9.963 Tf 9.963 0 Td[(F+g2:FromthisitfollowsthatF+soextendedislinear,andjF+fjF+jfjkFkpkfkpsoF+isbounded.DeneF)]TJ/F8 9.963 Tf 10.046 -3.615 Td[(byF)]TJ/F8 9.963 Tf 6.725 -4.114 Td[(f:=F+f)]TJ/F11 9.963 Tf 9.963 0 Td[(Ff:AsF)]TJ/F8 9.963 Tf 9.316 -3.616 Td[(isthedierenceoftwolinearfunctionsitislinear.Sincebyitsdenition,F+fFfiff0,weseethatF)]TJ/F8 9.963 Tf 6.725 -3.615 Td[(f0iff0.ClearlykF)]TJ/F14 9.963 Tf 6.724 -3.615 Td[(kkF+kp+kFk2kFkp.Wehaveproved:Proposition6.7.1EverylinearfunctiononLwhichisboundedwithrespecttothekkpnormcanbewrittenasthedierenceF=F+)]TJ/F11 9.963 Tf 10.462 0 Td[(F)]TJ/F35 9.963 Tf 10.978 -3.615 Td[(oftwolin-earfunctionswhichareboundedandtakenon-negativevaluesonnon-negativefunctions.Infact,wecouldformulatethispropositionmoreabstractlyasdealingwithanormedvectorspacewhichhasanorderrelationconsistentwithitsmetricbutweshallrefrainfromthismoreabstractformulation.6.7.2DualityofLpandLqwhenS<1.Theorem6.7.1SupposethatS<1andthatFisaboundedlinearfunc-tiononLpwith1p<1.Thenthereexistsauniqueg2LqsuchthatFf=f;g=Ifg:Hereq=p=p)]TJ/F8 9.963 Tf 9.962 0 Td[(1ifp>1andq=1ifp=1.Proof.ConsidertherestrictionofFtoL.WeknowthatF=F+)]TJ/F11 9.963 Tf 9.617 0 Td[(F)]TJ/F8 9.963 Tf 9.873 -3.616 Td[(wherebothF+andF)]TJ/F8 9.963 Tf 10.086 -3.615 Td[(arelinearandnon-negativeandareboundedwithrespecttothekkpnormonL.Themonotoneconvergencetheoremimpliesthatiffn&0thenkfnkp!0andtheboundednessofF+withrespecttothekkpsaysthatkfnkp!0F+fn!0:SoF+satisesalltheaxiomsforanintegral,andsodoesF)]TJ/F8 9.963 Tf 6.725 -3.616 Td[(.IffvanishesoutsideasetofImeasurezero,thenkfkp=0.Appliedtoafunctionoftheformf=1AweconcludethatifAhas=Imeasurezero,thenAhasmeasurezerowithrespecttothemeasuresdeterminedbyF+orF)]TJ/F8 9.963 Tf 6.725 -3.615 Td[(.Wecanapplythe

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6.7.THEDUALSPACEOFLP.173Radon-Nikodymtheoremtoconcludethattherearefunctionsg+andg)]TJ/F8 9.963 Tf 9.665 -3.616 Td[(whichbelongtoL1IandsuchthatFf=IfgforeveryfwhichbelongstoL1F.Inparticular,ifwesetg:=g+)]TJ/F11 9.963 Tf 9.414 0 Td[(g)]TJ/F8 9.963 Tf 9.772 -3.615 Td[(thenFf=IfgforeveryfunctionfwhichisintegrablewithrespecttobothF+andF)]TJ/F8 9.963 Tf 6.725 -3.615 Td[(,inparticularforanyf2LpI.Wemustshowthatg2Lq.Wersttreatthecasewherep>1.Supposethat0fjgjandthatfisbounded.ThenIfqIfq)]TJ/F7 6.974 Tf 6.227 0 Td[(1sgngg=Ffq)]TJ/F7 6.974 Tf 6.227 0 Td[(1sgngkFkpkfq)]TJ/F7 6.974 Tf 6.227 0 Td[(1kp:SoIfqkFkpIfq)]TJ/F7 6.974 Tf 6.226 0 Td[(1p1 p:Nowq)]TJ/F8 9.963 Tf 9.963 0 Td[(1p=qsowehaveIfqkFkpIfq1 p=kFkpIfq1)]TJ/F6 4.981 Tf 7.458 2.678 Td[(1 q:ThisgiveskfkqkFkpforall0fjgjwithfbounded.Wecanchoosesuchfunctionsfnwithfn%jgj.Itfollowsfromthemonotoneconvergencetheoremthatjgjandhenceg2LqI.Thisprovesthetheoremforp>1.Letusnowgivetheargumentforp=1.Wewanttoshowthatkgk1kFk1.Supposethatkgk1kFk1+where>0.Considerthefunction1AwhereA:=fx:jgxkFk1+ 2g:ThenkFk1+ 2AI1Ajgj=I1Asgngg=F1AsgngkFk1k1Asgngk1=kFk1AwhichisimpossibleunlessA=0,contrarytoourassumption.QED6.7.3ThecasewhereS=1.Herethecasesp>1andp=1maybedierent,dependingonhowinniteSis".Letusrstconsiderthecasewherep>1.IfwerestrictthefunctionalFtoanysubspaceofLpitsnormcanonlydecrease.ConsiderasubspaceconsistingofallfunctionswhichvanishoutsideasubsetS1whereS1<1.Weget

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174CHAPTER6.THEDANIELLINTEGRAL.acorrespondingfunctiong1denedonS1andsetequaltozerooS1withkg1kqkFkpandFf=Ifg1forallfbelongingtothissubspace.IfS2;g2isasecondsuchpair,thentheuniquenesspartofthetheoremshowsthatg1=g2almosteverywhereonS1S2.Thuswecanconsistentlydeneg12onS1[S2.Letb:=lubfkgkqgtakenoverallsuchg.SincethissetofnumbersisboundedbykFkpthisleastupperboundisnite.WecanthereforendanestedsequenceofsetsSnandcorrespondingfunctionsgnsuchthatkgnkq%b:Bythetriangleinequality,ifn>mthenkgn)]TJ/F11 9.963 Tf 9.963 0 Td[(gmkqkgnkq)-222(kgmkqandso,asinyourproofoftheL2Martingaleconvergencetheorem,thissequenceisCauchyinthekkqnorm.Hencethereisalimitg2LqandgissupportedonS0:=[Sn:TherecanbenopairS0;g0withSdisjointfromS0andg06=0onasubsetofpositivemeasureofS0.Indeed,ifthiswerethecase,thenwecouldconsiderg+g0onS[S0andthiswouldhaveastrictlylargerkkqnormthankgkq=b,contradictingthedenitionofb.Itisatthispointintheargumentthatweuseq<1whichisthesameasp>1.ThusFvanishesonanyfunctionwhichissupportedoutsideS0.WehavethusreducedthetheoremtothecasewhereSis-nite.IfSis-nite,decomposeSintoadisjointunionofsetsAiofnitemeasure.Letfmdenotetherestrictionoff2LptoAmandlethmdenotetherestrictionofgtoAm.Then1Xm=1fm=fasaconvergentseriesinLpandsoFf=XmFfm=XmZAmfmhmandthislastseriesconvergestoIfginL1.SowehaveprovedthatLp=Lqincompletegeneralitywhenp>1,andfor-niteSwhenp=1.ItmayhappenandwillhappenwhenweconsidertheHaarintegralonthemostgenerallocallycompactgroupthatwedon'tevenhave-niteness.Butwewillhavethefollowingmorecomplicatedcondition:RecallthatasetAiscalledintegrablebyLoomisif1A2B.NowsupposethatS=[S

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6.8.INTEGRATIONONLOCALLYCOMPACTHAUSDORFFSPACES.175wherethisunionisdisjoint,butpossiblyuncountable,ofintegrablesets,andwiththepropertythateveryintegrablesetiscontainedinatmostacountableunionoftheS.AsetAiscalledmeasurableiftheintersectionsASareallintegrable,andafunctioniscalledmeasurableifitsrestrictiontoeachShasthepropertythattherestrictionofftoeachSbelongstoB,andfurther,thateithertherestrictionoff+toeverySortherestrictionoff)]TJ/F8 9.963 Tf 10.618 -3.615 Td[(toeverySbelongstoL1.Ifwendourselvesinthissituation,thenwecanndagoneachSsinceSis-nite,andpiecethesealltogethertogetagdenedonallofS.Iff2L1thenthesetwheref6=0canhaveintersectionswithpositivemeasurewithonlycountablymanyoftheSandsowecanapplytheresultforthe-nitecaseforp=1tothismoregeneralcaseaswell.6.8IntegrationonlocallycompactHausdorspaces.SupposethatSisalocallycompactHausdorspace.AsinthecaseofRn,wecanandwilltakeLtobethespaceofcontinuousfunctionsofcompactsupport.Dini'slemmathensaysthatiffn2L&0thenfn!0intheuniformtopology.IfAisanysubsetofSwewilldenotethesetoff2LwhosesupportiscontainedinAbyLA.Lemma6.8.1Anon-negativelinearfunctionIisboundedintheuniformnormonLCwheneverCiscompact.Proof.Chooseg02Lsothatgx1forx2C.Iff2LCthenjfjkfk1gsojIfjIjfjIgkfk1:QED:6.8.1Rieszrepresentationtheorems.ThisisthesameRiesz,buttwomoretheorems.Theorem6.8.1Everynon-negativelinearfunctionalIonLisanintegral.Proof.ThisisDini'slemmatogetherwiththeprecedinglemma.Indeed,byDiniweknowthatfn2L&0impliesthatkfnk1&0.Sincef1hascompactsupport,letCbeitssupport,acompactset.AllthesucceedingfnarethenalsosupportedinCandsobytheprecedinglemmaIfn&0.QEDTheorem6.8.2LetFbeaboundedlinearfunctiononLwithrespecttotheuniformnorm.ThentherearetwointegralsI+andI)]TJ/F35 9.963 Tf 10.289 -3.615 Td[(suchthatFf=I+f)]TJ/F11 9.963 Tf 9.963 0 Td[(I)]TJ/F8 9.963 Tf 6.725 -4.114 Td[(f:

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176CHAPTER6.THEDANIELLINTEGRAL.Proof.WeapplyProposition6.7.1tothecaseofourLandwiththeuniformnorm,kk1.WegetF=F+)]TJ/F11 9.963 Tf 9.963 0 Td[(F)]TJ/F8 9.963 Tf -196.216 -23.161 Td[(andanexaminationofheproofwillshowthatinfactkFk1kFk1:Bytheprecedingtheorem,Farebothintegrals.QED6.8.2Fubini'stheorem.Theorem6.8.3LetS1andS2belocallycompactHausdorspacesandletIandJbenon-negativelinearfunctionalsonLS1andLS2respectively.ThenIxJyhx;y=JyIxhx;yforeveryh2LS1S2intheobviousnotation,andthiscommonvalueisanintegralonLS1S2.ProofviaStone-Weierstrass.Theequationinthetheoremisclearlytrueifhx;y=fxgywheref2LS1andg2LS2andsoitistrueforanyhwhichcanbewrittenasanitesumofsuchfunctions.LethbeageneralelementofLS1S2.thenwecanndcompactsubsetsC1S1andC2S2suchthathissupportedinthecompactsetC1C2.ThefunctionsoftheformXfixgiywherethefiareallsupportedinC1andthegiinC2,andthesumisnite,formanalgebrawhichseparatespoints.Soforany>0wecanndakoftheaboveformwithkh)]TJ/F11 9.963 Tf 9.963 0 Td[(kk1<:LetB1andB2beboundsforIonLC1andJonLC2asprovidedbyLemma6.8.1.ThenjJyhx;y)]TJ/F1 9.963 Tf 9.963 9.465 Td[(XJgifixj=j[Jyf)]TJ/F11 9.963 Tf 9.962 0 Td[(k]xj
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6.9.THERIESZREPRESENTATIONTHEOREMREDUX.177Sinceisarbitrary,thisgivestheequalityinthetheorem.Sincethissamefunctionalisnon-negative,itisanintegralbytherstoftheRieszrepresentationtheoremsabove.QEDLetXbealocallycompactHausdorspace,andletLdenotethespaceofcontinuousfunctionsofcompactsupportonX.RecallthattheRieszrepresen-tationtheoremoneofthemassertsthatanynon-negativelinearfunctionIonLsatisesthestartingaxiomsfortheDaniellintegral,andhencecorrespondstoameasuredenedona-eld,andsuchthatIfisgivenbyintegrationoffrelativetothismeasureforanyf2L.6.9TheRieszrepresentationtheoremredux.IwanttogiveanalternativeproofoftheRieszrepresentationtheoremwhichwillgivesomeinformationaboutthepossible-eldsonwhichisdened.Inparticular,IwanttoshowthatwecanndawhichispossiblyanextensionofthegivenbyourpreviousproofoftheRieszrepresentationtheoremwhichisdenedona-eldwhichcontainstheBoreleldBX.RecallthatBXisthesmallest-eldwhichcontainstheopensets.LetFbea-eldwhichcontainsBX.Anon-negativevaluedmeasureonFiscalledregularif1.K<1foranycompactsubsetKX.2.ForanyA2FA=inffU:AU;Uopeng3.IfUXisopenthenU=supfK:KU;Kcompactg:Thesecondconditioniscalledouterregularityandthethirdconditioniscalledinnerregularity.6.9.1Statementofthetheorem.HereistheimprovedversionoftheRieszrepresentationtheorem:Theorem6.9.1LetXbealocallycompactHausdorspace,LthespaceofcontinuousfunctionsofcompactsupportonX,andIanon-negativelinearfunctionalonL.Thenthereexistsa-eldFcontainingBXandanon-negativeregularmeasureonFsuchthatIf=Zfd.4forallf2L.Furthermore,therestrictionoftoBXisunique.

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178CHAPTER6.THEDANIELLINTEGRAL.Theproofofthistheoremhingesonsometopologicalfactswhosetrueplaceisinthechapteronmetricspaces,butIwillprovethemhere.Theimportanceofthetheoremisthatitwillallowustoderivesomeconclusionsaboutspaceswhichareveryhugesuchasthespaceofall"pathsinRnbutareneverthelesslocallycompactinfactcompactHausdorspaces.Itisbecausewewanttoconsidersuchspaces,thattheearlierproof,whichhingedontakinglimitsofsequencesintheverydenitionoftheDaniellintegral,isinsucienttogetattheresultswewant.6.9.2Propositionsintopology.Proposition6.9.1LetXbeaHausdorspace,andletHandKbedisjointcompactsubsetsofX.ThenthereexistdisjointopensubsetsUandVofXsuchthatHUandKV.Thisweactuallydidproveinthechapteronmetricspaces.Proposition6.9.2LetXbealocallycompactHausdorspace,x2X,andUanopensetcontainingx.ThenthereexistsanopensetOsuchthatx2O Oiscompact,and OU.Proof.ChooseanopenneighborhoodWofxwhoseclosureiscompact,whichispossiblesinceweareassumingthatXislocallycompact.LetZ=UWsothat ZiscompactandhencesoisH:= ZnZ.TakeK:=fxgintheprecedingproposition.WethengetanopensetVcontainingxwhichisdisjointfromanopensetGcontaining ZnZ.TakeO:=VZ.Thenx2Oand O ZiscompactandOhasemptyintersectionwith ZnZ,andhenceisiscontainedinZU.QEDProposition6.9.3LetXbealocallycompactHausdorspace,KUwithKcompactandUopensubsetsofX.ThenthereexistsaVwithKV VUwithVopenand Vcompact.Proof.Eachx2KhasaneighborhoodOwithcompactclosurecontainedinU,bytheprecedingproposition.ThesetoftheseOcoverK,soanitesubcollectionofthemcoverKandtheunionofthisnitesubcollectiongivesthedesiredV.Proposition6.9.4LetXbealocallycompactHausdorspace,KUwithKcompactandUopen.Thenthereexistsacontinuousfunctionhwithcompactsupportsuchthat1Kh1U

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6.9.THERIESZREPRESENTATIONTHEOREMREDUX.179andSupphU:Proof.ChooseVasinProposition6.9.3.ByUrysohn'slemmaappliedtothecompactspace Vwecanndafunctionh: V![0;1]suchthath=1onKandf=0on VnV.Extendhtobezeroonthecomplementof V.Thenhdoesthetrick.Proposition6.9.5LetXbealocallycompactHausdorspace,f2L,i.e.fisacontinuousfunctionofcompactsupportonX.SupposethatthereareopensubsetsU1;:::UnsuchthatSuppfn[i=1Ui:Thentherearef1;:::;fn2LsuchthatSuppfiUiandf=f1++fn:Iffisnon-negative,theficanbechosensoastobenon-negative.Proof.Byinduction,itisenoughtoconsiderthecasen=2.LetK:=Suppf,soKU1[U2.LetL1:=KnU1;L2:=KnU2:SoL1andL2aredisjointcompactsets.ByProposition6.9.1wecannddisjointopensetsV1;V2withL1V1;L2V2:SetK1:=KnV1;K2:=KnV2:ThenK1andK2arecompact,andK=K1[K2;K1U1;K2U2:Chooseh1andh2asinProposition6.9.4.Thenset1:=h1;2:=h2)]TJ/F11 9.963 Tf 9.963 0 Td[(h1^h2:ThenSupp1=Supph1U1byconstruction,andSupp2Supph2U2,theitakevaluesin[0;1],and,ifx2K=Suppf1x+2x=h1_h2x=1:Thensetf1:=1f;f2:=2f:QED

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180CHAPTER6.THEDANIELLINTEGRAL.6.9.3ProofoftheuniquenessoftherestrictedtoBX.ItisenoughtoprovethatU=supfIf:f2L;0f1Ug.5=supfIf:f2L;0f1U;SuppfUg.6foranyopensetU,sinceeitheroftheseequationsdeterminesonanyopensetUandhencefortheBoreleld.Sincef1Uandbotharemeasurablefunctions,itisclearthatU=R1Uisatleastaslargeastheexpressionontherighthandsideof.5.Thisinturnisasleastaslargeastherighthandsideof.6sincethesupremumin.6istakenoverassmallersetoffunctionsthatthatof6.5.SoitisenoughtoprovethatUistherighthandsideof.6.Leta
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6.10.EXISTENCE.181toprovethatmisanoutermeasurewemustprovecountablesubadditivity.Wewillrstprovecountablesubadditivityonopensets,andthenusethe=2nargumenttoconcludecountablesubadditivityonallsets:SupposefUngisasequenceofopensets.Wewishtoprovethatm[nUn!XnmUn:.9SetU:=[nUn;andsupposethatf2L;0f1U;SuppfU:SinceSuppfiscompactandcontainedinU,itiscoveredbynitelymanyoftheUi.Inotherwords,thereissomeniteintegerNsuchthatSuppfN[n=1Un:ByProposition6.9.5wecanwritef=f1++fN;SuppfiUi;i=1;:::;N:ThenIf=XIfiXmUi;usingthedenition6.7.Replacingthenitesumontherighthandsideofthisinequalitybytheinnitesum,andthentakingthesupremumoverfproves.9,whereweusethedenition6.7onceagain.NextletfAngbeanysequenceofsubsetsofX.Wewishtoprovethatm[nAn!XnmAn:Thisisautomaticiftherighthandsideisinnite.SoassumethatXnmAn<1andchooseopensetsUnAnsothatmUnmAn+ 2n:ThenU:=SUnisanopensetcontainingA:=SAnandmAmUXmUiXnmAn+:Sinceisarbitrary,wehaveprovedcountablesubadditivity.

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182CHAPTER6.THEDANIELLINTEGRAL.6.10.2MeasurabilityoftheBorelsets.LetFdenotethecollectionofsubsetswhicharemeasurableinthesenseofCaratheodoryfortheoutermeasurem.WewishtoprovethatFBX.SinceBXisthe-eldgeneratedbytheopensets,itisenoughtoshowthateveryopensetismeasurableinthesenseofCaratheodory,i.e.thatmAmAU+mAUc.10foranyopensetUandanysetAwithmA<1:If>0,chooseanopensetVAwithmVmA+whichispossiblebythedenition.8.WewillshowthatmVmVU+mVUc)]TJ/F8 9.963 Tf 9.962 0 Td[(2:.11ThiswillthenimplythatmAmAU+mAUc)]TJ/F8 9.963 Tf 9.963 0 Td[(3andsince>0isarbitrary,thiswillimply.10.Usingthedenition6.7,wecanndanf12Lsuchthatf11VUandSuppf1VUwithIf1mVU)]TJ/F11 9.963 Tf 9.962 0 Td[(:LetK:=Suppf1.ThenKUandsoKcUcandKcisopen.HenceVKcisanopensetandVKcVUc:Usingthedenition6.7,wecanndanf22Lsuchthatf21VKcandSuppf2VKcwithIf2mVKc)]TJ/F11 9.963 Tf 9.962 0 Td[(:ButmVKcmVUcsinceVKcVUc.SoIf2mVUc)]TJ/F11 9.963 Tf 9.963 0 Td[(:Sof1+f21K+1VKc1VsinceK=Suppf1VandSuppf2VKc.AlsoSuppf1+f2K[VKc=V:Thusf=f1+f22Landsoby.7,If1+f2mV:Thisproves.11andhencethatallBorelsetsaremeasurable.

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6.10.EXISTENCE.1836.10.3Compactsetshavenitemeasure.Letbethemeasureassociatedtomonthe-eldFofmeasurablesets.Wewillnowprovethatisregular.Theconditionofouterregularityisautomatic,sincethiswashowwedenedA=mAforageneralset.IfKisacompactsubsetofX,wecanndanf2Lsuchthat1KfbyProposition6.9.4.Let0<<1andsetU:=fx:fx>1)]TJ/F11 9.963 Tf 9.962 0 Td[(g:ThenUisanopensetcontainingK.If0g2Lsatisesg1U,theng=0onUc,andforx2U,gx1whilefx>1)]TJ/F11 9.963 Tf 9.962 0 Td[(.Sog1 1)]TJ/F11 9.963 Tf 9.962 0 Td[(fandhence,by.7mU1 1)]TJ/F11 9.963 Tf 9.962 0 Td[(If:So,by.8KmU1 1)]TJ/F11 9.963 Tf 9.963 0 Td[(If<1:Reviewingtheprecedingargument,weseethatwehaveinfactprovedthemoregeneralstatementProposition6.10.1IfAisanysubsetofXandf2Lissuchthat1AfthenmAIf:6.10.4Interiorregularity.Wenowproveinteriorregularity,whichwillbeveryimportantforus.WewishtoprovethatU=supfK:KU;Kcompactg;foranyopensetU,where,accordingto.7,mU=supfIf:f2L;0f1U;SuppfUg:SinceSuppfiscompact,andcontainedinU,wewillbedoneifweshowthatf2L;0f1IfSuppf:.12SoletVbeanopensetcontainingSuppf.Bydenition.7,VIf

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184CHAPTER6.THEDANIELLINTEGRAL.and,sinceVisanarbitraryopensetcontainingSuppf,wehaveSuppfIfusingthedenition.8ofmSuppf.InthecourseofthisargumentwehaveprovedProposition6.10.2Ifg2L;0g1KwhereKiscompact,thenIgK:6.10.5Conclusionoftheproof.Finally,wemustshowthatalltheelementsofLareintegrablewithrespecttoandIf=Zfd:.13SincetheelementsofLarecontinuous,theyareBorelmeasurable.Aseveryf2Lcanbewrittenasthedierenceoftwonon-negativeelementsofL,andasbothsidesof.13arelinearinf,itisenoughtoprove6.13fornon-negativefunctions.FollowingLebesgue,dividethey-axis"upintointervalsofsize.Thatis,letbeapositivenumber,and,foreverypositiveintegernsetfnx:=8<:0iffxn)]TJ/F8 9.963 Tf 9.963 0 Td[(1fx)]TJ/F8 9.963 Tf 9.963 0 Td[(n)]TJ/F8 9.963 Tf 9.963 0 Td[(1ifn)]TJ/F8 9.963 Tf 9.963 0 Td[(1
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6.10.EXISTENCE.185Ontheotherhand,themonotonicityoftheintegralanditsdenitionimplythatKnZfndKn)]TJ/F7 6.974 Tf 6.227 0 Td[(1:SummingtheseinequalitiesgivesNXi=1KnIfN)]TJ/F7 6.974 Tf 6.227 0 Td[(1Xi=0KnNXi=1KnRfdN)]TJ/F7 6.974 Tf 6.227 0 Td[(1Xi=0KnwhereNissucientlylarge.ThusIfandRfdliewithinadistanceN)]TJ/F7 6.974 Tf 6.226 0 Td[(1Xi=0Kn)]TJ/F11 9.963 Tf 9.963 0 Td[(NXi=1Kn=K0)]TJ/F11 9.963 Tf 9.963 0 Td[(KNSuppfofoneanother.Sinceisarbitrary,wehaveproved.13andcompletedtheproofoftheRieszrepresentationtheorem.

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186CHAPTER6.THEDANIELLINTEGRAL.

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Chapter7Wienermeasure,Brownianmotionandwhitenoise.7.1Wienermeasure.WebeginbyconstructingWienermeasurefollowingapaperbyNelson,JournalofMathematicalPhysics5332-343.7.1.1TheBigPathSpace.Let_RndenotetheonepointcompacticationofRn.Let:=Y0t<1_Rn.1betheproductofcopiesof_Rn,oneforeachnon-negativet.Thisisanuncount-ableproduct,andsoahugespace,butbyTychono'stheorem,itiscompactandHausdor.Wecanthinkofapoint!ofasbeingafunctionfromR+to_Rn,i.e.asasacurve"withnorestrictionswhatsoever.LetFbeacontinuousfunctiononthem-foldproduct:F:mYi=1_Rn!R;andlett1t2tmbexedtimes".Dene=F;t1;:::;tm:!Rby!:=F!t1;:::;!tm:Wecancallsuchafunctionanitefunctionsinceitsvalueat!dependsonlyonthevaluesof!atnitelymanypoints.Thesetofsuchfunctionssatisesour187

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188CHAPTER7.WIENERMEASURE,BROWNIANMOTIONANDWHITENOISE.abstractaxiomsforaspaceonwhichwecandeneintegration.Furthermore,thesetofsuchfunctionsisanalgebracontaining1andwhichseparatespoints,soisdenseinCbytheStone-Weierstrasstheorem.LetuscallthespaceofsuchfunctionsCfin.IfwedeneanintegralIonCfinthen,bytheStone-WeierstrasstheoremitextendstoCandtherefore,bytheRieszrepresentationtheorem,givesusaregularBorelmeasureon.Foreachx2Rnwearegoingtodenesuchanintegral,IxbyIx=ZZFx1;x2;:::;xmpx;x1;t1px1;x2;t2)]TJ/F11 9.963 Tf 7.749 0 Td[(t1pxm)]TJ/F7 6.974 Tf 6.226 0 Td[(1;xm;tm)]TJ/F11 9.963 Tf 7.749 0 Td[(tm)]TJ/F7 6.974 Tf 6.227 0 Td[(1dx1:::dxmwhen=F;t1;:::;tmwherepx;y;t=1 tn=2e)]TJ/F7 6.974 Tf 6.227 0 Td[(x)]TJ/F10 6.974 Tf 6.227 0 Td[(y2=2t.2withpx;1=0andallintegrationsareover_Rn.Inordertocheckthatthisiswelldened,wehavetoverifythatifFdoesnotdependonagivenxithenwegetthesameanswerifwedeneintermsofthecorrespondingfunctionoftheremainingm)]TJ/F8 9.963 Tf 9.962 0 Td[(1variables.ThisamountstothecomputationZpx;y;spy;z;tdy=px;z;s+t:Ifn=1thisisthecomputation1 2tZRe)]TJ/F7 6.974 Tf 6.227 0 Td[(x)]TJ/F10 6.974 Tf 6.226 0 Td[(y2=2se)]TJ/F7 6.974 Tf 6.227 0 Td[(y)]TJ/F10 6.974 Tf 6.226 0 Td[(z22tdy=1 2s+tex)]TJ/F10 6.974 Tf 6.226 0 Td[(z2=2s+t:Ifwemakethechangeofvariablesu=x)]TJ/F11 9.963 Tf 9.963 0 Td[(ythisbecomesnt?ns=nt+swherenrx:=1 p re)]TJ/F10 6.974 Tf 6.227 0 Td[(x2=2r:Intermsofourscalingoperator"SagivenbySafx=faxwecanwritenr=r)]TJ/F6 4.981 Tf 7.422 2.677 Td[(1 2Sr)]TJ/F6 4.981 Tf 6.592 1.93 Td[(1 2nwherenistheunitGaussiannx=e)]TJ/F10 6.974 Tf 6.227 0 Td[(x2=2.NowtheFouriertransformtakesconvolutionintomultiplication,satisesSaf^==aS1=a^f;

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7.1.WIENERMEASURE.189andtakestheunitGaussianintotheunitGaussian.ThusuponFouriertrans-form,theequationnt?ns=nt+sbecomestheobviousfactthate)]TJ/F10 6.974 Tf 6.227 0 Td[(s2=2e)]TJ/F10 6.974 Tf 6.227 0 Td[(t2=2=e)]TJ/F7 6.974 Tf 6.227 0 Td[(s+t2=2:Thesameprooforaniteratedversionoftheonedimensionalresultappliesinn-dimensions.So,foreachx2Rnwehavedenedameasureon.WedenotethemeasurecorrespondingtoIxbyprx.Itisaprobabilitymeasureinthesensethatprx=1.TheintuitiveideabehindthedenitionofprxisthatitassignsprobabilityprxE:=ZE1ZEmpx;x1;t1px1;x2;t2)]TJ/F11 9.963 Tf 9.962 0 Td[(t1pxm)]TJ/F7 6.974 Tf 6.226 0 Td[(1;xm;tm)]TJ/F11 9.963 Tf 9.963 0 Td[(tm)]TJ/F7 6.974 Tf 6.227 0 Td[(1dx1:::dxmtothesetofallpaths!whichstartatxandpassthroughthesetE1attimet1,thesetE2attimet2etc.andwehavedenotedthissetofpathsbyE.7.1.2Theheatequation.Wepausetoreectuponthecomputationwedidintheprecedingsection.DenetheoperatorTtonthespaceSoronS0byTtfx=ZRnpx;y;tfydy:.3Inotherwords,Ttistheoperationofconvolutionwitht)]TJ/F10 6.974 Tf 6.227 0 Td[(n=2e)]TJ/F10 6.974 Tf 6.227 0 Td[(x2=2t:WehaveveriedthatTtTs=Tt+s:.4Also,wehaveveriedthatwhenwetakeFouriertransforms,Ttf^=e)]TJ/F10 6.974 Tf 6.227 0 Td[(t2=2^f:.5Ifwelett!0inthisequationwegetlimt!0Tt=Identity:.6Usingsomelanguagewewillintroducelater,conditions.4and.6saythattheTtformacontinuoussemi-groupofoperators.Ifwedierentiate.5withrespecttot,andletut;x:=Ttfxweseethatuisasolutionoftheheatequation"@2u @t2=@2u @x12++@2u @xn2

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190CHAPTER7.WIENERMEASURE,BROWNIANMOTIONANDWHITENOISE.withtheinitialconditionsu;x=fx.Intermsoftheoperator:=)]TJ/F1 9.963 Tf 9.41 14.048 Td[(@2 @x12++@2 @xn2wearetemptedtowriteTt=e)]TJ/F10 6.974 Tf 6.226 0 Td[(t;inanalogytoourstudyofellipticoperatorsoncompactmanifolds.Wewillspendlotoftimejustifyingthesekindofformulasinthenon-compactsettinglateroninthecourse.7.1.3Pathsarecontinuouswithprobabilityone.Thepurposeofthissubsectionistoprovethatifweusethemeasureprx,thenthesetofdiscontinuouspathshasmeasurezero.Webeginwithsometechnicalissues.WerecallthatthestatementthatameasureisregularmeansthatforanyBorelsetAA=inffG:AG;GopengandforanyopensetUU=supfK:KU;Kcompactg:Thissecondconditionhasthefollowingconsequence:Supposethat)-436(isanycollectionofopensetswhichisclosedunderniteunion.IfO=[G2)]TJ/F11 9.963 Tf 6.586 11.905 Td[(GthenO=supG2)]TJ/F11 9.963 Tf 6.586 8.363 Td[(GsinceanycompactsubsetofOiscoveredbynitelymanysetsbelongingto.Theimportanceofthisstemsfromthefactthatwecanallow)-395(toconsistofuncountablymanyopensets,andwewillneedtoimposeuncountablymanyconditionsinsinglingoutthespaceofcontinuouspaths,forexample.Indeed,ourrsttaskwillbetoshowthatthemeasureprxisconcentratedonthespaceofcontinuouspathsinRnwhichdonotgotoinnitytoofast.Webeginwiththefollowingcomputationinonedimension:pr0fj!tj>rg=21 2t1=2Z1re)]TJ/F10 6.974 Tf 6.226 0 Td[(x2=2tdx2 t1=2Z1rx re)]TJ/F10 6.974 Tf 6.226 0 Td[(x2=2tdx=2 t1=2t rZ1rx te)]TJ/F10 6.974 Tf 6.227 0 Td[(x2=2tdx=2t 1=2e)]TJ/F10 6.974 Tf 6.227 0 Td[(r2=2t r:

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7.1.WIENERMEASURE.191Forxedrthistendstozeroveryfastast!0.Inn-dimensionskyk>intheEuclideannormimpliesthatatleastoneofitscoordinatesyisatisesjyij>=p nsowendthatprxfj!t)]TJ/F11 9.963 Tf 9.963 0 Td[(xj>gce)]TJ/F10 6.974 Tf 6.227 0 Td[(2=2ntforasuitableconstantdependingonlyonn.Inparticular,ifwelet;denotethesupremumoftheaboveprobabilityoverall0forsomej=1;:::mg:ThenprxA21 2;.8independentlyofthenumbermofsteps.Proof.LetB:=f!jj!t1)]TJ/F11 9.963 Tf 9.962 0 Td[(!tmj>1 2gletCi:=f!jj!ti)]TJ/F11 9.963 Tf 9.962 0 Td[(!tmj>1 2gandletDi:=f!jj!t1)]TJ/F11 9.963 Tf 9.962 0 Td[(!tij>andj!t1)]TJ/F11 9.963 Tf 9.962 0 Td[(!tkjk=1;:::i)]TJ/F8 9.963 Tf 9.963 0 Td[(1g:If!2A,then!2DiforsomeibythedenitionofA,bytakingitobetherstjthatworksinthedenitionofA.If!62Band!2Dithen!2Cisinceithastomoveadistanceofatleast1 2togetbackfromoutsidetheballofradiustoinsidetheballofradius1 2.SowehaveAB[m[i=1CiDiandhenceprxAprxB+mXi=1prxCiDi:.9NowwecanestimateprxCiDiasfollows.For!tobelongtothisintersection,wemusthave!2Diandthenthepathmovesadistanceatleast 2intimetn)]TJ/F11 9.963 Tf 9.333 0 Td[(tiandthesetwoeventsareindependent,soprxCiDi 2;prxDi.Hereisthisargumentinmoredetail:LetF=1fy;zjjy)]TJ/F10 6.974 Tf 6.226 0 Td[(zj>1 2g

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192CHAPTER7.WIENERMEASURE,BROWNIANMOTIONANDWHITENOISE.sothat1Ci=F;ti;tn:Similarly,letGbetheindicatorfunctionofthesubsetof_Rn_Rn_Rnicopiesconsistingofallpointswithjxk)]TJ/F11 9.963 Tf 9.963 0 Td[(x1j;k=1;:::;i)]TJ/F8 9.963 Tf 9.962 0 Td[(1;jx1)]TJ/F11 9.963 Tf 9.962 0 Td[(xij>sothat1Di=G;t1;:::;tj:ThenprxCiDi=Z:::Zpx;x1;t1pxi)]TJ/F7 6.974 Tf 6.227 0 Td[(1;xi;ti)]TJ/F11 9.963 Tf 7.749 0 Td[(ti)]TJ/F7 6.974 Tf 6.227 0 Td[(1Fx1;:::;xiGxi;xnpxi;xn;tn)]TJ/F11 9.963 Tf 7.749 0 Td[(tidx1dxn:Thelastintegralwithrespecttoxnis1 2;.ThusprxCiDi 2;prxDi:TheDiaredisjointbydenition,soXprxDiprx[Di1:SoprxAprxB+1 2;21 2;:QEDLetE:f!jj!ti)]TJ/F11 9.963 Tf 9.963 0 Td[(!tjj>2forsome1j2theneitherj!t1)]TJ/F11 9.963 Tf 10.648 0 Td[(!tjj>orj!t1)]TJ/F11 9.963 Tf 9.963 0 Td[(!tkj>orboth.SoprxE21 2;:.10Lemma7.1.2Let0a2forsomes;t2[a;b]g:ThenprxEa;b:21 2;:Proof.Hereiswherewearegoingtousetheregularityofthemeasure.LetSdenoteanitesubsetof[a;b]andandletEa;b;;S:=f!jj!s)]TJ/F11 9.963 Tf 9.962 0 Td[(!tj>2forsomes;t2Sg:

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7.1.WIENERMEASURE.193ThenEa;b;;SisanopensetandprxEa;b;;S<21 2;foranyS.TheunionoverallSoftheEa;b;;SisEa;b;.Theregularityofthemeasurenowimpliesthelemma.QEDLetkandnbeintegers,andset:=1 n:LetFk;;:=f!jj!t)]TJ/F11 9.963 Tf 9.962 0 Td[(!sj>4forsomet;s2[0;k];withjt)]TJ/F11 9.963 Tf 9.963 0 Td[(sj4forsomesandtwhichlieineitherthesameorinadjacentsubintervals.So!mustlieinEa;b;foroneofthesesubintervals,andthereareknofthem.QEDLet!2beacontinuouspathinRn.Restrictedtoanyinterval[0;k]itisuniformlycontinuous.Thismeansthatforany>0itbelongstothecomplementofthesetFk;;forsome.Wecanlet=1=pforsomeintegerp.LetCdenotethesetofcontinuouspathsfrom[0;1toRn.ThenC=k[Fk;;csothecomplementCcofthesetofcontinuouspathsis[k[Fk;;;acountableunionofsetsofmeasurezerosinceprxFk;;!lim!02k1 2;==0:WehavethusprovedafamoustheoremofWiener:Theorem7.1.1[Wiener.]Themeasureprxisconcentratedonthespaceofcontinuouspaths,i.e.prxC=1.Inparticular,thereisaprobabilitymeasureonthespaceofcontinuouspathsstartingattheoriginwhichassignsprobabilitypr0E=ZE1ZEmp;x1;t1px1;x2;t2)]TJ/F11 9.963 Tf 9.963 0 Td[(t1pxm)]TJ/F7 6.974 Tf 6.226 0 Td[(1;xm;tm)]TJ/F11 9.963 Tf 9.963 0 Td[(tm)]TJ/F7 6.974 Tf 6.227 0 Td[(1dx1:::dxmtothesetofallpaths!whichstartat0andpassthroughthesetE1attimet1,thesetE2attimet2etc.andwehavedenotedthissetofpathsbyE.

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194CHAPTER7.WIENERMEASURE,BROWNIANMOTIONANDWHITENOISE.7.1.4EmbeddinginS0.Forconvenienceinnotationletmenowspecializetothecasen=1.LetWCconsistofthosepaths!with!=0andZ10+t)]TJ/F7 6.974 Tf 6.227 0 Td[(2wtdt<1:Proposition7.1.1[Stroock]TheWienermeasurepr0isconcentratedonW.Indeed,weletEj!tjdenotetheexpectationofthefunctionj!tjof!withrespecttoWienermeasure,soEj!tj=1 p 2tZRjxje)]TJ/F10 6.974 Tf 6.227 0 Td[(x2=2tdx=1 p 2ttZ10x te)]TJ/F10 6.974 Tf 6.227 0 Td[(x2=tdx=Ct1=2:Thus,byFubini,EZ10+t)]TJ/F7 6.974 Tf 6.227 0 Td[(2jwtjdt=Z10+t)]TJ/F7 6.974 Tf 6.227 0 Td[(2Ejwtj<1:Hencethesetof!withR10+t)]TJ/F7 6.974 Tf 6.226 0 Td[(2jwtjdt=1musthavemeasurezero.QEDNoweachelementofWdenesatempereddistribution,i.e.anelementofS0accordingtotheruleh!;i=Z10!ttdt:.12WeclaimthatthismapfromWtoS0ismeasurableandhencetheWienermeasurepushesforwardtogiveameasureonS0.Toseethis,letusrstputadierenttopologyofuniformconvergenceonW.Inotherwords,foreach!2WletU!consistofall!1suchthatsupt0j!1t)]TJ/F11 9.963 Tf 9.963 0 Td[(!tj<;andtakethesetoformabasisforatopologyonW.SinceweputtheweaktopologyonS0itisclearthatthemap.12iscontinuousrelativetothisnewtopology.SoitwillbesucienttoshowthateachsetU!isoftheformAWwhereAisinB,theBoreleldassociatedtotheproducttopologyon.SorstconsiderthesubsetsVn;!ofWconsistingofall!12Wsuchthatsupt0j!1t)]TJ/F11 9.963 Tf 9.963 0 Td[(!tj)]TJ/F8 9.963 Tf 11.658 6.74 Td[(1 n:

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7.2.STOCHASTICPROCESSESANDGENERALIZEDSTOCHASTICPROCESSES.195ClearlyU!=[nVn;!;acountableunion,soitisenoughtoshowthateachVn;!isoftheformAnWwhereAn2BX.Nowbythedenitionofthetopologyon,ifrisanyrealnumber,thesetAn;r:=f!1jj!1r)]TJ/F11 9.963 Tf 9.962 0 Td[(!rj)]TJ/F8 9.963 Tf 11.657 6.74 Td[(1 nisclosed.Soifweletrrangeoverthenon-negativerationalnumbersQ+,thenAn=r2Q+An;rbelongstoB.Butif!1iscontinuous,thenif!12Anthensupt2R+j!1t)]TJ/F11 9.963 Tf -335.963 -13.675 Td[(!tj)]TJ/F7 6.974 Tf 11.635 3.923 Td[(1 n,andsoAnW=Vn;!aswastobeproved.7.2Stochasticprocessesandgeneralizedstochas-ticprocesses.Inthestandardprobabilityliteratureastochasticprocessisdenedasfollows:oneisgivenanindexsetTandforeacht2TonehasarandomvariableXt.Moreprecisely,onehassomeprobabilitytriple;F;Pandforeacht2Tarealvaluedmeasurablefunctionon;F.SoastochasticprocessXisjustacollectionX=fXt;t2Tgofrandomvariables.UsuallyT=ZorZ+inwhichcasewecallXadiscretetimerandomprocessorT=RorR+inwhichcasewecallXacontinuoustimerandomprocess.ThusthewordprocessmeansthatwearethinkingofTasrepresentingthesetofalltimes.ArealizationofX,thatisthesetXt!forsome!2iscalledasamplepath.IfTisoneoftheabovechoices,thenXissaidtohaveindependentincrementsifforallt0
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196CHAPTER7.WIENERMEASURE,BROWNIANMOTIONANDWHITENOISE.SupposethatXisacontinuoustimerandomvariablewiththepropertythatforalmostall!,thesamplepathXt!iscontinuous.Letbeacontinuousfunctionofcompactsupport.ThentheRiemannapproximatingsumstotheintegralZTXt!tdtwillconvergeforalmostall!andhencewegetarandomvariablehX;iwherehX;i!=ZTXt!tdt;therighthandsidebeingdenedalmosteverywhereasthelimitoftheRie-mannapproximatingsums.Thesamewillbetrueifvanishesrapidlyatinnityandthesamplepathssatisfya.e.aslowgrowthconditionsuchasgivenbyProposition7.1.1inadditiontobeingcontinuousa.e.ThenotationhX;iisjustiedsincehX;iclearlydependslinearlyon.ButnowwecanmakethefollowingdenitionduetoGelfand.WemayrestrictfurtherbyrequiringthatbelongtoDorS.WethenconsideraruleZwhichassignstoeachsucharandomvariablewhichwemightdenotebyZorhZ;iandwhichdependslinearlyonandsatisesappropriatecontinuityconditions.Suchanobjectiscalledageneralizedrandomprocess.TheideaisthatjustasinthecaseofgeneralizedfunctionswemaynotbeabletoevaluatedZtatagiventimet,butmaybeabletoevaluateasmearedoutversion"Z.Thepurposeofthenextfewsectionsistodothefollowingcomputation:WewishtoshowthatforthecaseBrownianmotion,hX;iisaGaussianrandomvariablewithmeanzeroandwithvarianceZ10Z10mins;tstdsdt:FirstweneedsomeresultsaboutGaussianrandomvariables.7.3Gaussianmeasures.7.3.1Generalitiesaboutexpectationandvariance.LetVbeavectorspacesayovertherealsandnitedimensional.LetXbeaV-valuedrandomvariable.Thatis,wehavesomemeasurespaceM;F;whichwillbexedandhiddeninthissectionwhereisaprobabilitymeasureonM,andX:M!Visameasurablefunction.IfXisintegrable,thenEX:=ZMXd

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7.3.GAUSSIANMEASURES.197iscalledtheexpectationofXandisanelementofV.ThefunctionXXisaVVvaluedfunction,andifitisintegrable,thenVarX=EXX)]TJ/F11 9.963 Tf 9.962 0 Td[(EXEX=EX)]TJ/F11 9.963 Tf 9.962 0 Td[(EXX)]TJ/F11 9.963 Tf 9.963 0 Td[(EXiscalledthevarianceofXandisanelementofVV.Itisbyitsdenitionasymmetrictensor,andsocanbethoughtofasaquadraticformonV.IfA:V!Wisalinearmap,thenAXisaWvaluedrandomvariable,andEAX=AEX;VarAX=AAVarX.13assumingthatEXandVarXexist.WecanalsowritethislastequationasVarAX=VarXA;2W.14ifwethinkofthevarianceasquadraticfunctiononthedualspace.ThefunctiononVgivenby7!EeiXiscalledthecharacteristicfunctionassociatedtoXandisdenotedbyX.Herewehaveusedthenotationvtodenotethevalueof2Vonv2V.ItisaversionoftheFouriertransformwiththeconventionsusedbytheprobabilists.Moreprecisely,letXdenotethepushforwardofthemeasurebythemapX,sothatXisaprobabilitymeasureonV.ThenXistheFouriertransformofthismeasureexceptthattherearenopowersof2infrontoftheintegralandaplusratherthanaminussignisbeforetheiintheexponent.Thesearetheconventionsoftheprobabilists.WhatisimportantforusisthefactthattheFouriertransformdeterminesthemeasure,i.e.XdeterminesX.TheprobabilistswouldsaythatthelawoftherandomvariablemeaningXisdeterminedbyitscharacteristicfunction.Togetafeelingfor.14considerthecasewhereA=isalinearmapfromVtoR.ThenVarX=VarXistheusualvarianceofthescalarvaluedrandomvariableX.ThusweseethatVarX0,soVarXisnon-negativedenitesymmetricbilinearformonV.Thevarianceofascalarvaluedrandomvariablevanishesifandonlyifitisaconstant.ThusVarXispositivedeniteunlessXisconcentratedonhyperplane.SupposethatA:V!Wisanisomorphism,andthatXisabsolutelycontinuouswithrespecttoLebesguemeasure,soX=dvwhereissomefunctiononVcalledtheprobabilitydensityofX.ThenAXisabsolutelycontinuouswithrespecttoLebesguemeasureonWanditsdensityisgivenbyw=A)]TJ/F7 6.974 Tf 6.227 0 Td[(1wjdetAj)]TJ/F7 6.974 Tf 6.227 0 Td[(1.15asfollowsfromthechangeofvariablesformulaformultipleintegrals.

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198CHAPTER7.WIENERMEASURE,BROWNIANMOTIONANDWHITENOISE.7.3.2Gaussianmeasuresandtheirvariances.Letdbeapositiveinteger.WesaythatNisaunitd-dimensionalGaussianrandomvariableifNisarandomvariablewithvaluesinRdwithdensity)]TJ/F10 6.974 Tf 6.226 0 Td[(d=2e)]TJ/F7 6.974 Tf 6.226 0 Td[(x21+x2d=2:ItisclearthatEN=0and,since)]TJ/F10 6.974 Tf 6.227 0 Td[(d=2Zxixje)]TJ/F7 6.974 Tf 6.227 0 Td[(x21+x2d=2dx=ij;thatVarN=Xiii.16where1;:::;disthestandardbasisofRd.WewillsometimesdenotethistensorbyId.IngeneralwehavetheidenticationVVwithHomV;V,sowecanthinkoftheVarXasanelementofHomV;VifXisaV-valuedrandomvariable.IfweidentifyRdwithitsdualspaceusingthestandardbasis,thenIdcanbethoughtofastheidentitymatrix.WecancomputethecharacteristicfunctionofNbyreducingthecomputa-tiontoaproductofonedimensionalintegralsyieldingNt1;:::;td=e)]TJ/F7 6.974 Tf 6.227 0 Td[(t21++t2d=2:.17AV-valuedrandomvariableXiscalledGaussianifitisequalinlawtoarandomvariableoftheformAN+awhereA:Rd!Visalinearmap,wherea2V,andwhereNisaunitGaussianrandomvariable.ClearlyEX=a;VarX=AAIdor,putanotherway,VarX=IdAandhenceX=NAeia=e)]TJ/F6 4.981 Tf 7.422 2.677 Td[(1 2IdAeiaorX=e)]TJ/F7 6.974 Tf 7.587 0 Td[(VarX=2+iEX:.18

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7.3.GAUSSIANMEASURES.199ItisabitofanuisancetocarryalongtheEXinallthecomputations,soweshallrestrictourselvestocenteredGaussianrandomvariablesmeaningthatEX=0.ThusforacenteredGaussianrandomvariablewehaveX=e)]TJ/F7 6.974 Tf 7.587 0 Td[(VarX=2:.19Conversely,supposethatXisaVvaluedrandomvariablewhosecharacteristicfunctionisoftheformX=e)]TJ/F10 6.974 Tf 6.227 0 Td[(Q=2;whereQisaquadraticform.SincejXj1weseethatQmustbenon-negativedenite.SupposethatwehavechosenabasisofVsothatVisidentiedwithRqwhereq=dimV.BytheprincipalaxistheoremwecanalwaysndanorthogonaltransformationcijwhichbringsQtodiagonalform.Inotherwords,ifwesetj:=XicijithenQ=Xjj2j:Thejareallnon-negativesinceQisnon-negativedenite.Soifwesetaij:=1 2jcij;andA=aijwendthatQ=IqA.HenceXhasthesamecharacteristicfunctionasaGaussianrandomvariablehencemustbeGaussian.AsacorollarytothisargumentweseethatArandomvariableXiscenteredGaussianifandonlyifXisarealvaluedGaussianrandomvariablewithmeanzeroforeach2V.7.3.3ThevarianceofaGaussianwithdensity.InourdenitionofacenteredGaussianrandomvariablewewerecarefulnottodemandthatthemapAbeanisomorphism.Forexample,ifAwerethezeromapthenwewouldendupwiththefunctionattheoriginforcenteredGaussianswhichforreasonsofpassingtothelimitwewanttoconsiderasaGaussianrandomvariable.ButsupposethatAisanisomorphism.Thenby.15,Xwillhaveadensitywhichisproportionaltoe)]TJ/F10 6.974 Tf 6.226 0 Td[(Sv=2whereSisthequadraticformonVgivenbySv=JdA)]TJ/F7 6.974 Tf 6.227 0 Td[(1vandJdistheunitquadraticformonRd:Jdx=x21+x2d

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200CHAPTER7.WIENERMEASURE,BROWNIANMOTIONANDWHITENOISE.or,intermsofthebasisfigofthedualspacetoRd,Jd=Xiii:HereJd2RdRd=HomRd;Rd.ItistheinverseofthemapId.WecanregardSasbelongingtoHomV;VwhilewealsoregardVarX=AAIdasanelementofHomV;V.IclaimthatVarXandSareinversestooneanother.Indeed,droppingthesubscriptdwhichisxedinthiscomputation,VarX;=IA;A=AIAwhenthoughtofasabilinearformonVV,andhenceVarX=AIAwhenthoughtofasanelementofHomV;V.SimilarlythinkingofSasabilinearformonVwehaveSv;w=JA)]TJ/F7 6.974 Tf 6.227 0 Td[(1v;A)]TJ/F7 6.974 Tf 6.226 0 Td[(1w=JA)]TJ/F7 6.974 Tf 6.227 0 Td[(1vA)]TJ/F7 6.974 Tf 6.227 0 Td[(1wsoS=A)]TJ/F7 6.974 Tf 6.227 0 Td[(1JA)]TJ/F7 6.974 Tf 6.226 0 Td[(1whenSisthoughtofasanelementofHomV;V.SinceIandJareinversesofoneanother,thetwoabovedisplayedexpressionsforSandVarXshowthattheseareinversesononeanother.Thishasthefollowingveryimportantcomputationalconsequence:SupposewearegivenarandomvariableXwithwhoselawhasadensityproportionaltoe)]TJ/F10 6.974 Tf 6.226 0 Td[(Sv=2whereSisaquadraticformwhichisgivenasamatrix"S=SijintermsofabasisofV.ThenVarXisgivenbyS)]TJ/F7 6.974 Tf 6.227 0 Td[(1intermsofthedualbasisofV.7.3.4ThevarianceofBrownianmotion.Forexample,considerthetwodimensionalvectorspacewithcoordinatesx1;x2andprobabilitydensityproportionaltoexp)]TJ/F8 9.963 Tf 8.944 6.74 Td[(1 2x21 s+x2)]TJ/F11 9.963 Tf 9.962 0 Td[(x12 t)]TJ/F11 9.963 Tf 9.963 0 Td[(swhere0
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7.3.GAUSSIANMEASURES.201wecanwritetheabovevarianceasBs;sBs;tBt;sBt;t:Nowsupposethatwehavepickedsomenitesetoftimes0
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202CHAPTER7.WIENERMEASURE,BROWNIANMOTIONANDWHITENOISE.we,followingStroock,areusingSinsteadofDasourspaceoftestfunctionsthisnotionwasintroducedbyGelfandsomeftyyearsago.SeeGelfandandVilenkin,GeneralizedFunctionsvolumeIV.IfwehavegeneralizedrandomprocessZasabove,wecanconsideritsderiva-tiveinthesenseofgeneralizedfunctions,i.e._Z:=Z)]TJ/F8 9.963 Tf 10.164 2.629 Td[(_:7.4ThederivativeofBrownianmotioniswhitenoise.Toseehowthisderivativeworks,letusconsiderwhathappensforBrownianmotion.Let!beacontinuouspathofslowgrowth,andset!ht:=1 h!t+h)]TJ/F11 9.963 Tf 9.963 0 Td[(!t:Thepaths!arenotdierentiablewithprobabilityonesothislimitdoesnotexistasafunction.Butthelimitdoesexistasageneralizedfunction,assigningthevalueZ10)]TJ/F8 9.963 Tf 10.164 2.629 Td[(_t!tdtto.Nowifs
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7.4.THEDERIVATIVEOFBROWNIANMOTIONISWHITENOISE.203Weconcludethatthevarianceof_ZisgivenbyZ10t2dtwhichwecanwriteasZ10Z10s)]TJ/F11 9.963 Tf 9.963 0 Td[(tstdsdt:Noticethatnowthecovariancefunction"isthegeneralizedfunctions)]TJ/F11 9.963 Tf 10.099 0 Td[(t.ThegeneralizedprocessextendedtothewholelinewiththiscovarianceiscalledwhitenoisebecauseitisaGaussianprocesswhichisstationaryundertranslationsintimeanditscovariancefunction"iss)]TJ/F11 9.963 Tf 8.983 0 Td[(t,signifyingindepen-dentvariationatalltimes,andtheFouriertransformofthedeltafunctionisaconstant,i.e.assignsequalweighttoallfrequencies.

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204CHAPTER7.WIENERMEASURE,BROWNIANMOTIONANDWHITENOISE.

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Chapter8Haarmeasure.AtopologicalgroupisagroupGwhichisalsoatopologicalspacesuchthatthemapsGG!G;x;y7!xyandG!G;x7!x)]TJ/F7 6.974 Tf 6.227 0 Td[(1arecontinuous.IfthetopologyonGislocallycompactandHausdor,wesaythatGisalocallycompact,Hausdor,topologicalgroup.Ifa2Gisxed,thenthemap`a`a:G!G;`ax=axisthecompositeofthemultiplicationmapGG!GandthecontinuousmapG!GG;x7!a;x:So`aiscontinuous,onetoone,andwithinverse`a)]TJ/F6 4.981 Tf 5.396 0 Td[(1.IfisameasureG,thenwecanpushitforwardby`a,thatis,considerthepushedforwardmeasure`a.Wesaythatthemeasureisleftinvariantif`a=8a2G:Thebasictheoremonthesubject,provedbyHaarin1933isTheorem8.0.1IfGisalocallycompactHausdortopologicalgroupthereex-istsanon-zeroregularBorelmeasurewhichisleftinvariant.Anyothersuchmeasurediersfrombymultiplicationbyapositiveconstant.Thischapterisdevotedtotheproofofthistheoremandsomeexamplesandconsequences.205

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206CHAPTER8.HAARMEASURE.8.1Examples.8.1.1Rn.Rnisagroupunderaddition,andLebesguemeasureisclearlyleftinvariant.SimilarlyTn.8.1.2Discretegroups.IfGhasthediscretetopologythenthecountingmeasurewhichassignsthevalueonetoeveryoneelementsetfxgisHaarmeasure.8.1.3Liegroups.Wecanreformulatetheconditionofleftinvarianceasfollows:LetIdenotetheintegralassociatedtothemeasure:If=Zfd:ThenZfd`a=I`afwhere`afx=fax:.1Indeed,thisismosteasilycheckedonindicatorfunctionsofsets,where`a1A=1`)]TJ/F6 4.981 Tf 5.396 0 Td[(1aAandZ1Ad`a=`aA:=`)]TJ/F7 6.974 Tf 6.226 0 Td[(1aA=Z1`)]TJ/F6 4.981 Tf 5.396 0 Td[(1aAd:SotheleftinvarianceconditionisI`af=If8a2G:.2SupposethatGisadierentiablemanifoldandthatthemultiplicationmapGG!Gandtheinversemapx7!x)]TJ/F7 6.974 Tf 6.226 0 Td[(1aredierentiable.NowifGisn-dimensional,andwecouldndann-formwhichdoesnotvanishanywhere,andsuchthat`a=inthesenseofpull-backonformsthenwecanchooseanorientationrelativetowhichbecomesidentiedwithadensity,andthenIf=ZGf

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8.1.EXAMPLES.207isthedesiredintegral.Indeed,I`af=Z`af=Z`af`asince`a==Z`af=Zf=If:Weshallreplacetheproblemofndingaleftinvariantn-formbytheap-parentlyharderlookingproblemofndingnleftinvariantone-forms!1;:::;!nonGandthensetting:=!1^^!n:ThegeneraltheoryofLiegroupssaysthatsuchoneformstheMaurer-Cartanformsalwaysexist,butIwanttosowhowtocomputetheminimportantspecialcases.SupposethatwecanndahomomorphismM:G!GldwhereGldisthegroupofddinvertiblematriceseitherrealorcomplex.SoMisamatrixvaluedfunctiononGsatisfyingMe=idwhereeistheidentityelementofGandidistheidentitymatrix,andMxy=MxMywheremultiplicationontheleftisgroupmultiplicationandmultiplicationontherightismatrixmultiplication.WecanthinkofMasamatrixvaluedfunctionorasamatrixMx=Mijxofrealorcomplexvaluedfunctions.Supposethatallofthesefunctionsaredierentiable.ThenwecanformdM:=dMijwhichisamatrixoflineardierentialformsonG,or,equivalently,amatrixvaluedlineardierentialformonG.Finally,considerM)]TJ/F7 6.974 Tf 6.227 0 Td[(1dM:Again,thisisamatrixvaluedlineardierentialformonGorwhatisthesamethingamatrixoflineardierentialformsonG.ExplicitlyitisthematrixwhoseikentryisXjMx)]TJ/F7 6.974 Tf 6.227 0 Td[(1ijdMjk:

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208CHAPTER8.HAARMEASURE.Iclaimthateveryentryofthismatrixisaleftinvariantlineardierentialform.Indeed,`aMx=Max=MaMx:LetuswriteA=Ma:Sinceaisxed,Aisaconstantmatrix,andso`aM)]TJ/F7 6.974 Tf 6.227 0 Td[(1=AM)]TJ/F7 6.974 Tf 6.226 0 Td[(1=M)]TJ/F7 6.974 Tf 6.227 0 Td[(1A)]TJ/F7 6.974 Tf 6.227 0 Td[(1while`adM=dAM=AdMsinceAisaconstant.So`aM)]TJ/F7 6.974 Tf 6.226 0 Td[(1dM=M)]TJ/F7 6.974 Tf 6.227 0 Td[(1A)]TJ/F7 6.974 Tf 6.227 0 Td[(1AdM=M)]TJ/F7 6.974 Tf 6.226 0 Td[(1dM:Ofcourse,ifthesizeofMistoosmall,theremightnotbeenoughlinearlyindependententries.Inthecomplexcasewewanttobeabletochoosetherealandimaginarypartsoftheseentriestobelinearlyindependent.Butif,forexample,themapx7!Mxisanimmersion,thentherewillbeenoughlinearlyindependententriestogoaround.Forexample,considerthegroupofalltwobytworealmatricesoftheformab01;a6=0:Thisgroupissometimesknownastheax+bgroup"sinceab01x1=ax+b1:Inotherwords,Gisthegroupofalltranslationsandrescalingsandre-orientationsoftherealline.Wehaveab01)]TJ/F7 6.974 Tf 6.227 0 Td[(1=a)]TJ/F7 6.974 Tf 6.227 0 Td[(1)]TJ/F11 9.963 Tf 7.748 0 Td[(a)]TJ/F7 6.974 Tf 6.226 0 Td[(1b01anddab01=dadb00soab01)]TJ/F7 6.974 Tf 6.227 0 Td[(1dab01=a)]TJ/F7 6.974 Tf 6.227 0 Td[(1daa)]TJ/F7 6.974 Tf 6.227 0 Td[(1db00andtheHaarmeasureisproportionaltodadb a2.3

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8.1.EXAMPLES.209Asasecondexample,considerthegroupSUofallunitarytwobytwomatriceswithdeterminantone.Eachcolumnofaunitarymatrixisaunitvector,andthecolumnsareorthogonal.Wecanwritetherstcolumnofthematrixas whereandarecomplexnumberswithjj2+jj2=1:.4Thesecondcolumnmustthenbeproportionalto)]TJ/F11 9.963 Tf 7.749 0 Td[(andtheconditionthatthedeterminantbeonexesthisconstantofproportion-alitytobeone.SowecanwriteM= )]TJ/F11 9.963 Tf 7.748 0 Td[( where.4issatised.SowecanthinkofMasacomplexmatrixvaluedfunctiononthegroupSU.SinceMisunitary,M)]TJ/F7 6.974 Tf 6.227 0 Td[(1=MsoM)]TJ/F7 6.974 Tf 6.226 0 Td[(1=)]TJET1 0 0 1 12.972 -57.844 cmq[]0 d0 J0.398 w0 0.199 m6.161 0.199 lSQ1 0 0 1 0 -8.114 cmBT/F11 9.963 Tf 0 0 Td[( andM)]TJ/F7 6.974 Tf 6.226 0 Td[(1dM=)]TJET1 0 0 1 -123.216 -43.116 cmq[]0 d0 J0.398 w0 0.199 m6.161 0.199 lSQ1 0 0 1 0 -8.114 cmBT/F11 9.963 Tf 0 0 Td[( d )]TJ/F11 9.963 Tf 7.748 0 Td[(dd d=d +d )]TJ/F11 9.963 Tf 7.748 0 Td[(d+d)]TJET1 0 0 1 -37.654 -4.383 cmq[]0 d0 J0.398 w0 0.199 m6.161 0.199 lSQ1 0 0 1 0 -8.114 cmBT/F11 9.963 Tf 0 0 Td[(d + d d+ d:Eachoftherealandimaginarypartsoftheentriesisaleftinvariantoneform.Butletusmultiplythreeoftheseentriesdirectly: d+ d^)]TJ/F11 9.963 Tf 7.749 0 Td[(d+d^)]TJET1 0 0 1 118.33 8.114 cmq[]0 d0 J0.398 w0 0.199 m6.161 0.199 lSQ1 0 0 1 0 -8.114 cmBT/F11 9.963 Tf 0 0 Td[(d + d =)]TJ/F8 9.963 Tf 7.748 0 Td[(jj2+jj2d^d^)]TJET1 0 0 1 -53.985 -12.639 cmq[]0 d0 J0.398 w0 0.199 m6.161 0.199 lSQ1 0 0 1 0 -8.114 cmBT/F11 9.963 Tf 0 0 Td[(d + d )]TJ/F11 9.963 Tf 7.749 0 Td[(d^d^)]TJET1 0 0 1 -74.422 -9.12 cmq[]0 d0 J0.398 w0 0.199 m6.161 0.199 lSQ1 0 0 1 0 -8.114 cmBT/F11 9.963 Tf 0 0 Td[(d + d :Wecansimplifythisexpressionbydierentiatingtheequation + =1togetd + d+d + d=0:Sofor6=0wecansolveford :d =)]TJ/F8 9.963 Tf 9.534 6.739 Td[(1 d + d+ d:

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210CHAPTER8.HAARMEASURE.Whenwemultiplybyd^dthetermsinvolvingdandddisappear.Wethusget)]TJ/F11 9.963 Tf 7.749 0 Td[(d^d^)]TJET1 0 0 1 115.491 -55.268 cmq[]0 d0 J0.398 w0 0.199 m6.161 0.199 lSQ1 0 0 1 0 -8.114 cmBT/F11 9.963 Tf 0 0 Td[(d + d =d^d^ d + d :Ifwewrite d = dandusejj2+jj2=1theaboveexpressionsimpliesfurtherto1 d^d^d .5asaleftinvariantthreeformonSU.Youmightthinkthatthisthreeformiscomplexvalued,butweshallnowgiveanalternativeexpressionforitwhichwillshowthatitisinfactrealvalued.Forthisintroducepolarcoordinatesinfourdimensionsasfollows:Write=w+iz=x+iysox2+y2+z2+w2=1;w=cosz=sincosx=sinsincosy=sinsinsin0;0;02:Thend^d =dw+idz^dw)]TJ/F11 9.963 Tf 9.962 0 Td[(idz=)]TJ/F8 9.963 Tf 7.749 0 Td[(2idw^dz=)]TJ/F8 9.963 Tf 7.749 0 Td[(2idcos^dsincos=)]TJ/F8 9.963 Tf 7.749 0 Td[(2isin2d^d:Now=sinsineisod=id+wherethemissingtermsinvolvedanddandsowilldisappearwhenmultipliedbyd^d .Henced^d^d =)]TJ/F8 9.963 Tf 7.749 0 Td[(2sin2sind^d^d:Finally,weseethatthethreeform.5whenexpressedinpolarcoordinatesis)]TJ/F8 9.963 Tf 7.748 0 Td[(2sin2sind^d^d:Ofcoursewecanmultiplythisbyanyconstant.IfwenormalizesothatG=1theHaarmeasureis1 22sin2sinddd:

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8.2.TOPOLOGICALFACTS.2118.2Topologicalfacts.Since`aisahomeomorphism,ifVisaneighborhoodoftheidentityelemente,thenaVisaneighborhoodofa,andifUisaneighborhoodofUthena)]TJ/F7 6.974 Tf 6.226 0 Td[(1Uisaneighborhoodofe.HereweareusingtheobviousnotationaV=`aVetc.SupposethatUisaneighborhoodofe.ThensoisU)]TJ/F7 6.974 Tf 6.227 0 Td[(1:=fx)]TJ/F7 6.974 Tf 6.227 0 Td[(1:x2UgandhencesoisW=UU)]TJ/F7 6.974 Tf 6.227 0 Td[(1:ButW)]TJ/F7 6.974 Tf 6.226 0 Td[(1=W:Proposition8.2.1Everyneighborhoodofecontainsasymmetricneighbor-hood,i.e.onethatsatisesW)]TJ/F7 6.974 Tf 6.226 0 Td[(1=W.LetUbeaneighborhoodofe.TheinverseimageofUunderthemultiplicationmapGG!Gisaneighborhoodofe;einGGandhencecontainsanopensetoftheformVV.HenceProposition8.2.2EveryneighborhoodUofecontainsaneighborhoodVofesuchthatV2=VVU.HereweareusingthenotationAB=fxy:x2A;y2BgwhereAandBaresubsetsofG.IfAandBarecompact,soisABasasubsetofGG,andsincetheimageofacompactsetunderacontinuousmapiscompact,wehaveProposition8.2.3IfAandBarecompact,soisAB.Proposition8.2.4IfAGthen A,theclosureofA,isgivenby A=VAVwhereVrangesoverallneighborhoodsofe.Proof.Ifa2 AandVisaneighborhoodofe,thenaV)]TJ/F7 6.974 Tf 6.227 0 Td[(1isanopensetcontaininga,andhencecontainingapointofA.Soa2AV,andthelefthandsideoftheequationinthepropositioniscontainedintherighthandside.Toshowthereverseinclusion,supposethatxbelongstotherighthandside.ThenxV)]TJ/F7 6.974 Tf 6.226 0 Td[(1intersectsAforeveryV.ButthesetsxV)]TJ/F7 6.974 Tf 6.227 0 Td[(1rangeoverallneighborhoodsofx.Sox2 A.QEDRecallthatfollowingLoomisasweareLdenotesthespaceofcontinuousfunctionsofcompactsupportonG.

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212CHAPTER8.HAARMEASURE.Proposition8.2.5SupposethatGislocallycompact.Iff2Lthenfisuni-formlyleftandrightcontinuous.Thatis,given>0thereisaneighborhoodVofesuchthats2Vjfsx)]TJ/F11 9.963 Tf 9.963 0 Td[(fxj<:Equivalently,thissaysthatxy)]TJ/F7 6.974 Tf 6.226 0 Td[(12Vjfx)]TJ/F11 9.963 Tf 9.963 0 Td[(fyj<:Proof.LetC:=SuppfandletUbeasymmetriccompactneighborhoodofe.ConsiderthesetZofpointsssuchthatjfsx)]TJ/F11 9.963 Tf 9.963 0 Td[(fxj<8x2UC:IclaimthatthiscontainsanopenneighborhoodWofe.Indeed,foreachxedy2UCthesetofssatisfyingthisconditionatyisanopenneighborhoodWyofe,andthisWyworksinsomeneighborhoodOyofy.SinceUCiscom-pact,nitelymanyoftheseOycoverUC,andhencetheintersectionofthecorrespondingWyformanopenneighborhoodWofe.NowtakeV:=UW:Ifs2Vandx2UCthenjfsx)]TJ/F11 9.963 Tf 10.833 0 Td[(fxj<.Ifx62UC,thensx62CsincewechoseUtobesymmetricandx62C,sofsx=0andfx=0,sojfsx)]TJ/F11 9.963 Tf 9.963 0 Td[(fxj=0<:QEDIntheconstructionoftheHaarintegral,wewillneedthisproposition.SoitisexactlyatthispointwheretheassumptionthatGislocallycompactcomesin.8.3ConstructionoftheHaarintegral.Letfandgbenon-zeroelementsofL+andletmfandmgbetheirrespectivemaxima.Ateachx2Suppf;wehavefxmf mgmgsoifc>mf mgandsischosensothatgachievesitsmaximumatsx,thenfycgsx

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8.3.CONSTRUCTIONOFTHEHAARINTEGRAL.213inaneighborhoodofx.SinceSuppfiscompact,wecancoveritbynitelymanysuchneighborhoods,sothatthereexistnitelymanyciandsisuchthatfxXcigsix8x:.6Ifwechoosexsothatfx=mf,thentherighthandsideisatmostPicimgandthusweseethatXicimf=mg>0:Soletusdenethesizeoffrelativetog"byf;g:=g.l.b.fXci:9sisuchthat.6holdsg:.7Wehaveveriedthatf;gmf mg:.8Itisclearthat`af;g=f;g8a2G.9f1+f2;gf1;g+f2;g.10cf;g=cf;g8c>0.11f1f2f1;gf2;g:.12IffxPcigsixforallxandgyPdjhtjyforallythenfxXijcidjhtjsix8x:Takinggreatestlowerboundsgivesf;hf;gg;h:.13Tonormalizeourintegral,xsomef02L+;f06=0:DeneIgf:=f;g f0;g:Since,accordingto.13wehavef0;gf0;ff;g;weseethat1 f0;fIgf:

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214CHAPTER8.HAARMEASURE.Since.13saysthatf;gf;f0f0;gweseethatIgff;f0:Soforeachnon-zerof2L+letSfdenotetheclosedintervalSf:=1 f0;f;f;f0;andletS:=Yf2L+;f6=0Sf:ThisspaceiscompactbyTychono.Eachnon-zerog2L+determinesapointIg2SwhosecoordinateinSfisIgf.ForanyneighborhoodVofe,letCVdenotetheclosureinSofthesetIg;g2V.WehaveCV1CVn=CV1Vn6=;:TheCVareallcompact,andsothereisapointIintheintersectionofalltheCV:I2C:=VCV:TheideaisthatIsomehowisthelimitoftheIgaswerestrictthesupportofgtolieinsmallerandsmallerneighborhoodsoftheidentity.Weshallprovethataswemaketheseneighborhoodssmallerandsmaller,theIgarecloserandclosertobeingadditive,andsotheirlimitIsatisestheconditionsforbeinganinvariantintegral.Herearethedetails:Lemma8.3.1Givenf1andf2inL+and>0thereexistsaneighborhoodVofesuchthatIgf1+Igf2Igf1+f2+forallgwithSuppgV.Proof.Choose2Lsuchthat=1onSuppf1+f2.Foragiven>0tobechosenlater,letf:=f1+f2+;h1:=f1 f;h2:=f2 f:Hereh1andh2weredenedonSuppfandvanishoutsideSuppf1+f2,soextendthemtobezerooutsideSupp.Foran>0and>0tobechosenlater,ndaneighborhoodV=V;sothatjh1x)]TJ/F11 9.963 Tf 9.963 0 Td[(h1yj
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8.3.CONSTRUCTIONOFTHEHAARINTEGRAL.215Letgbeanon-zeroelementofL+withSuppgV.IffxXcjgsjxthengsjx6=0impliesthatjhix)]TJ/F11 9.963 Tf 9.962 0 Td[(his)]TJ/F7 6.974 Tf 6.227 0 Td[(1jj<;i=1;2sofix=fxhixXcjgsjxhixXcjgsjx[his)]TJ/F7 6.974 Tf 6.227 0 Td[(1j+];i=1;2:Thisimpliesthatfi;gXjcj[his)]TJ/F7 6.974 Tf 6.226 0 Td[(1j+]andsince0hi1bydenition,f1;g+f2;gXcj[1+2]:WecanchoosethecjandsjsothatPcjisascloseasweliketof;g.Hencef1;g+f2;gf;g[1+2]:Dividingbyf0;ggivesIgf1+Igf2Igf[1+2][Igf1+f2+Ig][1+2];where,ingoingfromthesecondtothethirdinequalitywehaveusedthede-nitionoff,.10appliedtof1+f2andand.11.NowIgf1+f2f1+f2;f0andIg;f0.Sochooseandsothat2f1+f2;f0++2;f0<:Thiscompletestheproofofthelemma.Foranynitenumberoffi2L+andanyneighborhoodVoftheidentity,thereisanon-zerogwithSuppg2VandjIfi)]TJ/F11 9.963 Tf 9.963 0 Td[(Igfij<;i=1;:::;n:Applyingthistof1;f2andf3=f1+f2andtheVsuppliedbythelemma,wegetIf1+f2)]TJ/F11 9.963 Tf 9.963 0 Td[(Igf1+f2Igf1+Igf2If1+If2+2

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216CHAPTER8.HAARMEASURE.andIf1+If2Igf1+Igf2+2Igf1+f2+3If1+f2+4:Inshort,IsatisesIf1+f2=If1+If2forallf1;f2inL+,isleftinvariant,andIcf=cIfforc0.Asusual,extendItoallofLbyIf1)]TJ/F11 9.963 Tf 9.963 0 Td[(f2=If1)]TJ/F11 9.963 Tf 9.963 0 Td[(If2andthisiswelldened.Sinceforf2L+wehaveIff;f0mf=mf0=kfk1=mf0weseethatIisboundedinthesupnorm.SoitisanintegralbyDini'slemma.Hence,bytheRieszrepresentationtheorem,ifGisHausdor,wegetaregularleftinvariantBorelmeasure.Thiscompletestheexistencepartofthemaintheorem.Fromthefactthatisregular,andnotthezeromeasure,weconcludethatthereissomecompactsetKwithK>0.LetUbeanynon-emptyopenset.ThetranslatesxU;x2KcoverK,andsinceKiscompact,anitenumber,saynofthem,coverK.Buttheyallhavethesamemeasure,Usinceisleftinvariant.ThusKnUimplyingU>0foranynon-emptyopensetU.14ifisaleftinvariantregularBorelmeasure.Iff2L+andf6=0,thenf>>0onsomenon-emptyopensetU,andhenceitsintegralis>U.Sof2L+;f6=0Zfd>0.15foranyleftinvariantregularBorelmeasure.8.4Uniqueness.LetandbetwoleftinvariantregularBorelmeasuresonG.Picksomeg2L+;g6=0sothatbothRgdandRgdarepositive.WearegoingtouseFubinitoprovethatforanyf2LwehaveRfd Rgd=Rfd Rgd:.16

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8.4.UNIQUENESS.217Thisclearlyimpliesthat=cwherec=Rgd Rgd:Toprove.16,itisenoughtoshowthattherighthandsidecanbeexpressedintermsofanyleftinvariantregularBorelmeasuresaybecausethisimpliesthatbothsidesdonotdependonthechoiceofHaarmeasure.Denehx;y:=fxgyx Rgtxdt:Theintegralinthedenominatorispositiveforallxandbytheleftuniformcontinuitytheintegralisacontinuousfunctionofx.Thushiscontinuousfunctionofcompactsupportinx;ysobyFubini,ZZhx;ydydx=ZZhx;ydxdy:Intheinnerintegralontherightreplacexbyy)]TJ/F7 6.974 Tf 6.227 0 Td[(1xusingtheleftinvarianceof.TherighthandsidebecomesZhy)]TJ/F7 6.974 Tf 6.227 0 Td[(1x;ydxdy:UseFubiniagainsothatthisbecomesZhy)]TJ/F7 6.974 Tf 6.227 0 Td[(1x;ydydx:Nowusetheleftinvarianceoftoreplaceybyxy.ThislastiteratedintegralbecomesZhy)]TJ/F7 6.974 Tf 6.227 0 Td[(1;xydydx:SowehaveZZhx;ydydx=Zhy)]TJ/F7 6.974 Tf 6.226 0 Td[(1;xydydx:FromthedenitionofhthelefthandsideisRfxdx.Fortherighthandsidehy)]TJ/F7 6.974 Tf 6.227 0 Td[(1;xy=fy)]TJ/F7 6.974 Tf 6.226 0 Td[(1gx Rgty)]TJ/F7 6.974 Tf 6.227 0 Td[(1dt:Integratingthisrstwithrespecttodygiveskgxwherekistheconstantk=Zfy)]TJ/F7 6.974 Tf 6.226 0 Td[(1 Rgty)]TJ/F7 6.974 Tf 6.227 0 Td[(1dtdy:

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218CHAPTER8.HAARMEASURE.Nowintegratewithrespectto.WegetRfd=kRgdsoRfd Rgd;therighthandsideof8.16,doesnotdependon,sinceitequalskwhichisexpressedintermsof.QED8.5G<1ifandonlyifGiscompact.Sinceisregular,themeasureofanycompactsetisnite,soifGiscompactthenG<1.Wewanttoprovetheconverse.LetUbeanopenneighborhoodofewithcompactclosure,K.SoK>0.ThefactthatG<1impliesthatonecannothavemdisjointsetsoftheformxiKifm>G K:LetnbesuchthatwecanndndisjointsetsoftheformxiKbutnon+1disjointsetsofthisform.Thissaysthatforanyx2G;xKcannotbedisjointfromallthexiK.ThusG=[ixiK!K)]TJ/F7 6.974 Tf 6.227 0 Td[(1whichiscompact.QEDIfGiscompact,theHaarmeasureisusuallynormalizedsothatG=1.8.6Thegroupalgebra.Iff;g2Ldenetheirconvolutionbyf?gx:=Zfxygy)]TJ/F7 6.974 Tf 6.227 0 Td[(1dy;.17wherewehavexed,onceandforall,aleftHaarmeasure.Theleftinvarianceunderleftmultiplicationbyx)]TJ/F7 6.974 Tf 6.226 0 Td[(1impliesthatf?gx=Zfygy)]TJ/F7 6.974 Tf 6.226 0 Td[(1xdy:.18InwhatfollowswewillwritedyinsteadofdysincewehavechosenaxedHaarmeasure.IfA:=SuppfandB:=Suppgthenfygy)]TJ/F7 6.974 Tf 6.227 0 Td[(1xiscontinuousasafunctionofyforeachxedxandvanishesunlessy2Aandy)]TJ/F7 6.974 Tf 6.227 0 Td[(1x2B.Thusf?gvanishesunlessx2AB.Alsojf?gx1)]TJ/F11 9.963 Tf 9.963 0 Td[(f?gx2jk`x1f)]TJ/F11 9.963 Tf 9.963 0 Td[(`x2k1Zjgy)]TJ/F7 6.974 Tf 6.226 0 Td[(1jdy:

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8.6.THEGROUPALGEBRA.219Sincex7!`xfiscontinuousintheuniformnorm,weconcludethatf;g2Lf?g2LandSuppf?gSuppfSuppg:.19Iclaimthatwehavetheassociativelaw:If,f;g;h2Lthentheclaimisthatf?g?h=f?g?h.20Indeed,usingtheleftinvarianceoftheHaarmeasureandFubiniwehavef?g?hx:=Zf?gxyhy)]TJ/F7 6.974 Tf 6.227 0 Td[(1dy=ZZfxyzgz)]TJ/F7 6.974 Tf 6.227 0 Td[(1hy)]TJ/F7 6.974 Tf 6.227 0 Td[(1dzdy=ZZfxzgz)]TJ/F7 6.974 Tf 6.227 0 Td[(1yhy)]TJ/F7 6.974 Tf 6.227 0 Td[(1dzdy=ZZfxzgz)]TJ/F7 6.974 Tf 6.227 0 Td[(1yhy)]TJ/F7 6.974 Tf 6.227 0 Td[(1dydz=Zfxzg?hz)]TJ/F7 6.974 Tf 6.227 0 Td[(1dz=f?g?hx:Itiseasytocheckthat?iscommutativeifandonlyifGiscommutative.Inowwanttoextendthedenitionof?toallofL1,andhereIwillfollowLoomisandrestrictourdenitionofL1sothatourintegrablefunctionsbelongtoB,thesmallestmonotoneclasscontainingL.WhenweweredoingtheWienerintegral,weneededallBorelsets.Hereitismoreconvenienttooperatewiththissmallerclass,fortechnicalreasonswhichwillbepracticallyinvisible.FormostgroupsoneencountersinreallifethereisnodierencebetweentheBorelsetsandtheBairesets.Forexample,iftheHaarmeasureis-niteonecanforgetabouttheseconsiderations.IffandgarefunctionsonGdenethefunctionfgonGGbyfgx;y:=fygy)]TJ/F7 6.974 Tf 6.227 0 Td[(1x:Theorem8.6.1[31AinLoomis]Iff;g2B+thenfg2B+GGandkf?gkpkfk1kgkp.21foranypwith1p1.Proof.Iff2L+thenthesetofg2B+suchthatfg2B+GGisL-monotoneandincludesL+,soincludesB+.SoifgisanL-boundedfunctioninB+,thesetoff2B+suchthatfg2B+GGincludesL+andisL-monotone,andsoincludesB+.Sofg2B+GGwheneverfandgare

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220CHAPTER8.HAARMEASURE.L-boundedelementsofB+.ButthemostgeneralelementofB+canbewrittenasthelimitofanincreasingsequenceofLboundedelementsofB+,andsotherstassertioninthetheoremfollows.Asfandgarenon-negative,Fubiniassertsthatthefunctiony7!fygy)]TJ/F7 6.974 Tf 6.226 0 Td[(1xisintegrableforeachxedx,thatf?gisintegrableasafunctionofxandthatkf?gk1=ZZfygy)]TJ/F7 6.974 Tf 6.227 0 Td[(1xdxdy=kfk1kgk1:Thisproves.21withequalityforp=1.Forp=1.21isobviousfromthedenitions.For1
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8.7.THEINVOLUTION.221Indeed,bothsidessendx2Gtoaxb)]TJ/F7 6.974 Tf 6.227 0 Td[(1:IfisachoiceofleftHaarmeasure,thenitfollowsfrom.22thatrbisanotherchoiceofHaarmeasure,andsomustbesomepositivemultipleof.ThefunctiononGdenedbyrb=biscalledthemodularfunctionofG.Itisimmediatethatthisdenitiondoesnotdependonthechoiceof.Example.Inthecasethatwearedealingwithmanifoldsandintegrationofn-forms,If=Zf;thenthepush-forwardofthemeasureassociatedtoIunderadieomorphismassignstoanyfunctionftheintegralIf=Zf=Zf)]TJ/F7 6.974 Tf 6.226 0 Td[(1=Zf)]TJ/F7 6.974 Tf 6.226 0 Td[(1:Sothepushforwardmeasurecorrespondstotheform)]TJ/F7 6.974 Tf 6.226 0 Td[(1:Thusincomputingzusingdierentialforms,wehavetocomputethepull-backunderrightmultiplicationbyz,notz)]TJ/F7 6.974 Tf 6.226 0 Td[(1.Forexample,intheax+bgroup,wehaveab01xy01=axay+b01soda^db a27!xda^db x2a2andhencethemodularfunctionisgivenbyxy01=1 x:Inallcasesthemodularfunctioniscontinuousasfollowsfromtheuniformrightcontinuity,Proposition8.2.5,andfromitsdenition,itfollowsthatst=st:Inotherwords,isacontinuoushomomorphismfromGtothemultiplicativegroupofpositiverealnumbers.ThegroupGiscalledunimodularif1.Forexample,acommutativegroupisobviouslyunimodular.Also,acompactgroupisunimodular,becauseGhasnitemeasure,andiscarriedintoitselfbyrightmultiplicationsosG=rsG=r)]TJ/F7 6.974 Tf 6.227 0 Td[(1sG=G:

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222CHAPTER8.HAARMEASURE.8.7.2Denitionoftheinvolution.Foranycomplexvaluedcontinuousfunctionfofcompactsupportdene~fby~fx:= fx)]TJ/F7 6.974 Tf 6.227 0 Td[(1x)]TJ/F7 6.974 Tf 6.227 0 Td[(1:.23Itfollowsimmediatelyfromthedenitionthat~f~=f:Thatis,applying~twiceistheidentitytransformation.Also,rs~fx= fsx)]TJ/F7 6.974 Tf 6.227 0 Td[(1x)]TJ/F7 6.974 Tf 6.227 0 Td[(1ssors~f=s`sf~:.24Similarly,rsf~x= fx)]TJ/F7 6.974 Tf 6.227 0 Td[(1s)]TJ/F7 6.974 Tf 6.227 0 Td[(1x)]TJ/F7 6.974 Tf 6.227 0 Td[(1=s`s~forrsf~=s`s~f:.25Supposethatfisrealvalued,andconsiderthefunctionalJf:=I~f:Thenfrom.24andthedenitionofwehaveJ`sf=s)]TJ/F7 6.974 Tf 6.227 0 Td[(1Irs~f=s)]TJ/F7 6.974 Tf 6.226 0 Td[(1sI~f=I~f=Jf:Inotherwords,Jisaleftinvariantintegralonrealvaluedfunctions,andhencemustbesomeconstantmultipleofI,J=cI:LetVbeasymmetricneighborhoodofechosensosmallthatj1)]TJ/F8 9.963 Tf 10.353 0 Td[(sj0,sincee=1andiscontinuous.Ifwetakef=1Vthenfx=fx)]TJ/F7 6.974 Tf 6.227 0 Td[(1andjJf)]TJ/F11 9.963 Tf 9.962 0 Td[(IfjIf:DividingbyIfshowsthatjc)]TJ/F8 9.963 Tf 9.274 0 Td[(1j<,andsinceisarbitrary,wehaveprovedthatI~f= If:.26Wecanderivetwoimmediateconsequences:Proposition8.7.1HaarmeasureisinverseinvariantifandonlyifGisuni-modular,andProposition8.7.2Theinvolutionf7!~fextendstoananti-linearisometryofL1C.

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8.8.THEALGEBRAOFFINITEMEASURES.2238.7.3Relationtoconvolution.Weclaimthatf?g~=~g?~f.27Proof.f?g~x=Z fx)]TJ/F7 6.974 Tf 6.227 0 Td[(1ygy)]TJ/F7 6.974 Tf 6.227 0 Td[(1dyx)]TJ/F7 6.974 Tf 6.227 0 Td[(1=Z gy)]TJ/F7 6.974 Tf 6.226 0 Td[(1y)]TJ/F7 6.974 Tf 6.226 0 Td[(1 fy)]TJ/F7 6.974 Tf 6.226 0 Td[(1x)]TJ/F7 6.974 Tf 6.227 0 Td[(1y)]TJ/F7 6.974 Tf 6.227 0 Td[(1x)]TJ/F7 6.974 Tf 6.226 0 Td[(1dy=~g?~fx:QED8.7.4Banachalgebraswithinvolutions.ForageneralBanachalgebraBoverthecomplexnumbersamapx7!xyiscalledaninvolutionifitisantilinearandanti-multiplicative,i.e.satisesxyy=yyxyanditssquareistheidentity.Thusthemapf7!~fisaninvolutiononL1G.8.8Thealgebraofnitemeasures.Ingeneral,thealgebraL1Gwillnothaveanidentityelement,sincetheonlycandidatefortheidentityelementwouldbethe-function"h;fi=fe;andthiswillnotbeanhonestfunctionunlessthetopologyofGisdiscrete.Soweneedtointroduceadierentalgebraifwewanttohaveanalgebrawithidentity.IfGwereaLiegroupwecouldconsiderthealgebraofalldistributions.ForagenerallocallycompactHausdorgroupwecanproceedasfollows:LetMGdenotethespaceofallnitecomplexmeasuresonG:Anon-negativemeasureiscalledniteifG<1.Arealvaluedmeasureiscalledniteifitspositiveandnegativepartsarenite,andacomplexvaluedmeasureiscalledniteifitsrealandimaginarypartsarenite.GiventwonitemeasuresandonGwecanformtheproductmeasureonGGandthenpushthismeasureforwardunderthemultiplicationmapm:GG!G;andsodenetheirconvolutionby?:=m:

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224CHAPTER8.HAARMEASURE.OnechecksthattheconvolutionoftworegularBorelmeasuresisagainaregularBorelmeasure,andthatonmeasureswhichareabsolutelycontinuouswithre-specttoHaarmeasure,thiscoincideswiththeconvolutionaspreviouslydened.OnecanalsomakethealgebraofregularniteBorelmeasuresunderconvolu-tionintoaBanachalgebraunderthetotalvariationnorm".Thisalgebradoesincludethe-functionwhichisameasure!andsohasanidentity.Iwillnotgointothismatterhereexcepttomakeanumberofvaguebutimportantpoints.8.8.1Algebrasandcoalgebras.AnalgebraAisavectorspaceoverthecomplexnumberstogetherwithamapm:AA!Awhichissubjecttovariousconditionsperhapstheassociativelaw,perhapsthecommutativelaw,perhapstheexistenceoftheidentity,etc..Thedual"objectwouldbeaco-algebra,consistingofavectorspaceCandamapc:C!CCsubjectstoaseriesofconditionsdualtothoselistedabove.IfAisnitedi-mensional,thenwehaveanidenticationofAAwithA?A?,andsothedualspaceofanitedimensionalalgebraisacoalgebraandviceversa.Forinnitedimensionalalgebrasorcoalgebraswehavetopasstocertaintopologicalcompletions.Forexample,considerthespaceCbGdenotethespaceofcontinuousboundedfunctionsonGendowedwiththeuniformnormkfk1=l.u.b.x2Gfjfxjg:Wehaveaboundedlinearmapc:CbG!CbGGgivenbycfx;y:=fxy:InthecasethatGisnite,andendowedwiththediscretetopology,thespaceCbGisjustthespaceofallfunctionsonG,andCbGG=CbGCbGwhereCbGCbGcanbeidentiedwiththespaceofallfunctionsonGGoftheformx;y7!Xifixgiywherethesumisnite.Inthegeneralcase,noteveryboundedcontinuousfunctiononGGcanbewrittenintheaboveform,but,byStone-Weierstrass,thespaceofsuchfunctionsisdenseinCbGG.SowecansaythatCbGisalmost"aco-algebra,oraco-algebrainthetopologicalsense",inthatthe

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8.9.INVARIANTANDRELATIVELYINVARIANTMEASURESONHOMOGENEOUSSPACES.225mapcdoesnotcarryCintoCCbutratherintothecompletionofCC.IfAdenotesthedualspaceofC,thenAbecomesanhonestalgebra.Tomakeallthisworkinthecaseathand,weneedyetanotherversionoftheRieszrepresentationtheorem.Iwillstateandprovetheappropriatetheorem,butnotgointothefurtherdetails:LetXbeatopologicalspace,letCb:=CbX;RbethespaceofboundedcontinuousrealvaluedfunctionsonX.Foranyf2CbandanysubsetAXletkfk1;A:=l.u.b.x2Afjfxjg:Sokfk1=kfk1;X:Acontinuouslinearfunction`iscalledtightifforevery>0thereisacompactsetKandapositivenumberAsuchthatj`fjAkfk1;K+kfk1:Theorem8.8.1[YetanotherRieszrepresentationtheorem.]If`2Cbisatightnon-negativelinearfunctional,thenthereisanitenon-negativemea-sureonX;BXsuchthath`;fi=ZXfdforallf2Cb.Proof.Weneedtoshowthatfn&0h`;fni&0.Given>0,choose:= 1+2kf1k1:Sokf1k11 2:Thissameinequalitythenholdswithf1replacedbyfnsincethefnaremono-tonedecreasing.WehavetheKasinthedenitionoftightness,andbyDini'slemma,wecanchooseNsothatkfnk1;K 2A8n>N:Thenjh`;fnijforalln>N.QED8.9Invariantandrelativelyinvariantmeasuresonhomogeneousspaces.LetGbealocallycompactHausdortopologicalgroup,andletHbeaclosedsubgroup.ThenGactsonthequotientspaceG=Hbyleftmultiplication,the

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226CHAPTER8.HAARMEASURE.elementasendingthecosetxHintoaxH.Byabuseoflanguage,wewillcontinuetodenotethisactionby`a.So`axH:=axH:Wecanconsiderthecorrespondingactiononmeasures7!`a:Themeasureissaidtobeinvariantif`a=8a2G:ThemeasureonG=HissaidtoberelativelyinvariantwithmodulusDifDisafunctiononGsuchthat`a=Da8a2G:FromitsdenitionitfollowsthatDab=DaDb;anditisnothardtoseefromtheensuingdiscussionthatDiscontinuous.Wewillonlydealwithpositivemeasureshere,soDiscontinuoushomomorphismofGintothemultiplicativegroupofrealnumbers.Wecallsuchanobjectapositivecharacter.ThequestionswewanttoaddressinthissectionarewhatarethepossibleinvariantmeasuresorrelativelyinvariantmeasuresonG=H,andwhataretheirmodularfunctions.Forexample,considertheax+bgroupactingontherealline.SoGistheax+bgroup,andHisthesubgroupconsistingofthoseelementswithb=0,thepurerescalings".SoHisthesubgroupxingtheoriginintherealline,andwecanidentifyG=Hwiththerealline.LetNGbethesubgroupconsistingofpuretranslations,soNconsistsofthoseelementsofGwitha=1.ThegroupNactsastranslationsoftheline,anduptoscalarmultipletheonlymeasureonthereallineinvariantunderalltranslationsisLebesguemeasure,dx.Buth=a001actsonthereallinebysendingx7!axandhence`hdx=a)]TJ/F7 6.974 Tf 6.227 0 Td[(1dx:Thepushforwardofthemeasureunderthemapassignsthemeasure)]TJ/F7 6.974 Tf 6.226 0 Td[(1AtothesetA.SothereisnomeasureonthereallineinvariantunderG.Ontheotherhand,theaboveformulashowsthatdxisrelativelyinvariantwithmodularfunctionDab01=a)]TJ/F7 6.974 Tf 6.227 0 Td[(1:

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8.9.INVARIANTANDRELATIVELYINVARIANTMEASURESONHOMOGENEOUSSPACES.227Noticethatthisisthesameasthemodularfunction,ofthegroupG,andthatthemodularfunctionforthesubgroupHisthetrivialfunction1sinceHiscommutative.Wenowturntothegeneralstudy,andwillfollowLoomisindealingwiththeintegralsratherthanthemeasures,andsodenotetheHaarintegralofGbyI,withitsmodularfunction,denotetheHaarintegralofHbyJanditsmodularfunctionby.WewillletKdenoteanintegralonG=H,andDapositivecharacteronG.Iff2C0G,thenwewillletJtfxtdenotethatfunctionofxobtainedbyintegratingthefunctiont7!fxtonHwithrespecttoJ.BytheleftinvarianceofJ,weseethatifs2HthenJtxst=Jtxt:Inotherwords,thefunctionJtfxtisconstantoncosetsofHandhencedenesafunctiononG=Hwhichiseasilyseentobecontinuousandofcompactsupport.ThusJdenesamapJ:C0G!C0G=H:Wewillprovebelowthatthismapissurjective.ThemainresultweareaimingforinthissectionduetoA.WeilisTheorem8.9.1InorderthatapositivecharacterDbethemodularfunctionofarelativelyinvariantintegralKonG=HitisnecessaryandsucientthatDs=s s8s2H:.28Ifthishappens,thenKisuniquelydetermineduptoscalarmultiple,infact,KJfD=cIf8f2C0G.29wherecissomepositiveconstant.Webeginwithsomepreliminaries.Let:G!G=Hdenotetheprojectionmapwhichsendseachelementx2Gintoitsrightcosetx=xH:ThetopologyonG=HisdenedbydeclaringasetUG=Htobeopenifandonlyif)]TJ/F7 6.974 Tf 6.226 0 Td[(1Uisopen.Themapisthennotonlycontinuousbydenitionbutalsoopen,i.e.sendsopensetsintoopensets.Indeed,ifOGisanopensubset,then)]TJ/F7 6.974 Tf 6.227 0 Td[(1O=[h2HOhwhichisaunionofopensets,henceopen,henceOisopen.

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228CHAPTER8.HAARMEASURE.Lemma8.9.1IfBisacompactsubsetofG=HthenthereexistsacompactsetABsuchthatA=B;Proof.SinceweareassumingthatGislocallycompact,wecanndanopenneighborhoodOofeinGwhoseclosureCiscompact.ThesetsxO;x2GareallopensubsetsofG=Hsinceisopen,andtheirimagescoverallofG=H.Inparticular,sinceBiscompact,nitelymanyofthemcoverB,soB[ixiO[ixiC=[ixiC!theunionsbeingnite.ThesetK=[ixiCiscompact,beingtheniteunionofcompactsets.Theset)]TJ/F7 6.974 Tf 6.227 0 Td[(1Bisclosedsinceitscomplementistheinverseimageofanopenset,henceopen.SoA:=K)]TJ/F7 6.974 Tf 6.226 0 Td[(1Biscompact,anditsimageisB.QEDProposition8.9.1Jissurjective.LetF2C0G=HandletB=SuppF.ChooseacompactsetAGwithA=Basinthelemma.Choose2C0Gwith0and>0onA.Ifx2AH=)]TJ/F7 6.974 Tf 6.226 0 Td[(1Bthenxh>0forsomeh2H,andsoJ>0onB.Sowemayextendthefunctionz7!Fz Jztoacontinuousfunction,callit,bydeningittobezerooutsideB=SuppF.Thefunctiong=,i.e.gx=xishenceacontinuousfunctiononG,andhencef:=gisacontinuousfunctionofcompactsupportonG.SincegisconstantonHcosets,Jfz=zJhz=Fz:QED

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8.9.INVARIANTANDRELATIVELYINVARIANTMEASURESONHOMOGENEOUSSPACES.229Nowtotheproofofthetheorem.SupposethatKisanintegralonC0G=HwithmodularfunctionD.DeneMf=KJfDforf2C0G.ByapplyingthemonotoneconvergencetheoremforJandKweseethatMisanintegral.Wemustcheckthatitisleftinvariant,andhencedeterminesaHaarmeasurewhichisamultipleofthegivenHaarmeasure.M`af=KJ`afD=Da)]TJ/F7 6.974 Tf 6.227 0 Td[(1KJ`afD=Da)]TJ/F7 6.974 Tf 6.227 0 Td[(1K`aJfD=Da)]TJ/F7 6.974 Tf 6.227 0 Td[(1DaKJfD=Mf:ThisshowsthatifKisrelativelyinvariantwithmodularfunctionDthenKisuniqueuptoscalarfactor.LetusmultiplyKbeascalarifnecessarywhichdoesnotchangeDsothatIf=KJfD:Wenowarguemoreorlessasbefore:Leth2H.ThenhIf=Irhf=KJrhfD=DhKJrhfD=DhhKJfd=DhhIf;provingthat.28holds.Conversely,supposethat.28holds,andtrytodeneKbyKJf=IfD)]TJ/F7 6.974 Tf 6.227 0 Td[(1:SinceJissurjective,thiswilldeneanintegralonC0G=Honceweshowthatitiswelldened,i.e.onceweshowthatJf=0IfD)]TJ/F7 6.974 Tf 6.227 0 Td[(1=0:SupposethatJf=0,andlet2C0G.ThenxDx)]TJ/F7 6.974 Tf 6.227 0 Td[(1Jfx=0forallx2G,andsotakingIoftheaboveexpressionwillalsovanish.WewillnowuseFubini:Wehave0=Ix)]TJ/F11 9.963 Tf 4.566 -8.069 Td[(xD)]TJ/F7 6.974 Tf 6.227 0 Td[(1xJhfxh=IxJh)]TJ/F11 9.963 Tf 4.566 -8.069 Td[(xD)]TJ/F7 6.974 Tf 6.227 0 Td[(1xfxh=

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230CHAPTER8.HAARMEASURE.JhIx)]TJ/F11 9.963 Tf 4.566 -8.07 Td[(xD)]TJ/F7 6.974 Tf 6.227 0 Td[(1xfxhWecanwritetheexpressionthatisinsidethelastIxintegralasrh)]TJ/F6 4.981 Tf 5.396 0 Td[(1xh)]TJ/F7 6.974 Tf 6.227 0 Td[(1D)]TJ/F7 6.974 Tf 6.226 0 Td[(1xh)]TJ/F7 6.974 Tf 6.227 0 Td[(1fxandhenceJhIx)]TJ/F11 9.963 Tf 4.567 -8.07 Td[(xD)]TJ/F7 6.974 Tf 6.227 0 Td[(1xfxh=JhIx)]TJ/F11 9.963 Tf 4.566 -8.07 Td[(xh)]TJ/F7 6.974 Tf 6.227 0 Td[(1D)]TJ/F7 6.974 Tf 6.226 0 Td[(1xh)]TJ/F7 6.974 Tf 6.227 0 Td[(1fxh)]TJ/F7 6.974 Tf 6.227 0 Td[(1bythedeningpropertiesof.Nowusethehypothesisthat=DtogetJhIx)]TJ/F11 9.963 Tf 4.566 -8.07 Td[(h)]TJ/F7 6.974 Tf 6.227 0 Td[(1xh)]TJ/F7 6.974 Tf 6.226 0 Td[(1D)]TJ/F7 6.974 Tf 6.227 0 Td[(1xfxandapplyFubiniagaintowritethisasIx)]TJ/F11 9.963 Tf 4.566 -8.07 Td[(D)]TJ/F7 6.974 Tf 6.227 0 Td[(1xfxJhh)]TJ/F7 6.974 Tf 6.227 0 Td[(1xh)]TJ/F7 6.974 Tf 6.227 0 Td[(1:Byequation.26appliedtothegroupH,wecanreplacetheJintegralabovebyJhxhsonallyweconcludethatIx)]TJ/F11 9.963 Tf 4.566 -8.07 Td[(D)]TJ/F7 6.974 Tf 6.227 0 Td[(1xfxJhxh=0forany2C0G.Nowchoose2C0G=Hwhichisnon-negativeandidenticallyoneonSuppf,andchoose2C0GwithJ=.ThentheaboveexpressionisID)]TJ/F7 6.974 Tf 6.226 0 Td[(1f.SowehaveprovedthatJf=0IfD)]TJ/F7 6.974 Tf 6.227 0 Td[(1=0;andhencethatK:C0G=H!CdenedbyKF=ID)]TJ/F7 6.974 Tf 6.226 0 Td[(1fifJf=Fiswelldened.WestillmustshowthatKdenedthiswayisrelativelyinvariantwithmodularfunctionK.WecomputeK`aF=ID)]TJ/F7 6.974 Tf 6.227 0 Td[(1`af=DaI`aD)]TJ/F7 6.974 Tf 6.227 0 Td[(1f=DaID)]TJ/F7 6.974 Tf 6.227 0 Td[(1f=DaKF:QEDOfparticularimportanceisthecasewhereGandHareunimodular,forexamplecompact.Then,uptoscalarfactor,thereisauniquemeasureonG=HinvariantundertheactionofG.

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Chapter9Banachalgebrasandthespectraltheorem.Inthischapter,allringswillbeassumedtobeassociativeandtohaveanidentityelement,usuallydenotedbye.Ifanelementxintheringissuchthate)]TJ/F11 9.963 Tf 10.088 0 Td[(xhasarightinverse,thenwemaywritethisinversease)]TJ/F11 9.963 Tf 9.834 0 Td[(y,andtheequatione)]TJ/F11 9.963 Tf 9.962 0 Td[(xe)]TJ/F11 9.963 Tf 9.962 0 Td[(y=eexpandsouttox+y)]TJ/F11 9.963 Tf 9.962 0 Td[(xy=0:FollowingLoomis,wecallytherightadverseofxandxtheleftadverseofy.Loomisintroducesthistermbecausehewantstoconsideralgebraswithoutidentityelements.Butitwillbeconvenienttouseitevenunderourassumptionthatallouralgebrashaveanidentity.Ifanelementhasbotharightandleftinversethentheymustbeequalbytheassociativelaw,soifxhasarightandleftadversethesemustbeequal.Whenwesaythatanelementhasordoesnothaveaninverse,wewillmeanthatithasordoesnothaveatwosidedinverse.Similarlyforadverse.Allalgebraswillbeoverthecomplexnumbers.Thespectrumofanelementxinanalgebraisthesetofall2Csuchthatx)]TJ/F11 9.963 Tf 10.288 0 Td[(ehasnoinverse.WedenotethespectrumofxbySpecx.Proposition9.0.2IfPisapolynomialthenPSpecx=SpecPx:.1Proof.Theproductofinvertibleelementsisinvertible.Forany2CwritePt)]TJ/F11 9.963 Tf 9.963 0 Td[(asaproductoflinearfactors:Pt)]TJ/F11 9.963 Tf 9.963 0 Td[(=cYt)]TJ/F11 9.963 Tf 9.963 0 Td[(i:ThusPx)]TJ/F11 9.963 Tf 9.962 0 Td[(e=cYx)]TJ/F11 9.963 Tf 9.963 0 Td[(ie231

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232CHAPTER9.BANACHALGEBRASANDTHESPECTRALTHEOREM.inAandhencePx)]TJ/F11 9.963 Tf 10.335 0 Td[(e)]TJ/F7 6.974 Tf 6.227 0 Td[(1failstoexistifandonlyifx)]TJ/F11 9.963 Tf 10.335 0 Td[(ie)]TJ/F7 6.974 Tf 6.227 0 Td[(1failstoexistforsomei,i.e.i2Specx.ButtheseiarepreciselythesolutionsofP=:Thus2SpecPxifandonlyif=Pforsome2Specxwhichispreciselytheassertionoftheproposition.QED9.1Maximalideals.9.1.1Existence.Theorem9.1.1Everyproperrightidealinaringiscontainedinamaximalproperrightideal.Similarlyforleftideals.Alsoanypropertwosidedidealiscontainedinamaximalpropertwosidedideal.ProofbyZorn'slemma.Theproofisthesameinallthreecases:LetIbetheidealinquestionrightleftortwosidedandFbethesetofallproperidealsoftheappropriatetypecontainingIorderedbyinclusion.Sinceedoesnotbelongtoanyproperideal,theunionofanylinearlyorderedfamilyofproperidealsisagainproper,andsohasanupperbound.NowZornguaranteestheexistenceofamaximalelement.QED9.1.2Themaximalspectrumofaring.ForanyringRweletMspecRdenotethesetofmaximalpropertwosidedidealsofR.ForanytwosidedidealIweletSuppI:=fM2MspecR:IMg:NoticethatSuppf0g=MspecRandSuppR=;:ForanyfamilyIoftwosidedideals,amaximalidealcontainsalloftheIifandonlyifitcontainsthetwosidedidealPI.InsymbolsSuppI=SuppXI!:ThustheintersectionofanycollectionofsetsoftheformSuppIisagainofthisform.NoticealsothatifA=SuppIthenA=SuppJwhereJ=M2AM:

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9.1.MAXIMALIDEALS.233HereIJ,butJmightbeastrictlylargerideal.WeclaimthatA=SuppM2AM!andB=SuppM2BM!A[B=SuppM2A[BM!:.2Indeed,ifNisamaximalidealbelongingtoA[Bthenitcontainstheinter-sectionontherighthandsideof.2sothelefthandsidecontainstheright.Wemustshowthereverseinclusion.Sosupposethecontrary.ThismeansthatthereisamaximalidealNwhichcontainstheintersectionontherightbutdoesnotbelongtoeitherAorB.SinceNdoesnotbelongtoA,theidealJA:=TM2AMisnotcontainedinN,soJA+N=R,andhencethereexista2JAandm2Nsuchthata+m=e.Similarly,thereexistb2JBandn2Nsuchthatb+n=e.Butthene=e2=a+mb+n=ab+an+mb+mn:EachofthelastthreetermsontherightbelongtoNsinceitisatwosidedideal,andsodoesabsinceab2M2AM!M2BM!=M2A[BM!N:Thuse2Nwhichisacontradiction.TheabovefactsshowthatthesetsoftheformSuppIgivetheclosedsetsofatopology.IfAMspecRisanarbitrarysubset,itsclosureisgivenby A=SuppM2AM!:Forthecaseofcommutativerings,amajoradvancewastoreplacemaximalidealsbyprimeidealsintheprecedingconstruction-givingrisetothenotionofSpecR-theprimespectrumofacommutativering.ButthemotivationforthisdevelopmentincommutativealgebracamefromtheseconstructionsinthetheoryofBanachalgebras.9.1.3Maximalidealsinacommutativealgebra.Proposition9.1.1AnidealMinacommutativealgebraismaximalifandonlyifR=Misaeld.Proof.IfJisanidealinR=M,itsinverseimageundertheprojectionR!R=MisanidealinR.IfJisproper,soisthisinverseimage.ThusMismaximalif

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234CHAPTER9.BANACHALGEBRASANDTHESPECTRALTHEOREM.andonlyifF:=R=Mhasnoidealsotherthan0andF.Thusif06=X2F,thesetofallmultiplesofXmustbeallofFifMismaximal.Inparticulareverynon-zeroelementhasaninverse.Conversely,ifeverynon-zeroelementofFhasaninverse,thenFhasnoproperideals.QED9.1.4Maximalidealsintheringofcontinuousfunctions.LetSbeacompactHausdorspace,andletCSdenotetheringofcontinuouscomplexvaluedfunctionsonS.Foreachp2S,themapofCS!Cgivenbyf7!fpisasurjectivehomomorphism.Thekernelofthismapconsistsofallfwhichvanishatp.Bytheprecedingproposition,thisisthenamaximalideal,whichweshalldenotebyMp.Theorem9.1.2IfIisaproperidealofCS,thenthereisapointp2SsuchthatIMp:InparticulareverymaximalidealinCSisoftheformMpsowemayidentifyMspecCSwithSasaset.ThisidenticationisahomeomorphismbetweentheoriginaltopologyofSandthetopologygivenaboveonMspecCS.Proof.Supposethatforeveryp2Sthereisanf2Isuchthatfp6=0.Thenjfj2=f f2Iandjfpj2>0andjfj20everywhere.ThuseachpointofSiscontainedinaneighborhoodUforwhichthereexistsag2Iwithg0everywhere,andg>0onU.SinceSiscompact,wecancoverSwithnitelymanysuchneighborhoods.Ifwetakehtobethesumofthecorrespondingg's,thenh2Iandh>0everywhere.Soh)]TJ/F7 6.974 Tf 6.227 0 Td[(12CSande=1=hh)]TJ/F7 6.974 Tf 6.227 0 Td[(12IsoI=CS,acontradiction.Thisprovestherstpartofthethetheorem.Toprovethelaststatement,wemustshowthattheclosureofanysubsetASintheoriginaltopologycoincideswithitsclosureinthetopologyderivedfromthemaximalidealstructure.Thatis,wemustshowthatclosureofAinthetopologyofS=SuppM2AM!:NowM2AMconsistsexactlyofallcontinuousfunctionswhichvanishatallpointsofA.AnysuchfunctionmustvanishontheclosureofAinthetopologyofS.Sothelefthandsideoftheaboveequationiscontainedintherighthandside.Wemustshowthereverseinclusion.Supposep2SdoesnotbelongtotheclosureofAinthetopologyofS.ThenUrysohn'sLemmaassertsthatthereisanf2CSwhichvanishesonAandfp6=0.Thusp62SuppTM2AM.QED

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9.2.NORMEDALGEBRAS.235Theorem9.1.3LetIbeanidealinCSwhichisclosedintheuniformtopol-ogyonCS.ThenI=M2SuppIM:Proof.SuppIconsistsofallpointspsuchthatfp=0forallf2I.Sincefiscontinuous,thesetofzerosoffisclosed,andhenceSuppIbeingtheintersectionofsuchsetsisclosed.LetObethecomplementofSuppIinS.ThenOisalocallycompactspace,andtheelementsofTM2SuppIMwhenrestrictedtoOconsistofallfunctionswhichvanishatinnity.I,whenrestrictedtoOisauniformlyclosedsubalgebraofthisalgebra.IfwecouldshowthattheelementsofIseparatepointsinOthentheStone-WeierstrasstheoremwouldtellusthatIconsistsofallcontinuousfunctionsonOwhichvanishatinnity",i.e.allcontinuousfunctionswhichvanishonSuppI,whichistheassertionofthetheorem.SoletpandqbedistinctpointsofO,andletf2CSvanishonSuppIandatqwithfp=1.SuchafunctionexistsbyUrysohn'sLemma,again.Letg2Ibesuchthatgp6=0.SuchagexistsbythedenitionofSuppI.Thengf2I;gfq=0;andgfp6=0.QED9.2Normedalgebras.Anormedalgebraisanalgebraoverthecomplexnumberswhichhasanormasavectorspacewhichsatiseskxykkxkkyk:.3Sincee=eethisimpliesthatkekkek2sokek1:ConsiderthenewnormkykN:=lubkxk6=0kyxk=kxk:Thisstillsatises.3.Indeed,ifx;y,andzaresuchthatyz6=0thenkxyzk kzk=kxyzk kyzkkyzk kzkkxkNkykNandtheinequalitykxyzk kzkkxkNkykNiscertainlytrueifyz=0.Sotakingthesupoverallz6=0weseethatkxykNkxkNkykN:

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236CHAPTER9.BANACHALGEBRASANDTHESPECTRALTHEOREM.From.3wehavekykNkyk:UnderthenewnormwehavekekN=1:Ontheotherhand,fromitsdenitionkyk=kekkykN:Combiningthiswiththepreviousinequalitygiveskyk=kekkykNkyk:InotherwordsthenormskkandkkNareequivalent.Sowithnolossofgeneralitywecanaddtherequirementkek=1.4toouraxiomsforanormedalgebra.Supposeweweakenourconditionandallowkktobeonlyapseudo-norm.thismeansthatweallowthepossibleexistenceofnon-zeroelementsxwithkxk=0.Then9.3impliesthatthesetofallsuchelementsisanideal,callitI.ThenkkdescendstoA=I.Furthermore,anycontinuousi.e.boundedlinearfunctionmustvanishonIsoalsodescendstoA=Iwithnochangeinnorm.Inotherwords,AcanbeidentiedwithA=I.IfAisanormedalgebrawhichiscompletei.e.AisaBanachspaceasanormedspacethenwesaythatAisaBanachalgebra.9.3TheGelfandrepresentation.LetAbeanormedvectorspace.ThespaceAofcontinuouslinearfunctionsonAbecomesanormedvectorspaceunderthenormk`k:=supkxk6=0j`xj=kxk:Eachx2AdenesalinearfunctiononAbyx`:=`xandjx`jk`kkxksoxisacontinuousfunctionof`relativetothenormintroducedaboveonA.LetB=B1AdenotetheunitballinA.InotherwordsB=f`:k`k1g.ThefunctionsxonBinduceatopologyonBcalledtheweaktopology.Proposition9.3.1Biscompactundertheweaktopology.

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9.3.THEGELFANDREPRESENTATION.237Proof.Foreachx2A,thevaluesassumedbythesetof`2BatxlieinthecloseddiskDkxkofradiuskxkinC.ThusBYx2ADkxkwhichiscompactbyTychono'stheorem-beingtheproductofcompactspaces.ToprovethatBiscompact,ifissucienttoshowthatBisaclosedsubsetofthisproductspace.SupposethatfisintheclosureofB.ForanyxandyinAandany>0,wecanndan`2Bsuchthatjfx)]TJ/F11 9.963 Tf 9.963 0 Td[(`xj<;jfy)]TJ/F11 9.963 Tf 9.962 0 Td[(`yj<;andjfx+y)]TJ/F11 9.963 Tf 9.963 0 Td[(`x+yj<:Since`x+y=`x+`ythisimpliesthatjfx+y)]TJ/F11 9.963 Tf 9.963 0 Td[(fx)]TJ/F11 9.963 Tf 9.963 0 Td[(fyj<3:Sinceisarbitrary,weconcludethatfx+y=fx+fy:Similarly,fx=fx.Inotherwords,f2B.QEDNowletAbeanormedalgebra.LetAdenotethesetofallcontinuoushomomorphismsofAontothecomplexnumbers.Inotherwords,inadditiontobeinglinear,wedemandofh2thathxy=hxhyandhe=1:LetE:=h)]TJ/F7 6.974 Tf 6.227 0 Td[(1.ThenEisclosedundermultiplication.Inparticular,ifx2Ewecannothavekxk<1forotherwisexnisasequenceofelementsinEtendingto0,andsobythecontinuityofhwewouldhaveh=1whichisimpossible.Sokxk1forallx2E.Ifyissuchthathy=6=0,thenx:=y=2Esojhyjkyk;andthisclearlyalsoholdsifhy=0.Inotherwords,B:SincetheconditionsforbeingahomomorphismwillholdforanyweaklimitofhomomorphismsthesameproofasgivenaboveforthecompactnessofB,weconcludethatiscompact.Onceagainwecanturnthetablesandthinkofy2Aasafunction^yonvia^yh:=hy:ThismapfromAintoanalgebraoffunctionsoniscalledtheGelfandrepresentation.Theinequalityjhyjkykforallhtranslatesintok^yk1kyk:.5Puttingitalltogetherweget

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238CHAPTER9.BANACHALGEBRASANDTHESPECTRALTHEOREM.Theorem9.3.1isacompactsubsetofAandtheGelfandrepresentationy7!^yisanormdecreasinghomomorphismofAontoasubalgebra^AofC.Theabovetheoremistrueforanynormedalgebra-wehavenotusedanycompletenesscondition.ForBanachalgebras,i.e.completenormedalgebras,wecanproceedfurtherandrelatetoMspecA.RecallthatanelementofMspecAcorrespondstoahomomorphismofAontosomeeld.InthecommutativeBanachalgebracasewewillshowthatthiseldisCandthatanysuchhomomorphismisautomaticallycontinuous.SoforcommutativeBanachalgebraswecanidentifywithMspecA.9.3.1InvertibleelementsinaBanachalgebraformanopenset.InthissectionAwillbeaBanachalgebra.Proposition9.3.2Ifkxk<1thenxhasanadversex0givenbyx0=)]TJ/F13 6.974 Tf 12.944 12.453 Td[(1Xn=1xnsothate)]TJ/F11 9.963 Tf 9.963 0 Td[(xhasaninversegivenbye)]TJ/F11 9.963 Tf 9.962 0 Td[(x0=e+1Xn=1xn:BotharecontinuousfunctionsofxProof.Letsn:=)]TJ/F10 6.974 Tf 14.142 12.453 Td[(nX1xi:Thenifm
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9.3.THEGELFANDREPRESENTATION.239Theorem9.3.2LetybeaninvertibleelementofAandseta:=1 ky)]TJ/F7 6.974 Tf 6.227 0 Td[(1k:Theny+xisinvertiblewheneverkxk
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240CHAPTER9.BANACHALGEBRASANDTHESPECTRALTHEOREM.whereXisacosetofIinA.TheproductoftwocosetsXandYisthecosetcontainingxyforanyx2X;y2Y.ThuskXYk=minx2X;y2Ykxykminx2X;y2Ykxkkyk=kXkkYk:Also,ifEisthecosetcontainingethenEistheidentityelementforA=IandsokEk1:ButweknowthatthisimpliesthatkEk=1.QEDSupposethatAiscommutativeandMisamaximalidealofA.WeknowthatA=Misaeld,andtheprecedingpropositionimpliesthatA=Misanormedeldcontainingthecomplexnumbers.ThefollowingfamousresultimpliesthatA=MisinfactnormisomorphictoC.Itdeservesasubsectionofitsown:TheGelfand-Mazurtheorem.Adivisionalgebraisapossiblynotcommutativealgebrainwhicheverynon-zeroelementhasaninverse.Theorem9.3.3Everynormeddivisionalgebraoverthecomplexnumbersisisometricallyisomorphictotheeldofcomplexnumbers.LetAbethenormeddivisionalgebraandx2A.Wemustshowthatx=eforsomecomplexnumber.Supposenot.Thenbythedenitionofadivisionalgebra,x)]TJ/F11 9.963 Tf 9.963 0 Td[(e)]TJ/F7 6.974 Tf 6.226 0 Td[(1existsforall2Candalltheseelementscommute.Thusx)]TJ/F8 9.963 Tf 9.962 0 Td[(+he)]TJ/F7 6.974 Tf 6.227 0 Td[(1)]TJ/F8 9.963 Tf 9.963 0 Td[(x)]TJ/F11 9.963 Tf 9.963 0 Td[(e)]TJ/F7 6.974 Tf 6.226 0 Td[(1=hx)]TJ/F8 9.963 Tf 9.963 0 Td[(+he)]TJ/F7 6.974 Tf 6.227 0 Td[(1x)]TJ/F11 9.963 Tf 9.963 0 Td[(e)]TJ/F7 6.974 Tf 6.227 0 Td[(1ascanbecheckedbymultiplyingbothsidesofthisequationontherightbyx)]TJ/F11 9.963 Tf 8.848 0 Td[(eandontheleftbyx)]TJ/F8 9.963 Tf 8.847 0 Td[(+he.Thusthestrongderivativeofthefunction7!x)]TJ/F11 9.963 Tf 9.962 0 Td[(e)]TJ/F7 6.974 Tf 6.227 0 Td[(1existsandisgivenbytheusualformulax)]TJ/F11 9.963 Tf 8.668 0 Td[(e)]TJ/F7 6.974 Tf 6.226 0 Td[(2.Inparticular,forany`2Athefunction7!`x)]TJ/F11 9.963 Tf 9.963 0 Td[(e)]TJ/F7 6.974 Tf 6.226 0 Td[(1isanalyticontheentirecomplexplane.Ontheotherhandfor6=0wehavex)]TJ/F11 9.963 Tf 9.962 0 Td[(e)]TJ/F7 6.974 Tf 6.227 0 Td[(1=)]TJ/F7 6.974 Tf 6.227 0 Td[(11 x)]TJ/F11 9.963 Tf 9.962 0 Td[(e)]TJ/F7 6.974 Tf 6.226 0 Td[(1andthisapproacheszeroas!1.Henceforany`2Athefunction7!`x)]TJ/F11 9.963 Tf 9.53 0 Td[(e)]TJ/F7 6.974 Tf 6.227 0 Td[(1isaneverywhereanalyticfunctionwhichvanishesatinnity,andhenceisidenticallyzerobyLiouville'stheorem.Butthisimpliesthatx)]TJ/F11 9.963 Tf -335.963 -11.955 Td[(e)]TJ/F7 6.974 Tf 6.227 0 Td[(10bytheHahnBanachtheorem,acontradiction.QED

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9.3.THEGELFANDREPRESENTATION.2419.3.2TheGelfandrepresentationforcommutativeBanachalgebras.LetAbeacommutativeBanachalgebra.WeknowthateverymaximalidealisthekernelofahomomorphismhofAontothecomplexnumbers.Conversely,supposethathissuchahomomorphism.Weclaimthatjhxjkxkforanyx2A.Indeed,supposethatjhxj>kxkforsomex.Thenkx=hxk<1soe)]TJ/F11 9.963 Tf 10.166 0 Td[(x=hxisinvertible;inparticularhe)]TJ/F11 9.963 Tf 10.166 0 Td[(x=hx6=0whichimpliesthat1=he6=hx=hx,acontradiction.Inshort,wecanidentifyMspecAwithandthemapx7!^xisanormdecreasingmapofAontoasubalgebra^AofCMspecAwhereweusetheuniformnormkk1onCMspecA.Acomplexnumberisinthespectrumofanx2Aifandonlyifx)]TJ/F11 9.963 Tf 10.361 0 Td[(ebelongstosomemaximalidealM,inwhichcase^xM=.Thusk^xk1=l.u.b.fjj:2Specxg:.89.3.3Thespectralradius.Therighthandsideof.8makessenseinanyalgebra,andiscalledthespec-tralradiusofxandisdenotedbyjxjsp.WeclaimthatTheorem9.3.4InanyBanachalgebrawehavejxjsp=limn!1kxnk1 n:.9Proof.Ifjj>kxkthene)]TJ/F11 9.963 Tf 10.442 0 Td[(x=isinvertible,andthereforesoisx)]TJ/F11 9.963 Tf 10.442 0 Td[(eso62Specx.Thusjxjspkxk:Weknowfrom.1that2Specxn2Specxn,sothepreviousinequalityappliedtoxngivesjxjspkxnk1 nandsojxjspliminfkxnk1 n:Wemustprovethereverseinequalitywithlimsup.Supposethatjj<1=jxjspsothat:=1=satisesjj>jxjspandhencee)]TJ/F11 9.963 Tf 9.124 0 Td[(xisinvertible.Theformulafortheadversegivesx0=)]TJ/F13 6.974 Tf 12.633 12.453 Td[(1X1xn

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242CHAPTER9.BANACHALGEBRASANDTHESPECTRALTHEOREM.whereweknowthatthisconvergesintheopendiskofradius1=kxk.However,weknowthate)]TJ/F11 9.963 Tf 10.062 0 Td[(x)]TJ/F7 6.974 Tf 6.226 0 Td[(1existsforjj<1=jxjsp.Inparticular,forany`2Athefunction7!`x0isanalyticandhenceitsTaylorseries)]TJ/F1 9.963 Tf 9.409 9.465 Td[(X`xnnconvergesonthisdisk.HereweusethefactthattheTaylorseriesofafunctionofacomplexvariableconvergesonanydiskcontainedintheregionwhereitisanalytic.Thusj`nxnj!0foreachxed`2Aifjj<1=jxjsp.Consideredasafamilyoflinearfunctionsof`,weseethat`7!`nxnisboundedforeachxed`,andhencebytheuniformboundednessprinciple,thereexistsaconstantKsuchthatknxnkjxjsp:QEDInacommutativeBanachalgebrawecancombine9.9with.8tocon-cludethatk^xk1=limn!1kxnk1 n:.10Wesaythatxisageneralizednilpotentelementiflimkxnk1 n=0.From.9weseethatxisageneralizednilpotentelementifandonlyif^x0.Thismeansthatxbelongstoallmaximalideals.Theintersectionofallmaximalidealsiscalledtheradicalofthealgebra.ABanachalgebraiscalledsemi-simpleifitsradicalconsistsonlyofthe0element.9.3.4ThegeneralizedWienertheorem.Theorem9.3.5LetAbeacommutativeBanachalgebra.Thenx2Ahasaninverseifandonlyif^xnevervanishes.Proof.Ifxy=ethen^x^y1.Soifxhasaninverse,then^xcannotvanishanywhere.Conversely,supposexdoesnothaveaninverse.ThenAxisaproperideal.SoxiscontainedinsomemaximalidealMbyZorn'slemma.So^xM=0.

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9.3.THEGELFANDREPRESENTATION.243Example.LetGbeacountablecommutativegroupgiventhediscretetopology.ThenwemaychooseitsHaarmeasuretobethecountingmeasure.ThusL1GconsistsofallcomplexvaluedfunctionsonGwhichareabsolutelysummable,i.e.suchthatXa2Gjfaj<1:RecallthatL1GisaBanachalgebraunderconvolution:f?gx:=Xy2Gfy)]TJ/F7 6.974 Tf 6.226 0 Td[(1xgy:Werepeattheproof:SinceL1GL2GthissumconvergesandXx2Gjf?gxjXx;y2Gjfxy)]TJ/F7 6.974 Tf 6.226 0 Td[(1jjgyj=Xy2GjgyjXx2Gjfxy)]TJ/F7 6.974 Tf 6.226 0 Td[(1j=Xy2GjgyjXw2Gjfwji.e.kf?gkkfkkgk:Ifx2L1Gisdenedbyxt=1ift=x0otherwisethenx?y=xy:WeknowthatthemostgeneralcontinuouslinearfunctiononL1Gisob-tainedfrommultiplicationbyanelementofL1Gandthenintegrating=summing.Thatisitisgivenbyf7!Xx2Gfxxwhereissomeboundedfunction.Underthislinearfunctionwehavex7!xandso,ifthislinearfunctionistobemultiplicative,wemusthavexy=xy:Sincexn=xnandjxjistobebounded,wemusthavejxj1.Afunctionsatisfyingthesetwoconditions:xy=xyandjxj1

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244CHAPTER9.BANACHALGEBRASANDTHESPECTRALTHEOREM.iscalledacharacterofthecommutativegroupG.Thespaceofcharactersisitselfagroup,denotedby^G.WehaveshownthatMspecL1G=^G.Inparticularwehaveatopologyon^GandtheGelfandtransformf7!^fsendseveryelementofL1Gtoacontinuousfunctionon^G.Forexample,ifG=Zunderaddition,theconditiontobeacharactersaysthatm+n=mn;jj1:Son=nwhere=eiforsome2R=Z.Thus^f=Xn2ZfneinisjusttheFourierserieswithcoecientsfn.TheimageoftheGelfandtrans-formisjustthesetofFourierserieswhichconvergeabsolutely.WeconcludefromTheorem9.3.5thatifFisanabsolutelyconvergentFourierserieswhichvanishesnowhere,then1=FhasanabsolutelyconvergentFourierseries.BeforeGelfand,thiswasadeeptheoremofWiener.TodealwiththeversionofthistheoremwhichtreatstheFouriertransformratherthanFourierseries,wewouldhavetoconsideralgebraswhichdonothaveanidentityelement.Mostofwhatwedidgoesthroughwithonlymildmodications,butIdonottogointothis,asmygoalsareelsewhere.9.4Self-adjointalgebras.LetAbeasemi-simplecommutativeBanachalgebra.Sincesemi-simple"meansthattheradicalisf0g,weknowthattheGelfandtransformisinjec-tive.Aiscalledselfadjointifforeveryx2Athereexistsanx2Asuchthatx^= ^x:BytheinjectivityoftheGelfandtransform,theelementxisuniquelyspeciedbythisequation.Ingeneral,foranyBanachalgebra,amapf7!fyiscalledaninvolutoryanti-automorphismiffgy=gyfyf+gy=fy+gyfy= fyand

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9.4.SELF-ADJOINTALGEBRAS.245fyy=f.Forexample,ifAisthealgebraofboundedoperatorsonaHilbertspace,thenthemapT7!Tsendingeveryoperatortoitsadjointisanexampleofaninvolutoryanti-automorphism.AnotherexampleisL1Gunderconvolution,foralocallycompactHausdorgroupGwheretheinvolutionwasthemapf7!~f.IfAisasemi-simpleself-adjointcommutativeBanachalgebra,themapx7!xisaninvolutoryanti-automorphism.Ithasthisfurtherproperty:f=f1+f2isinvertible:indeed,iff=fthen^fisrealvalued,so1+^f2vanishesnowhere,andso1+f2isinvertiblebyTheorem9.3.5.ConverselyTheorem9.4.1LetAbeacommutativesemi-simpleBanachalgebrawithaninvolutoryanti-automorphismf7!fysuchthat1+f2isinvertiblewheneverf=fy.ThenAisself-adjointandy=.Proof.Wemustshowthatfy^= ^f.Werstprovethatifwesetg:=f+fythen^gisrealvalued.Supposethecontrary,that^gM=a+ib;b6=0forsomeM2MspecA:Nowgy=fy+fyy=gandhenceg2y=g2soh:=ag2)]TJ/F8 9.963 Tf 9.962 0 Td[(a2)]TJ/F11 9.963 Tf 9.963 0 Td[(b2g ba2+b2satiseshy=h:WehavehM=aa+ib2)]TJ/F8 9.963 Tf 9.962 0 Td[(a2)]TJ/F11 9.963 Tf 9.963 0 Td[(b2a+ib ba2+b2=i:So1+hM2=0contradictingthehypothesisthat1+h2isinvertible.Nowletusapplythisresultto1 2fandto1 2if.Wehavef=g+ihwhereg=1 2f+fy;h=1 2if)]TJ/F11 9.963 Tf 9.963 0 Td[(fyandweknowthat^gand^harerealandsatisfygy=gandhy=h.So ^f= ^g+i^h=^g)]TJ/F11 9.963 Tf 9.962 0 Td[(i^h=fy^:QED

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246CHAPTER9.BANACHALGEBRASANDTHESPECTRALTHEOREM.Theorem9.4.2LetAbeacommutativeBanachalgebrawithaninvolutoryanti-automorphismywhichsatisestheconditionkffyk=kfk28f2A:.11ThentheGelfandtransformf7!^fisanormpreservingsurjectiveisomorphismwhichsatisesfy^= ^f:Inparticular,Aissemi-simpleandself-adjoint.Proof.kfk2=kffykkfkkfyksokfkkfyk.Replacingfbyfygiveskfykkfk.sokfk=kfykandkffyk=kfk2=kfkkfyk:.12NowsinceAiscommutative,ffy=fyfandf2f2y=f2fy2=ffyffyy.13andsoapplying.12tof2andthenapplyingitonceagaintofwegetkf2kkfy2k=kffyffyyk=kffykkffyk=kfk2kfyk2orkf2k2=kfk4:Thuskf2k=kfk2;andthereforekf4k=kf2k2=kfk4andbyinductionkf2kk=kfk2kforallnon-negativeintegersk.Hencelettingn=2kintherighthandsideof.10weseethatk^fk1=kfksotheGelfandtransformisnormpreserving,andhenceinjective.Toshowthaty=itisenoughtoshowthatiff=fythen^fisrealvalued,asintheproofoftheprecedingtheorem.Supposenot,so^fM=a+ib;b6=0.Foranyrealnumbercwehavef+ice^M=a+ib+csojf+ice^Mj2=a2+b+c2kf+icek2=kf+icef)]TJ/F11 9.963 Tf 9.962 0 Td[(icek

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9.4.SELF-ADJOINTALGEBRAS.247=kf2+c2ekkfk2+c2:Thissaysthata2+b2+2bc+c2kfk2+c2whichisimpossibleifwechoosecsothat2bc>kfk2.Sowehaveprovedthaty=.Nowbydenition,iffM=fNforallf2A,themaximalidealsMandNcoincide.SotheimageofelementsofAundertheGelfandtransformseparatepointsofMspecA.Buteveryf2Acanbewrittenasf=1 2f+f+i1 2if)]TJ/F11 9.963 Tf 9.963 0 Td[(fi.e.asasumg+ihwhere^gand^farerealvalued.Hencetherealvaluedfunctionsoftheform^gseparatepointsofMspecA.HencebytheStoneWeierstrassthe-oremweknowthattheimageoftheGelfandtransformisdenseinCMspecA.SinceAiscompleteandtheGelfandtransformisnormpreserving,weconcludethattheGelfandtransformissurjective.QED9.4.1Animportantgeneralization.ABanachalgebrawithaninvolutionysuchthat.11holdsiscalledaC-algebra.NoticethatwearenotassumingthatthisBanachalgebraiscommu-tative.Butanelementxofsuchanalgebraiscallednormalifxxy=xyx;inotherwordsifxdoescommutewithxy.ThenwecanrepeattheargumentatthebeginningoftheproofofTheorem9.4.2toconcludethatifxisanormalelementofaCalgebra,thenkx2kk=kxk2kandhenceby.9jxjsp=kxk:.14Anelementxofanalgebrawithinvolutioniscalledself-adjointifxy=x.Inparticular,everyself-adjointelementisnormal.Again,arerunofapreviousargumentshowsthatifxisself-adjoint,meaningthatxy=xthenSpecxR.15Indeed,supposethata+ib2Specxwithb6=0,andlety:=1 bx)]TJ/F11 9.963 Tf 9.963 0 Td[(aesothaty=yyandi2Specy.Soe+iyisnotinvertible.Soforanyrealnumberr,r+1e)]TJ/F8 9.963 Tf 9.963 0 Td[(re)]TJ/F11 9.963 Tf 9.962 0 Td[(iy=e+iy

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248CHAPTER9.BANACHALGEBRASANDTHESPECTRALTHEOREM.isnotinvertible.Thisimpliesthatjr+1jkre)]TJ/F11 9.963 Tf 9.963 0 Td[(iykandsor+12kre)]TJ/F11 9.963 Tf 9.963 0 Td[(iyk2=kre)]TJ/F11 9.963 Tf 9.962 0 Td[(iyre+iyk:by.11.Thusr+12kr2e+y2kr2+kyk2whichisnotpossibleif2r)]TJ/F8 9.963 Tf 9.963 0 Td[(1>kyk2.Sowehaveproved:Theorem9.4.3LetAbeaCalgebra.Ifx2Aisnormal,thenjxjsp=kxk:Ifx2Aisself-adjoint,thenSpecxR:9.4.2Animportantapplication.Proposition9.4.1IfTisaboundedlinearoperatoronaHilbertspace,thenkTTk=kTk2:.16Inotherwords,thealgebraofallboundedoperatorsonaHilbertspaceisaC-algebraundertheinvolutionT7!T.Proof.kTTk=supkk=1kTTk=supkk=1;kk=1jTT;j=supkk=1;kk=1jT;Tjsupkk=1T;T=kTk2sokTk2kTTkkTkkTksokTkkTk:ReversingtheroleofTandTgivesthereverseinequalitysokTk=kTk.InsertingintotheprecedinginequalitygiveskTk2kTTkkTk2

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9.5.THESPECTRALTHEOREMFORBOUNDEDNORMALOPERATORS,FUNCTIONALCALCULUSFORM.249sowehavetheequality.16.QEDThusthemapT7!TsendingeveryboundedoperatoronaHilbertspaceintoitsadjointisananti-involutionontheBanachalgebraofallboundedop-erators,anditsatises.11.WecanthusapplyTheorem9.4.2toconclude:Theorem9.4.4LetBbeanycommutativesubalgebraofthealgebraofboundedoperatorsonaHilbertspacewhichisclosedinthestrongtopologyandwiththepropertythatT2BT2B.ThentheGelfandtransformT7!^TgivesanormpreservingisomorphismofBwithCMwhereM=MspecB.Furthermore,T^= ^TforallT2B.Inparticular,ifTisself-adjoint,then^Tisrealvalued.9.5TheSpectralTheoremforBoundedNormalOperators,FunctionalCalculusForm.Asaspecialcaseofthedenitionwegaveearlier,aboundedoperatorTonaHilbertspaceHiscallednormalifTT=TT:Wecanthenconsiderthesubalgebraofthealgebraofallboundedoperatorswhichisgeneratedbye,theidentityoperator,TandT.Taketheclosure,B,ofthisalgebrainthestrongtopology.WecanapplytheprecedingtheoremtoBtoconcludethatBisisometricallyisomorphictothealgebraofallcontinuousfunctionsonthecompactHausdorspaceM=MspecB:RememberthatapointofMisahomomorphismh:B!CandthathT=T^h= ^Th= hT:Sincehisahomomorphism,weseethathisdeterminedonthealgebrageneratedbye;TandTbythevaluehT,andsinceitiscontinuous,itisdeterminedonallofBbytheknowledgeofhT.WethusgetamapM!C;M3h7!hT;.17andweknowthatthismapisinjective.NowhT)]TJ/F11 9.963 Tf 9.962 0 Td[(hTe=hT)]TJ/F11 9.963 Tf 9.962 0 Td[(hT=0soT)]TJ/F11 9.963 Tf 9.713 0 Td[(hTebelongstothemaximalidealh,andhenceisnotinvertibleinthealgebraB.ThustheimageofourmapliesinSpecBT.HereIhaveaddedthesubscriptB,becauseourgeneraldenitionofthespectrumofanelementTinanalgebraBconsistsofthosecomplexnumberszsuchthatT)]TJ/F11 9.963 Tf 10.36 0 Td[(zedoesnothaveaninverseinB.IfweletAdenotethealgebraofallboundedoperatorson

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250CHAPTER9.BANACHALGEBRASANDTHESPECTRALTHEOREM.H,itislogicallyconceivablethatT)]TJ/F11 9.963 Tf 9.714 0 Td[(zehasaninverseinAwhichdoesnotlieinB.Infact,thiscannothappen.Butthisrequiresaproof,whichIwillgivelater.SoforthetimebeingIwillstickwiththetemporarynotationSpecB.Sothemaph7!hTactuallymapsMtothesubsetSpecBTofthecomplexplane.If2SpecBT,thenbydenition,T)]TJ/F11 9.963 Tf 10.014 0 Td[(eisnotinvertibleinB,soliesinsomemaximalidealh,so2SpecBT.Sothemap9.17mapsMontoSpecBT.SincethetopologyonMisinheritedfromtheweaktopology,themaph7!hTiscontinuous.SinceMiscompactandCisHausdor,thefactthathisbijectiveandcontinuousimpliesthath)]TJ/F7 6.974 Tf 6.226 0 Td[(1isalsocontinuous.IndeedwemustshowthatifUisanopensubsetofM,thenhUisopeninSpecBT.ButMnUiscompact,henceitsimageunderthecontinuousmaphiscompact,henceclosed,andsothecomplementofthisimagewhichishUisopen.Thushisahomeomorphism,andhencewehaveanorm-preserving-isomorphismT7!^TofBwiththealgebraofcontinuousfunctionsonSpecBT.FurthermoretheelementTcorrespondsthefunctionz7!zrestrictedtoSpecBT.SinceBisdeterminedbyT,letmeusethenotationTforSpecBT,postponinguntillatertheproofthatT=SpecAT.Letmeset=h)]TJ/F7 6.974 Tf 6.227 0 Td[(1andnowfollowthecustomarynotationinHilbertspacetheoryanduseItodenotetheidentityoperator.9.5.1Statementofthetheorem.Theorem9.5.1LetTbeaboundednormaloperatoronaHilbertspaceH.LetBTdenotetheclosureinthestrongtopologyofthealgebrageneratedbyI;TandT.Thenthereisauniquecontinuousisomorphism:CT!BTsuchthat1=Iandz7!z=T:Furthermore,T=;2Hf=f:.18Iff2CTisrealvaluedthenfisself-adjoint,andiff0thenf0asanoperator.TheonlyfactsthatwehavenotyetprovedasidefromthebigissueofprovingthatT=SpecATare.18andtheassertionswhichfollowit.Now.18isclearlytrueifwetakeftobeapolynomial,inwhichcasef=fT.Then

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9.5.THESPECTRALTHEOREMFORBOUNDEDNORMALOPERATORS,FUNCTIONALCALCULUSFORM.251justapplytheStone-Weierstrasstheoremtoconclude.18forallf.Iffisrealthenf= fandthereforef=f.Iff0thenwecanndarealvaluedg2CTsuchthatf=g2andthesquareofaself-adjointoperatorisnon-negative.QEDInviewofthistheorem,thereisamoresuggestivenotationforthemap.SincetheimageofthemonomialzisT,andsincetheimageofanypolynomialPthoughtofasafunctiononTisPT,wearesafeinusingthenotationfT:=fforanyf2CT.9.5.2SpecBT=SpecAT:Hereisthemainresultofthissection:Theorem9.5.2LetAbeaCalgebraandletBbeasubalgebraofAwhichisclosedundertheinvolution.Thenforanyx2BwehaveSpecBx=SpecAx:.19Remarks:1.AppliedtothecasewhereAisthealgebraofallboundedoperatorsonaHilbertspace,andwhereBistheclosedsubalgebrabyI;TandTwegetthespectraltheoremfornormaloperatorsaspromised.2.Ifx)]TJ/F11 9.963 Tf 9.962 0 Td[(zehasnoinverseinAithasnoinverseinB.SoSpecAxSpecBx:Wemustshowthereverseinclusion.Webeginbyformulatingsomegeneralresultsandintroducingsomenotation.ForanyassociativealgebraAweletGAdenotethesetofelementsofAwhichareinvertiblethegroup-like"elements.Proposition9.5.1LetBbeaBanachalgebra,andletxn2GBbesuchthatxn!xandx62GB.Thenkx)]TJ/F7 6.974 Tf 6.227 0 Td[(1nk!1:Proof.Supposenot.ThenthereissomeC>0andasubsequenceofelementswhichwewillrelabelasxnsuchthatkx)]TJ/F7 6.974 Tf 6.227 0 Td[(1nk
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252CHAPTER9.BANACHALGEBRASANDTHESPECTRALTHEOREM.Proposition9.5.2LetBbeaclosedsubalgebraofaBanachalgebraAcon-tainingtheunite.ThenGBistheunionofsomeofthecomponentsofBGA.Ifx2BthenSpecBxistheunionofSpecAxandapossiblyemptycollectionofboundedcomponentsofCnSpecAx.IfSpecAxdoesnotseparatethecomplexplanethenSpecBx=SpecAx:Proof.WeknowthatGBGABandbothareopensubsetsofB.WeclaimthatGABcontainsnoboundarypointsofGB.Ifxweresuchaboundarypoint,thenx=limxnforxn2GB,andbythecontinuityofthemapy7!y)]TJ/F7 6.974 Tf 6.227 0 Td[(1inAweconcludethatx)]TJ/F7 6.974 Tf 6.226 0 Td[(1n!x)]TJ/F7 6.974 Tf 6.226 0 Td[(1inA,soinparticularthekxnk)]TJ/F7 6.974 Tf 6.226 0 Td[(1areboundedwhichisimpossiblebyProposition9.5.1.LetObeacomponentofBGAwhichintersectsGB,andletUbethecomplementof GB.ThenOGBandOUareopensubsetsofBandsinceOdoesnotintersect@GB,theunionofthesetwodisjointopensetsisO.SoOUisemptysinceweassumedthatOisconnected.HenceOGB.Thisprovestherstassertion.Forthesecondassertion,letusxtheelementx,andletGAx=CnSpecAxsothatGAxconsistsofthosecomplexnumberszforwhichx)]TJ/F11 9.963 Tf 8.985 0 Td[(zeisinvertibleinA,withasimilarnotationforGBx.BothoftheseareopensubsetsofCandGBxGAx.Furthermore,asbefore,GBxGAxandGAxcannotcontainanyboundarypointsofGBx.Soagain,GBxisaunionofsomeoftheconnectedcomponentsofGAx.ThereforeSpecBxistheunionofSpecAxandtheremainingcomponents.SinceSpecBxisbounded,itwillnotcontainanyunboundedcomponents.Thethirdassertionfollowsimmediatelyfromthesecond.QEDButnowwecanproveTheorem9.5.2.Weneedtoshowthatifx2BisinvertibleinAthenitisinvertibleinB.IfxisinvertibleinAthensoarexandxx.Butxxisself-adjoint,henceitsspectrumisaboundedsubsetofR,sodoesnotseparateC.Since062SpecAxxweconcludefromthelastassertionofthepropositionthat062SpecBxxsoxxhasaninverseinB.Butthenxxx)]TJ/F7 6.974 Tf 6.227 0 Td[(12Bandx)]TJ/F11 9.963 Tf 4.566 -8.07 Td[(xxx)]TJ/F7 6.974 Tf 6.226 0 Td[(1=e:QED.

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9.5.THESPECTRALTHEOREMFORBOUNDEDNORMALOPERATORS,FUNCTIONALCALCULUSFORM.2539.5.3Adirectproofofthespectraltheorem.IstartedoutthischaperwiththegeneraltheoryofBanachalgebras,wenttotheGelfandrepresentationtheorem,thespecialpropertiesofCalgebras,andthensomegeneralfactsabouthowthespectrumofanelementcanvarywiththealgebracontainingit.ItookthisroutebecauseoftheimpacttheGelfandrepresentationtheoremhadonthecourseofmathematics,especiallyinalgebraicgeometry.Butthekeyideasare.9,which,foraboundedoperatorTonaBanachspacesaysthatmax2SpecTjj=limn!1kTnk1 n,.16whichsaysthatifTisaboundedoperatoronaHilbertspacethenkTTk=kTk2,andIfTisaboundedoperatoronaHilbertspaceandTT=TTthenitfollowsfrom.16thatkT2kk=kTk2k:WecouldprovethesefactsbytheargumentsgivenaboveandconcludethatifTisanormalboundedoperatoronaHilbertspacethenmax2SpecTjj=kTk:.20SupposeforsimplicitythatTisself-adjoint:T=T.ThentheargumentgivenseveraltimesaboveshowsthatSpecTR.LetPbeapolynomial.Then.1combinedwiththeprecedingequationsaysthatkPTk=max2SpecTjPj:.21Thenormontherightistherestrictiontopolynomialsoftheuniformnormkk1onthespaceCSpecT.NowthemapP7!PTisahomomorphismoftheringofpolynomialsintoboundednormaloperatorsonourHilbertspacesatisfying P7!PTandkPTk=kPk1;SpecT:TheWeierstrassapproximationtheoremthenallowsustoconcludethatthishomomorphismextendstotheringofcontinuousfunctionsonSpecTwithallthepropertiesstatedinTheorem9.5.1.

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254CHAPTER9.BANACHALGEBRASANDTHESPECTRALTHEOREM.

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Chapter10Thespectraltheorem.Thepurposeofthischapteristogiveamorethoroughdiscussionofthespectraltheorem,especiallyforunboundedself-adjointoperators,WebeginbygivingaslightlydierentdiscussionshowinghowtheGelfandrepresentationtheorem,especiallyforalgebraswithinvolution,impliesthespec-traltheoremforboundedself-adjointor,moregenerallynormaloperatorsonaHilbertspace.RecallthatanoperatorTiscallednormalifitcommuteswithT.Moregenerally,anelementTofaBanachalgebraAwithinvolutioniscallednormalifTT=TT.LetBbetheclosedcommutativesubalgebrageneratedbye;TandT.WeknowthatBisisometricallyisomorphictotheringofcontinuousfunctionsonM,thespaceofmaximalidealsofB,whichisthesameasthespaceofcontinuoushomomorphismsofBintothecomplexnumbers.WeshallshowagainthatMcanalsobeidentiedwithSpecAT,thesetofall2CsuchthatT)]TJ/F11 9.963 Tf 8.551 0 Td[(eisnotinvertible.Thismeansthateverycontinuouscontinuousfunction^fonM=SpecATcorrespondstoanelementfofB.InthecasewhereAisthealgebraofboundedoperatorsonaHilbertspace,wewillshowthatthishomomorphismextendstothespaceofBorelfunctionsonSpecAT.IngeneraltheimageoftheextendedhomomorphismwilllieinA,butnotnecessarilyinB.WenowrestrictattentiontothecasewhereAisthealgebraofboundedoperatorsonaHilbertspace.IfUisaBorelsubsetofSpecAT,letusdenotetheelementofAcorre-spondingto1UbyPU.ThenPU2=PUandPU=PUsoPUisaselfadjointi.e.orthogonal"projection.Also,ifUV=;thenPUPV=0andPU[V=PU+PV:ThusU7!PUisnitelyadditive.Infact,itiscountablyadditiveintheweaksensethatforanypairofvectorsx;yinourHilbertspaceHthemapx;y:U7!PUx;y255

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256CHAPTER10.THESPECTRALTHEOREM.isacomplexvaluedmeasureonM.Weshallprovetheseresultsinpartiallyreversedorder,inthatwerstprovetheexistenceofthecomplexvaluedmeasurex;yusingtheRieszrepresentationtheoremdescribingallcontinuouslinearfunctionsonCM,andthendeducetheexistenceoftheresolutionoftheidentityorprojectionvaluedmeasureU7!PU;moreprecisedenitionbelowfromwhichwecanrecoverT.Thekeytool,inadditiontotheGelfandrepresentationtheoremandtheRiesztheoremdescrib-ingallcontinuouslinearfunctionsonCMasbeingsignedmeasuresisouroldfriend,theRieszrepresentationtheoremforcontinuouslinearfunctionsonaHilbertspace.10.1Resolutionsoftheidentity.InthissectionBdenotesaclosedcommutativeself-adjointsubalgebraofthealgebraofallboundedlinearoperatorsonaHilbertspaceH.Self-adjointmeansthatT2BT2B.BytheGelfandrepresentationtheoremweknowthatBisisometricallyisomorphictoCMunderamapwedenotebyT7!^TandweknowthatT7! ^T:Fixx;y2H:Themap^T7!Tx;yisalinearfunctiononCMwithjTx;yjkTkkxkkyk=k^Tk1kxkkyk:InparticularitisacontinuouslinearfunctiononCM.Hence,bytheRieszrepresentationtheorem,thereexistsauniquecomplexvaluedboundedmeasurex;ysuchthatTx;y=ZM^Tdx;y8^T2CM:When^Tisreal,T=TsoTx;y=x;Ty= Ty;x.Theuniquenessofthemeasureimpliesthaty;x= x;y:

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10.1.RESOLUTIONSOFTHEIDENTITY.257Thus,foreachxedBorelsetUMitsmeasurex;yUdependslinearlyonxandanti-linearlyony.WehaveM=ZM1dx;y=ex;y=x;ysojx;yMjkxkkyk:SoiffisanyboundedBorelfunctiononM,theintegralZMfdx;yiswelldened,andisboundedinabsolutevaluebykfk1kxkkyk.Ifweholdfandxxed,thisintegralisaboundedanti-linearfunctionofy,andhencebytheRieszrepresentationtheoremthereexistsaw2Hsuchthatthisintegralisgivenbyw;y.Thewinquestiondependslinearlyonfandonxbecausetheintegraldoes,andsowehavedenedalinearmapOfromboundedBorelfunctionsonMtoboundedoperatorsonHsuchthatOfx;y=ZMfdx;yandkOfkkfk1:OncontinuousfunctionswehaveO^T=TsoOisanextensionoftheinverseoftheGelfandtransformfromcontinuousfunctionstoboundedBorelfunctions.SoweknowthatOismultiplicativeandtakescomplexconjugationintoadjointwhenrestrictedtocontinuousfunctions.LetusprovethesefactsforallBorelfunctions.IffisrealweknowthatOfy;xisthecomplexconjugateofOfx;ysincey;x= x;y.HenceOfisself-adjointiffisrealfromwhichwededucethatO f=Of:Nowtothemultiplicativity:ForS;T2BwehaveZM^S^Tdx;y=STx;y=ZM^SdTx;y:Sincethisholdsforall^S2CMforxedT;x;yweconcludebytheunique-nessofthemeasurethatTx;y=^Tx;y:Therefore,foranyboundedBorelfunctionfwehaveTx;Ofy=OfTx;y=ZMfdTx;y=ZM^Tfdx;y:

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258CHAPTER10.THESPECTRALTHEOREM.Thisholdsforall^T2CMandsobytheuniquenessofthemeasureagain,weconcludethatx;Ofy=fx;yandhenceOfgx;y=ZMgfdx;y=ZMgdx;Ofy=Ogx;Ofy=OfOgx;yorOfg=OfOgasdesired.WehavenowextendedthehomomorphismfromCMtoAtoahomo-morphismfromtheboundedBorelfunctionsonMtoboundedoperatorsonH.Nowdene:PU:=O1UforanyBorelsetU.Thefollowingfactsareimmediate:1.P;=02.PM=etheidentity3.PUV=PUPVandPU=PU.Inparticular,PUisaself-adjointprojectionoperator.4.IfUV=;thenPU[V=PU+PV.5.Foreachxedx;y2HthesetfunctionPx;y:U7!PUx;yisacomplexvaluedmeasure.SuchaPiscalledaresolutionoftheidentity.Itfollowsfromthelastitemthatforanyxedx2H,themapU7!PUxisanHvaluedmeasure.Wehaveshownthatanycommutativeclosedself-adjointsubalgebraBofthealgebraofboundedoperatorsonaHilbertspaceHgivesrisetoauniqueresolutionoftheidentityonM=MspecBsuchthatT=ZM^TdP.1intheweak"sensethatTx;y=ZM^Tdx;yx;yU=PUx;y:Actually,givenanyresolutionoftheidentitywecangiveameaningtotheintegralZMfdP

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10.1.RESOLUTIONSOFTHEIDENTITY.259foranyboundedBorelfunctionfinthestrongsenseasfollows:ifs=Xi1UiisasimplefunctionwhereM=U1[[Un;UiUj=;;i6=jand1;:::;n2C,deneOs:=XiPUi=:ZMsdP:ThisiswelldenedonsimplefunctionsisindependentoftheexpressionandismultiplicativeOst=OsOt:Also,sincethePUareselfadjoint,O s=Os:ItisalsoclearthatOislinearandOsx;y=ZMsdPx;y:Asaconsequence,wegetkOsxk2=OsOsx;x=ZMjsj2dPx;xsokOsxk2jsk1kxk2:Ifwechooseisuchthatjij=ksk1andtakex=PUiy6=0,thenweseethatkOsk=ksk1providedwenowtakekfk1todenotetheessentialsupremumwhichmeansthefollowing:ItfollowsfromthepropertiesofaresolutionoftheidentitythatifUnisasequenceofBorelsetssuchthatPUn=0,thenPU=0ifU=SUn.SoiffisanycomplexvaluedBorelfunctiononM,therewillexistalargestopensubsetVCsuchthatPf)]TJ/F7 6.974 Tf 6.226 0 Td[(1V=0.WedenetheessentialrangeofftobethecomplementofV,saythatfisessentiallyboundedifitsessentialrangeiscompact,andthendeneitsessentialsupremumkfk1tobethesupremumofjjforintheessentialrangeoff.Furthermoreweidentifytwoessentiallyboundedfunctionsfandgifkf)]TJ/F11 9.963 Tf 10.325 0 Td[(gk1=0andcallthecorrespondingspaceL1P.

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260CHAPTER10.THESPECTRALTHEOREM.EveryelementofL1Pcanbeapproximatedinthekk1normbysimplefunctions,andhencetheintegralOf=ZMfdPisdenedasthestronglimitoftheintegralsofthecorrespondingsimplefunc-tions.Themapf7!Ofislinear,multiplicative,andsatisesO f=OfandkOfk=kfk1asbefore.IfSisaboundedoperatoronHwhichcommuteswithalltheOfthenitcommuteswithallthePU=O1U.Conversely,ifScommuteswithallthePUitcommuteswithalltheOsforssimpleandhencewithalltheOf.Puttingitalltogetherwehave:Theorem10.1.1LetBbeacommutativeclosedselfadjointsubalgebraofthealgebraofallboundedoperatorsonaHilbertspaceH.ThenthereexistsaresolutionoftheidentityPdenedonM=MspecBsuchthat.1holds.Themap^T7!TofCM!BgivenbytheinverseoftheGelfandtransformextendstoamapOfromL1PtothespaceofboundedoperatorsonHOf=ZMfdP:Furthermore,PU6=0foranynon-emptyopensetUandanoperatorScom-muteswitheveryelementofBifandonlyifitcommuteswithallthePUinwhichcaseitcommuteswithalltheOf.Wemustprovethelasttwostatements.IfUisopen,wemaychooseT6=0suchthat^TissupportedinUbyUrysohn'slemma.Butthen.1impliesthatT=0,acontradiction.ForanyboundedoperatorSandanyx;y2HandT2BwehaveSTx;y=Tx;Sy=Z^TdPx;SywhileTSx;y=Z^TdPSx;y:IfST=TSforallT2BthismeansthatthemeasuresPSx;yandPx;Syarethesame,whichmeansthatPUSx;y=PUx;Sy=SPUx;yforallxandywhichmeansthatSPU=PUSforallU.WealreadyknowthatSPU=PUSforallSimpliesthatSOf=OfSforallf2L1P.QED

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10.2.THESPECTRALTHEOREMFORBOUNDEDNORMALOPERATORS.26110.2Thespectraltheoremforboundednormaloperators.LetTbeaboundedoperatoronaHilbertspaceHsatisfyingTT=TT:Recallthatsuchanoperatoriscallednormal.LetBbetheclosureofthealgebrageneratedbye;TandT.Wecanapplythetheoremoftheprecedingsectiontothisalgebra.TheoneusefuladditionalfactisthatwemayidentifyMwithSpecT.Indeed,denethemapM!SpecTbyh7!hT:WeknowthathT)]TJ/F11 9.963 Tf 10.425 0 Td[(hTe=0soT)]TJ/F11 9.963 Tf 10.425 0 Td[(hTeliesinthemaximalidealcor-respondingtohandsoisnotinvertible,consequentlyhT2SpecT.SothemapisindeedintoSpecT.If2SpecTthenbydenitionT)]TJ/F11 9.963 Tf 10.992 0 Td[(eisnotinvertible,hencelieinsomemaximalideal,hence=hTforsomehsothismapissurjective.Ifh1T=h2Tthenh1T= h1T=h2T.Sinceh1agreeswithh2onTandTtheyagreeonallofB,henceh1=h2.Inotherwordsthemaph7!hTisinjective.FromthedenitionofthetopologyonMitiscontinuous.SinceMiscompact,thisimpliesthatitisahomeomorphism.QEDThusinthetheoremoftheprecedingsection,wemayreplaceMbySpecTwhenBistheclosedalgebrageneratedbyTandTwhereTisanormaloperator.InthecasethatTisaself-adjointoperator,weknowthatSpecTR,soourresolutionoftheidentityPisdenedasaprojectionvaluedmeasureonRand.1givesaboundedselfadjointoperatorasT=ZdPrelativetoaresolutionoftheidentitydenedonR.10.3Stone'sformula.LetTbeaboundedself-adjointoperator.WeknowthatT=ZRdPforsomeprojectionvaluedmeasurePonR.WealsoknowthateveryboundedBorelfunctiononRgivesrisetoanoperator.Inparticular,ifzisacomplexnumberwhichisnotreal,thefunction7!1 )]TJ/F11 9.963 Tf 9.963 0 Td[(z

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262CHAPTER10.THESPECTRALTHEOREM.isbounded,andhencecorrespondstoaboundedoperatorRz;T=ZRz)]TJ/F11 9.963 Tf 9.962 0 Td[()]TJ/F7 6.974 Tf 6.227 0 Td[(1dP:Sinceze)]TJ/F11 9.963 Tf 9.962 0 Td[(T=ZRz)]TJ/F11 9.963 Tf 9.962 0 Td[(dPandourhomomorphismismultiplicative,wehaveRz;T=z)]TJ/F11 9.963 Tf 9.963 0 Td[(T)]TJ/F7 6.974 Tf 6.226 0 Td[(1:Aconclusionoftheaboveargumentisthatthisinversedoesindeedexistforallnon-realz.TheoperatorvaluedfunctionRz;TiscalledtheresolventofT.Stone'sformulagivesanexpressionfortheprojectionvaluedmeasureintermsoftheresolvent.Itsaysthatforanyrealnumbersa
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10.5.OPERATORSANDTHEIRDOMAINS.263followedbyStoneandvonNeumannwastoproveaversionofthespectraltheoremforunboundedself-adjointoperators.Therearetwoormoreapproacheswecouldtaketotheproofofthisthe-orem.BothinvolvetheresolventRz=Rz;T=zI)]TJ/F11 9.963 Tf 9.963 0 Td[(T)]TJ/F7 6.974 Tf 6.227 0 Td[(1:.3Afterspendingsometimeexplainingwhatanunboundedoperatorisandgivingtheverysubtledenitionofwhatanunboundedself-adjointoperatoris,wewillprovethattheresolventofaself-adjointoperatorexistsandisaboundednormaloperatorforallnon-realz.Wecouldthenapplythespectraltheoremforboundednormaloperatorstoderivethespectraltheoremforunboundedself-adjointoperators.Thisisthefastestapproach,butdependsonthewholemachineryoftheGelfandrep-resentationtheoremthatwehavedevelopedsofar.Or,wecouldcouldprovethespectraltheoremforunboundedself-adjointoperatorsdirectlyusingamildmodicationofStone'sformula.Wewillpresentbothmethods.InthesecondmethodwewillfollowthetreatmentbyLorch.10.5Operatorsandtheirdomains.LetBandCbeBanachspaces.WemakeBCintoaBanachspaceviakfx;ygk=kxk+kyk:Hereweareusingfx;ygtodenotetheorderedpairofelementsx2Bandy2CsoastoavoidanyconictwithournotationforscalarproductinaHilbertspace.Sofx;ygisjustanotherwayofwritingxy.Asubspace)]TJ/F14 9.963 Tf 8.994 0 Td[(BCwillbecalledagraphmorepreciselyagraphofalineartransformationiff0;yg2)]TJ/F14 9.963 Tf 15.635 0 Td[(y=0:Anotherwayofsayingthesamethingisfx;y1g2)-667(andfx;y2g2)]TJ/F14 9.963 Tf 15.636 0 Td[(y1=y2:Inotherwords,iffx;yg2)-333(thenyisdeterminedbyx.SoletD\051denotethesetofallx2Bsuchthatthereisay2Cwithfx;yg2)]TJ/F11 9.963 Tf 6.227 0 Td[(:ThenD\051isalinearsubspaceofB,but,andthisisveryimportant,D\051isnotnecessarilyaclosedsubspace.WehavealinearmapT\051:D\051!C;Tx=ywherefx;yg2)]TJ/F11 9.963 Tf 6.226 0 Td[(:

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264CHAPTER10.THESPECTRALTHEOREM.Equallywell,wecouldstartwiththelineartransformation:SupposewearegivenanotnecessarilyclosedsubspaceDTBandalineartransformationT:DT!C:Wecanthenconsideritsgraph\050TBCwhichconsistsofallfx;Txg:Thusthenotionofagraph,andthenotionofalineartransformationdenedonlyonasubspaceofBarelogicallyequivalent.WhenwestartwithTasusuallywillbethecasewewillwriteDTforthedomainofTand\050Tforthecorrespondinggraph.Thereisacertainamountofabuseoflanguagehere,inthatwhenwewriteT,wemeantoincludeDTandhence\050Taspartofthedenition.AlineartransformationissaidtobeclosedifitsgraphisaclosedsubspaceofBC.LetusdisentanglewhatthissaysfortheoperatorT.Itsaysthatiffn2DTthenfn!fandTfn!gf2DTandTf=g:Thisisamuchweakerrequirementthancontinuity.ContinuityofTwouldsaythatfn!falonewouldimplythatTfnconvergestoTf.Closednesssaysthatifweknowthatbothfnconvergesandgn=Tfnconvergesthenwecanconcludethatf=limfnliesinDTandthatTf=g.Animportanttheorem,knownastheclosedgraphtheoremsaysthatifTisclosedandDTisallofBthenTisbounded.Aswewillnotneedtousethistheoreminthislecture,wewillnotpresentitsproofhere.10.6Theadjoint.SupposethatwehavealinearoperatorT:DT!CandletusmakethehypothesisthatDTisdenseinB.AnyelementofBisthencompletelydeterminedbyitsrestrictiontoDT.Nowconsider\050TCBdenedbyf`;mg2\050T,h`;Txi=hm;xi8x2DT:.4SincemisdeterminedbyitsrestrictiontoDT,weseethat)]TJ/F13 6.974 Tf 72.173 3.615 Td[(=\050Tisindeedagraph.ItiseasytocheckthatitisalinearsubspaceofCB.InotherwordswehavedenedalineartransformationT:=T\050T

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10.7.SELF-ADJOINTOPERATORS.265whosedomainconsistsofall`2Csuchthatthereexistsanm2Bforwhichh`;Txi=hm;xi8x2DT.If`n!`andmn!mthenthedenitionofconvergenceinthesespacesimpliesthatforanyx2DTwehaveh`;Txi=limh`n;Txi=limhmn;xi=hm;xi:IfweletxrangeoverallofDTweconcludethat)]TJ/F13 6.974 Tf 92.165 3.616 Td[(isaclosedsubspaceofCB.InotherwordswehaveprovedTheorem10.6.1IfT:DT!CisalineartransformationwhosedomainDTisdenseinB,ithasawelldenedadjointTwhosegraphisgivenby.4.FurthermoreTisaclosedoperator.10.7Self-adjointoperators.NowletusrestricttothecasewhereB=C=HisaHilbertspace,sowemayidentifyB=C=HwithHviatheRieszrepresentationtheorem.IfT:DT!HisanoperatorwithDTdenseinHwemayidentifythedomainofTasconsistingofallfg;hg2HHsuchthatTx;g=x;h8x2DTandthenwriteTx;g=x;Tg8x2DT;g2DT:Wenowcometothecentraldenition:AnoperatorAdenedonadomainDAHiscalledself-adjointifDAisdenseinH,DA=DA,andAx=Ax8x2DA.TheconditionsaboutthedomainDAarerathersubtle,andweshallgointosomeoftheirsubtletiesinalaterlecture.Forthemomentwerecordoneimme-diateconsequenceofthetheoremoftheprecedingsection:Proposition10.7.1Anyselfadjointoperatorisclosed.Ifwecombinethispropositionwiththeclosedgraphtheoremwhichassertsthataclosedoperatordenedonthewholespacemustbebounded,wederiveafamoustheoremofHellingerandToeplitzwhichassertsthatanyselfadjointoperatordenedonthewholeHilbertspacemustbebounded.Thisshowsthatforself-adjointoperators,beinggloballydenedandbeingboundedamounttothesamething.Atthetimeofitsappearanceintheseconddecadeofthetwentiethcentury,thistheoremofHellingerandToeplitzwasconsideredanastoundingresult.ItwasonlyaftertheworkofBanach,inparticulartheproofoftheclosedgraphtheorem,thatthisresultcouldbeputinproperperspective.

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266CHAPTER10.THESPECTRALTHEOREM.10.8Theresolvent.Thefollowingtheoremwillbecentralforus.Theorem10.8.1LetAbeaself-adjointoperatoronaHilbertspaceHwithdomainD=DA.Letc=+i;6=0beacomplexnumberwithnon-zeroimaginarypart.ThencI)]TJ/F11 9.963 Tf 9.963 0 Td[(A:DA!Hisbijective.FurthermoretheinversetransformationcI)]TJ/F11 9.963 Tf 9.962 0 Td[(A)]TJ/F7 6.974 Tf 6.227 0 Td[(1:H!DAisboundedandinfactkcI)]TJ/F11 9.963 Tf 9.963 0 Td[(A)]TJ/F7 6.974 Tf 6.227 0 Td[(1k1 jj:.5Remark.Inthecaseofaboundedselfadjointoperatorthisisanimmediateconsequenceofthespectraltheorem,morepreciselyofthefactthatGelfandtransformisanisometricisomorphismoftheclosedalgebrageneratedbyAwiththealgebraCSpecA.Indeed,thefunction7!1=c)]TJ/F11 9.963 Tf 10.528 0 Td[(isboundedonthewholerealaxiswithsupremum1=jj.SinceSpecARweconcludethatcI)]TJ/F11 9.963 Tf 10.152 0 Td[(A)]TJ/F7 6.974 Tf 6.227 0 Td[(1existsanditsnormsatises.5.Wewillnowgiveadirectproofofthistheoremvalidforgeneralself-adjointoperators,andwillusethistheoremfortheproofofthespectraltheoreminthegeneralcase.Proof.Letg2DAandsetf=cI)]TJ/F11 9.963 Tf 9.963 0 Td[(Ag=[I)]TJ/F11 9.963 Tf 9.963 0 Td[(A]g+ig:Thenkfk2=f;f=k[cI)]TJ/F11 9.963 Tf 9.962 0 Td[(A]gk2+2kgk2+[I)]TJ/F11 9.963 Tf 9.963 0 Td[(A]g;ig+ig;[I)]TJ/F11 9.963 Tf 9.963 0 Td[(A]g:Iclaimthattheselasttwotermscancel.Indeed,sinceg2DAandAisselfadjointwehaveg;[I)]TJ/F11 9.963 Tf 9.962 0 Td[(A]g=[I)]TJ/F11 9.963 Tf 9.963 0 Td[(A]g;g=[I)]TJ/F11 9.963 Tf 9.962 0 Td[(A]g;gsinceisreal.Hence[I)]TJ/F11 9.963 Tf 9.963 0 Td[(A]g;ig=)]TJ/F11 9.963 Tf 7.748 0 Td[(ig;[I)]TJ/F11 9.963 Tf 9.963 0 Td[(A]g:Wehavethusprovedthatkfk2=kI)]TJ/F11 9.963 Tf 9.962 0 Td[(Agk2+2kgk2:.6

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10.8.THERESOLVENT.267Inparticularkfk22kgk2forallg2DA.Sincejj>0,weseethatf=0g=0socI)]TJ/F11 9.963 Tf 10.531 0 Td[(AisinjectiveonDA,andfurthermorethatthatcI)]TJ/F11 9.963 Tf 10.209 0 Td[(A)]TJ/F7 6.974 Tf 6.226 0 Td[(1whichisdenedonimcI)]TJ/F11 9.963 Tf 9.962 0 Td[(Asatises.5.WemustshowthatthisimageisallofH.Firstweshowthattheimageisdense.Forthisitisenoughtoshowthatthereisnoh6=02HwhichisorthogonaltoimcI)]TJ/F11 9.963 Tf 9.963 0 Td[(A.Sosupposethat[cI)]TJ/F11 9.963 Tf 9.962 0 Td[(A]g;h=08g2DA:Theng; ch=cg;h=Ag;h8g2DAwhichsaysthath2DAandAh= ch.ButAisselfadjointsoh2DAandAh= ch.Thus ch;h= ch;h=Ah;h=h;Ah=h; ch=ch;h:Sincec6= cthisisimpossibleunlessh=0.WehavenowestablishedthattheimageofcI)]TJ/F11 9.963 Tf 9.963 0 Td[(AisdenseinH.WenowprovethatitisallofH.Soletf2H.Weknowthatwecanndfn=cI)]TJ/F11 9.963 Tf 9.963 0 Td[(Agn;gn2DAwithfn!f:Thesequencefnisconvergent,henceCauchy,andfrom.5appliedtoele-mentsofDAweknowthatkgm)]TJ/F11 9.963 Tf 9.963 0 Td[(gnkjj)]TJ/F7 6.974 Tf 6.227 0 Td[(1kfn)]TJ/F11 9.963 Tf 9.962 0 Td[(fmk:HencethesequencefgngisCauchy,sogn!gforsomeg2H.ButweknowthatAisaclosedoperator.Henceg2DAandcI)]TJ/F11 9.963 Tf 9.963 0 Td[(Ag=f.QEDTheoperatorRz=RzA=zI)]TJ/F11 9.963 Tf 9.963 0 Td[(A)]TJ/F7 6.974 Tf 6.227 0 Td[(1iscalledtheresolventofAwhenitexistsasaboundedoperator.Thesetofz2Cforwhichtheresolventexistsiscalledtheresolventsetandthecomplementoftheresolventsetiscalledthespectrumoftheoperator.Theprecedingtheoremassertsthatthespectrumofaself-adjointoperatorisasubsetoftherealnumbers.Letzandwbothbelongtotheresolventset.WehavewI)]TJ/F11 9.963 Tf 9.962 0 Td[(A=w)]TJ/F11 9.963 Tf 9.963 0 Td[(zI+zI)]TJ/F11 9.963 Tf 9.963 0 Td[(A:MultiplyingthisequationontheleftbyRwgivesI=w)]TJ/F11 9.963 Tf 9.962 0 Td[(zRw+RwzI)]TJ/F11 9.963 Tf 9.962 0 Td[(A;

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268CHAPTER10.THESPECTRALTHEOREM.andmultiplyingthisontherightbyRzgivesRz)]TJ/F11 9.963 Tf 9.963 0 Td[(Rw=w)]TJ/F11 9.963 Tf 9.963 0 Td[(zRwRz:FromthisitfollowsinterchangingzandwthatRzRw=RwRz,inotherwordsallresolventsRzcommutewithoneanotherandwecanalsowritetheprecedingequationasRz)]TJ/F11 9.963 Tf 9.962 0 Td[(Rw=w)]TJ/F11 9.963 Tf 9.963 0 Td[(zRzRw:.7Thisequation,whichisknownastheresolventequationdatesbacktothetheoryofintegralequationsinthenineteenthcentury.Itfollowsfromtheresolventequationthatz7!RzforxedAisacontin-uousfunctionofz.Onceweknowthattheresolventisacontinuousfunctionofz,wemaydividetheresolventequationbyz)]TJ/F11 9.963 Tf 9.46 0 Td[(wifz6=wand,ifwisinteriortotheresolventset,concludethatlimz!wRz)]TJ/F11 9.963 Tf 9.962 0 Td[(Rw z)]TJ/F11 9.963 Tf 9.962 0 Td[(w=)]TJ/F11 9.963 Tf 7.749 0 Td[(R2w:Thissaysthatthederivativeinthecomplexsense"oftheresolventexistsandisgivenby)]TJ/F11 9.963 Tf 7.748 0 Td[(R2z.Inotherwords,theresolventisaholomorphicoperatorvalued"functionofz.Toemphasizethisholomorphiccharacteroftheresolvent,wehaveProposition10.8.1Letzbelongtotheresolventset.ThetheopendiskofradiuskRzk)]TJ/F7 6.974 Tf 6.227 0 Td[(1aboutzbelongstotheresolventsetandonthisdiskwehaveRw=RzI+z)]TJ/F11 9.963 Tf 9.962 0 Td[(wRz+z)]TJ/F11 9.963 Tf 9.963 0 Td[(w2R2z+:.8Proof.Theseriesontherightconvergesintheuniformtopologysincejz)]TJ/F11 9.963 Tf -335.963 -11.955 Td[(wj
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10.9.THEMULTIPLICATIONOPERATORFORMOFTHESPECTRALTHEOREM.269Theorem10.9.1LetBbeacommutativeclosedself-adjointsubalgebraofthealgebraofallboundedoperatorsonaseparableHilbertspaceH.ThenthereexistsameasurespaceM;F;withM<1,aunitaryisomorphismW:H!L2M;;andamapB!boundedmeasurablefunctionsonM;T7!~Tsuchthat[WTW)]TJ/F7 6.974 Tf 6.227 0 Td[(1f]m=~Tmfm:Infact,McanbetakentobeaniteorcountabledisjointunionofM=MspecBM=N[1Mi;Mi=MN2Z+[1and~Tm=^Tmifm2Mi=M:Inshort,thetheoremsaysthatanysuchBisisomorphictoanalgebraofmultiplicationoperatorsonanL2space.Weprovethetheoremintwostages.10.9.1Cyclicvectors.Anelementx2HiscalledacyclicvectorforBifBx=H.InmoremundanetermsthissaysthatthespaceoflinearcombinationsofthevectorsTx;T2BaredenseinH.Forexample,ifBconsistsofallmultiplesoftheidentityoperator,thenBxconsistsofallmultiplesofx,soBcannothaveacyclicvectorunlessHisonedimensional.Moregenerally,ifHisnitedimensionalandBisthealgebrageneratedbyaself-adjointoperator,thenBcannothaveacyclicvectorifAhasarepeatedeigenvalue.Proposition10.9.1SupposethatxisacyclicvectorforB.ThenitisacyclicvectorfortheprojectionvaluedmeasurePonMassociatedtoBinthesensethatthelinearcombinationsofthevectorsPUxaredenseinHasUrangesovertheBorelsetsonM.Proof.Supposenot.Thenthereexistsanon-zeroy2HsuchthatPUx;y=0forallBorelsubsetUofM.ThenTx;y=ZM^TdPUx;y=0

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270CHAPTER10.THESPECTRALTHEOREM.whichcontradictstheassumptionthatthelinearcombinationsoftheTxaredenseinH.QEDLetuscontinuewiththeassumptionthatxisacyclicvectorforB.Let=x;xsoU=PUx;x:ThisisanitemeasureonM,infactM=kxk2:.9WewillconstructaunitaryisomorphismofHwithL2M;startingwiththeassignmentWx=1=1M:WewouldlikethistobeaBmorphism,evenamorphismfortheactionofmultiplicationbyboundedBorelfunctions.ThisforcesthedenitionWPUx=1U1=1U:ThisthenforcesW[c1PU1x+cnPUnx]=sforanysimplefunctions=c11U1+cn1Un:Adirectcheckshowsthatthisiswelldenedforsimplefunctions.WecanwritethismapasW[Osx]=s;andanotherdirectcheckshowsthatkW[Osx]k=ksk2wherethenormontherightistheL2normrelativetothemeasure.SincethesimplefunctionsaredenseinL2M;andthevectorsOsxaredenseinHthisextendstoaunitaryisomorphismofHontoL2M;.Furthermore,W)]TJ/F7 6.974 Tf 6.227 0 Td[(1f=Ofxforanyf2L2M;.Forsimplefunctions,andthereforeforallf2L2M;wehaveW)]TJ/F7 6.974 Tf 6.226 0 Td[(1^Tf=O^Tfx=TOfx=TW)]TJ/F7 6.974 Tf 6.227 0 Td[(1forWTW)]TJ/F7 6.974 Tf 6.226 0 Td[(1f=^Tfwhichistheassertionofthetheorem.Inotherwordswehaveprovedthetheoremundertheassumptionoftheexistenceofacyclicvector.

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10.9.THEMULTIPLICATIONOPERATORFORMOFTHESPECTRALTHEOREM.27110.9.2Thegeneralcase.Startwithanynon-zerovectorx1andconsiderH1=Bx1=theclosureoflinearcombinationsofTx1;T2B.ThespaceH1isaclosedsubspaceofHwhichisinvariantunderB,i.e.TH1H18T2B.ThereforethespaceH?1isalsoinvariantunderBsinceifx1;y=0thenx1;Ty=Tx1;y=0sinceT2B:NowifH1=Hwearedone,sincex1isacyclicvectorforBactingonH1.Ifnotchooseanon-zerox22H2andrepeattheprocess.Wecanchooseacollectionofnon-zerovectorsziwhoselinearcombinationsaredenseinH-thisistheseparabilityassumption.Sowemaychooseourxitobeobtainedfromorthogonalprojectionsappliedtothezi.InotherwordswehaveH=H1H2H3wherethisiseitheraniteoracountableHilbertspacecompleteddirectsum.LetusalsotakecaretochooseourxnsothatXkxnk2<1whichwecando,sincecnxnisjustasgoodasxnforanycn6=0.WehaveaunitaryisomorphismofHnwithL2M;nwherenU=PUxn;xn.Inparticular,nM=kxnk2:SoifwetakeMtobethedisjointunionofcopiesMnofMeachwithmeasurenthenthetotalmeasureofMisniteandL2M=ML2Mn;nwherethisiseitheranitedirectsumoraHilbertspacecompletionofacountabledirectsum.Thusthetheoremforthecycliccaseimpliesthetheoremforthegeneralcase.QED10.9.3Thespectraltheoremforunboundedself-adjointoperators,multiplicationoperatorform.WenowletAbeapossiblyunboundedself-adjointoperator,andweapplytheprevioustheoremtothealgebrageneratedbytheboundedoperatorsiI)]TJ/F11 9.963 Tf 7.879 0 Td[(A)]TJ/F7 6.974 Tf 6.227 0 Td[(1whicharetheadjointsofoneanother.Observethatthereisnonon-zerovectory2HsuchthatA+iI)]TJ/F7 6.974 Tf 6.227 0 Td[(1y=0:Indeedifsuchay2Hexisted,wewouldhave0=x;A+iI)]TJ/F7 6.974 Tf 6.226 0 Td[(1y=A)]TJ/F11 9.963 Tf 9.962 0 Td[(iI)]TJ/F7 6.974 Tf 6.227 0 Td[(1x;y=)]TJ/F8 9.963 Tf 7.749 0 Td[(iI)]TJ/F11 9.963 Tf 9.962 0 Td[(A)]TJ/F7 6.974 Tf 6.226 0 Td[(1x;y8x2H

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272CHAPTER10.THESPECTRALTHEOREM.andweknowthattheimageofiI)]TJ/F11 9.963 Tf 9.963 0 Td[(A)]TJ/F7 6.974 Tf 6.227 0 Td[(1isDAwhichisdenseinH.Nowconsiderthefunction)]TJ/F8 9.963 Tf 4.566 -8.07 Td[(A+iI)]TJ/F7 6.974 Tf 6.226 0 Td[(1~onMgivenbyTheorem10.9.1.Itcannotvanishonanysetofpositivemeasure,sinceanyfunctionsupportedonsuchasetwouldbeinthekerneloftheoperatorconsistingofmultiplicationby)]TJ/F8 9.963 Tf 4.566 -8.07 Td[(A+iI)]TJ/F7 6.974 Tf 6.227 0 Td[(1~.Thusthefunction~A:=)]TJ/F8 9.963 Tf 8.717 -8.07 Td[(A+iI)]TJ/F7 6.974 Tf 6.226 0 Td[(1~)]TJ/F7 6.974 Tf 6.227 0 Td[(1)]TJ/F11 9.963 Tf 9.963 0 Td[(iisnitealmosteverywhereonMrelativetothemeasurealthoughitmightandgenerallywillbeunbounded.OurplanistoshowthatundertheunitaryisomorphismWtheoperatorAgoesoverintomultiplicationby~A.FirstweshowProposition10.9.2x2DAifandonlyif~AWx2L2M;.Proof.Supposex2DA.Thenx=A+iI)]TJ/F7 6.974 Tf 6.227 0 Td[(1yforsomey2HandsoWx=)]TJ/F8 9.963 Tf 4.566 -8.07 Td[(A+iI)]TJ/F7 6.974 Tf 6.227 0 Td[(1~f;f=Wy:But~A)]TJ/F8 9.963 Tf 4.566 -8.07 Td[(A+iI)]TJ/F7 6.974 Tf 6.227 0 Td[(1~=1)]TJ/F11 9.963 Tf 9.963 0 Td[(ihwhereh=)]TJ/F8 9.963 Tf 4.567 -8.07 Td[(A+iI)]TJ/F7 6.974 Tf 6.227 0 Td[(1~isaboundedfunction.Thus~AWx2L2M;:Conversely,if~AWx2L2M;,then~A+iIWx2L2M;,whichmeansthatthereisay2HsuchthatWy=~A+iIWx.Therefore)]TJ/F8 9.963 Tf 4.566 -8.069 Td[(A+iI)]TJ/F7 6.974 Tf 6.226 0 Td[(1~~A+iIWx=Wxandhencex=A+iI)]TJ/F7 6.974 Tf 6.227 0 Td[(1y2DA:QEDProposition10.9.3Ifh2WDAthen~Ah=WAW)]TJ/F7 6.974 Tf 6.227 0 Td[(1h.Proof.Letx=W)]TJ/F7 6.974 Tf 6.227 0 Td[(1hwhichweknowbelongstoDAsowemaywritex=A+iI)]TJ/F7 6.974 Tf 6.227 0 Td[(1yforsomey2H,andhenceAx=y)]TJ/F11 9.963 Tf 9.963 0 Td[(ixandWy=)]TJ/F8 9.963 Tf 8.717 -8.07 Td[(A+iI)]TJ/F7 6.974 Tf 6.226 0 Td[(1~)]TJ/F7 6.974 Tf 6.227 0 Td[(1h:SoWAx=Wy)]TJ/F11 9.963 Tf 9.962 0 Td[(iWx=)]TJ/F8 9.963 Tf 8.717 -8.07 Td[(A+iI)]TJ/F7 6.974 Tf 6.227 0 Td[(1~)]TJ/F7 6.974 Tf 6.226 0 Td[(1h)]TJ/F11 9.963 Tf 9.963 0 Td[(ih=~AhQED

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10.9.THEMULTIPLICATIONOPERATORFORMOFTHESPECTRALTHEOREM.273Thefunction~AmustberealvaluedalmosteverywheresinceifitsimaginarypartwerepositiveornegativeonasetUofpositivemeasure,then~A1U;1U2wouldhavenon-zeroimaginarypartcontradictingthefactthatmultiplicationby~Aisaselfadjointoperator,beingunitarilyequivalenttotheselfadjointoperatorA.PuttingallthistogetherwegetTheorem10.9.2LetAbeaselfadjointoperatoronaseparableHilbertspaceH.ThenthereexistsanitemeasurespaceM;,aunitaryisomorphismW:H!L2M;andarealvaluedmeasurablefunction~Awhichisnitealmosteverywheresuchthatx2DAifandonlyif~AWx2L2M;andifh2WDAthen~Ah=WAW)]TJ/F7 6.974 Tf 6.226 0 Td[(1h.10.9.4Thefunctionalcalculus.LetfbeanyboundedBorelfunctiondenedonR.Thenf~AisaboundedfunctiondenedonM.MultiplicationbythisfunctionisaboundedoperatoronL2M;andhencecorrespondstoaboundedself-adjointoperatoronH.WithaslightabuseoflanguagewemightdenotethisoperatorbyOf~A.HoweverwewillusethemoresuggestivenotationfA:Themapf7!fAisanalgebraichomomorphism, fA=fA,kfAkkfk1wherethenormontheleftistheuniformoperatornormandthenormontherightisthesupnormonRifAx=xthenfAx=fx,iff0thenfA0intheoperatorsense,iffn!fpointwiseandifkfnk1isbounded,thenfnA!fAstrongly,andiffnisasequenceofBorelfunctionsonthelinesuchthatjfnjjjforallnandforall2R,andiffn!foreachxed2Rthenforeachx2DAfnAx!Ax:

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274CHAPTER10.THESPECTRALTHEOREM.Alloftheabovestatementsareobviousexceptperhapsforthelasttwowhichfollowfromthedominatedconvergencetheorem.Itisalsoclearfromthepre-cedingdiscussionthatthemapf7!fAisuniquelydeterminedbytheaboveproperties.Multiplicationbythefunctioneit~AisaunitaryoperatoronL2M;andeis~Aeit~A=eis+t~A:HencefromtheaboveweconcludeTheorem10.9.3[HalfofStone'stheorem.]ForanselfadjointoperatorAtheoperatoreitAgivenbythefunctionalcalculusasaboveisaunitaryoperatorandt7!eitAisaoneparametergroupofunitarytransformations.ThefullStone'stheoremassertsthatanyunitaryoneparametergroupsisofthisform.Wewilldiscussthislater.10.9.5Resolutionsoftheidentity.ForeachmeasurablesubsetXofthereallinewecanconsideritsindicatorfunction1Xandhence1XAwhichweshalldenotebyPX.InotherwordsPX:=1XA:ItfollowsfromtheabovethatPX=PXPXPY=PXYPX[Y=PX+PYifXY=0PX=s)]TJ/F8 9.963 Tf 9.963 0 Td[(limNX1PXiifXiXj=;ifi6=jandX=[XiP;=0PR=I:Foreachx;y2HwehavethecomplexvaluedmeasurePx;yX=PXx;yandforanyboundedBorelfunctionfwehavefAx;y=ZRfdPx;y:

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10.9.THEMULTIPLICATIONOPERATORFORMOFTHESPECTRALTHEOREM.275IfgisanunboundedcomplexvaluedBorelfunctiononRwedeneDgAtoconsistofthosex2HforwhichZRjgj2dPx;x<1:ThesetofsuchxisdenseinHandwedenegAonDAbygAx;y=ZRgdPx;yforx;y2DgAandtheRieszrepresentationtheorem.ThisiswrittensymbolicallyasgA=ZRgdP:Inthespecialcaseg=wewriteA=ZRdP:thisisthespectraltheoremforselfadjointoperators.IntheolderliteratureoneoftenseesthenotationE:=P;:AtranslationofthepropertiesofPintopropertiesofEisE2=E.10E=E.11
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276CHAPTER10.THESPECTRALTHEOREM.10.10TheRiesz-Dunfordcalculus.Supposethatwehaveacontinuousmapz7!Szdenedonsomeopensetofcomplexnumbers,whereSzisaboundedoperatoronsomexedBanachspaceandbycontinuity,wemeancontinuityrelativetotheuniformmetriconoperators.IfCisacontinuouspiecewisedierentiableormoregenerallyanyrectiablecurvelyinginthisopenset,andift7!ztisapiecewisesmoothorrectiableparametrizationofthiscurve,thenthemapt7!Sztiscontinuous.Foranypartition0=t0t1tn=1oftheunitintervalwecanformtheCauchyapproximatingsumnXi=1Sztizti)]TJ/F11 9.963 Tf 9.962 0 Td[(zti)]TJ/F7 6.974 Tf 6.227 0 Td[(1;andtheusualproofoftheexistenceoftheRiemannintegralshowsthatthistendstoalimitasthemeshbecomesmoreandmorerenedandthemeshdistancetendstozero.ThelimitisdenotedbyZCSzdzandthisnotationisjustiesbecausethechangeofvariablesformulaforanordinaryintegralshowsthatthisvaluedoesnotdependontheparametrization,butonlyontheorientationofthecurveC.WearegoingtoapplythistoSz=Rz,theresolventofanoperator,andthemainequationsweshallusearetheresolventequation.7andthepowerseriesfortheresolvent.8whichwerepeathere:Rz)]TJ/F11 9.963 Tf 9.962 0 Td[(Rw=w)]TJ/F11 9.963 Tf 9.963 0 Td[(zRzRwandRw=RzI+z)]TJ/F11 9.963 Tf 9.962 0 Td[(wRz+z)]TJ/F11 9.963 Tf 9.963 0 Td[(w2R2z+:Weprovedthattheresolventofaself-adjointoperatorexistsforallnon-realvaluesofz.ButalotofthetheorygoesoverfortheresolventRz=Rz;T=zI)]TJ/F11 9.963 Tf 9.963 0 Td[(T)]TJ/F7 6.974 Tf 6.227 0 Td[(1whereTisanarbitraryoperatoronaBanachspace,solongaswerestrictourselvestotheresolventset,i.e.thesetwheretheresolventexistsasaboundedoperator.So,followingLorchSpectralTheorywerstdevelopsomefactsaboutintegratingtheresolventinthemoregeneralBanachspacesettingwhereourprincipalapplicationwillbetothecasewhereTisaboundedoperator.Forexample,supposethatCisasimpleclosedcurvecontainedinthediskofconvergenceaboutzof.8i.e.oftheabovepowerseriesforRw.Thenwecanintegratetheseriestermbyterm.ButZCz)]TJ/F11 9.963 Tf 9.962 0 Td[(wndw=0

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10.10.THERIESZ-DUNFORDCALCULUS.277foralln6=)]TJ/F8 9.963 Tf 7.749 0 Td[(1soZCRwdw=0:Bytheusualmethodofbreakinganyanydeformationupintoasuccessionofsmalldeformationsandthenbreakinganysmalldeformationupintoasequenceofsmallrectangles"weconcludeTheorem10.10.1IftwocurvesC0andC1lieintheresolventsetandarehomotopicbyafamilyCtofcurveslyingentirelyintheresolventsetthenZC0Rzdz=ZC1Rzdz:Herearesomeimmediateconsequencesofthiselementaryresult.SupposethatTisaboundedoperatorandjzj>kTk.ThenzI)]TJ/F11 9.963 Tf 9.963 0 Td[(T)]TJ/F7 6.974 Tf 6.227 0 Td[(1=z)]TJ/F7 6.974 Tf 6.226 0 Td[(1I)]TJ/F11 9.963 Tf 9.963 0 Td[(z)]TJ/F7 6.974 Tf 6.227 0 Td[(1T)]TJ/F7 6.974 Tf 6.227 0 Td[(1=z)]TJ/F7 6.974 Tf 6.226 0 Td[(1I+z)]TJ/F7 6.974 Tf 6.227 0 Td[(1T+z)]TJ/F7 6.974 Tf 6.226 0 Td[(2T2+existsbecausetheseriesinparenthesesconvergesintheuniformmetric.Inotherwords,allpointsinthecomplexplaneoutsidethediskofradiuskTklieintheresolventsetofT.FromthisitfollowsthatthespectrumofanyboundedoperatorcannotbeemptyiftheBanachspaceisnotf0g.Recallthethespectrumisthecomplementoftheresolventset.Indeed,iftheresolventsetwerethewholeplane,thenthecircleofradiuszeroabouttheoriginwouldbehomotopictoacircleofradius>kTkviaahomotopylyingentirelyintheresolventset.IntegratingRzaroundthecircleofradiuszerogives0.Wecanintegratearoundalargecircleusingtheabovepowerseries.Inperformingthisintegration,alltermsvanishexcepttherstwhichgive2iIbytheusualCauchyintegralorbydirectcomputation.Thus2I=0whichisimpossibleinanon-zerovectorspace.Hereisanotherveryimportantandeasyconsequenceoftheprecedingtheorem:Theorem10.10.2LetCbeasimpleclosedrectiablecurvelyingentirelyintheresolventsetofT.ThenP:=1 2iZCRzdz.17isaprojectionwhichcommuteswithT,i.e.P2=PandPT=TP:Proof.ChooseasimpleclosedcurveC0disjointfromCbutsucientlyclosetoCsoastobehomotopictoCviaahomotopylyingintheresolventset.ThusP=1 2iZC0Rwdw

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278CHAPTER10.THESPECTRALTHEOREM.andsoi2P2=ZCRzdzZC0Rwdw=ZCZC0Rw)]TJ/F11 9.963 Tf 9.962 0 Td[(Rzz)]TJ/F11 9.963 Tf 9.963 0 Td[(w)]TJ/F7 6.974 Tf 6.226 0 Td[(1dwdzwherewehaveusedtheresolventequation.7.Wewritethislastexpressionasasumoftwoterms,ZC0RwZC1 z)]TJ/F11 9.963 Tf 9.963 0 Td[(wdzdw)]TJ/F1 9.963 Tf 9.963 13.56 Td[(ZCRzZC01 z)]TJ/F11 9.963 Tf 9.963 0 Td[(wdwdz:SupposethatwechooseC0tolieentirelyinsideC.ThentherstexpressionaboveisjustiRC0Rwdwwhilethesecondexpressionvanishes,allbytheelementaryCauchyintegralof1=z)]TJ/F11 9.963 Tf 9.963 0 Td[(w.Thuswegeti2P2=2i2PorP2=P.ThisprovesthatPisaprojection.ItcommuteswithTbecauseitisanintegralwhoseintegrandRzcommuteswithTforallz.QEDThesameargumentprovesTheorem10.10.3LetCandC0besimpleclosedcurveseachlyinginthere-solventset,andletPandP0bethecorrespondingprojectionsgivenby.17.ThenPP0=0ifthecurveslieexteriortooneanotherwhilePP0=P0ifC0isinteriortoC.LetuswriteB0:=PB;B00=I)]TJ/F11 9.963 Tf 9.962 0 Td[(PBfortheimagesoftheprojectionsPandI)]TJ/F11 9.963 Tf 8.985 0 Td[(PwherePisgivenby.17.EachofthesespacesisinvariantunderTandhenceunderRzbecausePT=TPandhencePRz=RzP.ForanytransformationScommutingwithPletuswriteS0:=PS=SP=PSPandS00=I)]TJ/F11 9.963 Tf 9.962 0 Td[(PS=SI)]TJ/F11 9.963 Tf 9.962 0 Td[(P=I)]TJ/F11 9.963 Tf 9.963 0 Td[(PSI)]TJ/F11 9.963 Tf 9.963 0 Td[(PsothatS0andS0aretherestrictionsofStoB0andB00respectively.Forexample,wemayconsiderR0z=PRz=RzP.Forx02B0wehaveR0zzI)]TJ/F11 9.963 Tf 9.811 0 Td[(T0x0=RzPzI)]TJ/F11 9.963 Tf 9.811 0 Td[(TPx0=RzzI)]TJ/F11 9.963 Tf 9.812 0 Td[(TPx0=x0.InotherwordsR0zistheresolventofT0onB0andsimilarlyforR00z.SoifzisintheresolventsetforTitisintheresolventsetforT0andT00.Conversely,supposethatzisintheresolventsetforbothT0andT00.ThenthereexistsaninverseA1forzI0)]TJ/F11 9.963 Tf 9.654 0 Td[(T0onB0andaninverseA2forzI00)]TJ/F11 9.963 Tf 9.654 0 Td[(T00onB00andsoA1A2istheinverseofzI)]TJ/F11 9.963 Tf 9.963 0 Td[(TonB=B0B00.SoapointbelongstotheresolventsetofTifandonlyifitbelongstotheresolventsetofT0andofT00.Sincethespectrumisthecomplementoftheresolventset,wecansaythatapointbelongstothespectrumofTifandonlyifitbelongseithertothespectrumofT0orofT00:SpecT=SpecT0[SpecT00:

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10.11.LORCH'SPROOFOFTHESPECTRALTHEOREM.279WenowshowthatthisdecompositionisinfactthedecompositionofSpecTintothosepointswhichlieinsideCandoutsideC.SowemustshowthatifzliesexteriortoCthenitliesintheresolventsetofT0.ThiswillcertainlybetrueifwecanndatransformationAonBwhichcommuteswithTandsuchthatAzI)]TJ/F11 9.963 Tf 9.963 0 Td[(T=P.18forthenA0willbetheresolventatzofT0.NowzI)]TJ/F11 9.963 Tf 9.963 0 Td[(TRw=wI)]TJ/F11 9.963 Tf 9.963 0 Td[(TRw+z)]TJ/F11 9.963 Tf 9.963 0 Td[(wRw=I+z)]TJ/F11 9.963 Tf 9.963 0 Td[(wRwsozI)]TJ/F11 9.963 Tf 9.963 0 Td[(T1 2iZCRw1 z)]TJ/F11 9.963 Tf 9.963 0 Td[(wdw==1 2iZC1 z)]TJ/F11 9.963 Tf 9.963 0 Td[(wdwI+1 2iZCRwdw=0+P=P:WehavethusprovedTheorem10.10.4LetTbeaboundedlineartransformationonaBanachspaceandCasimpleclosedcurvelyinginitsresolventset.LetPbetheprojectiongivenby.17andB=B0B00;T=T0T00thecorrespondingdecompositionofBandofT.ThenSpecT0consistsofthosepointsofSpecTwhichlieinsideCandSpecT00consistsofthosepointsofSpecTwhichlieexteriortoC.WenowbegintohaveabetterunderstandingofStone'sformula:SupposeAisaself-adjointoperator.Weknowthatitsspectrumliesontherealaxis.Ifwedrawarectanglewhoseupperandlowersidesareparalleltotheaxis,andifitsverticalsidesdonotintersectSpecA,wewouldgetaprojectionontoasubspaceMofourHilbertspacewhichisinvariantunderA,andsuchthatthespectrumofAwhenrestrictedtoMliesintheintervalcutoutontherealaxisbyourrectangle.Theproblemishowtomakesenseofthisprocedurewhentheverticaledgesoftherectanglemightcutthroughthespectrum,inwhichcasetheintegral10.17mightnotevenbedened.ThisisresolvedbythemethodofLorchtheexpositionistakenfromhisbookwhichweexplaininthenextsection.10.11Lorch'sproofofthespectraltheorem.10.11.1Positiveoperators.RecallthatifAisaboundedself-adjointoperatoronaHilbertspaceHthenwewriteA0ifAx;x0forallx2Handbyaslightabuseoflanguagecall

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280CHAPTER10.THESPECTRALTHEOREM.suchanoperatorpositive.Clearlythesumoftwopositiveoperatorsispositiveasisthemultipleofapositiveoperatorbyanon-negativenumber.AlsowewriteA1A2fortwoselfadjointoperatorsifA1)]TJ/F11 9.963 Tf 9.963 0 Td[(A2ispositive.Proposition10.11.1IfAisaboundedself-adjointoperatorandAIthenA)]TJ/F7 6.974 Tf 6.227 0 Td[(1existsandkA)]TJ/F7 6.974 Tf 6.227 0 Td[(1k1:Proof.WehavekAxkkxkAx;xx;x=kxk2sokAxkkxk8x2H:SoAisinjective,andhenceA)]TJ/F7 6.974 Tf 6.226 0 Td[(1isdenedonimAandisboundedby1there.WemustshowthatthisimageisallofH.IfyisorthogonaltoimAwehavex;Ay=Ax;y=08x2HsoAy=0soy;yAy;y=0andhencey=0.ThusimAisdenseinH.SupposethatAxn!z.ThenthexnformaCauchysequencebytheestimateaboveonkA)]TJ/F7 6.974 Tf 6.227 0 Td[(1kandsoxn!xandthecontinuityofAimpliesthatAx=z.QEDSupposethatA0.Thenforany>0wehaveA+II,andbythepropositionA+I)]TJ/F7 6.974 Tf 6.227 0 Td[(1exists,i.e.)]TJ/F11 9.963 Tf 7.749 0 Td[(belongstotheresolventsetofA.Sowehaveproved.Proposition10.11.2IfA0thenSpecA[0;1.Theorem10.11.1IfAisaself-adjointtransformationthenkAk1,)]TJ/F11 9.963 Tf 27.608 0 Td[(IAI:.19Proof.SupposekAk1.ThenusingCauchy-SchwarzandthenthedenitionofkAkweget[I)]TJ/F11 9.963 Tf 9.963 0 Td[(A]x;x=x;x)]TJ/F8 9.963 Tf 9.962 0 Td[(Ax;xkxk2)-222(kAxkkxkkxk2)-222(kAkkxk20soI)]TJ/F11 9.963 Tf 9.963 0 Td[(A0andappliedto)]TJ/F11 9.963 Tf 7.748 0 Td[(AgivesI+A0or)]TJ/F11 9.963 Tf 7.749 0 Td[(IAI.Conversely,supposethat)]TJ/F11 9.963 Tf 7.749 0 Td[(IAI.SinceI)]TJ/F11 9.963 Tf 10.787 0 Td[(A0weknowthatSpecA;1]andsinceI+A0wehaveSpecA)]TJ/F8 9.963 Tf 7.749 0 Td[(1;1].SoSpecA[)]TJ/F8 9.963 Tf 7.749 0 Td[(1;1]sothatthespectralradiusofAis1.ButforselfadjointoperatorswehavekA2k=kAk2andhencetheformulaforthespectralradiusgiveskAk1.QED

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10.11.LORCH'SPROOFOFTHESPECTRALTHEOREM.281Animmediatecorollaryofthetheoremisthefollowing:Supposethatisarealnumber.ThenkA)]TJ/F11 9.963 Tf 10.095 0 Td[(Ikisequivalentto)]TJ/F11 9.963 Tf 10.095 0 Td[(IA+I.SoonewayofinterpretingthespectraltheoremA=Z1dEistosaythatforanydoublyinnitesequence<)]TJ/F7 6.974 Tf 6.227 0 Td[(2<)]TJ/F7 6.974 Tf 6.227 0 Td[(1<0<1<2
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282CHAPTER10.THESPECTRALTHEOREM.10.11.3Partitionintopuretypes.Nowconsiderageneralself-adjointoperatorA,andletH1:=MNHilbertspacedirectsumandsetH2:=H?1:ThespaceH1andhencethespaceH2areinvariantunderAinthesensethatAmapsDAH1toH1andsimilarlyforH2.WeletPdenoteorthogonalprojectionontoH1soI)]TJ/F11 9.963 Tf 9.966 0 Td[(Pisorthogonalpro-jectionontoH2.WeclaimthatP[DA]=DAH1andI)]TJ/F11 9.963 Tf 9.963 0 Td[(P[DA]=DAH2:.20Supposethatx2DA.WemustshowthatPx2DAforthenx=Px+I)]TJ/F11 9.963 Tf 9.438 0 Td[(PxisadecompositionofeveryelementofDAintoasumofelementsofDAH1andDAH2.Bydenition,wecanndanorthonormalbasisofH1consistingofeigen-vectorsuiofA,andthenPx=Xaiuiai:=x;ui:Thesumontherightisingeneralinnite.Letydenoteanynitepartialsum.SinceeigenvectorsbelongtoDAweknowthaty2DA.WehaveA[x)]TJ/F11 9.963 Tf 9.962 0 Td[(y];Ay)]TJ/F8 9.963 Tf 9.963 0 Td[([x)]TJ/F11 9.963 Tf 9.962 0 Td[(y];A2y=0sincex)]TJ/F11 9.963 Tf 10.031 0 Td[(yisorthogonaltoalltheeigenvectorsoccurringintheexpressionfory.WethushavekAxk2=kAx)]TJ/F11 9.963 Tf 9.963 0 Td[(yk2+kAyk2FromthisweseeasweletthenumberoftermsinyincreasethatbothyconvergestoPxandtheAyconverge.HencePx2DAproving.20.LetA1denotetheoperatorArestrictedtoP[DA]=DAH1withsimilarnotationforA2.WeclaimthatA1isselfadjointasisA2.ClearlyDA1:=PDAisdenseinH1,foriftherewereavectory2H1orthogonaltoDA1itwouldbeorthogonaltoDAinHwhichisimpossible.SimilarlyDA2:=DAH2isdenseinH2.Nowsupposethaty1andz1areelementsofH1suchthatA1x1;y1=x1;z18x12DA1:SinceA1x1=Ax1andx1=x)]TJ/F11 9.963 Tf 9.672 0 Td[(x2forsomex2DA,andsincey1andz1areorthogonaltox2,wecanwritetheaboveequationasAx;y1=x;z18x2DAwhichimpliesthaty12DAH1=DA1andA1y1=Ay1=z1.Inotherwords,A1isself-adjoint.Similarly,soisA2.Wehavethusproved

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10.11.LORCH'SPROOFOFTHESPECTRALTHEOREM.283Theorem10.11.2LetAbeaself-adjointtransformationonaHilbertspaceH.ThenH=H1H2withself-adjointtransformationsA1onH1havingpurepointspectrumandA2onH2havingnopointspectrumsuchthatDA=DA1DA2andA=A1A2:Wehaveprovedthespectraltheoremforaselfadjointoperatorwithpurepointspectrum.Ourproofofthefullspectraltheoremwillbecompleteonceweproveitforoperatorswithnopointspectrum.10.11.4Completionoftheproof.InthissubsectionwewillassumethatAisaself-adjointoperatorwithnopointspectrum,i.e.noeigenvalues.Let0andn>0bepositiveintegers,andletKm;n:=1 2iZCz)]TJ/F11 9.963 Tf 9.963 0 Td[(mz)]TJ/F11 9.963 Tf 9.962 0 Td[(nRzdz:.21Infact,wewouldliketobeabletoconsidertheaboveintegralwhenm=n=0,inwhichcaseitshouldgiveusaprojectionontoasubspacewhereIAI.ButunfortunatelyiforbelongtoSpecAtheaboveintegralneednotconvergewithm=n=0.HoweverwedoknowthatkRzkjimzj)]TJ/F7 6.974 Tf 6.227 0 Td[(1sothattheblowupintheintegrandatandiskilledbyz)]TJ/F11 9.963 Tf 10.848 0 Td[(mand)]TJ/F11 9.963 Tf 10.666 0 Td[(znsincethecurvemakesnon-zeroanglewiththerealaxis.Sincethecurveissymmetricabouttherealaxis,theboundedoperatorKm;nisself-adjoint.Furthermore,modifyingthecurveCtoacurveC0lyinginsideC,againintersectingtherealaxisonlyatthepointsandandhavingtheseintersectionsatnon-zeroanglesdoesnotchangethevalue:Km;n.WewillnowproveasuccessionoffactsaboutKm;n:Km;nKm0;n0=Km+m0;n+n0:.22Proof.CalculatetheproductusingacurveC0forKm0;n0asindicatedabove.ThenusethefunctionalequationfortheresolventandCauchy'sin-tegralformulaexactlyasintheproofofTheorem10.10.2:i2Km;nKm0;n0=ZCZC0z)]TJ/F11 9.963 Tf 9.963 0 Td[(m)]TJ/F11 9.963 Tf 9.962 0 Td[(znw)]TJ/F11 9.963 Tf 9.962 0 Td[(m0)]TJ/F11 9.963 Tf 9.963 0 Td[(wn01 z)]TJ/F11 9.963 Tf 9.963 0 Td[(w[Rw)]TJ/F11 9.963 Tf 9.962 0 Td[(Rz]dzdw

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284CHAPTER10.THESPECTRALTHEOREM.whichwewriteasasumoftwointegrals,therstgivingi2Km+m0;n+n0andthesecondgivingzero.QEDAsimilarargumentsimilartotheproofofTheorem10.10.3showsthatKm;nK00m0;n0=0if;0;0=;:.23Proposition10.11.3Thereexistsaboundedself-adjointoperatorLm;nsuchthatLm;n2=Km;n:Proof.Thefunctionz7!z)]TJ/F11 9.963 Tf 9.662 0 Td[(m=2)]TJ/F11 9.963 Tf 9.661 0 Td[(zn=2isdenedandholomorphiconthecomplexplanewiththeclosedintervals;]and[;1removed.TheintegralLm;n=1 2iZCz)]TJ/F11 9.963 Tf 9.963 0 Td[(m=2)]TJ/F11 9.963 Tf 9.963 0 Td[(zn=2Rzdziswelldenedsince,ifm=1orn=1thesingularityisoftheformjimzj)]TJ/F6 4.981 Tf 7.423 2.677 Td[(1 2atworstwhichisintegrable.Thentheproofof.22appliestoprovetheproposition.QEDForeachcomplexzweknowthatRzx2DA.HenceA)]TJ/F11 9.963 Tf 9.963 0 Td[(IRzx=A)]TJ/F11 9.963 Tf 9.963 0 Td[(zIRzx+z)]TJ/F11 9.963 Tf 9.962 0 Td[(Rzx=x+z)]TJ/F11 9.963 Tf 9.963 0 Td[(Rzx:BywritingtheintegraldeningKm;nasalimitofapproximatingsums,weseethatA)]TJ/F11 9.963 Tf 9.333 0 Td[(IKm;nisdenedandthatitisgivenbythesumoftwointegrals,therstofwhichvanishesandthesecondgivesKm+1;n.WehavethusshownthatKm;nmapsHintoDAandA)]TJ/F11 9.963 Tf 9.963 0 Td[(IKm;n=Km+1;n:.24SimilarlyI)]TJ/F11 9.963 Tf 9.963 0 Td[(AKm;n=Km;n+1:.25Wealsohavex;xAx;xx;xforx2imKm;n:.26Proof.Wehave[A)]TJ/F11 9.963 Tf 9.962 0 Td[(I]Km;ny;Km;ny=Km+1;ny;Km;ny=Km;nKm+1;ny;y=Km+1;2ny;y=Lm+1;2n2y;y=Lm+1;2ny;Lm+1;2ny0:AIonimKm;n:

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10.11.LORCH'SPROOFOFTHESPECTRALTHEOREM.285AsimilarargumentshowsthatAIthere.QEDThusifwedeneMm;ntobetheclosureofimKm;nweseethatAisboundedwhenrestrictedtoMm;nandIAIthere.WeletNm;ndenotethekernelofKm;nsothatMm;nandNm;naretheorthogonalcomplementsofoneanother.SofarwehavenotmadeuseoftheassumptionthatAhasnopointspectrum.Hereiswherewewillusethisassumption:SinceA)]TJ/F11 9.963 Tf 9.962 0 Td[(IKm;n=Km+1;nweseethatifKm+1;nx=0wemusthaveA)]TJ/F11 9.963 Tf 8.903 0 Td[(IKm;nx=0which,byourassumptionimpliesthatKm;nx=0.Inotherwords,Proposition10.11.4ThespaceNm;n;andhenceitsorthogonalcom-plementMm;nisindependentofmandn.WewilldenotethecommonspaceMm;nbyM.WehaveprovedthatAisaboundedoperatorwhenrestrictedtoMandsatisesIAIonMthere.WenowclaimthatIf<
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286CHAPTER10.THESPECTRALTHEOREM.Ifwenowhaveadoublyinnitesequenceasinourreformulationofthespectraltheorem,andwesetMi:=Mii+1wehaveprovedthespectraltheoreminthenopointspectrumcase-andhenceinthegeneralcaseifweshowthatMMi=H:Inviewof.27itisenoughtoprovethattheclosureofthelimitofM)]TJ/F10 6.974 Tf 6.226 0 Td[(rrisallofHasr!1,or,whatamountstothesamething,ifyisperpendiculartoallK)]TJ/F10 6.974 Tf 6.227 0 Td[(rr;1xthenymustbezero.NowK)]TJ/F10 6.974 Tf 6.227 0 Td[(rr;1x;y=x;K)]TJ/F10 6.974 Tf 6.227 0 Td[(rr;1ysowemustshowthatifK)]TJ/F10 6.974 Tf 6.226 0 Td[(rry=0forallrtheny=0.NowK)]TJ/F10 6.974 Tf 6.226 0 Td[(rr=1 2iZCz+rr)]TJ/F11 9.963 Tf 9.962 0 Td[(zRzdz=)]TJ/F8 9.963 Tf 13.679 6.74 Td[(1 2iZz2)]TJ/F11 9.963 Tf 9.963 0 Td[(r2RzwherewemaytakeCtobethecircleofradiusrcenteredattheorigin.Wealsohave1=1 2ir2ZCr2)]TJ/F11 9.963 Tf 9.963 0 Td[(z2 zdz:Soy=1 2ir2ZCr2)]TJ/F11 9.963 Tf 9.963 0 Td[(z2[z)]TJ/F7 6.974 Tf 6.226 0 Td[(1I)]TJ/F11 9.963 Tf 9.962 0 Td[(Rz]dzy:NowzI)]TJ/F11 9.963 Tf 9.963 0 Td[(ARz=Iso)]TJ/F11 9.963 Tf 7.749 0 Td[(ARz=I)]TJ/F11 9.963 Tf 9.963 0 Td[(zRzorz)]TJ/F7 6.974 Tf 6.227 0 Td[(1I)]TJ/F11 9.963 Tf 9.963 0 Td[(Rz=)]TJ/F11 9.963 Tf 7.749 0 Td[(z)]TJ/F7 6.974 Tf 6.227 0 Td[(1ARzsopullingtheAoutfromundertheintegralsignwecanwritetheaboveequationasy=Agrwheregr=1 2ir2ZCr2)]TJ/F11 9.963 Tf 9.962 0 Td[(z2z)]TJ/F7 6.974 Tf 6.226 0 Td[(1Rzdzy:NowonCwehavez=reisoz2=r2e2i=r2cos2+isin2andhencez2)]TJ/F11 9.963 Tf 9.962 0 Td[(r2=r2cos2)]TJ/F8 9.963 Tf 9.962 0 Td[(1+isin2=2r2)]TJ/F8 9.963 Tf 9.41 0 Td[(sin2+isincos:NowkRzkjrsinj)]TJ/F7 6.974 Tf 6.227 0 Td[(1soweseethatkz2)]TJ/F11 9.963 Tf 9.963 0 Td[(r2Rzk4r:Sincejz)]TJ/F7 6.974 Tf 6.227 0 Td[(1j=r)]TJ/F7 6.974 Tf 6.227 0 Td[(1onC,wecanboundkgrkbykgrjr2)]TJ/F7 6.974 Tf 6.227 0 Td[(1r)]TJ/F7 6.974 Tf 6.226 0 Td[(14r2rkyk=4r)]TJ/F7 6.974 Tf 6.227 0 Td[(1kyk!0asr!1.Sincey=AgrandAisclosedbeingself-adjointweconcludethaty=0.ThisconcludesLorch'sproofofthespectraltheorem.

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10.12.CHARACTERIZINGOPERATORSWITHPURELYCONTINUOUSSPECTRUM.28710.12Characterizingoperatorswithpurelycon-tinuousspectrum.SupposethatAisaself-adjointoperatorwithonlycontinuousspectrum.LetE:=P;beitsspectralresolution.Forany2Hthefunction7!E;iscontinuous.Itisalsoamonotoneincreasingfunctionof.Forany>0wecanndasucientlynegativeasuchthatjEa;j<=2andasucientlylargebsuchthatkk2)]TJ/F8 9.963 Tf 8.785 0 Td[(En;<=2.Onthecompactinterval[a;b]anycon-tinuousfunctionisuniformlycontinuous.Thereforethefunction7!E;isuniformlycontinuousonR.NowletandbeelementsofHandconsidertheproductmeasuredE;dE;ontheplaneR2,the;plane.Lemma10.12.1Thediagonalline=hasmeasurezerorelativetotheaboveproductmeasure.Proof.Wemayassumethat6=0.Forany>0wecannda>0suchthatE+;)]TJ/F11 9.963 Tf 9.962 0 Td[(E)]TJ/F10 6.974 Tf 6.227 0 Td[(;< kk2forall2R.SoZRdE;Z+)]TJ/F10 6.974 Tf 6.227 0 Td[(dE;<:Thissaysthatthemeasureofthebandofwidthaboutthediagonalhasmeasurelessthat.Lettingshrinkto0showsthatthediagonallinehasmeasurezero.Wecanrestatethislemmamoreabstractlyasfollows:ConsidertheHilbertspaceH^HthecompletionofthetensorproductHH.TheEandEdetermineaprojectionvaluedmeasureQontheplanewithvaluesinH^H.ThespectralmeasureassociatedwiththeoperatorAI)]TJ/F11 9.963 Tf 9.722 0 Td[(IAisthenF:=Qf;j)]TJ/F11 9.963 Tf 9.962 0 Td[(
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288CHAPTER10.THESPECTRALTHEOREM.10.13Appendix.Theclosedgraphtheorem.LurkinginthebackgroundofourentirediscussionistheclosedgraphtheoremwhichsaysthatifaclosedlineartransformationfromoneBanachspacetoanotheriseverywheredened,itisinfactbounded.Wedidnotactuallyusethistheorem,butitsstatementandproofbyBanachgreatlyclariedthenotionofawhatanunboundedself-adjointoperatoris,andexplainedtheHellingerToeplitztheoremasImentionedearlier.SohereIwillgivethestandardproofofthistheoremessentiallyaBairecategorystyleargumenttakenfromLoomis.InwhatfollowsXandYwilldenoteBanachspaces,Bn:=BnX=fx2X;kxkngdenotestheballofradiusnabouttheorigininXandUr=BrY=fy2Y:kykrgtheballofradiusrabouttheorigininY.Lemma10.13.1LetT:X!Ybeaboundedeverywheredenedlineartransformation.IfT[B1]UrisdenseinUrthenUrT[B1]:Proof.ThesetT[B1]isclosed,soitwillbeenoughtoshowthatUr)]TJ/F10 6.974 Tf 6.227 0 Td[(T[B1]forany>0,or,whatisthesamething,thatUr1 1)]TJ/F11 9.963 Tf 9.962 0 Td[(T[B1]=T[B1 1)]TJ/F9 4.981 Tf 5.397 0 Td[(]:Sox>0.Letz2Ur.Sety0:=0,andchoosey12T[B1]Ursuchthatkz)]TJ/F11 9.963 Tf 10.079 0 Td[(y1k
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10.13.APPENDIX.THECLOSEDGRAPHTHEOREM.289Lemma10.13.2IfT[B1]isdenseinnoballofpositiveradiusinY,thenT[X]containsnoballofpositiveradiusinY.Proof.Underthehypothesesofthelemma,T[Bn]isalsodenseinnoballofpositiveradiusofY.SogivenanyballUY,wecanndaclosedballUr1;y1ofradiusr1abouty1suchthatUr1;y1UandisdisjointfromT[B1].Byinduction,wecanndanestedsequenceofballsUrn;ynUrn)]TJ/F6 4.981 Tf 5.396 0 Td[(1yn)]TJ/F6 4.981 Tf 5.397 0 Td[(1suchthatUrn;ynisdisjointformT[Bn]andcanalsoarrangethatrn!0.ChoosingapointineachoftheseballswegetaCauchysequencewhichconvergestoapointy2UwhichliesinnoneoftheT[Bn],i.e.y6inT[X].Sou6T[X].QEDTheorem10.13.1[Theboundedinversetheorem.]IfT:X!Yisboundedandbijective,thenT)]TJ/F7 6.974 Tf 6.227 0 Td[(1isbounded.Proof.ByLemma10.13.2,T[B1]isdenseinsomeballUr;y1andhenceT[B1+B1]=T[B1)]TJ/F11 9.963 Tf 9.963 0 Td[(B1]isdenseinaballofradiusrabouttheorigin.SinceB1+B1B2soT[B2]UrisdenseinUr.ByLemma10.13.1,thisimpliesthatT[B2]Uri.e.thatT)]TJ/F7 6.974 Tf 6.227 0 Td[(1[Ur]B2whichsaysthatkT)]TJ/F7 6.974 Tf 6.227 0 Td[(1k2 r:QEDTheorem10.13.2IfT:X!YisdenedonallofXandissuchthatgraphTisaclosedsubspaceofXY,thenTisbounded.Proof.Let)]TJ/F14 9.963 Tf 28.021 0 Td[(XYdenotethegraphofT.Byassumption,itisaclosedsubspaceoftheBanachspaceXYunderthenormkfx;ygk=kxk+kyk.So)-333(isaBanachspaceandtheprojection)]TJ/F14 9.963 Tf 8.994 0 Td[(!X;fx;yg7!xisbijectivebythedenitionofagraph,andhasnorm1.Soitsinverseisbounded.Similarlytheprojectionontothesecondfactorisbounded.SothecompositemapX!Yx7!fx;yg7!y=Txisbounded.QED

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290CHAPTER10.THESPECTRALTHEOREM.

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Chapter11Stone'stheoremRecallthatifAisaself-adjointoperatoronaHilbertspaceHwecanformtheoneparametergroupofunitaryoperatorsUt=eiAtbyvirtueofafunctionalcalculuswhichallowsustoconstructfAforanyboundedBorelfunctiondenedonRifweuseourrstproofofthespectraltheoremusingtheGelfandrepresentationtheoremorforanyfunctionholomor-phiconSpecAifweuseoursecondproof.Inanyevent,thespectraltheoremallowsustowriteUt=Z1eitdEandtoverifythatU=I;Us+t=UsUtandthatUdependscontinuouslyont.WecalledthisassertionthersthalfofStone'stheorem.Thesecondhalftobestatedmorepreciselybelowassertstheconverse:thatanyoneparametergroupofunitarytransformationscanbewrittenineither,henceboth,oftheaboveforms.TheideathatwewillfollowhingesonthefollowingelementarycomputationZ10e)]TJ/F10 6.974 Tf 6.226 0 Td[(z+ixtdt=e)]TJ/F10 6.974 Tf 6.226 0 Td[(z+ixt )]TJ/F11 9.963 Tf 7.748 0 Td[(z+ix1t=0=1 z)]TJ/F11 9.963 Tf 9.962 0 Td[(ixifRez>0validforanyrealnumberx.IfwesubstituteAforxandwriteUtinsteadofeiAtthissuggeststhatRz;iA=zI)]TJ/F11 9.963 Tf 9.962 0 Td[(iA)]TJ/F7 6.974 Tf 6.227 0 Td[(1=Z10e)]TJ/F10 6.974 Tf 6.227 0 Td[(ztUtdtifRez>0:SinceAisself-adjoint,itsspectrumisreal.SothespectrumofiAispurelyimaginary,andhenceanyznotontheimaginaryaxisisintheresolventsetofiA.TheaboveformulagivesusanexpressionfortheresolventintermsofUt291

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292CHAPTER11.STONE'STHEOREMforzlyingintherighthalfplane.Wecanobtainasimilarformulaforthelefthalfplane.Ourpreviousstudiesencourageustobelievethatoncewehavefoundalltheseputativeresolvents,itshouldnotbesohardtoreconstructAandthentheone-parametergroupUt=eiAt.Thisprogramworks!Butbecauseofsomeofthesubtletiesinvolvedinthedenitionofaself-adjointoperator,wewillbeginwithanimportanttheoremofvon-Neumannwhichwewillneed,andwhichwillalsogreatlyclarifyexactlywhatitmeanstobeself-adjoint.Asecondmatterwhichwilllengthentheseproceedingsisthatwhileweareatit,wewillproveamoregeneralversionofStone'stheoremvalidinanarbitraryFrechetspaceFandforuniformlyboundedsemigroups"ratherthanunitarygroups.Stoneprovedhistheoremtomeettheneedsofquantummechanics,whereaunitaryoneparametergroupcorresponds,viaWigner'stheoremtoaoneparametergroupofsymmetriesofthelogicofquantummechanics.Inmorepedestrianterms,unitaryoneparametergroupsarisefromsolutionsofSchrodinger'sequation.Butmanyotherimportantequations,forexampletheheatequationsinvarioussettings,requirethemoregeneralresult.ThetreatmentherewillessentiallyfollowthatofYosida,FunctionalAnalysisespeciallyChapterIX,Nelson,TopicsindynamicsI:Flows,andReedandSimonMethodsofMathematicalPhysics,II.FourierAnalysis,Self-Adjointness.11.1vonNeumann'sCayleytransform.ThegroupGl;Cofallinvertiblecomplextwobytwomatricesactsasfrac-tionallineartransformations"ontheplane:thematrixabcdsendsz7!az+b cz+d:TwodierentmatricesM1andM2givethesamefractionallineartransformationifandonlyifM1=M2forsomenon-zerocomplexnumberasisclearfromthedenition.Since1)]TJ/F11 9.963 Tf 7.748 0 Td[(i1iii)]TJ/F8 9.963 Tf 7.749 0 Td[(11=2i1001;thefractionallineartransformationscorrespondingto1)]TJ/F11 9.963 Tf 7.749 0 Td[(i1iandii)]TJ/F8 9.963 Tf 7.749 0 Td[(11areinversetooneanother.Itisatheoremintheelementarytheoryofcomplexvariablesthatfractionallineartransformationsaretheonlyorientationpreservingtransformationsoftheplanewhichcarrycirclesandlinesintocirclesandlines.Evenwithoutthisgeneraltheory,animmediatecomputationshowsthat1)]TJ/F11 9.963 Tf 7.749 0 Td[(i1icarriestheextendedrealaxisontotheunitcircle,andhenceitsinversecarriestheunit

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11.1.VONNEUMANN'SCAYLEYTRANSFORM.293circleontotheextendedrealaxis.Extended"meanswiththepoint1added.Indeedintheexpressionz=x)]TJ/F11 9.963 Tf 9.962 0 Td[(i x+iwhenxisreal,thenumeratoristhecomplexconjugateofthedenominatorandhencejzj=1.Underthistransformation,thecardinalpoints0;1;1oftheextendedrealaxisaremappedasfollows:07!)]TJ/F8 9.963 Tf 20.478 0 Td[(1;17!)]TJ/F11 9.963 Tf 20.479 0 Td[(i;and17!1:Wemightthinkofmultiplicationbyarealnumberasaself-adjointtrans-formationonaonedimensionalHilbertspace,andmultiplicationbyanumberofabsolutevalueoneasaunitaryoperatoronaonedimensionalHilbertspace.ThissuggestsingeneralthatifAisaselfadjointoperator,thenA)]TJ/F11 9.963 Tf 9.963 0 Td[(iIA+iI)]TJ/F7 6.974 Tf 6.227 0 Td[(1shouldbeunitary.Infact,wecanbemuchmoreprecise.Firstsomedenitions:AnoperatorU,possiblydenedonlyonasubspaceofaHilbertspaceHiscalledisometricifkUxk=kxkforallxinitsdomainofdenition.RecallthatinordertodenetheadjointTofanoperatorTitisnecessarythatitsdomainDTbedenseinH.OtherwisetheequationTx;y=x;Ty8x2DTdoesnotdetermineTy.AtransformationTinaHilbertspaceHiscalledsymmetricifDTisdenseinHsothatTisdenedandDTDTandTx=Tx8x2DT:AnotherwayofsayingthesamethingisTissymmetricifDTisdenseandTx;y=x;Ty8x;y2DT:Aself-adjointtransformationissymmetricsinceDT=DTisoneoftherequirementsofbeingself-adjoint.Exactlyhowandwhyasymmetricoperatorcanfailtobeself-adjointwillbeclariedintheensuingdiscussion.AlloftheresultsofthissectionareduetovonNeumann.Theorem11.1.1LetTbeaclosedsymmetricoperator.ThenT+iIx=0impliesthatx=0foranyx2DTsoT+iI)]TJ/F7 6.974 Tf 6.226 0 Td[(1existsasanoperatoronitsdomainDT+iI)]TJ/F7 6.974 Tf 6.227 0 Td[(1=imT+iI:ThisoperatorisboundedonitsdomainandtheoperatorUT:=T)]TJ/F11 9.963 Tf 9.963 0 Td[(iIT+iI)]TJ/F7 6.974 Tf 6.227 0 Td[(1withDUT=DT+iI)]TJ/F7 6.974 Tf 6.226 0 Td[(1=imT+iI

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294CHAPTER11.STONE'STHEOREMisisometricandclosed.TheoperatorI)]TJ/F11 9.963 Tf 9.963 0 Td[(UT)]TJ/F7 6.974 Tf 6.226 0 Td[(1existsandT=iUT+IUT)]TJ/F11 9.963 Tf 9.962 0 Td[(I)]TJ/F7 6.974 Tf 6.226 0 Td[(1:Inparticular,DT=imI)]TJ/F11 9.963 Tf 9.963 0 Td[(UTisdenseinH.Conversely,ifUisaclosedisometricoperatorsuchthatimI)]TJ/F11 9.963 Tf 9.597 0 Td[(UisdenseinHthenT=iU+II)]TJ/F11 9.963 Tf 9.962 0 Td[(U)]TJ/F7 6.974 Tf 6.227 0 Td[(1isasymmetricoperatorwithU=UT.Proof.Foranyx2DTwehave[TiI]x;[TiI]x=Tx;TxTx;ixix;Tx+x;x:ThemiddletermscancelbecauseTissymmetric.Hencek[TiI]xk2=kTxk2+kxk2:.1TakingtheplussignshowsthatT+iIx=0x=0andalsoshowsthatk[T+iI]xkkxksok[T+iI])]TJ/F7 6.974 Tf 6.226 0 Td[(1ykkykfory2[T+iI]DT:Ifwewritex=[T+iI])]TJ/F7 6.974 Tf 6.227 0 Td[(1ythen.1showsthatkUTyk2=kTxk2+kxk2=kyk2soUTisanisometrywithdomainconsistingofally=T+iIx,i.e.withdomainD[T+iI])]TJ/F7 6.974 Tf 6.227 0 Td[(1=im[T+iI].WenowshowthatUTisclosed.Sowemustshowthatifyn!yandzn!zwherezn=UTyntheny2DUTandUTy=z.TheynformaCauchysequenceandyn=[T+iI]xnsinceyn2imT+iI.From1.1weseethatthexnandtheTxnformaCauchysequence,soxn!xandTxn!wwhichimpliesthatx2DTandTx=wsinceTisassumedtobeclosed.ButthenT+iIx=w+ix=ysoy2DUTandw)]TJ/F11 9.963 Tf 9.19 0 Td[(ix=z=UTy.SowehaveshownthatUTisclosed.Subtractandaddtheequationsy=T+iIxUTy=T)]TJ/F11 9.963 Tf 9.962 0 Td[(iIxtoget1 2I)]TJ/F11 9.963 Tf 9.963 0 Td[(UTy=ixand1 2I+UTy=Tx:ThethirdequationshowsthatI)]TJ/F11 9.963 Tf 9.962 0 Td[(UTy=0x=0Tx=0I+UTy=0bythefourthequation.Soy=1 2[I)]TJ/F11 9.963 Tf 9.963 0 Td[(UT]y+[I+UT]y=0:

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11.1.VONNEUMANN'SCAYLEYTRANSFORM.295ThusI)]TJ/F11 9.963 Tf 10.475 0 Td[(UT)]TJ/F7 6.974 Tf 6.227 0 Td[(1exists,andy=I)]TJ/F11 9.963 Tf 10.474 0 Td[(UT)]TJ/F7 6.974 Tf 6.227 0 Td[(1ixfromthethirdofthefourequationsabove,andthelastequationgivesTx=1 2I+UTy=1 2I+UTI)]TJ/F11 9.963 Tf 9.963 0 Td[(UT)]TJ/F7 6.974 Tf 6.226 0 Td[(12ixorT=iI+UTI)]TJ/F11 9.963 Tf 9.962 0 Td[(UT)]TJ/F7 6.974 Tf 6.227 0 Td[(1asrequired.Furthermore,everyx2DTisinimI)]TJ/F11 9.963 Tf 9.375 0 Td[(UT.Thiscompletestheproofofthersthalfofthetheorem.NowsupposewestartwithanisometryUandsupposethatI)]TJ/F11 9.963 Tf 10.245 0 Td[(Uy=0forsomey2DU.Letz2imI)]TJ/F11 9.963 Tf 9.963 0 Td[(Usoz=w)]TJ/F11 9.963 Tf 9.963 0 Td[(Uwforsomew.Wehavey;z=y;w)]TJ/F8 9.963 Tf 9.963 0 Td[(y;Uw=Uy;Uw)]TJ/F8 9.963 Tf 9.963 0 Td[(y;Uw=Uy)]TJ/F11 9.963 Tf 9.962 0 Td[(y;Uw=0:SinceweareassumingthatimI)]TJ/F11 9.963 Tf 10.68 0 Td[(UisdenseinH,theconditiony;z=08z2imI)]TJ/F11 9.963 Tf 9.152 0 Td[(Uimpliesthaty=0.ThusI)]TJ/F11 9.963 Tf 9.152 0 Td[(U)]TJ/F7 6.974 Tf 6.227 0 Td[(1exists,andwemaydeneT=iI+UI)]TJ/F11 9.963 Tf 9.962 0 Td[(U)]TJ/F7 6.974 Tf 6.227 0 Td[(1withDT=D)]TJ/F8 9.963 Tf 4.566 -8.07 Td[(I)]TJ/F11 9.963 Tf 9.962 0 Td[(U)]TJ/F7 6.974 Tf 6.227 0 Td[(1=imI)]TJ/F11 9.963 Tf 9.963 0 Td[(UdenseinH.Supposethatx=I)]TJ/F11 9.963 Tf 10.138 0 Td[(Uu;y=I)]TJ/F11 9.963 Tf 10.138 0 Td[(Uv2DT=imI)]TJ/F11 9.963 Tf 10.138 0 Td[(U.ThenTx;y=iI+Uu;I)]TJ/F11 9.963 Tf 9.963 0 Td[(Uv=i[Uu;v)]TJ/F8 9.963 Tf 9.963 0 Td[(u;Uv]+i[u;v)]TJ/F8 9.963 Tf 9.962 0 Td[(Uu;Uv]:ThesecondexpressioninbracketsvanishessinceUisanisometry.SoTx;y=iUu;v)]TJ/F11 9.963 Tf 9.962 0 Td[(iu;Uv=)]TJ/F11 9.963 Tf 7.749 0 Td[(Uu;iv+u;iUv=[I)]TJ/F11 9.963 Tf 9.963 0 Td[(U]u;i[I+U]v=x;Ty:ThisshowsthatTissymmetric.ToseethatUT=Uweagainwritex=I)]TJ/F11 9.963 Tf 9.963 0 Td[(Uu.WehaveTx=iI+UusoT+iIx=2iuandT)]TJ/F11 9.963 Tf 9.963 0 Td[(iIx=2iUu:ThusDUT=f2iuu2DUg=DUandUTiu=2iUu=Uiu:ThusU=UT.WemuststillshowthatTisaclosedoperator.Tmapsxn=I)]TJ/F11 9.963 Tf 10.456 0 Td[(UuntoI+Uun.IfbothI)]TJ/F11 9.963 Tf 10.486 0 Td[(UunandI+Uunconverge,thenunandUunconverge.ThefactthatUisclosedimpliesthatifu=limunthenu2DUandUu=limUun.ButthisthatI)]TJ/F11 9.963 Tf 8.858 0 Td[(Uun!I)]TJ/F11 9.963 Tf 8.858 0 Td[(UuandiI+Uun!iI+UusoTisclosed.QEDThemapT7!UTfromsymmetricoperatorstoisometriesiscalledtheCayleytransform.

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296CHAPTER11.STONE'STHEOREMRecallthatanisometryisunitaryifitsdomainandimageareallofH.IfUisaclosedisometry,thenxn2DUandxn!ximpliesthatUxnisconvergent,hencex2DUandUx=limUxn.Similarly,ifUxn!ythenthexnareCauchy,henceconvergenttoanxwithUx=y.SoforanyclosedisometryUthespacesDU?andimU?measurehowfarUisfrombeingunitary:IftheybothreducetothezerosubspacethenUisunitary.ForaclosedsymmetricoperatorTdeneH+T=fx2HjTx=ixgandH)]TJ/F10 6.974 Tf 0 -7.19 Td[(T=fx2HjTx=)]TJ/F11 9.963 Tf 7.749 0 Td[(ixg:.2ThemaintheoremofthissectionisTheorem11.1.2LetTbeaclosedsymmetricoperatorandU=UTitsCayleytransform.ThenH+T=DU?andH)]TJ/F10 6.974 Tf 0 -7.19 Td[(T=imU?:Everyx2DTisuniquelyexpressibleasx=x0+x++x)]TJ/F35 9.963 Tf -204.241 -19.361 Td[(withx02DT;x+2H+Tandx)]TJ/F14 9.963 Tf 9.492 1.494 Td[(2H)]TJ/F10 6.974 Tf 0 -7.19 Td[(T;soTx=Tx0+ix+)]TJ/F11 9.963 Tf 9.962 0 Td[(ix)]TJ/F11 9.963 Tf 6.725 1.494 Td[(:Inparticular,TisselfadjointifandonlyifUisunitary.Proof.Tosaythatx2DU?=D)]TJ/F8 9.963 Tf 4.566 -8.07 Td[(T+iI)]TJ/F7 6.974 Tf 6.226 0 Td[(1?saysthatx;T+iIy=08y2DT:Thissaysthatx;Ty=)]TJ/F8 9.963 Tf 7.749 0 Td[(x;iy=ix;y8y2DT:Thisispreciselytheassertionthatx2DTandTx=ix.WecanreadtheseequationsbackwardstoconcludethatH+T=DU?.Similarly,ifx2imU?thenx;T)]TJ/F11 9.963 Tf 9.962 0 Td[(iIz=08z2DTimplyingTx=)]TJ/F11 9.963 Tf 7.748 0 Td[(ixandconversely.WeknowthatDUandimUareclosedsubspacesofHsoanyw2HcanbewrittenasthesumofanelementofDUandanelementofDU?.Takingw=T+iIxforsomex2DTgivesT+iIx=y0+x1;y02DU=imT+iI;x12DU?:Wecanwritey0=T+iIx0;x02DTsoT+iIx=T+iIx0+x1:SinceT=TonDTandTx1=ix1asx12DU?wehaveTx1+ix1=2ix1:

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11.1.VONNEUMANN'SCAYLEYTRANSFORM.297Soifwesetx+=1 2ix1wehavex1=T+iIx+;x+2DU?:soT+iIx=T+iIx0+x+orTx)]TJ/F11 9.963 Tf 9.962 0 Td[(x0)]TJ/F11 9.963 Tf 9.962 0 Td[(x+=)]TJ/F11 9.963 Tf 7.749 0 Td[(ix)]TJ/F11 9.963 Tf 9.963 0 Td[(x0)]TJ/F11 9.963 Tf 9.963 0 Td[(x+:Thisimpliesthatx)]TJ/F11 9.963 Tf 9.962 0 Td[(x0)]TJ/F11 9.963 Tf 9.962 0 Td[(x+2H)]TJ/F10 6.974 Tf 0 -7.19 Td[(T=imU?:Soifwesetx)]TJ/F8 9.963 Tf 9.492 1.494 Td[(:=x)]TJ/F11 9.963 Tf 9.963 0 Td[(x0)]TJ/F11 9.963 Tf 9.962 0 Td[(x+wegetthedesireddecompositionx=x0+x++x)]TJ/F8 9.963 Tf 6.725 1.495 Td[(.Toshowthatthedecompositionisunique,supposethatx0+x++x)]TJ/F8 9.963 Tf 9.492 1.495 Td[(=0:ApplyingT+iIgives0=T+iIx0+2ix+:ButT+iIx02DUandx+2DU?sobothtermsabovemustbezero,sox+=0.Also,fromtheprecedingtheoremweknowthatT+iIx0=0x0=0.Hencesincex0=0andx+=0wemustalsohavex)]TJ/F8 9.963 Tf 9.492 1.495 Td[(=0.QED11.1.1Anelementaryexample.TakeH=L2[0;1]relativetothestandardLebesguemeasure.Considertheoperator1 id dtwhichisdenedonallelementsofHwhosederivative,inthesenseofdistributions,isagaininL2[0;1].Foranytwosuchelementswehavetheintegrationbypartsformula1 id dtx;y=x y)]TJ/F11 9.963 Tf 9.963 0 Td[(x y+x;1 id dty:EventhoughingeneralthevalueatapointofanelementinL2makesnosense,ifxissuchthatx02L2then1 hRh0xtdtmakessense,andintegrationbypartsusingacontinuousrepresentativeforxshowsthatthelimitofthisexpressioniswelldenedandequaltoxforourcontinuousrepresentative.SupposewetakeT=1 id dtbutwithDTconsistingofthoseelementswhosederivativesbelongtoL2asabove,butwhichinadditionsatisfyx=x=0:

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298CHAPTER11.STONE'STHEOREMThisspaceisdenseinH=L2butifyisanyfunctionwhosederivativeisinH,weseefromtheintegrationbypartsformulathatTx;y=x;1 id dty:Inotherwords,usingtheRieszrepresentationtheorem,weseethatT=1 id dtdenedonallywithderivativesinL2.NoticethatTet=ietsoinfactthespacesHTarebothonedimensional.Foreachcomplexnumbereiofabsolutevalueonewecanndaselfadjointextension"AofT,thatisanoperatorAsuchthatDTDADTwithDA=DA;A=AandA=TonDT.Indeed,letDAconsistofallxwithderivativesinL2andwhichsatisfytheboundarycondition"x=eix:LetuscomputeAanditsdomain.SinceDTDA,ifAx;y=x;Aywemusthavey2DTandAy=1 id dty.ButthentheintegrationbypartsformulagivesAx;y)]TJ/F8 9.963 Tf 9.963 0 Td[(x;1 id dty=eix y)]TJ/F11 9.963 Tf 9.963 0 Td[(x y:Thiswillvanishforallx2DAifandonlyify2DA.SoweseethatAisselfadjoint.Themoralisthattoconstructaselfadjointoperatorfromadierentialoperatorwhichissymmetric,wemayhavetosupplementitwithappropriateboundaryconditions.Ontheotherhand,considerthesameoperator1 id dtconsideredasanun-boundedoperatoronL2R.Wetakeasitsdomainthesetofallelementsofx2L2RwhosedistributionalderivativesbelongtoL2Randsuchthatlimt!x=0.ThefunctionsetdonotbelongtoL2Randsoouroperatorisinfactself-adjoint.Sotheissueofwhetherornotwemustaddboundaryconditionsdependsonthenatureofthedomainwherethedierentialoperatoristobedened.AdeepanalysisofthisphenomenonforsecondorderordinarydierentialequationswasprovidedbyHermannWeylinapaperpublishedin1911.Itissafetosaythatmuchoftheprogressinthetheoryofself-adjointoperatorswasinnosmallmeasureinuencedbyadesiretounderstandandgeneralizetheresultsofthisfundamentalpaper.

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11.2.EQUIBOUNDEDSEMI-GROUPSONAFRECHETSPACE.29911.2Equiboundedsemi-groupsonaFrechetspace.AFrechetspaceFisavectorspacewithatopologydenedbyasequenceofsemi-normsandwhichiscomplete.AnimportantexampleistheSchwartzspaceS.LetFbesuchaspace.WewanttoconsideraoneparameterfamilyofoperatorsTtonFdenedforallt0andwhichsatisfythefollowingconditions:T0=ITtTs=Tt+slimt!t0Ttx=Tt0x8t00andx2F.ForanydeningseminormpthereisadeningseminormqandaconstantKsuchthatpTtxKqxforallt0andallx2F.Wecallsuchafamilyanequiboundedcontinuoussemigroup.Wewillusuallydroptheadjectivecontinuous"andevenequibounded"sincewewillnotbeconsideringanyotherkindofsemigroup.11.2.1Theinnitesimalgenerator.Wearegoingtobeginbyshowingthateverysuchsemigrouphasaninnites-imalgenerator",i.e.canbewritteninsomesenseasTt=eAt.Itisimportanttoobservethatwehavemadeaseriouschangeofconventioninthatwearedroppingtheithatwehaveuseduntilnow.Withthisnewnotation,forex-ample,theinnitesimalgeneratorofagroupofunitarytransformationswillbeaskew-adjointoperatorratherthanaself-adjointoperator.Inquantumme-chanics,whereanobservable"isaself-adjointoperator,thereisagoodreasonforemphasizingtheself-adjointoperators,andhenceincludingthei.Therearemanygoodreasonsfordeviatingfromthephysicists'notation,nottheleasthavingtodowiththetheoryofLiealgebras.Idonotwanttogointothesereasonsnow.Somewillemergefromtheensuingnotation.Butthepresenceorabsenceoftheiisaculturaldividebetweenphysicistsandmathematicians.SowedenetheoperatorAasAx=limt&01 tTt)]TJ/F11 9.963 Tf 9.962 0 Td[(Ix:ThatisAistheoperatordenedonthedomainDAconsistingofthosexforwhichthelimitexists.OurrsttaskistoshowthatDAisdenseinF.ForthiswebeginaspromisedwiththeputativeresolventRz:=Z10e)]TJ/F10 6.974 Tf 6.226 0 Td[(ztTtdt.3whichisdenedbytheboundednessandcontinuitypropertiesofTtforallzwithRez>0.WebeginbycheckingthateveryelementofimRzbelongsto

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300CHAPTER11.STONE'STHEOREMDA:Wehave1 hTh)]TJ/F11 9.963 Tf 9.962 0 Td[(IRzx=1 hZ10e)]TJ/F10 6.974 Tf 6.226 0 Td[(ztTt+hxdt)]TJ/F8 9.963 Tf 11.537 6.74 Td[(1 hZ10e)]TJ/F10 6.974 Tf 6.226 0 Td[(ztTtxdt=1 hZ1he)]TJ/F10 6.974 Tf 6.227 0 Td[(zr)]TJ/F10 6.974 Tf 6.227 0 Td[(hTrxdr)]TJ/F8 9.963 Tf 9.324 6.74 Td[(1 hZ10e)]TJ/F10 6.974 Tf 6.227 0 Td[(ztTtxdt=ezh)]TJ/F8 9.963 Tf 9.963 0 Td[(1 hZ1he)]TJ/F10 6.974 Tf 6.226 0 Td[(ztTtxdt)]TJ/F8 9.963 Tf 9.324 6.74 Td[(1 hZh0e)]TJ/F10 6.974 Tf 6.226 0 Td[(ztTtxdt=ezh)]TJ/F8 9.963 Tf 9.962 0 Td[(1 h"Rzx)]TJ/F1 9.963 Tf 9.962 13.56 Td[(Zh0e)]TJ/F10 6.974 Tf 6.227 0 Td[(ztTtdt#)]TJ/F8 9.963 Tf 11.538 6.74 Td[(1 hZh0e)]TJ/F10 6.974 Tf 6.226 0 Td[(ztTtxdt:Ifwenowleth!0,theintegralinsidethebrackettendstozero,andtheexpressionontherighttendstoxsinceT0=I.WethusseethatRzx2DAandARz=zRz)]TJ/F11 9.963 Tf 9.963 0 Td[(I;or,rewritingthisinamorefamiliarform,zI)]TJ/F11 9.963 Tf 9.963 0 Td[(ARz=I:.4ThisequationsaysthatRzisarightinverseforzI)]TJ/F11 9.963 Tf 10.069 0 Td[(A.Itwillrequirealotmoreworktoshowthatitisalsoaleftinverse.WewillrstprovethatDAisdenseinFbyshowingthatimRzisdense.Infact,takingstobereal,wewillshowthatlims!1sRsx=x8x2F:.5Indeed,Z10se)]TJ/F10 6.974 Tf 6.227 0 Td[(stdt=1foranys>0.SowecanwritesRsx)]TJ/F11 9.963 Tf 9.963 0 Td[(x=sZ10e)]TJ/F10 6.974 Tf 6.227 0 Td[(st[Ttx)]TJ/F11 9.963 Tf 9.962 0 Td[(x]dt:ApplyinganyseminormpweobtainpsRsx)]TJ/F11 9.963 Tf 9.963 0 Td[(xsZ10e)]TJ/F10 6.974 Tf 6.226 0 Td[(stpTtx)]TJ/F11 9.963 Tf 9.963 0 Td[(xdt:Forany>0wecan,bythecontinuityofTt,nda>0suchthatpTtx)]TJ/F11 9.963 Tf 9.963 0 Td[(x<80t:NowletuswritesZ10e)]TJ/F10 6.974 Tf 6.227 0 Td[(stpTtx)]TJ/F11 9.963 Tf 9.962 0 Td[(xdt=sZ0e)]TJ/F10 6.974 Tf 6.227 0 Td[(stpTtx)]TJ/F11 9.963 Tf 9.963 0 Td[(xdt+sZ1e)]TJ/F10 6.974 Tf 6.226 0 Td[(stpTtx)]TJ/F11 9.963 Tf 9.963 0 Td[(xdt:

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11.3.THEDIFFERENTIALEQUATION301TherstintegralisboundedbysZ0e)]TJ/F10 6.974 Tf 6.227 0 Td[(stdtsZ10e)]TJ/F10 6.974 Tf 6.226 0 Td[(stdt=:Astothesecondintegral,letMbeaboundforpTtx+pxwhichexistsbytheuniformboundednessofTt.ThetriangleinequalitysaysthatpTtx)]TJ/F11 9.963 Tf 9.991 0 Td[(xpTtx+pxsothesecondintegralisboundedbyMZ1se)]TJ/F10 6.974 Tf 6.227 0 Td[(stdt=Me)]TJ/F10 6.974 Tf 6.227 0 Td[(s:Thistendsto0ass!1,completingtheproofthatsRsx!xandhencethatDAisdenseinF.11.3ThedierentialequationTheorem11.3.1Ifx2DAthenforanyt>0limh!01 h[Tt+h)]TJ/F11 9.963 Tf 9.962 0 Td[(Tt]x=ATtx=TtAx:Incolloquialterms,wecanformulatethetheoremassayingthatd dtTt=ATt=TtAinthesensethattheappropriatelimitsexistwhenappliedtox2DA.Proof.SinceTtiscontinuousint,wehaveTtAx=Ttlimh&01 h[Th)]TJ/F11 9.963 Tf 9.963 0 Td[(I]x=limh&01 h[TtTh)]TJ/F11 9.963 Tf 9.962 0 Td[(Tt]x=limh&01 h[Tt+h)]TJ/F11 9.963 Tf 9.962 0 Td[(Tt]x=limh&01 h[Th)]TJ/F11 9.963 Tf 9.962 0 Td[(I]Ttxforx2DA.ThisshowsthatTtx2DAandlimh&01 h[Tt+h)]TJ/F11 9.963 Tf 9.962 0 Td[(Tt]x=ATtx=TtAx:Toprovethetheoremwemustshowthatwecanreplaceh&0byh!0.Ourstrategyistoshowthatwiththeinformationthatwealreadyhaveabouttheexistenceofrighthandedderivatives,wecanconcludethatTtx)]TJ/F11 9.963 Tf 9.962 0 Td[(x=Zt0TsAxds:

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302CHAPTER11.STONE'STHEOREMSinceTtiscontinuous,thisisenoughtogivethedesiredresult.Inordertoestablishtheaboveequality,itisenough,bytheHahn-Banachtheoremtoprovethatforany`2Fwehave`Ttx)]TJ/F11 9.963 Tf 9.963 0 Td[(`x=Zt0`TsAxds:Inturn,itisenoughtoprovethisequalityfortherealandimaginarypartsof`.Soitallboilsdowntoalemmainthetheoryoffunctionsofarealvariable:Lemma11.3.1Supposethatfisacontinuousrealvaluedfunctionoftwiththepropertythattherighthandderivatived+ dtf:=limh&0ft+h)]TJ/F11 9.963 Tf 9.963 0 Td[(ft h=gtexistsforalltandgtiscontinuous.Thenfisdierentiablewithf0=g.Proof.Werstprovethatd+ dtf0onaninterval[a;b]impliesthatfbfa.Supposenot.Thenthereexistsan>0suchthatfb)]TJ/F11 9.963 Tf 9.963 0 Td[(fa<)]TJ/F11 9.963 Tf 7.749 0 Td[(b)]TJ/F11 9.963 Tf 9.962 0 Td[(a:SetFt:=ft)]TJ/F11 9.963 Tf 9.962 0 Td[(fa+t)]TJ/F11 9.963 Tf 9.962 0 Td[(a:ThenFa=0andd+ dtF>0:Atathisimpliesthatthereissomec>anearawithFc>0.Ontheotherhand,sinceFb<0,andFiscontinuous,therewillbesomepoints0.Thusifd+ dtfmonaninterval[t1;t2]wemayapplytheaboveresulttoft)]TJ/F11 9.963 Tf 9.962 0 Td[(mttoconcludethatft2)]TJ/F11 9.963 Tf 9.963 0 Td[(ft1mt2)]TJ/F11 9.963 Tf 9.963 0 Td[(t1;andifd+ dtftMwecanapplytheaboveresulttoMt)]TJ/F11 9.963 Tf 9.281 0 Td[(fttoconcludethatft2)]TJ/F11 9.963 Tf 9.254 0 Td[(ft1Mt2)]TJ/F11 9.963 Tf 9.254 0 Td[(t1.Soifm=mingt=mind+ dtfontheinterval[t1;t2]andMisthemaximum,wehavemft2)]TJ/F11 9.963 Tf 9.962 0 Td[(ft1 t2)]TJ/F11 9.963 Tf 9.962 0 Td[(t1M:Sinceweareassumingthatgiscontinuous,thisisenoughtoprovethatfisindeeddierentiablewithderivativeg.QED

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11.3.THEDIFFERENTIALEQUATION30311.3.1Theresolvent.WehavealreadyveriedthatRz=Z10e)]TJ/F10 6.974 Tf 6.226 0 Td[(ztTtdtmapsFintoDAandsatiseszI)]TJ/F11 9.963 Tf 9.963 0 Td[(ARz=IforallzwithRez>0,cf.4.WeshallnowshowthatforthisrangeofzzI)]TJ/F11 9.963 Tf 9.962 0 Td[(Ax=0x=08x2DAsothatzI)]TJ/F11 9.963 Tf 9.963 0 Td[(A)]TJ/F7 6.974 Tf 6.227 0 Td[(1existsandthatitisgivenbyRz.SupposethatAx=zxx2DAandchoose`2Fwith`x=1.Considert:=`Ttx:Bytheresultoftheprecedingsectionweknowthatisadierentiablefunctionoftandsatisesthedierentialequation0t=`TtAx=`Ttzx=z`Ttx=zt;=1:Sot=eztwhichisimpossiblesincetisaboundedfunctionoftandtherighthandsideoftheaboveequationisnotboundedfort0sincetherealpartofzispositive.Wehavefrom.4thatzI)]TJ/F11 9.963 Tf 9.963 0 Td[(ARzzI)]TJ/F11 9.963 Tf 9.962 0 Td[(Ax=zI)]TJ/F11 9.963 Tf 9.962 0 Td[(AxandweknowthatRzzI)]TJ/F11 9.963 Tf 10.292 0 Td[(Ax2DA.FromtheinjectivityofzI)]TJ/F11 9.963 Tf 10.292 0 Td[(AweconcludethatRzzI)]TJ/F11 9.963 Tf 9.962 0 Td[(Ax=x.FromzI)]TJ/F11 9.963 Tf 9.485 0 Td[(ARz=IweseethatzI)]TJ/F11 9.963 Tf 9.485 0 Td[(AmapsimRzDAontoFsocertainlyzI)]TJ/F11 9.963 Tf 9.963 0 Td[(AmapsDAontoFbijectively.HenceimRz=DA;imzI)]TJ/F11 9.963 Tf 9.962 0 Td[(A=FandRz=zI)]TJ/F11 9.963 Tf 9.963 0 Td[(A)]TJ/F7 6.974 Tf 6.227 0 Td[(1:Wehavealreadyestablishedthefollowing:

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304CHAPTER11.STONE'STHEOREMTheresolventRz=Rz;A:=R10e)]TJ/F10 6.974 Tf 6.226 0 Td[(ztTtdtisdenedasastronglimitforRez>0and,forthisrangeofz:DA=imRz;A.6ARz;Ax=Rz;AAx=zRz;A)]TJ/F11 9.963 Tf 9.963 0 Td[(Ixx2DA.7ARz;Ax=zRz;A)]TJ/F11 9.963 Tf 9.963 0 Td[(Ix8x2F.8limz%1zRz;Ax=xforzreal8x2F:.9WealsohaveTheorem11.3.2TheoperatorAisclosed.Proof.Supposethatxn2DA;xn!xandyn!ywhereyn=Axn.Wemustshowthatx2DAandAx=y.Setzn:=I)]TJ/F11 9.963 Tf 9.963 0 Td[(Axnsozn!x)]TJ/F11 9.963 Tf 9.963 0 Td[(y:SinceR;A=I)]TJ/F11 9.963 Tf 9.963 0 Td[(A)]TJ/F7 6.974 Tf 6.227 0 Td[(1isaboundedoperator,weconcludethatx=limxn=limI)]TJ/F11 9.963 Tf 9.963 0 Td[(A)]TJ/F7 6.974 Tf 6.227 0 Td[(1zn=I)]TJ/F11 9.963 Tf 9.963 0 Td[(A)]TJ/F7 6.974 Tf 6.227 0 Td[(1x)]TJ/F11 9.963 Tf 9.963 0 Td[(y:From.6weseethatx2DAandfromtheprecedingequationthatI)]TJ/F11 9.963 Tf -335.963 -11.955 Td[(Ax=x)]TJ/F11 9.963 Tf 9.963 0 Td[(ysoAx=y.QEDApplicationtoStone'stheorem.WenowhaveenoughinformationtocompletetheproofofStone'stheorem:SupposethatUtisaone-parametergroupofunitarytransformationsonaHilbertspace.WehaveUtx;y=x;Ut)]TJ/F7 6.974 Tf 6.227 0 Td[(1y=x;U)]TJ/F11 9.963 Tf 7.748 0 Td[(tyandsodier-entiatingattheoriginshowsthattheinnitesimalgeneratorA,whichweknowtobeclosed,isskew-symmetric:Ax;y=x;Ay8x;y2DA:AlsotheresolventszI)]TJ/F11 9.963 Tf 10.291 0 Td[(A)]TJ/F7 6.974 Tf 6.226 0 Td[(1existforallzwhicharenotpurelyimaginary,andzI)]TJ/F11 9.963 Tf 9.963 0 Td[(AmapsDAontothewholeHilbertspaceH.WritingA=iTweseethatTissymmetricandthatitsCayleytransformUThaszerokernelandissurjective,i.e.isunitary.HenceTisself-adjoint.ThisprovesStone'stheoremthateveryoneparametergroupofunitarytrans-formationsisoftheformeiTtwithTself-adjoint.11.3.2Examples.Forr>0letJr:=I)]TJ/F11 9.963 Tf 9.963 0 Td[(r)]TJ/F7 6.974 Tf 6.227 0 Td[(1A)]TJ/F7 6.974 Tf 6.227 0 Td[(1=rRr;Asoby1.8wehaveAJr=rJr)]TJ/F11 9.963 Tf 9.962 0 Td[(I:.10

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11.3.THEDIFFERENTIALEQUATION305Translations.ConsidertheoneparametergroupoftranslationsactingonL2R:[Utx]s=xs)]TJ/F11 9.963 Tf 9.962 0 Td[(t:.11Thisisdenedforallx2Sandisanisometricisomorphismthere,soextendstoaunitaryoneparametergroupactingonL2R.Equallywell,wecantaketheaboveequationinthesenseofdistributions,whereitmakessenseforallelementsofS0,inparticularforallelementsofL2R.WeknowthatwecandierentiateinthedistributionalsensetoobtainA=)]TJ/F11 9.963 Tf 11.279 6.739 Td[(d dsastheinnitesimalgenerator"inthedistributionalsense.Letusseewhatthegeneraltheorygives.Letyr:=Jrxsoyrs=rZ10e)]TJ/F10 6.974 Tf 6.227 0 Td[(rtxs)]TJ/F11 9.963 Tf 9.963 0 Td[(tdt=rZse)]TJ/F10 6.974 Tf 6.227 0 Td[(rs)]TJ/F10 6.974 Tf 6.227 0 Td[(uxudu:Therighthandexpressionisadierentiablefunctionofsandy0rs=rxs)]TJ/F11 9.963 Tf 9.963 0 Td[(r2Zse)]TJ/F10 6.974 Tf 6.227 0 Td[(rs)]TJ/F10 6.974 Tf 6.227 0 Td[(uxudu=rxs)]TJ/F11 9.963 Tf 9.963 0 Td[(ryrs:Ontheotherhandweknowfrom11.10thatAyr=AJrx=ryr)]TJ/F11 9.963 Tf 9.963 0 Td[(x:PuttingthetwoequationstogethergivesA=)]TJ/F11 9.963 Tf 11.279 6.74 Td[(d dsasexpected.Thisisaskew-adjointoperatorinaccordancewithStone'stheorem.WecannowgobackandgiveanintuitiveexplanationofwhatgoeswrongwhenconsideringthissameoperatorAbutonL2[0;1]insteadofonL2R.Ifxisasmoothfunctionofcompactsupportlyingin0;1,thenxcannottellwhetheritistobethoughtofaslyinginL2[0;1]orL2R,sotheonlychoiceforaunitaryoneparametergroupactingonxatleastforsmallt>0iftheshifttotherightasgivenby.11.ButoncetislargeenoughthatthesupportofUtxhitstherightendpoint,1,thistransformationcannotcontinueasis.Theonlyhopeistohavewhatgoesout"therighthandsidecomein,insomeform,ontheleft,andunitaritynowrequiresthatZ10jxs)]TJ/F11 9.963 Tf 9.962 0 Td[(tj2dt=Z10jxtj2dtwherenowtheshiftin.11meansmod1.Thisstillallowsfreedominthechoiceofphasebetweentheexitingvalueofthexanditsincomingvalue.Thuswespecifyaunitaryoneparametergroupwhenwexachoiceofphaseastheeectofpassinggo".Thischoiceofphaseistheoriginofthethatareneededtointroduceinndingtheselfadjointextensionsof1 id dtactingonfunctionsvanishingattheboundary.

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306CHAPTER11.STONE'STHEOREMTheheatequation.LetFconsistoftheboundeduniformlycontinuousfunctionsonR.Fort>0dene[Ttx]s=1 p 2tZ1e)]TJ/F7 6.974 Tf 6.227 0 Td[(s)]TJ/F10 6.974 Tf 6.227 0 Td[(v=2txvdv:Inotherwords,Ttisconvolutionwithntu=1 p 2te)]TJ/F10 6.974 Tf 6.227 0 Td[(u2=2t:WehavealreadyveriedinourstudyoftheFouriertransformthatthisisacontinuoussemi-groupwhenwesetT0=IwhenactingonS.Infact,forx2S,wecantaketheFouriertransformandconcludethat[Ttx]^=e)]TJ/F10 6.974 Tf 6.227 0 Td[(i2t=2^x:Dierentiatingthiswithrespecttotandsettingt=0andtakingtheinverseFouriertransformshowsthatd dtTtxt=0=1 2d2 ds2xforx2S.WewishtoarriveatthesameresultforTtactingonF.ItiseasyenoughtoverifythattheoperatorsTtarecontinuousintheuniformnormandhenceextendtoanequiboundedsemigrouponF.WewillnowverifythattheinnitesimalgeneratorAofthissemigroupisA=1 2d2 ds2withdomainconsistingofalltwicedierentiablefunctions.Letussetyr=Jrxsoyrs=Z1xvZ10r1 p 2te)]TJ/F10 6.974 Tf 6.227 0 Td[(rt)]TJ/F7 6.974 Tf 6.227 0 Td[(s)]TJ/F10 6.974 Tf 6.227 0 Td[(v2=2tdtdv=Z1xvZ102p r1 p 2e)]TJ/F10 6.974 Tf 6.227 0 Td[(2)]TJ/F10 6.974 Tf 6.227 0 Td[(rs)]TJ/F10 6.974 Tf 6.226 0 Td[(v2=22ddvsettingt=2=r=Z1xvr=21 2e)]TJ 6.227 5.776 Td[(p 2rjs)]TJ/F10 6.974 Tf 6.227 0 Td[(vjdvsinceforanyc>0wehaveZ10e)]TJ/F7 6.974 Tf 6.226 0 Td[(2+c2=2d=p 2e)]TJ/F7 6.974 Tf 6.227 0 Td[(2c:.12Letmepostponethecalculationofthisintegraltotheendofthesubsection.Assumingtheevaluationofthisintegralwecanwriteyrs=r 21 2Z1sxve)]TJ 6.227 5.777 Td[(p 2rv)]TJ/F10 6.974 Tf 6.227 0 Td[(sdv+Zsxve)]TJ 6.226 5.777 Td[(p 2rs)]TJ/F10 6.974 Tf 6.226 0 Td[(vdv:

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11.3.THEDIFFERENTIALEQUATION307Thisisadierentiablefunctionofsandwecandierentiatetoobtainy0rs=rZ1sxve)]TJ 6.227 5.777 Td[(p 2rv)]TJ/F10 6.974 Tf 6.227 0 Td[(sdv)]TJ/F1 9.963 Tf 9.963 13.56 Td[(Zsxve)]TJ 6.227 5.777 Td[(p 2rs)]TJ/F10 6.974 Tf 6.227 0 Td[(vdv:Thisisalsodierentiableandcomputeitsderivativetoobtainy00rs=)]TJ/F8 9.963 Tf 7.749 0 Td[(2rxs+r3=2p 2Z1xve)]TJ 6.227 5.776 Td[(p 2rjv)]TJ/F10 6.974 Tf 6.227 0 Td[(sjdv;ory00r=2ryr)]TJ/F11 9.963 Tf 9.963 0 Td[(x:Comparingthiswith.10whichsaysthatAyr=ryr)]TJ/F11 9.963 Tf 8.823 0 Td[(xweseethatindeedA=1 2d2 ds2:Letusnowverifytheevaluationoftheintegralin.12:StartwiththeknownintegralZ10e)]TJ/F10 6.974 Tf 6.226 0 Td[(x2dx=p 2:Setx=)]TJ/F11 9.963 Tf 10.101 0 Td[(c=sothatdx=+c=2dandx=0correspondsto=p c.Thusp 2=Z1p ce)]TJ/F7 6.974 Tf 6.226 0 Td[()]TJ/F10 6.974 Tf 6.226 0 Td[(c=2+c=2d=e2cZ1p ce)]TJ/F7 6.974 Tf 6.226 0 Td[(2+c2=2+c=2d=e2cZ1p ce)]TJ/F7 6.974 Tf 6.227 0 Td[(2+c2=2d+Z1p ce)]TJ/F7 6.974 Tf 6.227 0 Td[(2+c2=2c 2d:Inthesecondintegralinsidethebracketssett=)]TJ/F11 9.963 Tf 7.749 0 Td[(c=sodt=c 2dandthissecondintegralbecomesZp c0e)]TJ/F7 6.974 Tf 6.227 0 Td[(t2+c2=t2dtandhencep 2=e2cZ10e)]TJ/F7 6.974 Tf 6.227 0 Td[(2+c2=2dwhichis11.12.Bochner'stheorem.AcomplexvaluedcontinuousfunctionFiscalledpositivedeniteifforeverycontinuousfunctionofcompactsupportwehaveZRZRFt)]TJ/F11 9.963 Tf 9.963 0 Td[(st sdtds0:.13

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308CHAPTER11.STONE'STHEOREMWecanwritethisasF? ; 0wheretheconvolutionistakeninthesenseofgeneralizedfunctions.IfwewriteF=^Gand =^thenbyPlancherelthisequationbecomesG;0orhG;jj2i0whichwillcertainlybetrueifGisanitenon-negativemeasure.Bochner'stheoremassertstheconverse:thatanypositivedenitefunctionistheFouriertransformofanitenon-negativemeasure.WeshallfollowYosidapp.346-347inshowingthatStone'stheoremimpliesBochner'stheorem.LetFdenotethespaceoffunctionsonRwhichhavenitesupport,i.e.vanishoutsideaniteset.Thisisacomplexvectorspace,andhasthesemi-scalarproductx;y:=Xt;sFt)]TJ/F11 9.963 Tf 9.963 0 Td[(sxt ys:ItiseasytoseethatthefactthatFisapositivedenitefunctionimpliesthatx;x0forallx2F.PassingtothequotientbythesubspaceofnullvectorsandcompletingweobtainaHilbertspaceH.LetUrbedenedby[Urx]t=xt)]TJ/F11 9.963 Tf 9.963 0 Td[(rasusual.ThenUrx;Ury=Xt;sFt)]TJ/F11 9.963 Tf 8.306 0 Td[(sxt)]TJ/F11 9.963 Tf 8.306 0 Td[(r ys)]TJ/F11 9.963 Tf 9.962 0 Td[(r=Xt;sFt+r)]TJ/F8 9.963 Tf 8.305 0 Td[(s+rxt ys=x;y:SoUrdescendstoHtodeneaunitaryoperatorwhichweshallcontinuetodenotebyUr.WethusobtainaoneparametergroupofunitarytransformationsonH.AccordingtoStone'stheoremthereexistsaresolutionEoftheidentitysuchthatUt=Z1eitdE:Nowchoose2Ftobedenedbyt=1ift=00ift6=0:LetxbetheimageofinH.ThenUrx;x=XFt)]TJ/F11 9.963 Tf 9.963 0 Td[(st)]TJ/F11 9.963 Tf 9.962 0 Td[(rs=Fr:ButbyStonewehaveFr=Z1eirdx;x=Z1eirdkExk2

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11.4.THEPOWERSERIESEXPANSIONOFTHEEXPONENTIAL.309sowehaverepresentedFastheFouriertransformofanitenon-negativemeasure.QEDThelogicofourargumenthasbeen-theSpectralTheoremimpliesStone'stheoremimpliesBochner'stheorem.Infact,assumingtheHille-Yosidatheoremontheexistenceofsemigroupstobeprovedbelow,onecangointheoppositedirection.GivenaoneparametergroupUtofunitarytransformations,itiseasytocheckthatforanyx2Hthefunctiont7!Utx;xispositivedenite,andthenuseBochner'stheoremtoderivethespectraltheoremonthecyclicsubspacegeneratedbyxunderUt.Onecanthengetthefullspectraltheoreminmultiplicationoperatorformaswedidinthehandoutonunboundedself-adjointoperators.11.4Thepowerseriesexpansionoftheexpo-nential.InnitedimensionswehavetheformulaetB=1X0tk k!Bkwithconvergenceguaranteedasaresultoftheconvergenceoftheusualexpo-nentialseriesinonevariable.Thereareseriousproblemswiththisdenitionfromthepointofviewofnumericalimplementationwhichwewillnotdiscusshere.IninnitedimensionalspacessomeadditionalassumptionshavetobeplacedonanoperatorBbeforewecanconcludethattheaboveseriesconverges.Hereisaverystringentconditionwhichneverthelesssucesforourpurposes.LetFbeaFrechetspaceandBacontinuousmapofF!F.WewillassumethattheBkareequiboundedinthesensethatforanydeningsemi-normpthereisaconstantKandadeningsemi-normqsuchthatpBkxKqx8k=1;2;:::8x2F:HeretheKandqarerequiredtobeindependentofkandx.ThenpnXmtk k!BkxnXmtk k!pBkxKqxnXntk k!andsonX0tk k!BkxisaCauchysequenceforeachxedtandxanduniformlyinanycompactintervaloft.Itthereforeconvergestoalimit.Wewilldenotethemapx7!P10tk k!BkxbyexptB:

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310CHAPTER11.STONE'STHEOREMThismapislinear,andthecomputationaboveshowsthatpexptBxKexptqx:Theusualproofusingthebinomialformulashowsthatt7!exptBisaoneparameterequiboundedsemi-group.Moregenerally,ifBandCaretwosuchoperatorsthenifBC=CBthenexptB+C=exptBexptC.Also,fromthepowerseriesitfollowsthattheinnitesimalgeneratorofexptBisB.11.5TheHilleYosidatheorem.LetusnowreturntothegeneralcaseofanequiboundedsemigroupTtwithinnitesimalgeneratorAonaFrechetspaceFwhereweknowthattheresolventRz;AforRez>0isgivenbyRz;Ax=Z10e)]TJ/F10 6.974 Tf 6.227 0 Td[(ztTtxdt:ThisformulashowsthatRz;Axiscontinuousinz.TheresolventequationRz;A)]TJ/F11 9.963 Tf 9.963 0 Td[(Rw;A=w)]TJ/F11 9.963 Tf 9.963 0 Td[(zRz;ARw;AthenshowsthatRz;Axiscomplexdierentiableinzwithderivative)]TJ/F11 9.963 Tf 7.749 0 Td[(Rz;A2x.ItthenfollowsthatRz;AxhascomplexderivativesofallordersgivenbydnRz;Ax dzn=)]TJ/F8 9.963 Tf 7.749 0 Td[(1nn!Rz;An+1x:Ontheotherhand,dierentiatingtheintegralformulafortheresolventn-timesgivesdnRz;Ax dzn=Z10e)]TJ/F10 6.974 Tf 6.226 0 Td[(zt)]TJ/F11 9.963 Tf 7.748 0 Td[(tnTtdtwheredierentiationundertheintegralsignisjustiedbythefactthattheTtareequicontinuousint.PuttingtheprevioustwoequationstogethergiveszRz;An+1x=zn+1 n!Z10e)]TJ/F10 6.974 Tf 6.226 0 Td[(zttnTtxdt:Thisimpliesthatforanysemi-normpwehavepzRz;An+1xzn+1 n!Z10e)]TJ/F10 6.974 Tf 6.227 0 Td[(zttnsupt0pTtxdt=supt0pTtxsinceZ10e)]TJ/F10 6.974 Tf 6.227 0 Td[(zttndt=n! zn+1:SincetheTtareequiboundedbyhypothesis,weconclude

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11.5.THEHILLEYOSIDATHEOREM.311Proposition11.5.1ThefamilyofoperatorsfzRz;AngisequiboundedinRez>0andn=0;1;2;:::.Wenowcometothemainresultofthissection:Theorem11.5.1[Hille-Yosida.]LetAbeanoperatorwithdensedomainDA,andsuchthattheresolventsRn;A=nI)]TJ/F11 9.963 Tf 9.962 0 Td[(A)]TJ/F7 6.974 Tf 6.226 0 Td[(1existandareboundedoperatorsforn=1;2;:::.ThenAistheinnitesimalgeneratorofauniquelydeterminedequiboundedsemigroupifandonlyiftheoperatorsfI)]TJ/F11 9.963 Tf 9.962 0 Td[(n)]TJ/F7 6.974 Tf 6.227 0 Td[(1A)]TJ/F10 6.974 Tf 6.226 0 Td[(mgareequiboundedinm=0;1;2:::andn=1;2;:::.Proof.IfAistheinnitesimalgeneratorofanequiboundedsemi-groupthenweknowthatthefI)]TJ/F11 9.963 Tf 9.924 0 Td[(n)]TJ/F7 6.974 Tf 6.227 0 Td[(1A)]TJ/F10 6.974 Tf 6.227 0 Td[(mgareequiboundedbyvirtueoftheprecedingproposition.Sowemustprovetheconverse.SetJn=I)]TJ/F11 9.963 Tf 9.963 0 Td[(n)]TJ/F7 6.974 Tf 6.226 0 Td[(1A)]TJ/F7 6.974 Tf 6.227 0 Td[(1soJn=nnI)]TJ/F11 9.963 Tf 9.963 0 Td[(A)]TJ/F7 6.974 Tf 6.227 0 Td[(1andsoforx2DAwehaveJnnI)]TJ/F11 9.963 Tf 9.963 0 Td[(Ax=nxorJnAx=nJn)]TJ/F11 9.963 Tf 9.963 0 Td[(Ix:SimilarlynI)]TJ/F11 9.963 Tf 9.963 0 Td[(AJn=nIsoAJn=nJn)]TJ/F11 9.963 Tf 9.962 0 Td[(I.ThuswehaveAJnx=JnAx=nJn)]TJ/F11 9.963 Tf 9.962 0 Td[(Ix8x2DA:.14Theideaoftheproofisnowthis:Bytheresultsoftheprecedingsection,wecanconstructtheoneparametersemigroups7!expsJn.Sets=nt.Wecanthenforme)]TJ/F10 6.974 Tf 6.227 0 Td[(ntexpntJnwhichwecanwriteasexptnJn)]TJ/F11 9.963 Tf 10.157 0 Td[(I=exptAJnbyvirtueof.14.Weexpectfrom.5thatlimn!1Jnx=x8x2F:.15ThisthensuggeststhatthelimitoftheexptAJnbethedesiredsemi-group.Sowebeginbyproving.15.Werstproveitforx2DA.ForsuchxwehaveJn)]TJ/F11 9.963 Tf 9.389 0 Td[(Ix=n)]TJ/F7 6.974 Tf 6.227 0 Td[(1JnAxby.14andthisapproacheszerosincetheJnareequibounded.ButsinceDAisdenseinFandtheJnareequiboundedweconcludethat11.15holdsforallx2F.NowdeneTnt=exptAJn:=expntJn)]TJ/F11 9.963 Tf 9.962 0 Td[(I=e)]TJ/F10 6.974 Tf 6.227 0 Td[(ntexpntJn:

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312CHAPTER11.STONE'STHEOREMWeknowfromtheprecedingsectionthatpexpntJnxXntk k!pJknxentKqxwhichimpliesthatpTntxKqx:.16ThusthefamilyofoperatorsfTntgisequiboundedforallt0andn=1;2;:::.WenextwanttoprovethatthefTntgconvergeasn!1uniformlyoneachcompactintervaloft:TheJncommutewithoneanotherbytheirdenition,andhenceJncom-muteswithTmt.Bythesemi-grouppropertywehaved dtTmtx=AJmTmtx=TmtAJmxsoTntx)]TJ/F11 9.963 Tf 9.962 0 Td[(Tmtx=Zt0d dsTmt)]TJ/F10 6.974 Tf 6.226 0 Td[(sTnsxds=Zt0Tmt)]TJ/F10 6.974 Tf 6.227 0 Td[(sAJn)]TJ/F11 9.963 Tf 9.963 0 Td[(AJmTnsxds:Applyingthesemi-normpandusingtheequiboundednessweseethatpTntx)]TJ/F11 9.963 Tf 9.963 0 Td[(TmtxKtqJn)]TJ/F11 9.963 Tf 9.963 0 Td[(JmAx:From.15thisimpliesthattheTntxconvergeuniformlyineverycompactintervaloftforx2DA,andhencesinceDAisdenseandtheTntareequicontinuousforallx2F.ThelimitingfamilyofoperatorsTtareequicon-tinuousandformasemi-groupbecausetheTnthavethisproperty.Wemustshowthattheinnitesimalgeneratorofthissemi-groupisA.Letustemporarilydenotetheinnitesimalgeneratorofthissemi-groupbyB,sothatwewanttoprovethatA=B.Letx2DA.Weclaimthatlimn!1TntAJnx=TtAx.17uniformlyininanycompactintervaloft.Indeed,foranysemi-normpwehavepTtAx)]TJ/F11 9.963 Tf 9.963 0 Td[(TntAJnxpTtAx)]TJ/F11 9.963 Tf 9.962 0 Td[(TntAx+pTntAx)]TJ/F11 9.963 Tf 9.963 0 Td[(TntAJnxpTt)]TJ/F11 9.963 Tf 9.963 0 Td[(TntAx+KqAx)]TJ/F11 9.963 Tf 9.963 0 Td[(JnAxwherewehaveused.16togetfromthesecondlinetothethird.Thesecondtermontherighttendstozeroasn!1andwehavealreadyprovedthatthersttermconvergestozerouniformlyoneverycompactintervaloft.Thisestablishes.17.

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11.6.CONTRACTIONSEMIGROUPS.313Wethushave,forx2DA,Ttx)]TJ/F11 9.963 Tf 9.962 0 Td[(x=limn!1Tntx)]TJ/F11 9.963 Tf 9.963 0 Td[(x=limn!1Zt0TnsAJnxds=Zt0limn!1TnsAJnxds=Zt0TsAxdswherethepassageofthelimitundertheintegralsignisjustiedbytheuniformconvergenceintoncompactsets.ItfollowsfromTtx)]TJ/F11 9.963 Tf 10.124 0 Td[(x=Rt0TsAxdsthatxisinthedomainoftheinnitesimaloperatorBofTtandthatBx=Ax.SoBisanextensionofAinthesensethatDBDAandBx=AxonDA.ButsinceBistheinnitesimalgeneratorofanequiboundedsemi-group,weknowthatI)]TJ/F11 9.963 Tf 10.514 0 Td[(BmapsDBontoFbijectively,andweareassumingthatI)]TJ/F11 9.963 Tf 9.962 0 Td[(AmapsDAontoFbijectively.HenceDA=DB.QEDIncaseFisaBanachspace,sothereisasinglenormp=kk,thehypothesesofthetheoremread:DAisdenseinF,theresolventsRn;Aexistforallintegersn=1;2;:::andthereisaconstantKindependentofnandmsuchthatkI)]TJ/F11 9.963 Tf 9.962 0 Td[(n)]TJ/F7 6.974 Tf 6.227 0 Td[(1A)]TJ/F10 6.974 Tf 6.226 0 Td[(mkK8n=1;2;:::;m=1;2;::::.1811.6Contractionsemigroups.Inparticular,ifAsatiseskI)]TJ/F11 9.963 Tf 9.962 0 Td[(n)]TJ/F7 6.974 Tf 6.227 0 Td[(1A)]TJ/F7 6.974 Tf 6.226 0 Td[(1k1.19condition.18issatised,andsuchanAthengeneratesasemi-group.Underthisstrongerhypothesiswecandrawastrongerconclusion:In11.16wenowhavep=q=kkandK=1.Sincelimn!1Tntx=Ttxweseethatunderthehypothesis1.19wecanconcludethatkTtk18t0:Asemi-groupTtsatisfyingthisconditioniscalledacontractionsemi-group.Wewillstudyanotherusefulconditionforrecognizingacontractionsemigroupinthefollowingsubsection.WehavealreadygivenadirectproofthatifSisaself-adjointoperatoronaHilbertspacethentheresolventexistsforallnon-realzandsatiseskRz;Sk1 jImzj:

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314CHAPTER11.STONE'STHEOREMThisimplies.19forA=iSand)]TJ/F11 9.963 Tf 7.749 0 Td[(iSgivingusanindependentproofoftheexistenceofUt=expiStforanyself-adjointoperatorS.Aswemen-tionedpreviously,wecouldthenuseBochner'stheoremtogiveathirdproofofthespectraltheoremforunboundedself-adjointoperators.ImightdiscussBochner'stheoreminthecontextofgeneralizedfunctionslaterprobablynextsemesterifatall.OncewegiveanindependentproofofBochner'stheoremthenindeedwewillgetathirdproofofthespectraltheorem.11.6.1Dissipationandcontraction.LetFbeaBanachspace.Recallthatasemi-groupTtiscalledacontractionsemi-groupifkTtk18t0;andthat.19isasucientconditiononoperatorwithdensedomaintogenerateacontractionsemi-group.TheLumer-Phillipstheoremtobestatedbelowgivesanecessaryandsu-cientconditionontheinnitesimalgeneratorofasemi-groupforthesemi-grouptobeacontractionsemi-group.Itisgeneralizationofthefactthattheresolventofaself-adjointoperatorhasiinitsresolventset.TherststepistointroduceasortoffakescalarproductintheBanachspaceF.Asemi-scalarproductonFisarulewhichassignsanumberhhx;ziitoeverypairofelementsx;z2Finsuchawaythathhx+y;zii=hhx;zii+hhy;ziihhx;zii=hhx;ziihhx;xii=kxk2jhhx;ziijkxkkzk:Wecanalwayschooseasemi-scalarproductasfollows:bytheHahn-Banachtheorem,foreachz2Fwecanndan`z2Fsuchthatk`zk=kzkand`zz=kzk2:Chooseonesuch`zforeachz2Fandsethhx;zii:=`zx:Clearlyalltheconditionsaresatised.Ofcoursethisdenitionishighlyunnat-ural,unlessthereissomereasonablewayofchoosingthe`zotherthanusingtheaxiomofchoice.InaHilbertspace,thescalarproductisasemi-scalarproduct.AnoperatorAwithdomainDAonFiscalleddissipativerelativetoagivensemi-scalarproducthh;iiifRehhAx;xii08x2DA:Forexample,ifAisasymmetricoperatoronaHilbertspacesuchthatAx;x08x2DA.20

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11.6.CONTRACTIONSEMIGROUPS.315thenAisdissipativerelativetothescalarproduct.Theorem11.6.1[Lumer-Phillips.]LetAbeanoperatoronaBanachspaceFwithDAdenseinF.ThenAgeneratesacontractionsemi-groupifandonlyifAisdissipativewithrespecttoanysemi-scalarproductandimI)]TJ/F11 9.963 Tf 9.962 0 Td[(A=F:Proof.SupposerstthatDAisdenseandthatimI)]TJ/F11 9.963 Tf 10.218 0 Td[(A=F.Wewishtoshowthat.19holds,whichwillguaranteethatAgeneratesacontractionsemi-group.Lets>0.Thenifx2DAandy=sx)]TJ/F11 9.963 Tf 9.963 0 Td[(Axthenskxk2=shhx;xiishhx;xii)]TJ/F8 9.963 Tf 19.925 0 Td[(RehhAx;xii=Rehhy;xiiimplyingskxk2kykkxk:.21WeareassumingthatimI)]TJ/F11 9.963 Tf 10.282 0 Td[(A=F.Thistogetherwith.21withs=1impliesthatR;AexistsandkR;Ak1:Inturn,thisimpliesthatforallzwithjz)]TJ/F8 9.963 Tf 10.077 0 Td[(1j<1theresolventRz;AexistsandisgivenbythepowerseriesRz;A=1Xn=0z)]TJ/F8 9.963 Tf 9.963 0 Td[(1nR;An+1byourgeneralpowerseriesformulafortheresolvent.Inparticular,forsrealandjs)]TJ/F8 9.963 Tf 9.997 0 Td[(1j<1theresolventexists,andthen.21impliesthatkRs;Aks)]TJ/F7 6.974 Tf 6.227 0 Td[(1.RepeatingtheprocesswekeepenlargingtheresolventsetAuntilitincludesthewholepositiverealaxisandconcludefrom.21thatkRs;Aks)]TJ/F7 6.974 Tf 6.227 0 Td[(1whichimplies.19.AsweareassumingthatDAisdenseweconcludethatAgeneratesacontractionsemigroup.Conversely,supposethatTtisacontractionsemi-groupwithinnitesimalgeneratorA.WeknowthatDomAisdense.Lethh;iibeanysemi-scalarproduct.ThenRehhTtx)]TJ/F11 9.963 Tf 9.962 0 Td[(x;xii=RehhTtx;xii)-222(kxk2kTtxkkxk)-222(kxk20:Dividingbytandlettingt&0weconcludethatRehhAx;xii0forallx2DA,i.e.Aisdissipativeforhh;ii.QEDOnceagain,thisgivesadirectproofoftheexistenceoftheunitarygroupgeneratedgeneratedbyaskewadjointoperator.AusefulwayofverifyingtheconditionimI)]TJ/F11 9.963 Tf 9.999 0 Td[(A=Fisthefollowing:LetA:F!FbetheadjointoperatorwhichisdenedifweassumethatDAisdense.

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316CHAPTER11.STONE'STHEOREMProposition11.6.1SupposethatAisdenselydenedandclosed,andsupposethatbothAandAaredissipative.ThenimI)]TJ/F11 9.963 Tf 9.332 0 Td[(A=FandhenceAgeneratesacontractionsemigroup.Proof.ThefactthatAisclosedimpliesthatI)]TJ/F11 9.963 Tf 9.545 0 Td[(A)]TJ/F7 6.974 Tf 6.226 0 Td[(1isclosed,andsinceweknowthatI)]TJ/F11 9.963 Tf 9.033 0 Td[(A)]TJ/F7 6.974 Tf 6.226 0 Td[(1isboundedfromthefactthatAisdissipative,weconcludethatimI)]TJ/F11 9.963 Tf 10.133 0 Td[(AisaclosedsubspaceofF.Ifitwerenotthewholespacetherewouldbean`2Fwhichvanishedonthissubspace,i.e.h`;x)]TJ/F11 9.963 Tf 9.962 0 Td[(Axi=08x2DA:Thisimpliesthatthat`2DAandA`=`whichcannothappenifAisdissipativeby.21appliedtoAands=1.QED11.6.2Aspecialcase:exptB)]TJ/F19 11.955 Tf 11.955 0 Td[(IwithkBk1.SupposethatB:F!FisaboundedoperatoronaBanachspacewithkBk1.Thenforanysemi-scalarproductwehaveRehhB)]TJ/F11 9.963 Tf 9.962 0 Td[(Ix;xii=RehhBx;xii)-222(kxk2kBxkkxk)-222(kxk20soB)]TJ/F11 9.963 Tf 8.468 0 Td[(IisdissipativeandhenceexptB)]TJ/F11 9.963 Tf 8.468 0 Td[(Iexistsasacontractionsemi-groupbytheLumer-Phillipstheorem.WecanprovethisdirectlysincewecanwriteexptB)]TJ/F11 9.963 Tf 9.963 0 Td[(I=e)]TJ/F10 6.974 Tf 6.227 0 Td[(t1Xk=0tkBk k!:TheseriesconvergesintheuniformnormandwehavekexptB)]TJ/F11 9.963 Tf 9.962 0 Td[(Ike)]TJ/F10 6.974 Tf 6.227 0 Td[(t1Xk=0tkkBkk k!1:ForfutureuseCherno'stheoremandtheTrotterproductformulawerecordandprovethefollowinginequality:k[expnB)]TJ/F11 9.963 Tf 9.963 0 Td[(I)]TJ/F11 9.963 Tf 9.963 0 Td[(Bn]xkp nkB)]TJ/F11 9.963 Tf 9.963 0 Td[(Ixk8x2F;and8n=1;2;3::::.22

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11.7.CONVERGENCEOFSEMIGROUPS.317Proof.k[expnB)]TJ/F11 9.963 Tf 9.962 0 Td[(I)]TJ/F11 9.963 Tf 9.963 0 Td[(Bn]xk=ke)]TJ/F10 6.974 Tf 6.227 0 Td[(n1Xk=0nk k!Bk)]TJ/F11 9.963 Tf 9.963 0 Td[(Bnxke)]TJ/F10 6.974 Tf 6.227 0 Td[(n1Xk=0nk k!kBk)]TJ/F11 9.963 Tf 9.963 0 Td[(Bnxke)]TJ/F10 6.974 Tf 6.227 0 Td[(n1Xk=0nk k!kBjk)]TJ/F10 6.974 Tf 6.227 0 Td[(nj)]TJ/F11 9.963 Tf 9.962 0 Td[(Ixk=e)]TJ/F10 6.974 Tf 6.227 0 Td[(n1Xk=0nk k!kB)]TJ/F11 9.963 Tf 9.963 0 Td[(II+B++Bjk)]TJ/F10 6.974 Tf 6.226 0 Td[(nj)]TJ/F7 6.974 Tf 8.593 0 Td[(1xke)]TJ/F10 6.974 Tf 6.227 0 Td[(n1Xk=0nk k!jk)]TJ/F11 9.963 Tf 9.962 0 Td[(njkB)]TJ/F11 9.963 Tf 9.962 0 Td[(Ixk:Sotoprove.22itisenoughestablishtheinequalitye)]TJ/F10 6.974 Tf 6.227 0 Td[(n1Xk=0nk k!jk)]TJ/F11 9.963 Tf 9.962 0 Td[(njp n:.23Considerthespaceofallsequencesa=fa0;a1;:::gwithnitenormrelativetoscalarproducta;b:=e)]TJ/F10 6.974 Tf 6.226 0 Td[(n1Xk=0nk k!ak bk:TheCauchy-Schwarzinequalityappliedtoawithak=jk)]TJ/F11 9.963 Tf 8.501 0 Td[(njandbwithbk1givese)]TJ/F10 6.974 Tf 6.227 0 Td[(n1Xk=0nk k!jk)]TJ/F11 9.963 Tf 9.962 0 Td[(njvuut e)]TJ/F10 6.974 Tf 6.227 0 Td[(n1Xk=0nk k!k)]TJ/F11 9.963 Tf 9.962 0 Td[(n2vuut e)]TJ/F10 6.974 Tf 6.227 0 Td[(n1Xk=0nk k!:Thesecondsquarerootisone,andwerecognizethesumundertherstsquarerootasthevarianceofthePoissondistributionwithparametern,andweknowthatthisvarianceisn.QED11.7Convergenceofsemigroups.Wearegoingtobeinterestedinthefollowingtypeofresult.WewouldliketoknowthatifAnisasequenceofoperatorsgeneratingequiboundedonepa-rametersemi-groupsexptAnandAn!AwhereAgeneratesanequiboundedsemi-groupexptAthenthesemi-groupsconverge,i.e.exptAn!exptA.Wewillprovesucharesultforthecaseofcontractions.Butbeforewecanevenformulatetheresult,wehavetodealwiththefactthateachAncomesequippedwithitsowndomainofdenition,DAn.Wedonotwanttomaketheoverly

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318CHAPTER11.STONE'STHEOREMrestrictivehypothesisthattheseallcoincide,sinceinmanyimportantapplica-tionstheywon't.Forthispurposewemakethefollowingdenition.LetusassumethatFisaBanachspaceandthatAisanoperatoronFdenedonadomainDA.WesaythatalinearsubspaceDDAisacoreforAiftheclosure AofAandtheclosureofArestrictedtoDarethesame: A= AjD.ThiscertainlyimpliesthatDAiscontainedintheclosureofAjD.InthecasesofinteresttousDAisdenseinF,sothateverycoreofAisdenseinF.Webeginwithanimportantpreliminaryresult:Proposition11.7.1SupposethatAnandAaredissipativeoperators,i.e.gen-eratorsofcontractionsemi-groups.LetDbeacoreofA.Supposethatforeachx2Dwehavethatx2DAnforsucientlylargendependingonxandthatAnx!Ax:.24ThenforanyzwithRez>0andforally2FRz;Any!Rz;Ay:.25Proof.WeknowthattheRz;AnandRz;Aareallboundedinnormby1=Rez.Soitisenoughforustoproveconvergenceonadenseset.SincezI)]TJ/F11 9.963 Tf 10.268 0 Td[(ADA=F,itfollowsthatzI)]TJ/F11 9.963 Tf 10.268 0 Td[(ADisdenseinFsinceAisclosed.Soinproving.25wemayassumethaty=zI)]TJ/F11 9.963 Tf 9.963 0 Td[(Axwithx2D.ThenkRz;Any)]TJ/F11 9.963 Tf 9.962 0 Td[(Rz;Ayk=kRz;AnzI)]TJ/F11 9.963 Tf 9.963 0 Td[(Ax)]TJ/F11 9.963 Tf 9.963 0 Td[(xk=kRz;AnzI)]TJ/F11 9.963 Tf 9.963 0 Td[(Anx+Rz;AnAnx)]TJ/F11 9.963 Tf 9.963 0 Td[(Ax)]TJ/F11 9.963 Tf 9.962 0 Td[(xk=kRz;AnAn)]TJ/F11 9.963 Tf 9.962 0 Td[(Axk1 RezkAn)]TJ/F11 9.963 Tf 9.963 0 Td[(Axk!0;where,inpassingfromtherstlinetothesecondweareassumingthatnischosensucientlylargethatx2DAn.QEDTheorem11.7.1Underthehypothesesoftheprecedingproposition,exptAnx!exptAxforeachx2Funiformlyoneverycompactintervaloft.Proof.Letnt:=e)]TJ/F10 6.974 Tf 6.226 0 Td[(t[exptAnx)]TJ/F8 9.963 Tf 9.962 0 Td[(exptAx]fort0andsett=0fort<0.ItwillbeenoughtoprovethattheseFvaluedfunctionsconvergeuniformlyintto0,andsinceDisdenseandsincetheoperatorsenteringintothedenitionofnareuniformlyboundedinn,itisenoughtoprovethisconvergenceforx2Dwhichisdense.Weclaimthat

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11.7.CONVERGENCEOFSEMIGROUPS.319forxedx2Dthefunctionsntareuniformlyequi-continuous.Toseethisobservethatd dtnt=e)]TJ/F10 6.974 Tf 6.227 0 Td[(t[exptAnAnx)]TJ/F8 9.963 Tf 9.807 0 Td[(exptAAx])]TJ/F11 9.963 Tf 9.807 0 Td[(e)]TJ/F10 6.974 Tf 6.226 0 Td[(t[exptAnx)]TJ/F8 9.963 Tf 9.806 0 Td[(exptAx]fort0andtherighthandsideisuniformlyboundedint0andn.Sotoprovethatntconvergesuniformlyintto0,itisenoughtoprovethisfactfortheconvolutionn?whereisanysmoothfunctionofcompactsupport,sincewecanchoosethetohavesmallsupportandintegralp 2,andthenntiscloseton?t.NowtheFouriertransformofn?istheproductoftheirFouriertransforms:^n^.Wehave^ns=1 p 2Z10e)]TJ/F7 6.974 Tf 6.227 0 Td[(1)]TJ/F10 6.974 Tf 6.227 0 Td[(ist[exptAnx)]TJ/F8 9.963 Tf 7.749 0 Td[(exptAx]dt=1 p 2[R+is;Anx)]TJ/F11 9.963 Tf 7.749 0 Td[(R+is;Ax]:Thusbytheproposition^ns!0;infactuniformlyins.HenceusingtheFourierinversionformulaand,say,thedominatedconvergencetheoremforBanachspacevaluedfunctions,n?t=1 p 2Z1^ns^seistds!0uniformlyint.QEDTheprecedingtheoremisthelimittheoremthatwewilluseinwhatfollows.However,thereisanimportanttheoremvalidinanarbitraryFrechetspace,andwhichdoesnotassumethattheAnconverge,ortheexistenceofthelimitA,butonlytheconvergenceoftheresolventatasinglepointz0intherighthandplane!InthefollowingFisaFrechetspaceandfexptAngisafamilyofofequi-boundedsemi-groupswhichisalsoequiboundedinn,soforeverysemi-normpthereisasemi-normqandaconstantKsuchthatpexptAnxKqx8x2FwhereKandqareindependentoftandn.Iwillstatethetheoremhere,andreferyoutoYosidapp.269-271fortheproof.Theorem11.7.2[Trotter-Kato.]SupposethatfexptAngisanequiboundedfamilyofsemi-groupsasabove,andsupposethatforsomez0withpositiverealpartthereexistanoperatorRz0suchthatlimn!1Rz0;An!Rz0andimRz0isdenseinF:

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320CHAPTER11.STONE'STHEOREMThenthereexistsanequiboundedsemi-groupexptAsuchthatRz0=Rz0;AandexptAn!exptAuniformlyoneverycompactintervaloft0.11.8TheTrotterproductformula.Inwhatfollows,FisaBanachspace.EventuallywewillrestrictattentiontoaHilbertspace.ButwewillbeginwithaclassicaltheoremofLie:11.8.1Lie'sformula.LetAandBbelinearoperatorsonanitedimensionalHilbertspace.Lie'sformulasaysthatexpA+B=limn!1[expA=nexpB=n]n:.26Proof.LetSn:=exp1 nA+BsothatSnn=expA+B:LetTn=expA=nexpB=n.WewishtoshowthatSnn)]TJ/F11 9.963 Tf 9.963 0 Td[(Tnn!0:NoticethattheconstantandthelineartermsinthepowerseriesexpansionsforSnandTnarethesame,sokSn)]TJ/F11 9.963 Tf 9.963 0 Td[(TnkC n2whereC=CA;B.WehavethetelescopingsumSnn)]TJ/F11 9.963 Tf 9.963 0 Td[(Tnn=n)]TJ/F7 6.974 Tf 6.226 0 Td[(1Xk=0Sk)]TJ/F7 6.974 Tf 6.227 0 Td[(1nSn)]TJ/F11 9.963 Tf 9.963 0 Td[(TnTn)]TJ/F7 6.974 Tf 6.226 0 Td[(1)]TJ/F10 6.974 Tf 6.227 0 Td[(ksokSnn)]TJ/F11 9.963 Tf 9.962 0 Td[(TnnknkSn)]TJ/F11 9.963 Tf 9.962 0 Td[(TnkmaxkSnk;kTnkn)]TJ/F7 6.974 Tf 6.227 0 Td[(1:ButkSnkexp1 nkAk+kBkandkTnkexp1 nkAk+kBkandexp1 nkAk+kBkn)]TJ/F7 6.974 Tf 6.227 0 Td[(1=expn)]TJ/F8 9.963 Tf 9.963 0 Td[(1 nkAk+kBkexpkAk+kBk

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11.8.THETROTTERPRODUCTFORMULA.321sokSnn)]TJ/F11 9.963 Tf 9.963 0 Td[(TnnkC nexpkAk+kBk:ThissameproofworksifAandBareself-adjointoperatorssuchthatA+Bisself-adjointontheintersectionoftheirdomains.ForaproofseeReed-Simonvol.Ipages295-296.Forapplicationsthisistoorestrictive.SowegiveamoregeneralformulationandprooffollowingCherno.11.8.2Cherno'stheorem.Theorem11.8.1[Cherno.]Letf:[0;1!boundedoperatorsonFbeacontinuousmapwithkftk18tandf=I:LetAbeadissipativeoperatorandexptAthecontractionsemi-groupitgener-ates.LetDbeacoreofA.Supposethatlimh&01 h[fh)]TJ/F11 9.963 Tf 9.963 0 Td[(I]x=Ax8x2D:Thenforally2Flimft nny=exptAy.27uniformlyinanycompactintervaloft0.Proof.Forxedt>0letCn:=n tft n)]TJ/F11 9.963 Tf 9.963 0 Td[(I:Thent nCngeneratesacontractionsemi-groupbythespecialcaseoftheLumer-PhillipstheoremdiscussedinSection11.6.2,andthereforebychangeofvari-able,sodoesCn.SoCnisthegeneratorofasemi-groupexptCnandthehypothesisofthetheoremisthatCnx!Axforx2D.HencebythelimittheoremintheprecedingsectionexptCny!exptAyforeachy2Funiformlyonanycompactintervaloft.NowexptCn=expnft n)]TJ/F11 9.963 Tf 9.962 0 Td[(I

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322CHAPTER11.STONE'STHEOREMsowemayapply.22toconcludethatkexptCn)]TJ/F11 9.963 Tf 9.963 0 Td[(ft nnxkp nkft n)]TJ/F11 9.963 Tf 9.963 0 Td[(Ixk=t p nkn tft n)]TJ/F11 9.963 Tf 9.962 0 Td[(Ixk:TheexpressioninsidethekkontherighttendstoAxsothewholeexpressiontendstozero.Thisproves.27forallxinD.ButsinceDisdenseinFandft=nandexptAareboundedinnormby1itfollowsthat.27holdsforally2F.QED11.8.3Theproductformula.LetAandBbetheinnitesimalgeneratorsofthecontractionsemi-groupsPt=exptAandQt=exptBontheBanachspaceF.ThenA+BisonlydenedonDADBandingeneralweknownothingaboutthisintersection.HoweverletusassumethatDADBissucientlylargethattheclosure A+Bisadenselydenedoperatorand A+Bisinfactthegeneratorofacontractionsemi-groupRt.SoD:=DADBisacorefor A+B.Theorem11.8.2[Trotter.]UndertheabovehypothesesRty=limPt nQt nny8y2F.28uniformlyonanycompactintervaloft0.Proof.Deneft=PtQt:Forx2Dwehaveftx=PtI+tB+otx=I+At+Bt+otxsothehypothesesofCherno'stheoremaresatised.TheconclusionofCher-no'stheoremasserts.28.QEDAsymmetricoperatoronaHilbertspaceiscalledessentiallyselfadjointifitsclosureisself-adjoint.Soareformulationoftheprecedingtheoreminthecaseofself-adjointoperatorsonaHilbertspacesaysTheorem11.8.3SupposethatSandTareself-adjointoperatorsonaHilbertspaceHandsupposethatS+TdenedonDSDTisessentiallyself-adjoint.Thenforeveryy2Hexpit S+Ty=limn!1expt niAexpt niBny.29wheretheconvergenceisuniformonanycompactintervaloft.

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11.8.THETROTTERPRODUCTFORMULA.32311.8.4Commutators.AnoperatorAonaHilbertspaceiscalledskew-symmetricifA=)]TJ/F11 9.963 Tf 7.749 0 Td[(AonDA.ThisisthesameassayingthatiAissymmetric.SowecallanoperatorskewadjointifiAisself-adjoint.WecallanoperatorAessentiallyskewadjointifiAisessentiallyself-adjoint.IfAandBareboundedskewadjointoperatorsthentheirLiebracket[A;B]:=AB)]TJ/F11 9.963 Tf 9.963 0 Td[(BAiswelldenedandagainskewadjoint.Ingeneral,wecanonlydenetheLiebracketonDABDBAsoweagainmustmakesomeratherstringenthypothesesinstatingthefollowingtheorem.Theorem11.8.4LetAandBbeskewadjointoperatorsonaHilbertspaceHandletD:=DA2DB2DABDBA:Supposethattherestrictionof[A;B]toDisessentiallyskew-adjoint.Thenforeveryy2Hexpt [A;B]y=limn!1exp)]TJ/F1 9.963 Tf 7.749 15.508 Td[(r t nAexp)]TJ/F1 9.963 Tf 7.749 15.508 Td[(r t nBexpr t nAexpr t nB!ny.30uniformlyinanycompactintervaloft0.Proof.Therestrictionof[A;B]toDisassumedtobeessentiallyskew-adjoint,so[A;B]itselfwhichhasthesameclosureisalsoessentiallyskewadjoint.WehaveexptAx=I+tA+t2 2A2x+ot2forx2Dwithsimilarformulasforexp)]TJ/F11 9.963 Tf 7.748 0 Td[(tAetc.Letft:=exp)]TJ/F11 9.963 Tf 7.749 0 Td[(tAexp)]TJ/F11 9.963 Tf 7.749 0 Td[(tBexptAexptB:Multiplyingoutftxforx2Dgivesawholelotofcancellationsandyieldsfsx=I+s2[A;B]x+os2so.30isaconsequenceofCherno'stheoremwiths=p t.QEDWestillneedtodevelopsomemethodswhichallowustocheckthehypothe-sesofthelastthreetheorems.11.8.5TheKato-Rellichtheorem.ThisisthestartingpointofaclassoftheoremswhichassertsthatthatifAisself-adjointandifBisasymmetricoperatorwhichissmall"incomparisontoAthenA+Bisselfadjoint.

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324CHAPTER11.STONE'STHEOREMTheorem11.8.5[Kato-Rellich.]LetAbeaself-adjointoperatorandBasymmetricoperatorwithDBDAandkBxkakAxk+bkxk0a<1;8x2DA:ThenA+Bisself-adjoint,andisessentiallyself-adjointonanycoreofA.Proof.[FollowingReedandSimonIIpage162.]ToprovethatA+Bisselfadjoint,itisenoughtoprovethatimA+Bi0=H.WedothisforA+B+i0.TheproofforA+B)]TJ/F11 9.963 Tf 9.963 0 Td[(i0isidentical.Let>0.SinceAisself-adjoint,weknowthatkA+ixk2=kAxk2+2kxk2fromwhichweconcludedthatA+i)]TJ/F7 6.974 Tf 6.227 0 Td[(1mapsHontoDAandkA+i)]TJ/F7 6.974 Tf 6.227 0 Td[(1k1 ;kAA+i)]TJ/F7 6.974 Tf 6.227 0 Td[(1k1:Applyingthehypothesisofthetheoremtox=A+i)]TJ/F7 6.974 Tf 6.226 0 Td[(1yweconcludethatkBA+i)]TJ/F7 6.974 Tf 6.226 0 Td[(1ykakAA+i)]TJ/F7 6.974 Tf 6.226 0 Td[(1yk+bkA+i)]TJ/F7 6.974 Tf 6.226 0 Td[(1yka+b kyk:Thusfor>>1,theoperatorC:=BA+i)]TJ/F7 6.974 Tf 6.227 0 Td[(1satiseskCk<1sincea<1.Thus)]TJ/F8 9.963 Tf 7.749 0 Td[(162SpecAsoimI+C=H.WeknowthatimA+I=HhenceH=imI+CA+iI=imA+B+iIprovingthatA+Bisself-adjoint.IfDisanycoreforA,itfollowsimmediatelyfromtheinequalityinthehypothesisofthetheoremthattheclosureofA+BrestrictedtoDcontainsDAinitsdomain.ThusA+Bisessentiallyself-adjointonanycoreofA.QED11.8.6Feynmanpathintegrals.ConsidertheoperatorH0:L2R3!L2R3givenbyH0:=)]TJ/F1 9.963 Tf 9.409 14.047 Td[(@2 @x21+@2 @x22+@2 @x23:

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11.8.THETROTTERPRODUCTFORMULA.325HerethedomainofH0istakentobethose2L2R3forwhichthedierentialoperatorontheright,takeninthedistributionalsense,whenappliedtogivesanelementofL2R3.TheoperatorH0iscalledthefreeHamiltonianofnon-relativisticquantummechanics".TheFouriertransformFisaunitaryisomorphismofL2R3intoL2R3andcarriesH0intomultiplicationby2whosedomainconsistsofthose^2L2R3suchthat2^belongstoL2R3.Theoperatorconsistingofmultiplicationbye)]TJ/F10 6.974 Tf 6.227 0 Td[(it2isclearlyunitary,andprovidesuswithaunitaryoneparametergroup.TransferringthisoneparametergroupbacktoL2R3viatheFouriertransformgivesusaoneparametergroupofunitarytransformationswhoseinnitesimalgeneratoris)]TJ/F11 9.963 Tf 7.749 0 Td[(iH0.NowtheFouriertransformcarriesmultiplicationintoconvolution,andtheinverseFouriertransforminthedistributionalsenseofe)]TJ/F10 6.974 Tf 6.226 0 Td[(i2tisit)]TJ/F7 6.974 Tf 6.227 0 Td[(3=2eix2=4t.Hencewecanwrite,inaformalsense,exp)]TJ/F11 9.963 Tf 7.749 0 Td[(itH0fx=it)]TJ/F7 6.974 Tf 6.227 0 Td[(3=2ZR3expix)]TJ/F11 9.963 Tf 9.963 0 Td[(y2 4tfydy:Heretherighthandsideistobeunderstoodasalongwindedwayofwritingthelefthandsidewhichiswelldenedasamathematicalobject.Therighthandsidecanalsoberegardedasanactualintegralforcertainclassesoff,andastheL2limitofsuchsuchintegrals.WeshalldiscussthisinterpretationinSection11.10.LetVbeafunctiononR3.WedenotetheoperatoronL2R3consistingofmultiplicationbyValsobyV.SupposethatVissuchthatH0+Visagainself-adjoint.Forexample,ifVwerecontinuousandofcompactsupportthiswouldcertainlybethecasebytheKato-Rellichtheorem.Realisticpotentials"Vwillnotbeofcompactsupportorbebounded,butneverthelessinmanyimportantcasestheKato-Rellichtheoremdoesapply.ThentheTrotterproductformulasaysthatexp)]TJ/F11 9.963 Tf 7.748 0 Td[(itH0+V=limn!1exp)]TJ/F11 9.963 Tf 7.749 0 Td[(it nH0exp)]TJ/F11 9.963 Tf 7.749 0 Td[(it nVn:Wehaveexp)]TJ/F11 9.963 Tf 7.748 0 Td[(it nVfx=e)]TJ/F10 6.974 Tf 6.227 0 Td[(it nVxfx:HencewecanwritetheexpressionunderthelimitsignintheTrotterprod-uctformula,whenappliedtofandevaluatedatx0asthefollowingformalexpression:4it n)]TJ/F7 6.974 Tf 6.226 0 Td[(3n=2ZR3ZR3expiSnx0;:::;xnfxndxndx1whereSnx0;x1;:::;xn;t:=nXi=1t n"1 4xi)]TJ/F11 9.963 Tf 9.963 0 Td[(xi)]TJ/F7 6.974 Tf 6.226 0 Td[(1 t=n2)]TJ/F11 9.963 Tf 9.963 0 Td[(Vxi#:

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326CHAPTER11.STONE'STHEOREMIfX:s7!Xs;0stisapiecewisedierentiablecurve,thentheactionofaparticleofmassmmovingalongthiscurveisdenedinclassicalmechanicsasSX:=Zt0m 2_Xs2)]TJ/F11 9.963 Tf 9.963 0 Td[(VXsdswhere_Xisthevelocitydenedatallbutnitelymanypoints.Takem=1 2andletXbethepolygonalpathwhichgoesfromx0tox1,fromx1tox2etc.,eachintimet=nsothatthevelocityisjxi)]TJ/F11 9.963 Tf 10.218 0 Td[(xi)]TJ/F7 6.974 Tf 6.227 0 Td[(1j=t=nonthei-thsegment.Also,theintegralofVXsoverthissegmentisapproximatelyt nVxi.TheformalexpressionwrittenabovefortheTrotterproductformulacanbethoughtofasanintegraloverpolygonalpathswithsteplengtht=nofeiSnXfXtdnXwhereSnapproximatestheclassicalactionandwherednXisameasureonthisspaceofpolygonalpaths.ThissuggeststhatanintuitivewayofthinkingabouttheTrotterproductformulainthiscontextistoimaginethatthereissomekindofmeasure"dXonthespacex0ofallcontinuouspathsemanatingfromx0andsuchthatexp)]TJ/F11 9.963 Tf 7.748 0 Td[(itH0+Vfx=Zx0eiSXfXtdX:Thisformulawassuggestedin1942byFeynmaninhisthesisTrotter'spaperwasin1959,andhasbeenthebasisofanenormousnumberofimportantcalculationsinphysics,manyofwhichhavegivenrisetoexcitingmathematicaltheoremswhichwerethenprovedbyothermeans.Iamunawareofanygeneralmathematicaljusticationofthesepathintegral"methodsintheformthattheyareused.11.9TheFeynman-Kacformula.AnimportantadvancewasintroducedbyMarkKacin1951wheretheunitarygroupexp)]TJ/F11 9.963 Tf 7.749 0 Td[(itH0+Visreplacedbythecontractionsemi-groupexp)]TJ/F11 9.963 Tf 7.749 0 Td[(tH0+V.ThenthetechniquesofprobabilitytheoryinparticulartheexistenceofWienermeasureonthespaceofcontinuouspathscanbebroughttobeartojustifyaformulaforthecontractivesemi-groupasanintegraloverpathspace.Iwillstateandproveanelementaryversionofthisformulawhichfollowsdirectlyfromwhatwehavedone.Theassumptionsaboutthepotentialarephysicallyunrealistic,butIchoosetoregardtheextensiontoamorerealisticpotentialasatechnicalissueratherthanaconceptualone.LetVbeacontinuousrealvaluedfunctionofcompactsupport.Toeachcontinuouspath!onRnandforeachxedtimet0wecanconsidertheintegralZt0V!sds:Themap!7!Zt0V!sds.31

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11.9.THEFEYNMAN-KACFORMULA.327isacontinuousfunctiononthespaceofcontinuouspaths,andwehavet mmXj=1V!jt m!Zt0V!sds.32foreachxed!.Theorem11.9.1TheFeynman-Kacformula.LetVbeacontinuousrealvaluedfunctionofcompactsupportonRn.LetH=+VasanoperatoronH=L2Rn.ThenHisself-adjointandforeveryf2H)]TJ/F11 9.963 Tf 4.567 -8.07 Td[(e)]TJ/F10 6.974 Tf 6.226 0 Td[(tHfx=Zxf!texpZt0V!sdsdx!.33wherexisthespaceofcontinuouspathsemanatingfromxanddx!istheassociatedWienermeasure.Proof.[FromReed-SimonIIpage280.]SincemultiplicationbyVisaboundedself-adjointoperator,wecanapplytheKato-Rellichtheoremwitha=0!toconcludethatHisself-adjoint,andwiththesamedomainas.SowemayapplytheTrotterproductformulatoconcludethate)]TJ/F10 6.974 Tf 6.227 0 Td[(Htf=limm!1e)]TJ/F9 4.981 Tf 9.077 2.678 Td[(t me)]TJ/F9 4.981 Tf 9.077 2.678 Td[(t mVmf:ThisconvergenceisinL2,butbypassingtoasubsequencewemayalsoassumethattheconvergenceisalmosteverywhere.Nowhe)]TJ/F9 4.981 Tf 9.076 2.677 Td[(t me)]TJ/F9 4.981 Tf 9.077 2.677 Td[(t mVmfix=ZRnZRnpx;xm;t mpx;xm;t mfx1exp0@)]TJ/F10 6.974 Tf 13.07 12.453 Td[(mXj=1t mVxj1Adx1dxm:BytheverydenitionofWienermeasure,thislastexpressionisZxexp0@t mmXj=1V!jt m1Af!tdx!:TheintegrandwithrespecttotheWienermeasuredx!convergesonallcon-tinuouspaths,thatistosayalmosteverywherewithrespecttodxtotheintegrandonrighthandsideof.33.Sotojustify.33wemustprovethattheintegralofthelimitisthelimitoftheintegral.Wewilldothisbythedominatedconvergencetheorem:Zxexp0@t mmXj=1V!jt m1Af!tdx!

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328CHAPTER11.STONE'STHEOREMetmaxjVjZxjf!tjdx!=etmaxjVj)]TJ/F11 9.963 Tf 4.566 -8.069 Td[(e)]TJ/F10 6.974 Tf 6.227 0 Td[(tjfjx<1foralmostallx.Hence,bythedominatedconvergencetheorem,.33holdsforalmostallx.QED11.10ThefreeHamiltonianandtheYukawapo-tential.InthissectionIwanttodiscussthefollowingcircleofideas.ConsidertheoperatorH0:L2R3!L2R3givenbyH0:=)]TJ/F1 9.963 Tf 9.409 14.048 Td[(@2 @x21+@2 @x22+@2 @x23:HerethedomainofH0istakentobethose2L2R3forwhichthedierentialoperatorontheright,takeninthedistributionalsense,whenappliedtogivesanelementofL2R3.TheoperatorH0hasafancyname.ItiscalledthefreeHamiltonianofnon-relativisticquantummechanics".Strictlyspeakingweshouldaddforparticlesofmassone-halfinunitswherePlanck'sconstantisone".TheFouriertransformisaunitaryisomorphismofL2R3intoL2R3andcarriesH0intomultiplicationby2whosedomainconsistsofthose^2L2R3suchthat2^belongstoL2R3.TheoperatorsVt:L2R3!L2R3;^7!e)]TJ/F10 6.974 Tf 6.226 0 Td[(it2^formaoneparametergroupofunitarytransformationswhoseinnitesimalgen-eratorinthesenseofStone'stheoremisoperatorconsistingofmultiplicationby2withdomainasgivenabove.[Theminussignbeforetheiintheexponentialistheconventionusedinquantummechanics.Sowewriteexp)]TJ/F11 9.963 Tf 7.748 0 Td[(itAfortheone-parametergroupassociatedtotheself-adjointoperatorA.Iapologizeforthisratherirrelevantnotationalchange,butIwanttomakethenotationinthissectionconsistentwithwhatyouwillseeinphysicsbooks.]Thustheoperatorofmultiplicationby2,andhencetheoperatorH0isaself-adjointtransformation.Theoperatorofmultiplicationby2isclearlynon-negativeandsoeverypointonthenegativerealaxisbelongstoitsresolventset.Letuswriteapointonthenegativerealaxisas)]TJ/F11 9.963 Tf 7.748 0 Td[(2where>0.Thentheresolventofmultiplicationby2atsuchapointonthenegativerealaxisisgivenbymultiplicationby)]TJ/F11 9.963 Tf 7.749 0 Td[(fwheref=f:=1 2+2:Wecansummarizewhatweknow"sofarasfollows:

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11.10.THEFREEHAMILTONIANANDTHEYUKAWAPOTENTIAL.3291.TheoperatorH0isselfadjoint.2.TheoneparametergroupofunitarytransformationsitgeneratesviaStone'stheoremisUt=F)]TJ/F7 6.974 Tf 6.227 0 Td[(1VtFwhereVtismultiplicationbye)]TJ/F10 6.974 Tf 6.227 0 Td[(it2.3.Anypoint)]TJ/F11 9.963 Tf 7.748 0 Td[(2;>0liesintheresolventsetofH0andR)]TJ/F11 9.963 Tf 7.749 0 Td[(2;H0=F)]TJ/F7 6.974 Tf 6.227 0 Td[(1mfFwheremfdenotestheoperationofmultiplicationbyfandfisasgivenabove.4.Ifg2Sandmgdenotesmultiplicationbyg,thenthetheoperatorF)]TJ/F7 6.974 Tf 6.227 0 Td[(1mgFconsistsofconvolutionbyg.Neitherthefunctione)]TJ/F10 6.974 Tf 6.227 0 Td[(it2northefunctionfbelongstoS,sotheoperatorsUtandR)]TJ/F11 9.963 Tf 7.748 0 Td[(2;H0canonlybethoughtofasconvolutionsinthesenseofgeneralizedfunctions.11.10.1TheYukawapotentialandtheresolvent.Nevertheless,wewillbeabletogivesomeslightlymoreexplicitandveryin-structiverepresentationsoftheseoperatorsasconvolutions.Forexample,wewillusetheCauchyresiduecalculustocomputefandwewillnd,uptofactorsofpowersof2thatfisthefunctionYx:=e)]TJ/F10 6.974 Tf 6.226 0 Td[(r rwhererdenotesthedistancefromtheorigin,i.e.r2=x2.Thisfunctionhasanintegrablesingularityattheorigin,andvanishesrapidlyatinnity.SoconvolutionbyYwillbewelldenedandgivenbytheusualformulaonelementsofSandextendstoanoperatoronL2R3.ThefunctionYisknownastheYukawapotential.Yukawaintroducedthisfunctionin1934toexplaintheforcesthatholdthenucleustogether.Theexponentialdecaywithdistancecontrastswiththatoftheordinaryelectromag-neticorgravitationalpotential1=rand,inYukawa'stheory,accountsforthefactthatthenuclearforcesareshortrange.Infact,Yukawaintroducedaheavyboson"toaccountforthenuclearforces.Theroleofmesonsinnuclearphysicswaspredictedbybrillianttheoreticalspeculationwellbeforeanyexperimentaldiscovery.Herearethedetails:Sincef2L2wecancomputeitsinverseFouriertransformas)]TJ/F7 6.974 Tf 6.226 0 Td[(3=2f=limR!1)]TJ/F7 6.974 Tf 6.227 0 Td[(3ZjjReix 2+2d:.34

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330CHAPTER11.STONE'STHEOREM 0)]TJ/F11 9.963 Tf 7.749 0 Td[(RRR+ip R)]TJ/F11 9.963 Tf 7.749 0 Td[(R+ip R 6 ?HerelimmeanstheL2limitandjjdenotesthelengthofthevector,i.e.jj=p 2andwewillusesimilarnotationjxj=rforthelengthofx.Assumex6=0.Letu:=x jjjxjsouisthecosineoftheanglebetweenxand.Fixxandintroducesphericalcoordinatesinspacewithxatthenorthpoleands=jjsothat)]TJ/F7 6.974 Tf 6.227 0 Td[(3ZjjReix 2+2d=2)]TJ/F7 6.974 Tf 6.226 0 Td[(2ZR0Z1)]TJ/F7 6.974 Tf 6.226 0 Td[(1eisjxju s2+2s2duds=1 2ijxjZR)]TJ/F10 6.974 Tf 6.227 0 Td[(Rseisjxj s+is)]TJ/F11 9.963 Tf 9.962 0 Td[(ids:Thislastintegralisalongthebottomofthepathinthecomplexs-planecon-sistingoftheboundaryoftherectangleasdrawninthegure.Onthetwoverticalsidesoftherectangle,theintegrandisboundedbysomeconstanttime1=R,sothecontributionoftheverticalsidesisO=p R.OnthetoptheintegrandisOe)]TJ 6.227 5.912 Td[(p R.Sothelimitsoftheseintegralsarezero.Thereisonlyonepoleintheupperhalfplaneats=i,sotheintegralisgivenby2ithisresiduewhichequals2iie)]TJ/F10 6.974 Tf 6.227 0 Td[(jxj 2i=ie)]TJ/F10 6.974 Tf 6.227 0 Td[(jxj:Insertingthisbackinto11.34weseethatthelimitexistsandisequalto)]TJ/F7 6.974 Tf 6.227 0 Td[(3=2^f=1 4e)]TJ/F10 6.974 Tf 6.227 0 Td[(jxj jxj:Weconcludethatfor2S[H0+2)]TJ/F7 6.974 Tf 6.226 0 Td[(1]x=1 4ZR3e)]TJ/F10 6.974 Tf 6.227 0 Td[(jx)]TJ/F10 6.974 Tf 6.226 0 Td[(yj jx)]TJ/F11 9.963 Tf 9.963 0 Td[(yjydy;andsinceH0+2)]TJ/F7 6.974 Tf 6.227 0 Td[(1isaboundedoperatoronL2thisformulaextendsintheL2sensetoL2.

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11.10.THEFREEHAMILTONIANANDTHEYUKAWAPOTENTIAL.33111.10.2ThetimeevolutionofthefreeHamiltonian.Theexplicit"calculationoftheoperatorUtisslightlymoretricky.Thefunction7!e)]TJ/F10 6.974 Tf 6.227 0 Td[(it2isanimaginaryGaussian",soweexpectisinverseFouriertransformtoalsobeanimaginaryGaussian,andthenwewouldhavetomakesenseofconvolutionbyafunctionwhichhasabsolutevalueoneatallpoints.Thereareseveralwaystoproceed.Oneinvolvesintegrationbyparts,andIhopetoexplainhowthisworkslateroninthecourseinconjunctionwiththemethodofstationaryphase.HereIwillfollowReed-SimonvolIIp.59andaddalittlepositivetermtotandthenpasstothelimit.Inotherwords,letbeacomplexnumberwithpositiverealpartandconsiderthefunction7!e)]TJ/F10 6.974 Tf 6.226 0 Td[(2ThisfunctionbelongstoSanditsinverseFouriertransformisgivenbythefunctionx7!)]TJ/F7 6.974 Tf 6.227 0 Td[(3=2e)]TJ/F10 6.974 Tf 6.227 0 Td[(x2=4:Infact,weveriedthiswhenisreal,buttheintegraldeningtheinverseFouriertransformconvergesintheentirehalfplaneRe>0uniformlyinanyRe>andsoisholomorphicintherighthalfplane.Sotheformulaforrealpositiveimpliestheformulaforinthehalfplane.Wethushavee)]TJ/F10 6.974 Tf 6.227 0 Td[(H0x=1 43=2ZR3ejx)]TJ/F10 6.974 Tf 6.226 0 Td[(yj2=4ydy:Herethesquarerootinthecoecientinfrontoftheintegralisobtainedbycontinuationfromthepositivesquarerootonthepositiveaxis.Forexample,ifwetake=+itsothat)]TJ/F11 9.963 Tf 7.749 0 Td[(=)]TJ/F11 9.963 Tf 7.748 0 Td[(it)]TJ/F11 9.963 Tf 9.963 0 Td[(iwegetUtx=lim&0Ut)]TJ/F11 9.963 Tf 8.887 0 Td[(ix=lim&0it)]TJ/F11 9.963 Tf 8.887 0 Td[(i)]TJ/F6 4.981 Tf 7.422 2.677 Td[(3 2Zejx)]TJ/F10 6.974 Tf 6.227 0 Td[(yj2=4it)]TJ/F10 6.974 Tf 6.227 0 Td[(iydy:HerethelimitisinthesenseofL2.WethuscouldwriteUtx=i)]TJ/F7 6.974 Tf 6.227 0 Td[(3=2Zeijx)]TJ/F10 6.974 Tf 6.227 0 Td[(yj2=4tydyifweunderstandtherighthandsidetomeanthe&0limitoftheprecedingexpression.Actually,asReedandSimonpointout,if2L1theaboveintegralexistsforanyt6=0,soif2L1L2weshouldexpectthattheaboveintegralisindeedtheexpressionforUt.Hereistheirargument:Weknowthatexp)]TJ/F11 9.963 Tf 7.749 0 Td[(it)]TJ/F11 9.963 Tf 9.962 0 Td[(i!UtinthesenseofL2convergenceas&0.HereweuseatheoremfrommeasuretheorywhichsaysthatifyouhaveanL2convergentsequenceyoucanchoose

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332CHAPTER11.STONE'STHEOREMasubsequencewhichalsoconvergespointwisealmosteverywhere.Sochooseasubsequenceofforwhichthishappens.Butthenthedominatedconvergencetheoremkicksintoguaranteethattheintegralofthelimitisthelimitoftheintegrals.Tosumup:ThefunctionP0x;y;t:=it)]TJ/F7 6.974 Tf 6.226 0 Td[(3=2eijx)]TJ/F10 6.974 Tf 6.227 0 Td[(yj=4tiscalledthefreepropagator.For2L1L2[Ut]x=ZR3P0x;y;tydyandtheintegralconverges.ForgeneralelementsofL2theoperatorUtisobtainedbytakingtheL2limitoftheaboveexpressionforanysequenceofelementsofL1L2whichapproximateinL2.Alternatively,wecouldinterprettheaboveintegralasthe&limitofthecorrespondingexpressionwithtreplacedbyt)]TJ/F11 9.963 Tf 9.962 0 Td[(i.

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Chapter12MoreaboutthespectraltheoremInthischapterwepresentmoreapplicationsofthespectraltheoremandStone'stheorem,mainlytoproblemsarisingfromquantummechanics.MostofthematerialistakenfromthebookSpectraltheoryanddierentialoperatorsbyDaviesandthefourvolumeMethodsofModernMathematicalPhysicsbyReedandSimon.Thematerialintherstsectionistakefromthetwopapers:W.O.AmreinandV.Georgescu,HelveticaPhysicaActa46pp.636-658.andW.HunzikerandI.SigalJ.Math.Phys.41pp.3448-3510.12.1Boundstatesandscatteringstates.Itisatruisminatomicphysicsorquantumchemistrycoursesthattheeigen-statesoftheSchrodingeroperatorforatomicelectronsaretheboundstates,theonesthatremainboundtothenucleus,andthatthescatteringstates"whichyoinlargepositiveornegativetimescorrespondtothecontinuousspectrum.Thepurposeofthissectionistogiveamathematicaljusticationforthistruism.ThekeyresultisduetoRuelle,,usingergodictheorymethods.Themorestreamlinedversionpresentedherecomesfromthetwopa-persmentionedabove.Theergodictheoryusedislimitedtothemeanergodictheoremofvon-NeumannwhichhasaveryslickproofduetoF.RieszwhichIshallgive.12.1.1Schwartzschild'stheorem.ThereisaclassicalprecursortoRuelle'stheoremwhichisduetoSchwartzschild.ThisisthesameSchwartzschildwho,sometwentyyearslater,gavethefamousSchwartzschildsolutiontotheEinsteineldequations.Schwartzschild'stheoremsays,roughlyspeaking,thatinamechanicalsystemlikethesolarsys-tem,thetrajectorieswhicharecaptured",i.e.whichcomeinfrominnityat333

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334CHAPTER12.MOREABOUTTHESPECTRALTHEOREMnegativetimebutwhichremaininaniteregionforallfuturetimeconstituteasetofmeasurezero.SchwartzschildderivedhistheoremfromthePoincarerecurrencetheorem.IlearnedthesetwotheoremsfromacourseoncelestialmechanicsbyCarlLudwigSiegelthatIattendedin1953.Thesetheoremsap-pearinthelastfewpagesofSiegel'sfamousVorlesungenuberHimmelsmechanikwhichdeveloppedoutofthiscourse.ForproofsIrefertothetreatmentgiventhere.ThePoincarerecurrencetheorem.Thissaysthefollowing:Theorem12.1.1[ThePoincarerecurrencetheorem.]LetStbeameasurepreservingowonameasurespaceM;.LetAbeasubsetofMcontainedinaninvariantsetofnitemeasure.ThenoutsideofasubsetofAofmeasurezero,everypointpofAhasthepropertythatStp2Aforinnitelymanytimesinthefuture.Theideaisquitesimple.Ifthiswerenotthecase,therewouldbeaninnitesequenceofdisjointsetsofpositivemeasureallcontainedinaxedsetofnitemeasure.Schwartzschild'stheorem.ConsiderthesamesetupasinthePoincarerecurrencetheorem,butnowletMbeametricspacewitharegularmeasure,andassumethattheStarehomeomorphisms.LetAbeanopensetofnitemeasure,andletBconsistofallpsuchthatStp2Aforallt0.LetCconsistofallpsuchthattp2Aforallt2R.ClearlyCB.Theorem12.1.2[Schwartzchild'stheorem.]ThemeasureofBnCiszero.Again,theproofisstraightforwardandIrefertoSiegelfordetails.Phrasedinmoreintuitivelanguage,Schwartzschild'stheoremsaysthatoutsideofasetofmeasurezero,anypointp2AwhichhasthepropertythatStp2Aforallt0alsohasthepropertythatStp2Aforallt<0.Thecaptureorbits'havemeasurezero.Ofcourse,thecatchinthetheoremisthatoneneedstoprovethatBhaspositivemeasureforthetheoremtohaveanycontent.SiegelcallsthesetBthesetofpointswhichareweaklystableforthefuturewithrespecttoAandCthesetofpointswhichareweaklystableforalltime.Applicationtothesolarsystem?SupposeonehasamechanicalsystemwithkineticenergyTandpotentialenergyV.

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12.1.BOUNDSTATESANDSCATTERINGSTATES.335Ifthepotentialenergyisboundedfrombelow,thenwecanndaboundforthekineticenergyintermsofthetotalenergy:TaH+bwhichistheclassicalanalogueofthekeyoperatorboundinquantumversion,seeequation.7below.AregionoftheformkxkR;Hx;pEthenformsaboundedregionofnitemeasureinphasespace.Liouville'stheo-remassertsthattheowonphasespaceisvolumepreserving.SoSchwartzschild'stheoremappliestothisregion.IquoteSiegelastothepossibleapplicationtothesolarsystem:Undertheunprovedassumptionthattheplanetarysystemisweaklystablewithrespecttoalltime,wecandrawthefollowingconclusion:Iftheplanetarysystemcapturesaparticlecominginfrominnity,saysomeexternalmatter,thenthenewsystemwiththisadditionalparticleisnolongerweaklystablewithrespecttojustfuturetime,anditfollowsthattheparticle-oraplanetorthesun-mustagainbeexpelled,oracollisionmusttakeplace.Foraninterpretationofthesignicanceofthisresult,onemust,however,considerthatwedonotevenknowwhetherforn>2thesolutionsofthen-bodyproblemthatareweaklystablewithrespecttoalltimeformasetofpositivemeasure.12.1.2ThemeanergodictheoremWewillneedthecontinuoustimeversion:LetHbeaself-adjointoperatoronaHilbertspaceHandletVt=exp)]TJ/F11 9.963 Tf 7.748 0 Td[(itHbetheoneparametergroupitgeneratesbyStone'stheorem.vonNeumann'smeanergodictheoremassertsthatforanyf2HthelimitlimT!11 TZT0Vtfdtexists,andthelimitisaneigenvectorofHcorrespondingtotheeigenvalue0.Clearly,ifHf=0,thenVtf=fforalltandtheabovelimitexiststriviallyandisequaltof.IffisorthogonaltotheimageofH,i.e.ifHg=08g2DomHthenf2DomH=DomHandHf=Hf=0.SoifwedecomposeHintothezeroeigenspaceofHanditsorthogonalcomplement,wearereducedtothefollowingversionofthetheoremwhichistheonewewillactuallyuse:

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336CHAPTER12.MOREABOUTTHESPECTRALTHEOREMTheorem12.1.3LetHbeaself-adjointoperatoronaHilbertspaceH,andassumethatHhasnoeigenvectorswitheigenvalue0,sothattheimageofHisdenseinH.LetVt=exp)]TJ/F11 9.963 Tf 7.749 0 Td[(itHbetheoneparametergroupgeneratedbyH.ThenlimT!11 TZT0Vtfdt=0forallf2H.Proof.Ifh=)]TJ/F11 9.963 Tf 7.749 0 Td[(iHgthenVth=d dtVtgso1 TZT0Vthdt=1 TVtg)]TJ/F11 9.963 Tf 9.962 0 Td[(g!0:Byhypothesis,foranyf2Hwecan,forany>0,ndanhoftheaboveformsuchthatkf)]TJ/F11 9.963 Tf 9.963 0 Td[(hk<1 2so1 TZT0Vtfdt1 2+1 TZT0Vthdt:BythenchoosingTsucientlylargewecanmakethesecondtermlessthan1 2:12.1.3Generalconsiderations.LetHbeaself-adjointoperatoronaseparableHilbertspaceHandletVtbetheoneparametergroupgeneratedbyHsoVt:=exp)]TJ/F11 9.963 Tf 7.749 0 Td[(iHt:LetH=HpHcbethedecompositionofHintothesubspacescorrespondingtopurepointspec-trumandcontinuousspectrumofH.LetfFrg;r=1;2;:::beasequenceofself-adjointprojections.IntheapplicationwehaveinmindwewillletH=L2RnandtakeFrtobetheprojectionontothecompletionofthespaceofcontinuousfunctionssupportedintheballofradiusrcenteredattheorigin,butinthissectionourconsiderationswillbequitegeneral.WeletF0rbetheprojectionontothesubspaceorthogonaltotheimageofFrsoF0r:=I)]TJ/F11 9.963 Tf 9.963 0 Td[(Fr:LetM0:=ff2Hjlimr!1supt2RkI)]TJ/F11 9.963 Tf 9.963 0 Td[(FrVtfk2=0g;.1

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12.1.BOUNDSTATESANDSCATTERINGSTATES.337andM1:=f2HlimT!11 TZT0kFrVtfk2dt=0;forallr=1;2;::::.2Proposition12.1.1Thefollowinghold:1.M0andM1arelinearsubspacesofH.2.ThesubspacesM0andM1areclosed.3.M0isorthogonaltoM1.4.HpM0:5.M1Hc.Thefollowinginequalitywillbeusedrepeatedly:Foranyf;g2Hkf+gk2kf+gk2+kf)]TJ/F11 9.963 Tf 9.963 0 Td[(gk2=2kfk2+2kgk2.3wherethelastequalityisthetheoremofAppolonius.Proofof1.Letf1;f22M0.ThenforanyscalarsaandbandanyxedrandtwehavekI)]TJ/F11 9.963 Tf 9.963 0 Td[(FrVtaf1+bf2k22jaj2kI)]TJ/F11 9.963 Tf 9.963 0 Td[(FrVtf1k2+2jbj2kI)]TJ/F11 9.963 Tf 9.962 0 Td[(FrVtf2k2by.3.TakingseparatesupsovertontherightsideandthenovertontheleftshowsthatsuptkI)]TJ/F11 9.963 Tf 9.963 0 Td[(FrVtaf1+bf2k22jaj2suptkI)]TJ/F11 9.963 Tf 9.963 0 Td[(FrVtf1k2+2jbj2suptkI)]TJ/F11 9.963 Tf 9.963 0 Td[(FrVtf2k2forxedr.Lettingr!1thenshowsthataf1+bf22M0.Letf1;f22M1.Forxedrweuse.3toconcludethat1 TZT0kFrVtaf1+bf2k2dt2jaj2 TZT0kFrVtf1k2dt+2jbj2 TZT0kFrVtf2k2dt:Eachtermontherightconvergesto0asT!1provingthataf1+bf22M1.Thisproves1.Proofof2.Letfn2M0andsupposethatfn!f.Given>0chooseNsothatkfn)]TJ/F11 9.963 Tf 9.962 0 Td[(fk2<1 4foralln>N.ThisimpliesthatkI)]TJ/F11 9.963 Tf 9.963 0 Td[(FrVtf)]TJ/F11 9.963 Tf 9.962 0 Td[(fnk2<1 4

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338CHAPTER12.MOREABOUTTHESPECTRALTHEOREMforalltandnsinceVtisunitaryandI)]TJ/F11 9.963 Tf 9.963 0 Td[(Frisacontraction.ThensuptkI)]TJ/F11 9.963 Tf 9.962 0 Td[(FrVtfk21 2+2suptkI)]TJ/F11 9.963 Tf 9.963 0 Td[(FrVtfnk2foralln>Nandanyxedr.Wemaychoosersucientlylargesothatthesecondtermontherightisalsolessthat1 2.Thisprovesthatf2M0.Letfn2M1andsupposethatfn!f.Given>0chooseNsothatkfn)]TJ/F11 9.963 Tf 9.963 0 Td[(fk2<1 4foralln>N.Then1 TZT0kFrVrfk2dt2 TZT0kFrVrf)]TJ/F11 9.963 Tf 9.963 0 Td[(fnk2dt+2 TZT0kFrVrfnk2dt1 2+2 TZT0kFrVrfnk2dt:Fixn.ForanygivenrwecanchooseT0largeenoughsothatthesecondtermontherightis<1 2.ThisshowsthatforanyxedrwecanndaT0sothat1 TZT0kFrVrfk2dtT0,provingthatf2M1.Thisproves2.Proofof3.Letf2M0andg2M1both6=0.Thenjf;gj2=1 TZT0jf;gj2dt=1 TZT0jVtf;Vtgj2dt=1 TZT0jF0rVtf;g+Vtf;Frgj2dt2 TZT0jF0rVtf;Vtgj2dt+2 TZT0jVtf;Frgj2dt2 Tkgk2ZjF0rVtfk2dt+2 Tkfk2ZT0kFrVtgk2dtwhereweusedtheCauchy-Schwarzinequalityinthelaststep.Forany>0wemaychoosersothatkF0rVtfk2 4kgk2forallt.WecanchooseaTsuchthat1 TZT0kFrVtgk2dt< 4kfk2:

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12.1.BOUNDSTATESANDSCATTERINGSTATES.339Pluggingbackintothelastinequalityshowsthatjf;gj2<:Sincethisistrueforany>0weconcludethatf?g.Thisproves3.Proofof4.SupposeHf=Ef.ThenkF0rVtfk2=kF0re)]TJ/F10 6.974 Tf 6.227 0 Td[(iEtfk2=ke)]TJ/F10 6.974 Tf 6.227 0 Td[(iEtF0rfk2=kF0rfk2:ButweareassumingthatF0r!0inthestrongtopology.Sothislastexpressiontendsto0provingthatf2M0whichistheassertionof4.Proofof5.By3wehaveM1M?0.By4wehaveM?0H?p=Hc.Proposition12.1.1isvalidwithoutanyassumptionswhatsoeverrelatingHtotheFr.TheonlyplacewhereweusedHwasintheproofof4whereweusedthefactthatiffisaneigenvectorofHthenitisalsoaneigenvectorofofVtandsowecouldpulloutascalar.ThegoalistoimposesucientrelationsbetweenHandtheFrsothatHcM1:.4Ifweprovethisthenpart5ofProposition12.1.1impliesthatHc=M1andthenpart3saysthatM0M?1=H?c=Hp:Thenpart4givesM0=Hp:Asapreliminarywestateaconsequenceofthemeanergodictheorem:12.1.4Usingthemeanergodictheorem.RecallthatthemeanergodictheoremsaysthatifUtisaunitaryoneparametergroupactingwithoutnon-zeroxedvectorsonaHilbertspaceGthenlimT!11 TZT0Utdt=0forall2G:LetG=Hc^Hc:WeknowfromourdiscussionofthespectraltheoremProposition10.12.1thatHI)]TJ/F11 9.963 Tf 10.069 0 Td[(IHdoesnothavezeroasaneigenvalueactingonG.WemayapplythemeanergodictheoremtoconcludethatlimT!11 TZT0e)]TJ/F10 6.974 Tf 6.227 0 Td[(itHfeitHedt=0

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340CHAPTER12.MOREABOUTTHESPECTRALTHEOREMforanye;f2Hc.Wehaveje;e)]TJ/F10 6.974 Tf 6.227 0 Td[(itHfj2=ef;e)]TJ/F10 6.974 Tf 6.227 0 Td[(itHfeitHe:WeconcludethatlimT!11 TZT0je;Vtfj2dt=08e2H;f2Hc:.5Indeed,ife2Hcthisfollowsfromtheabove,whileife2Hptheintegrandisidenticallyzero.12.1.5TheAmrein-Georgescutheorem.Wecontinuewiththepreviousnotation,andletEc:H!Hcdenoteorthogonalprojection.WeletSnandSbeacollectionofboundedoperatorsonHsuchthat[Sn;H]=0,Sn!Sinthestrongtopology,TherangeofSisdenseinH,andFrSnEciscompactforallrandn.Theorem12.1.4[Armein-Georgescu.]Undertheabovehypotheses.4holds.Proof.SinceM1isaclosedsubspaceofH,toprovethat12.4holds,itisenoughtoprovethatDM1forsomesetDwhichisdenseinHc.SinceSleavesthespacesHpandHcinvariant,thefactthattherangeofSisdenseinHbyhypothesis,saysthatSHcisdenseinHc.Sowehavetoshowthatg=Sf;f2HclimT!11 TZT0kFrVtgk2dt=0foranyxedr.Wemayassumef6=0.Let>0bexed.ChoosensolargethatkS)]TJ/F11 9.963 Tf 9.963 0 Td[(Snfk2< 6:AnycompactoperatorinaseparableHilbertspaceisthenormlimitofniterankoperators.SowecanndaniterankoperatorTNsuchthatkFrSnEc)]TJ/F11 9.963 Tf 9.963 0 Td[(TNk2< 12kfk2:

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12.1.BOUNDSTATESANDSCATTERINGSTATES.341Writingg=S)]TJ/F11 9.963 Tf 9.962 0 Td[(Snf+Snfweconcludethat1 TZT0kFrVtgk2dt2 TZT0kFrVtS)]TJ/F11 9.963 Tf 9.962 0 Td[(Snfk2dt+2 TZT0kFrVtSnfk2dt 3+4 TZT0kFrSnEc)]TJ/F11 9.963 Tf 9.962 0 Td[(TNk2kVtfk2dt+4 TZT0kTNVtk2dt2 3+4 TZT0kTNVtk2dt:TosaythatTNisofniterankmeansthattherearegi;hi2H;i=1;:::N<1suchthatTNf=NXi=1f;higi:Substitutingthisinto4 TRT0kTNVtk2dt:gives4 TZT0kTNVtk2dt=4 TZT0NXi=0Vtf;higi2dt2N)]TJ/F7 6.974 Tf 6.227 0 Td[(14Xkgik21 TZT0jhi;Vtfj2dt:By.5wecanchooseT0solargethatthisexpressionis< 3forallT>T0.OfcourseaspecialcaseofthetheoremwillbewherealltheSn=SaswillbethecaseforRuelle'stheoremforKatopotentials.12.1.6Katopotentials.LetX=Rnforsomen.AlocallyL2realvaluedfunctiononXiscalledaKatopotentialifforany>0thereisa=suchthatkVkkk+kk.6forall2C10X.ClearlythesetofallKatopotentialsonXformarealvectorspace.ExamplesofKatopotentials.V2L2R3.Forexample,supposethatX=R3andV2L2X.WeclaimthatVisaKatopotential.Indeed,kVk:=kVk2kVk2kk1:Sowewillbedoneifweshowthatforanya>0thereisab>0suchthatkk1akk2+bkk2:

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342CHAPTER12.MOREABOUTTHESPECTRALTHEOREMBytheFourierinversionformulawehavekk1k^k1where^denotestheFouriertransformof.NowtheFouriertransformofisthefunction7!kk2^wherekkdenotestheEuclideannormof.Since^belongstotheSchwartzspaceS,thefunction7!+kk2^belongstoL2asdoesthefunction7!+kk2)]TJ/F7 6.974 Tf 6.227 0 Td[(1inthreedimensions.Letdenotethefunction7!kk:BytheCauchy-Schwarzinequalitywehavek^k1=j+2)]TJ/F7 6.974 Tf 6.227 0 Td[(1;+2^jck2+1^kck2^k2+ck^k2wherec2=k+2)]TJ/F7 6.974 Tf 6.226 0 Td[(1k2:Foranyr>0andanyfunction2Sletrbedenedby^r=r3^r:Thenk^rk1=k^k1;k^rk2=r3 2k^k2;andk2^rk2=r)]TJ/F6 4.981 Tf 7.423 2.678 Td[(1 2k2k2:Appliedtothisgivesk^k1cr)]TJ/F6 4.981 Tf 7.422 2.677 Td[(1 2k2^k2+cr3 2k^k2:ByPlancherelk2^k2=kk2andk^k2=kk2:ThisshowsthatanyV2L2R3isaKatopotential.V2L1X.IndeedkVk2kVk1kk2:IfweputthesetwoexamplestogetherweseethatifV=V1+V2whereV12L2R3andV22L1R3thenVisaKatopotential.

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12.1.BOUNDSTATESANDSCATTERINGSTATES.343TheCoulombpotential.ThefunctionVx=1 kxkonR3canbewrittenasasumV=V1+V2whereV12L2R3andV22L1R3andsoisKatopotential.Katopotentialsfromsubspaces.SupposethatX=X1X2andVdependsonlyontheX1componentwhereitisaKatopotential.ThenFubiniimpliesthatVisaKatopotentialifandonlyifVisaKatopotentialonX1.SoifX=R3Nandwewritex2Xasx=x1;:::;xNwherexi2R3thenVij=1 kxi)]TJ/F11 9.963 Tf 9.962 0 Td[(xjkareKatopotentialsasareanylinearcombinationofthem.SothetotalCoulombpotentialofanysystemofchargedparticlesisaKatopotential.Byexample12.1.6,therestrictionofthispotentialtothesubspacefxjPmixi=0gisaKatopotential.Thisistheatomicpotential"aboutthecenterofmass.12.1.7ApplyingtheKato-Rellichmethod.Theorem12.1.5LetVbeaKatopotential.ThenH=+Visself-adjointwithdomainD=Domandisboundedfrombelow.Further-more,wehaveanoperatorboundaH+b.7wherea=1 1)]TJ/F11 9.963 Tf 9.963 0 Td[(andb= 1)]TJ/F11 9.963 Tf 9.963 0 Td[(;0<<1:Proof.Asamultiplicationoperator,Visclosedonitsdomainofdenitionconsistingofall2L2suchthatV2L2.SinceC10Xisacorefor,wecanapplytheKatocondition.6toall2Dom.ThusHisdenedasasymmetricoperatoronDom.ForRez<0theoperatorz)]TJ/F8 9.963 Tf 10.16 0 Td[()]TJ/F7 6.974 Tf 6.226 0 Td[(1isbounded.SoforRez<0wecanwritezI)]TJ/F11 9.963 Tf 9.962 0 Td[(H=[I)]TJ/F11 9.963 Tf 9.963 0 Td[(VzI)]TJ/F8 9.963 Tf 9.963 0 Td[()]TJ/F7 6.974 Tf 6.227 0 Td[(1]zI)]TJ/F8 9.963 Tf 9.963 0 Td[(:BytheKatocondition.6wehavekVzI)]TJ/F8 9.963 Tf 9.963 0 Td[()]TJ/F7 6.974 Tf 6.226 0 Td[(1k+jRezj)]TJ/F7 6.974 Tf 6.227 0 Td[(1:

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344CHAPTER12.MOREABOUTTHESPECTRALTHEOREMIfwechoose<1andthenRezsucientlynegative,wecanmaketherighthandsideofthisinequality<1.ForthisrangeofzweseethatRz;H=zI)]TJ/F11 9.963 Tf 9.888 0 Td[(H)]TJ/F7 6.974 Tf 6.227 0 Td[(1isboundedsotherangeofzI)]TJ/F11 9.963 Tf 9.888 0 Td[(HisallofL2.ThisprovesthatHisself-adjointandthatitsresolventsetcontainsahalfplaneRez<<0andsoisboundedfrombelow.Also,for2Domwehave=H)]TJ/F11 9.963 Tf 9.962 0 Td[(VsokkkHk+kVkkHk+kk+kkwhichproves12.7.12.1.8Usingtheinequality12.7.Proposition12.1.2LetHbeaself-adjointoperatoronL2Xsatisfying.7forsomeconstantsaandb.Letf2L1Xbesuchthatfx!0asx!1.ThenforanyzintheresolventsetofHtheoperatorfRz;Hiscompact,where,asusual,fdenotestheoperatorofmultiplicationbyf,Proof.Letpj=1 i@ @xjasusual,andletg2L1XsotheoperatorgpisdenedastheoperatorwhichsendintothefunctionwhoseFouriertransformis7!g^.Theoperatorfxgpisthenormlimitoftheoperatorsfngnwherefnisobtainedfromfbysettingfn=1BnfwhereBnistheballofradius1abouttheorigin,andsimilarlyforg.TheoperatorfnxgnpisgivenbythesquareintegrablekernelKnx;y=fnx^gnx)]TJ/F11 9.963 Tf 9.962 0 Td[(yandsoiscompact.Hencefxgpiscompact.Wewilltakegp=1 1+p2=+)]TJ/F7 6.974 Tf 6.226 0 Td[(1:Theoperator+Rz;Hisbounded.Indeed,by.7k+zI)]TJ/F11 9.963 Tf 9.963 0 Td[(H)]TJ/F7 6.974 Tf 6.227 0 Td[(1k+akHzI)]TJ/F11 9.963 Tf 9.962 0 Td[(H)]TJ/F7 6.974 Tf 6.227 0 Td[(1k+bkRz;Hkk+ak+ajzj+bkRz;Hk:SofxRz;H=fx1 1+p2+Rz;Hiscompact,beingtheproductofacompactoperatorandaboundedoperator.

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12.2.NON-NEGATIVEOPERATORSANDQUADRATICFORMS.34512.1.9Ruelle'stheorem.LetustakeH=+VwhereVisaKatopotential.LetFrbetheoperatorofmultiplicationby1BrsoFrisprojectionontothespaceoffunctionssupportedintheballBrofradiusrcenteredattheorigin.TakeS=Rz;H,wherezhassucientlynegativerealpart.ThenFrSEciscompact,beingtheproductoftheoperatorFrRz;HwhichiscompactbyProposition12.1.2andtheboundedoperatorEc.AlsotheimageofSisallofH.SowemayapplytheAmreinGeorgescutheoremtoconcludethatM0=HpandM1=Hc.12.2Non-negativeoperatorsandquadraticforms.12.2.1Fractionalpowersofanon-negativeself-adjointop-erator.LetHbeaself-adjointoperatoronaseparableHilbertspaceHwithspectrumS.ThespectraltheoremtellsusthatthereisanitemeasureonSNandaunitaryisomorphismU:H!L2=L2SN;suchthatUHU)]TJ/F7 6.974 Tf 6.227 0 Td[(1ismultiplicationbythefunctionhs;n=sandsuchthat2HliesinDomHifandonlyifhU2L2.ClearlyH;0forall2Hifandonlyifassignsmeasurezerotothesetfs;n;s<0ginwhichcasethespectrumofmultiplicationbyh,whichisthesameassayingthatthespectrumofHiscontainedin[0;1.Whenthishappens,wesaythatHisnon-negative.WesaythatHcifH)]TJ/F11 9.963 Tf 9.962 0 Td[(cIisnon-negative.IfHisnon-negative,and>0,wewouldliketodeneHasbeingunitarilyequivalenttomultiplicationbyh.Asthespectraltheoremdoesnotsaythatthe,L2,andUareunique,sowehavetocheckthatthisiswelldened.ForthisconsiderthefunctionfonRdenedbyfx=1 jxj+1:Bythefunctionalcalculus,fHiswelldened,andinanyspectralrepresenta-tiongoesoverintomultiplicationbyfhwhichisinjective.SoK=fH)]TJ/F7 6.974 Tf 6.227 0 Td[(1)]TJ/F11 9.963 Tf 8.59 0 Td[(Iisawelldenedingeneralunboundedself-adjointoperatorwhosespectralrep-resentationismultiplicationbyh.ButtheexpressionforKisindependentofthespectralrepresentation.ThisshowsthatH=Kiswelldened.Proposition12.2.1LetHbeaself-adjointoperatoronaHilbertspaceHandletDomHbethedomainofH.Let0<<1.Thenf2DomHifandonlyiff2DomHandHf2DomH1)]TJ/F10 6.974 Tf 6.226 0 Td[(inwhichcaseHf=H1)]TJ/F10 6.974 Tf 6.226 0 Td[(Hf:

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346CHAPTER12.MOREABOUTTHESPECTRALTHEOREMInparticular,if=1 2,andwedeneBHf;gforf;g2DomH1 2byBHf;g:=H1 2f;H1 2g;thenf2DomHifandonlyiff2DomH1 2andalsothereexistsak2HsuchthatBHf;g=k;g8g2DomH1 2inwhichcaseHf=k:Proof.FortherstpartofthePropositionwemayusethespectralrepresen-tation:ThePropositionthenassertsthatf2L2satisesRjhj2jfj2d<1ifandonlyifZ+jhj2jfj2d<1andZ+jhj21)]TJ/F10 6.974 Tf 6.226 0 Td[(jhfj2d<1whichisobvious,asistheassertionthatthenhf=h1)]TJ/F10 6.974 Tf 6.227 0 Td[(hf:TheassertionthatthereexistsaksuchthatBHf;g=k;g8g2DomH1 2isthesameassayingthatH1 2f2DomH1 2andH1 2H1 2f=k.ButH1 2=H1 2sothesecondpartofthepropositionfollowsfromtherst.12.2.2Quadraticforms.ThesecondhalfofProposition12.2.1suggeststhatwestudynon-negativesesquilinearformsdenedonsomedensesubspaceDofaHilbertspaceH.SowewanttostudyB:DD!CsuchthatBf;gislinearinfforxedg,Bg;f= Bf;g;andBf;f0.Ofcourse,bytheusualpolarizationtricksuchaBisdeterminedbythecorre-spondingquadraticformQf:=Bf;f:WewouldliketondconditionsonBorQwhichguaranteethatB=BHforsomenon-negativeselfadjointoperatorHasgivenbyProposition12.2.1.Thatsomeconditionisnecessaryisexhibitedbythefollowing

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12.2.NON-NEGATIVEOPERATORSANDQUADRATICFORMS.347counterexample.LetH=L2RandletDconsistofallcontinuousfunctionsofcompactsupport.LetBf;g=f g:TheonlycandidateforanoperatorHwhichsatisesBf;g=Hf;gistheoperator"whichconsistsofmultiplicationbythedeltafunctionattheorigin.Butthereisnosuchoperator.Considerasequenceofuniformlyboundedcontinuousfunctionsfnofcom-pactsupportwhichareallidenticallyoneinsomeneighborhoodoftheori-ginandwhosesupportshrinkstotheorigin.Thenfn!0inthenormofH.Also,Qfn)]TJ/F11 9.963 Tf 10.878 0 Td[(fm;fn)]TJ/F11 9.963 Tf 10.879 0 Td[(fm0,soQfn)]TJ/F11 9.963 Tf 10.878 0 Td[(fm;fn)]TJ/F11 9.963 Tf 10.878 0 Td[(fm!0.ButQfn;fn16=0=Q;0.SoDisnotcompleteforthenormkk1kfk1:=Qf+kfk2H1 2:Considerafunctiong2Dwhichequalsoneontheinterval[)]TJ/F8 9.963 Tf 7.749 0 Td[(1;1]sothatg;g=1.Letgn:=g)]TJ/F11 9.963 Tf 9.038 0 Td[(fnwithfnasabove.Thengn!ginHyetQgn0.SoQisnotlowersemi-continuousasafunctiononD.Werecallthedenitionoflowersemi-continuity:12.2.3Lowersemi-continuousfunctions.LetXbeatopologicalspace,andletQ:X!Rbearealvaluedfunction.Letx02X.WesaythatQislowersemi-continuousatx0if,forevery>0thereisaneighborhoodU=Ux0;ofx0suchthatQx0.ThereexistsanindexsuchthatQx0>Qx0)]TJ/F7 6.974 Tf 12.018 3.923 Td[(1 2.ThenthereexistsaneighborhoodUofx0suchthatQx>Qx0)]TJ/F7 6.974 Tf 11.158 3.922 Td[(1 2forallx2UandhenceQxQx>Qx0)]TJ/F11 9.963 Tf 9.962 0 Td[(8x2U:Itiseasytocheckthatthesumandtheinfoftwolowersemi-continuousfunc-tionsislowersemi-continuous.

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348CHAPTER12.MOREABOUTTHESPECTRALTHEOREM12.2.4Themaintheoremaboutquadraticforms.LetHbeaseparableHilbertspaceandQanon-negativequadraticformdenedonadensedomainDH.WemayextendthedomainofdenitionofQbysettingitequalto+1atallpointsofHnD.ThenwecansaythatthedomainofQconsistsofthosefsuchthatQf<1.Thiswillbealittleconvenientintheformulationofthenexttheorem.Theorem12.2.1ThefollowingconditionsonQareequivalent:1.Thereisanon-negativeself-adjointoperatorHonHsuchthatD=DomH1 2andQf=kH1 2fk2:2.Qislowersemi-continuousasafunctiononH.3.D=DomQiscompleterelativetothenormkfk1:=)]TJ/F14 9.963 Tf 4.567 -8.07 Td[(kfk2+Qf1 2:Proof.1.implies2.AsHisnon-negative,theoperatorsnI+Hareinvertiblewithboundedinverse,andnI+H)]TJ/F7 6.974 Tf 6.227 0 Td[(1mapsHontothedomainofH.ConsiderthequadraticformsQnf:=)]TJ/F11 9.963 Tf 4.566 -8.07 Td[(nHnI+H)]TJ/F7 6.974 Tf 6.227 0 Td[(1f;f=)]TJ/F11 9.963 Tf 4.566 -8.07 Td[(HI+n)]TJ/F7 6.974 Tf 6.227 0 Td[(1H)]TJ/F7 6.974 Tf 6.227 0 Td[(1f;fwhichareboundedandcontinuousonallofH.InthespectralrepresentationofH,thespaceHisunitarilyequivalenttoL2S;whereS=SpecHNandHgoesoverintomultiplicationbythefunctionhwherehs;k=s:ThequadraticformsQnthusgooverintothequadraticforms~Qnwhere~Qng=Znh n+hg gdforanyg2L2S;.Thefunctionsnh n+hformanincreasingsequenceoffunctionsonS,andhencethefunctionsQnformanincreasingsequenceofcontinuousfunctionsonH.Hencetheirlimitislowersemi-continuous.Inthespectralrepresentation,thislimitisthequadraticformg7!Zhg gd

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12.2.NON-NEGATIVEOPERATORSANDQUADRATICFORMS.349whichisthespectralrepresentationofthequadraticformQ.2.implies3.LetffngbeaCauchysequenceofelementsofDrelativetokk1.Sincekkkk1,ffngisCauchywithrespecttothenormkkofHandsoconvergesinthisnormtoanelementf2H.Wemustshowthatf2Dandthatfn!finthekk1norm.Let>0.ChooseNsuchthatkfm)]TJ/F11 9.963 Tf 9.963 0 Td[(fnk21=Qfm)]TJ/F11 9.963 Tf 9.963 0 Td[(fn+kfm)]TJ/F11 9.963 Tf 9.962 0 Td[(fnk2<28m;n>N:Letm!1.Bythelowersemi-continuityofQweconcludethatQf)]TJ/F11 9.963 Tf 9.963 0 Td[(fn+kf)]TJ/F11 9.963 Tf 9.963 0 Td[(fnk22andhencef2Dandkf)]TJ/F11 9.963 Tf 9.963 0 Td[(fnk1<:3.implies1.LetH1denotetheHilbertspaceDequippedwiththekk1norm.NoticethatthescalarproductonthisHilbertspaceisf;g1=Bf;g+f;gwhereBf;f=Qf.Theoriginalscalarproduct;isaboundedquadraticformonH1,sothereisaboundedself-adjointoperatorAonH1suchthat0A1andf;g=Af;g18f;g2H1:NowapplythespectraltheoremtoA.SothereisaunitaryisomorphismUofH1withL2S;whereS=[0;1]NsuchthatUAU)]TJ/F7 6.974 Tf 6.227 0 Td[(1ismultiplicationbythefunctionawhereas;k=s.SinceAf;f1=0f=0weseethatthesetf0;kghasmeasurezerorelativetosoa>0exceptonasetofmeasurezero.Sothefunctionh=a)]TJ/F7 6.974 Tf 6.227 0 Td[(1)]TJ/F8 9.963 Tf 9.963 0 Td[(1iswelldenedandnon-negativealmosteverywhererelativeto.Wehavea=1+h)]TJ/F7 6.974 Tf 6.226 0 Td[(1andf;g=ZS1 1+h^f ^gdwhileQf;g+f;g=f;g1=ZSf gd:DenethenewmeasureonSby=1 1+h:ThenthetwopreviousequationsimplythatHisunitarilyequivalenttoL2S;,i.e.f;g=ZSf gd

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350CHAPTER12.MOREABOUTTHESPECTRALTHEOREMandQf;g=ZShf gd:ThislastequationsaysthatQisthequadraticformassociatedtotheoperatorHcorrespondingtomultiplicationbyh:12.2.5Extensionsandcores.AformQsatisfyingtheconditionsofTheorem12.2.1issaidtobeclosed.AformQ2issaidtobeanextensionofaformQ1ifithasalargerdomainbutcoincideswithQ1onthedomainofQ1.AformQissaidtobeclosableifithasaclosedextension,anditssmallestclosedextensioniscalleditsclosureandisdenotedby Q.IfQisclosable,thenthedomainof QisthecompletionofDomQrelativetothemetrickk1inTheorem12.2.1.Ingeneral,wecanconsiderthiscompletion;butonlyforclosableformscanweidentifythecompletionasasubsetofH.AsubsetDofDomQwhereQisclosediscalledacoreofQifQisthecompletionoftherestrictionofQtoD.Proposition12.2.3LetQ1andQ2bequadraticformswiththesamedensedomainDandsupposethatthereisaconstantc>1suchthatc)]TJ/F7 6.974 Tf 6.227 0 Td[(1Q1fQ2fcQ1f8f2D:IfQ1istheformassociatedtoanon-negativeself-adjointoperatorH1asinTheorem12.2.1thenQ2isassociatedwithaself-adjointoperatorH2andDomH1 21=DomH1 22=D:Proof.TheassumptionontherelationbetweentheformsimpliesthattheirassociatedmetricsonDareequivalent.SoifDiscompletewithrespecttoonemetricitiscompletewithrespecttotheother,andthedomainsoftheassociatedself-adjointoperatorsbothcoincidewithD.12.2.6TheFriedrichsextension.RecallthatanoperatorAdenedonadensedomainDiscalledsymmetricifAf;g=f;Ag8f;g2D:Asymmetricoperatoriscallednon-negativeifAf;f0Theorem12.2.2[Friedrichs.]LetQbetheformdenedonthedomainDofasymmetricoperatorAbyQf=Af;f:ThenQisclosableanditsclosureisassociatedwithaself-adjointextensionHofA.

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12.3.DIRICHLETBOUNDARYCONDITIONS.351Proof.LetH1bethecompletionofDrelativetothemetrickk1asgiveninTheorem12.2.1.TherststepistoshowthatwecanrealizeH1asasubspaceofH.Sincekfkkfk1,theidentitymapf7!fextendstoacontractionC:H1!H.Wewanttoshowthatthismapisinjective.Supposenot,sothatCf=0forsomef6=02H1.Thusthereexistsasequencefn2Dsuchthatkf)]TJ/F11 9.963 Tf 9.963 0 Td[(fnk1!0andkfnk!0:Sokfk21=limm!1limn!1fm;fn1=limm!1limn!1fAfm;fn+fm;fng=limm!1[Afm;0+fm;0]=0:SoCisinjectiveandhenceQisclosable.LetHbetheself-adjointoperatorassociatedwiththeclosureofQ.WemustshowthatHisanextensionofA.Forf;g2DDomHwehaveH1 2f;H1 2g=Qf;g=Hf;g:SinceDisdenseinH1,thisholdsforf2Dandg2H1.ByProposition12.2.1thisimpliesthatf2DomH.Inotherwords,HisanextensionofA:12.3Dirichletboundaryconditions.InthissectionwilldenoteaboundedopensetinRN,withpiecewisesmoothboundary,c>1isaconstant,bisacontinuousfunctiondenedontheclosure ofsatisfyingc)]TJ/F7 6.974 Tf 6.227 0 Td[(1
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352CHAPTER12.MOREABOUTTHESPECTRALTHEOREMForf2C10 wedeneAfbyAfx:=)]TJ/F11 9.963 Tf 7.749 0 Td[(bx)]TJ/F7 6.974 Tf 6.226 0 Td[(1NXi;j=1@ @xiaijx@f @xj:OfcoursethisoperatorisdenedonC1 butforf;g2C10 wehave,byGauss'stheoremintegrationbypartsAf;gb=)]TJ/F1 9.963 Tf 9.409 13.56 Td[(Z0@NXi;j=1@ @xiaijx@f @xj1A gdNx=ZXijaij@f @xi@ g @xjdNx=f;Agb:SoifwedenethequadraticformQf;g:=ZXijaij@f @xi@ g @xjdNx;.8thenQissymmetricandsodenesaquadraticformassociatedtothenon-negativesymmetricoperatorH.WemayapplytheFriedrichstheoremtocon-cludetheexistenceofaselfadjointextensionHofAwhichisassociatedtotheclosureofQ.TheclosureofQiscompleterelativetothemetricdeterminedbyTheorem12.2.1.Butourassumptionsaboutbandaguaranteethemetricsofquadraticformscomingfromdierentchoicesofbandaareequivalentandallequivalenttothemetriccomingfromthechoiceb1andaijwhichiskfk21=Z)]TJ/F14 9.963 Tf 4.566 -8.07 Td[(jfj2+jrfj2dNx;.9whererf=@1f@2f@Nfandjrfj2x=@1fx2++@Nfx2:TocomparethiswithProposition12.2.3,noticethatnowtheHilbertspacesHbwillalsovarybutareequivalentinnormaswellasthemetricsonthedomainoftheclosureofQ.12.3.1TheSobolevspacesW1;2andW1;20.LetusbemoreexplicitaboutthecompletionofC1 andC10 relativetothismetric.Iff2L2;dNxthenfdenesalinearfunctiononthespaceofsmoothfunctionsofcompactsupportcontainedinbytheusualrule`f:=ZfdNx82C1c:

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12.3.DIRICHLETBOUNDARYCONDITIONS.353Wecanthendenethepartialderivativesoffinthesenseofthetheoryofdistributions,forexample`@if=)]TJ/F1 9.963 Tf 9.409 13.561 Td[(Zf@idNx:ThesepartialderivativesmayormaynotcomefromelementsofL2;dNx.WedenethespaceW1;2toconsistofthosef2L2;dNxwhoserstpartialderivativesinthedistributionalsense@if=@f=@xiallcomeformelementsofL2;dNx.Wedeneascalarproduct;1onW1;2byf;g1:=Znfx gx+rfx rgxodNx:.10ItiseasytocheckthatW1;2isaHilbertspace,i.e.iscomplete.Indeed,iffnisaCauchysequenceforthecorrespondingmetrick_k1,thenfnandthe@ifnareCauchyrelativetothemetricofL2;dNx,andhenceconvergeinthismetrictolimits,i.e.fn!fand@ifn!gii=1;:::Nforsomeelementsfandg1;:::gNofL2;dNx.Wemustshowthatgi=@if.Butforany2C1cwehave`gi=gi; =limn!1@ifn; =)]TJ/F8 9.963 Tf 12.896 0 Td[(limn!1fn;@i =)]TJ/F8 9.963 Tf 7.749 0 Td[(f;@i whichsaysthatgi=@if.WedeneW1;20tobetheclosureinW1;2ofthesubspaceC1c.SinceC1cC10 thedomainof Q,theclosureoftheformQdenedby.8onC10 containsW1;20.WeclaimthatLemma12.3.1C1cisdenseinC10 relativetothemetrickk1givenby.9.Proof.Bytakingrealandimaginaryparts,itisenoughtoprovethistheoremforrealvaluedfunctions.Forany>0letFbeasmoothrealvaluedfunctiononRsuchthatFx=x8jxj>2Fx=08jxj
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354CHAPTER12.MOREABOUTTHESPECTRALTHEOREM0F0x38x2R.Forf2C10 denefx:=Ffx;soF2C1c.Also,jfxjjfxjandlim!0fx=fx8x2:Sothedominatedconvergencetheoremimpliesthatkf)]TJ/F11 9.963 Tf 9.403 0 Td[(fk2!0.WehavetoestablishconvergenceinL2ofthederivatives.ConsiderthesetBwheref=0andrf6=0.Bytheimplicitfunctiontheorem,thisisaunionofhypersurfaces,andsohasmeasurezero.WehaveZjrf)-222(rfj2dNx=ZnBjrf)-222(rfj2dNx:Onallofwehavej@ifj3j@ifjandonnBwehave@ifx!@ifx.SothedominatedconvergencetheoremprovestheL2convergenceofthepartialderivatives.Asaconsequence,weseethatthedomainof QispreciselyW1;20.12.3.2Generalizingthedomainandthecoecients.LetbeanyopensubsetofRn,letbbeanymeasurablefunctiondenedonandsatisfyingc)]TJ/F7 6.974 Tf 6.227 0 Td[(11andaameasurablematrixvaluedfunctiondenedonandsatisfyingc)]TJ/F7 6.974 Tf 6.227 0 Td[(1IxcI8x2:WecanstilldenetheHilbertspaceHb:=L2;bdNxasbefore,butcannotdenetheoperatorAasabove.Neverthelesswecandenetheclosedform Qf=ZXijaij@f @xi@ g @xjdNx;onW1;20whichweknowtobeclosedbecausethemetricitdeterminesbyTheorem12.2.1isequivalentasametrictothenormonW1;20.Therefore,byTheorem12.2.1,thereisanon-negativeself-adjointoperatorHsuchthatH1 2f;H1 2gb=Qf;g8f;g2W1;20:

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12.3.DIRICHLETBOUNDARYCONDITIONS.35512.3.3ASobolevversionofRademacher'stheorem.Rademacher'stheoremsaysthataLipschitzfunctiononRNisdierentiableal-mosteverywherewithaboundonitsderivativegivenbytheLipschitzconstant.Thefollowingisavariantofthistheoremwhichisusefulforourpurposes.Theorem12.3.1LetfbeacontinuousrealvaluedfunctiononRNwhichvan-ishesoutsideaboundedopensetandwhichsatisesjfx)]TJ/F11 9.963 Tf 9.963 0 Td[(fyjckx)]TJ/F11 9.963 Tf 9.962 0 Td[(yk8x;y2RN.11forsomec<1.Thenf2W1;20.Webreaktheproofupintoseveralsteps:Proposition12.3.1Supposethatfsatises.11andthesupportoffiscontainedinacompactsetK.Thenf2W1;2RNandkfk21=ZRN)]TJ/F14 9.963 Tf 4.566 -8.07 Td[(jfj2+jrfj2dNxjKjc2N+diamKwherejKjdenotestheLebesguemeasureofK.Proof.LetkbeaC1functiononRNsuchthatkx=0ifkxk1;kx>0ifkxk<1;andRRNkxdNx=1:Deneksbyksx=s)]TJ/F10 6.974 Tf 6.227 0 Td[(Nkx s:Soksx=0ifkxks;ksx>0ifkxk
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356CHAPTER12.MOREABOUTTHESPECTRALTHEOREMsojpsx)]TJ/F11 9.963 Tf 9.963 0 Td[(psyjckx)]TJ/F11 9.963 Tf 9.963 0 Td[(yk:Thisimpliesthatkrpsxkcsothemeanvaluetheoremimpliesthatsupx2RNjpsxjcdiamKsandsokpsk21jKsjc2diamK2s+N:ByPlancherelkpsk21=ZRN+kk2j^psj2dNandsinceconvolutiongoesoverintomultipication^ps=^fhswhereh=ZRNkxe)]TJ/F10 6.974 Tf 6.226 0 Td[(ixdNx:Thefunctionhissmoothwithh=1andjhj1forall.ByFatou'slemmakfk1=ZRN+kk2j^fj2dNliminfs!0ZRN+kk2jhsyj2j^fj2dN=liminfs!0kpsk21jKjc2N+diamK2:Thedominatedconvergencetheoremimpliesthatkf)]TJ/F11 9.963 Tf 9.963 0 Td[(psk21!0ass!0.Butthesupportofpsisslightlylargerthanthesupportoff,sowearenotabletoconcludedirectlythatf2W1;20.Sowerstmustcutfdowntozerowhereitissmall.WedothisbydeningtherealvaluedfunctionsonRbys=8>><>>:0ifjsjsifjsj22s)]TJ/F11 9.963 Tf 9.962 0 Td[(ifs22s+if)]TJ/F8 9.963 Tf 7.749 0 Td[(2s)]TJ/F11 9.963 Tf 18.265 0 Td[(:Thensetf=f.IfOistheopensetwherefx6=0thenfhasitssupportcontainedinthesetSconsistingofallpointswhosedistancefromthecomplementofOis>=c.Alsojfx)]TJ/F11 9.963 Tf 9.962 0 Td[(fyj2jfx)]TJ/F11 9.963 Tf 9.963 0 Td[(fyj2kx)]TJ/F11 9.963 Tf 9.962 0 Td[(yj:Sowemayapplytheprecedingresulttoftoconcludethatf2W1;2RNandkfk214jSjc2N+diamO2

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12.4.RAYLEIGH-RITZANDITSAPPLICATIONS.357andthenbyFatouappliedtotheFouriertransformsasbeforethatkfk214jOjc2N+diamO:Also,forsucientlysmallf2W1;20.Sowewillbedoneifweshowthatkf)]TJ/F11 9.963 Tf 10.023 0 Td[(fk!0as!0.ThesetLonwhichthisdierenceis6=0iscontainedinthesetofallxforwhich00suchthattheintersectionoftheinterval)]TJ/F11 9.963 Tf 10.093 0 Td[(;+withconsistsofthesinglepointfg.ThediscretespectrumofHwillbedenotedbydHorsimplybydwhenHisxedinthediscussion.ThecomplementindHinHiscalledtheessentialspectrumofHandisdenotedbyessHorsimplybyesswhenHisxedinthediscussion.12.4.2Characterizingthediscretespectrum.If2dHthenforsucientlysmall>0thespectralprojectionP=P)]TJ/F11 9.963 Tf 8.762 0 Td[(;+hasthepropertythatitisinvariantunderHandtherestrictionofHtotheimageofPhasonlyinitsspectrumandhencePHisnitedimensional,sincethemultiplicityofisnitebyassumption.Conversely,supposethat2HandthatP)]TJ/F11 9.963 Tf 10.738 0 Td[(;+isnitedi-mensional.ThismeansthatinthespectralrepresentationofH,thesubsetE)]TJ/F10 6.974 Tf 6.227 0 Td[(;+ofS=Nconsistingofalls;nj)]TJ/F11 9.963 Tf 9.962 0 Td[(
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358CHAPTER12.MOREABOUTTHESPECTRALTHEOREMthensinceL2E)]TJ/F10 6.974 Tf 6.227 0 Td[(;+=^MnL2)]TJ/F11 9.963 Tf 9.963 0 Td[(;+;fngweconcludethatallbutnitelymanyofthesummandsontherightarezero,whichimpliesthatforallbutnitelymanynwehave)]TJ/F11 9.963 Tf 9.962 0 Td[(;+fng=0:Foreachofthenitenon-zerosummands,wecanapplythecaseN=1ofthefollowinglemma:Lemma12.4.1LetbeameasureonRNsuchthatL2RN;isnitedimen-sional.Thenissupportedonanitesetinthesensethatthereissomenitesetofmdistinctpointsx1;:::;xmeachofpositivemeasureandsuchthatthecomplementoftheunionofthesepointshasmeasurezero.Proof.PartitionRNintocubeswhoseverticeshaveallcoordinatesoftheformt=2rforanintegerrandsothatthisisadisjointunion.ThecorrespondingdecompositionoftheL2spacesshowsthatonlynitemanyofthesecubeshavepositivemeasure,andasweincreaserthecubeswithpositivemeasurearenesteddownward,andcannotincreaseinnumberbeyondn=dimL2RN;.Hencetheyconvergeinmeasuretoatmostndistinctpointseachofpositivemeasureandthecomplementoftheirunionhasmeasurezero.Weconcludefromthislemmathatthereareatmostnitelymanypointssr;kwithsr2)]TJ/F11 9.963 Tf 10.072 0 Td[(;+whichhavenitemeasureinthespectralrepre-sentationofH,eachgivingrisetoaneigenvectorofHwitheigenvaluesr,andthecomplementofthesepointshasmeasurezero.Thisshowsthat2dH.WehaveprovedProposition12.4.12HbelongstodHifandonlyifthereissome>0suchthatP)]TJ/F11 9.963 Tf 9.963 0 Td[(;+Hisnitedimensional.12.4.3CharacterizingtheessentialspectrumThisissimplythecontrapositiveofProp.12.4.1:Proposition12.4.22HbelongstoessHifandonlyifforevery>0thespaceP)]TJ/F11 9.963 Tf 9.962 0 Td[(;+Hisinnitedimensional.12.4.4Operatorswithemptyessentialspectrum.Theorem12.4.1TheessentialspectrumofaselfadjointoperatorisemptyifandonlyifthereisacompletesetofeigenvectorsofHsuchthatthecorrespond-ingeigenvaluesnhavethepropertythatjnj!1asn!1.

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12.4.RAYLEIGH-RITZANDITSAPPLICATIONS.359Proof.Iftheessentialspectrumisempty,thenthespectrumconsistsofeigenvaluesofnitemultiplicitywhichhavenoaccumulationnitepoint,andsomustconvergeinabsolutevalueto1.Enumeratetheeigenvaluesaccordingtoincreasingabsolutevalue.Eachhasnitemultiplicityandsowecanndanorthonormalbasisofthenitedimensionaleigenspacecorrespondingtoeacheigenvalue.Theeigenvectorscorrespondingtodistincteigenvaluesareorthogo-nal.Sowhatwemustshowisthatthespacespannedbyalltheseeigenvectorsisdense.Supposenot.ThespaceLorthogonaltoalltheeigenvectorsisinvariantunderH.Ifthisspaceisnon-zero,thespectrumofHrestrictedtothissubspaceisnotempty,andisasubsetofthespectrumofH.SotherewillbeeigenvectorsinL,contrarytothedenitionofL.Conversely,supposethattheconditionshold.Letfnbethecompletesetofeigenvectors.Sincethesetofeigenvaluesisisolated,wewillbedoneifweshowthattheyconstitutetheentirespectrumofH.Supposethatzdoesnotcoincidewithanyofthen.WemustshowthattheoperatorzI)]TJ/F11 9.963 Tf 10.503 0 Td[(HhasaboundedinverseonthedomainofH,whichconsistsofallf=PnanfnsuchthatPjanj2<1andP2njanj2<1.ButforthesefkzI)]TJ/F11 9.963 Tf 9.963 0 Td[(Hfk2=Xnjz)]TJ/F11 9.963 Tf 9.962 0 Td[(nj2janj2c2kfk2wherec=minnjn)]TJ/F11 9.963 Tf 9.963 0 Td[(zj>0:Therearesomeimmediateconsequenceswhichareusefultostateexplicitly.Corollary12.4.1LetHbeanon-negativeselfadjointoperatoronaHilbertspaceH.ThefollowingconditionsonHareequivalent:1.TheessentialspectrumofHisempty.2.ThereexistsanorthonormalbasisofHconsistingofeigenvectorsfnofH,eachwithnitemultiplicitywitheigenvaluesn!1.3.TheoperatorI+H)]TJ/F7 6.974 Tf 6.227 0 Td[(1iscompact.Sincetherearenonegativeeigenvalues,weknowthat1and2areequivalent.Wemustshowthat2and3areequivalent.IfI+H)]TJ/F7 6.974 Tf 6.227 0 Td[(1iscompact,weknowthatthereisanorthonormalbasisffngofHconsistingofeigenvectorswitheigenvaluesn!0.WeknowfromthespectraltheoremthatI+H)]TJ/F7 6.974 Tf 6.227 0 Td[(1isunitarilyequivalenttomultiplicationbythepositivefunction1=+handsoI+H)]TJ/F7 6.974 Tf 6.227 0 Td[(1hasnokernel.ThentheffngconstituteanorthonormalbasisofHconsistingofeigenvectorswitheigenvaluesn=)]TJ/F7 6.974 Tf 6.226 0 Td[(1n!1.Conversely,supposethat2holds.ConsidertheniterankoperatorsAndenedbyAnf:=nXj=11 1+jf;fjfj:ThenkI+H)]TJ/F7 6.974 Tf 6.226 0 Td[(1f)]TJ/F11 9.963 Tf 9.962 0 Td[(Anfk=k1Xj=n+11 1+jf;fjfjk1 1+nkfk:

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360CHAPTER12.MOREABOUTTHESPECTRALTHEOREMThisshowsthatnI+H)]TJ/F7 6.974 Tf 6.226 0 Td[(1canbeapproximatedinoperatornormbyniterankoperators.Soweneedonlyapplythefollowingcharacterizationofcompactoperatorswhichweshouldhavestatedandprovedlastsemester:12.4.5Acharacterizationofcompactoperators.Theorem12.4.2AnoperatorAonaseparableHilbertspaceHiscompactifandonlyifthereexistsasequenceofoperatorsAnofniteranksuchthatAn!Ainoperatornorm.Proof.SupposethereareAn!Ainoperatornorm.WewillshowthattheimageABoftheunitballBofHistotallybounded:Given>0,choosensucientlylargethatkA)]TJ/F11 9.963 Tf 10.495 0 Td[(Ank<1 2,sotheimageofBunderA)]TJ/F11 9.963 Tf 10.495 0 Td[(Aniscontainedinaballofradius1 2.SinceAnisofniterank,AnBiscontainedinaboundedregioninanitedimensionalspace,sowecanndpointsx1;:::;xkwhicharewithinadistance1 2ofanypointofAnB.Thenthex1;:::;xkarewithindistanceofanypointofABwhichsaysthatABistotallybounded.Conversely,supposethatAiscompact.ChooseanorthonormalbasisffkgofH.LetPnbeorthogonalprojectionontothespacespannedbytherstnelementsofofthisbasis.ThenAn:=PnAisaniterankoperator,andwewillprovethatAn!A.ForthisitisenoughtoshowthatI)]TJ/F11 9.963 Tf 10.248 0 Td[(PnconvergestozerouniformlyonthecompactsetAB.Choosex1;:::;xkinthisimagewhicharewithin1 2distanceofanypointofAB.ForeachxedjwecanndannsuchthatkPnx)]TJ/F11 9.963 Tf 10.045 0 Td[(xk<1 2.Thisfollowsfromthefactthattheffkkformanorthonormalbasis.Choosenlargeenoughtoworkforallthexj.Thenforanyx2ABwehavekPnx)]TJ/F11 9.963 Tf 9.963 0 Td[(xkkx)]TJ/F11 9.963 Tf 9.962 0 Td[(xjk+kPnxj)]TJ/F11 9.963 Tf 9.962 0 Td[(xjk:Wecanchoosejsothatthersttermis<1 2andthesecondtermis<1 2foranyj:12.4.6Thevariationalmethod.LetHbeanon-negativeself-adjointoperatoronaHilbertspaceH.ForanynitedimensionalsubspaceLofHwithLD=DomHdeneL:=supfHf;fjf2Landkfk=1g:Denen=inffL;jLD;anddimL=ng:.12Thenareanincreasingfamilyofnumbers.WeshallshowthattheyconstitutethatpartofthediscretespectrumofHwhichliesbelowtheessentialspectrum:Theorem12.4.3LetHbeanon-negativeself-adjointoperatoronaHilbertspaceH.Denethenumbersn=nHby.12.Thenoneofthefollowingthreealternativesholds:

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12.4.RAYLEIGH-RITZANDITSAPPLICATIONS.3611.Hhasemptyessentialspectrum.Inthiscasethen!1andcoin-cidewiththeeigenvaluesofHrepeatedaccordingtomultiplicityandlistedinincreasingorder,orelseHisnitedimensionalandthencoincidewiththeeigenvaluesofHrepeatedaccordingtomultiplicityandlistedinincreasingorder.2.Thereexistsana<1suchthatnN.ThenaisthesmallestnumberintheessentialspectrumofHandH[0;aconsistsofthe1;:::;NwhichareeigenvaluesofHrepeatedaccordingtomultiplicityandlistedinincreasingorder.Proof.LetbbethesmallestpointintheessentialspectrumofHsob=1incase1..SoHhasonlyisolatedeigenvaluesofnitemultiplicityin[0;bandtheseconstitutetheentirespectrumofHinthisinterval.Letffkgbeanorthonormalsetoftheseeigenvectorscorrespondingtotheseeigenvaluesklistedwithmultiplicityinincreasingorder.LetMndenotethespacespannedbytherstnoftheseeigenvectors,andletf2Mn.Thenf=Pnj=1f;fjfjsoHf=nXj=1jf;fjfjandsoHf;f=nXj=1jjf;fjj2nnXj=1jf;fjj2=nkfk2sonn:Intheotherdirection,letLbeann-dimensionalsubspaceofDomHandletPdenoteorthogonalprojectionofHontoMn)]TJ/F7 6.974 Tf 6.227 0 Td[(1sothatPf=n)]TJ/F7 6.974 Tf 6.227 0 Td[(1Xj=1f;fjfj:TheimageofPrestrictedtoLhasdimensionn)]TJ/F8 9.963 Tf 8.799 0 Td[(1whileLhasdimensionn.Sotheremustbesomef2LwithPf=0.Bythespectraltheorem,thefunction~f=UfcorrespondingtofissupportedinthesetwherehnandhenceHf;fnkfk2sonn:Therearenowthreecasestoconsider:Ifb=+1i.e.theessentialspectrumofHisemptythen=ncanhavenoniteaccumulationpointsoweareincase

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362CHAPTER12.MOREABOUTTHESPECTRALTHEOREM1.Ifthereareinnitelymanynin[0;btheymusthaveniteaccumulationpointab,andbydenition,aisintheessentialspectrum.Thenwemusthavea=bandweareincase2.Theremainingpossibilityisthatthereareonlynitelymany1;:::MM.Sinceb2essH,thespaceK:=Pb)]TJ/F11 9.963 Tf 9.963 0 Td[(;b+Hisinnitedimensionalforall>0.Letff1;f2;:::;gbeanorthonormalbasisofK,andletLbethespacespannedbytherstmofthesebasiselements.Bythespectraltheorem,Hf;fb+kfk2foranyf2L.soforallmwehavemb+.Soweareincase3.Inapplicationssaytochemistryonedealswithself-adjointoperatorswhichareboundedfrombelow,ratherthanbeingnon-negative.Butthisrequiresjustatrivialshiftinstatingandapplyingtheprecedingtheorem.Insomeoftheseapplicationsthebottomoftheessentialspectrumisat0,andoneisinterestedinthelowesteigenvalue1whichisnegative.12.4.7Variationsonthevariationalformula.Analternativeformulationoftheformula.Insteadof.12wecandeterminethenasfollows:Wedene1asbefore:1=minf6=0Hf;f f;f:Supposethatf1isanfwhichattainsthisminimum.Wethenknowthatf1isaneigenvectorofHwitheigenvalue1.Nowdene2:=minf6=0;f?f1Hf;f f;f:This2coincideswiththe2givenby.12andanf2whichachievestheminimumisaneigenvectorofHwitheigenvalue2.Proceedingthisway,af-terndingtherstneigenvalues1;:::;nandcorrespondingeigenvectorsf1;:::;fnwedenen+1=minf6=0;f?f1;f?f2;:::;f?fnHf;f f;f:Thisgivesthesamekas.12.VariationsontheconditionLDomH.Insomeapplications,theconditionLDomHisundulyrestrictive,especiallywhenwewanttocompareeigenvaluesofdierentselfadjointoperators.Inthese

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12.4.RAYLEIGH-RITZANDITSAPPLICATIONS.363applications,onecanfrequentlyndacommoncoreDforthequadraticformsQassociatedtotheoperators.Thatis,DDomH1 2andDisdenseinDomH1 2forthemetrickk1givenbykjfk21=Qf;f+kfk2whereQf;f=Hf;f:Theorem12.4.4Denen=inffL;jLDomH0n=inffL;jLD00n=inffL;jLDomH1 2:Thenn=0n=00n:Proof.Werstprovethat0n=00n.SinceDDomH1 2theconditionLDimpliesLDomH1 2so0n00n8n:Conversely,given>0letLDomH1 2besuchthatLisn-dimensionalandL00n+:RestrictingQtoLL,wecanndanorthonormalbasisf1;:::;fnofLsuchthatQfi;fj=iij;01;n=L:Wecanthenndgi2Dsuchthatkgi)]TJ/F11 9.963 Tf 10.539 0 Td[(fik1
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364CHAPTER12.MOREABOUTTHESPECTRALTHEOREMwherec00ndependsonlyonn.Letting!0showsthat0n00n.Tocompletetheproofofthetheorem,itsucestoshowthatDomHisacoreforDomH1 2.Thisfollowsfromthespectraltheorem:ThedomainofHisunitarilyequivalenttothespaceofallfsuchthatZS+hs;n2jfs;nj2d<1wherehs;n=s.Thisisclearlydenseinthespaceoffforwhichkfk1=ZS+hjfj2d<1sincehisnon-negativeandnitealmosteverywhere.12.4.8Thesecularequation.Thedenition.12makessenseinarealnitedimensionalvectorspace.IfQisarealquadraticformonanitedimensionalrealHilbertspaceV,thenwecanwriteQf=Hf;fwhereHisaself-adjoint=symmetricoperator,andthenndanorthonormalbasisaccordingto12.12.Intermsofsuchabasisf1;:::;fn,wehaveQf=Xkkr2kwheref=Xrkfk:IfweconsidertheproblemofndinganextremepointofQfsubjecttotheconstraintthatf;f=1,thisbecomesbyLagrangemultipliers,theproblemofndingandfsuchthatdQf=dSf;whereSf=f;f:Intermsofthecoordinatesr1;:::;rnwehave1 2dQf=1r1;:::;nrnwhile1 2dSf=r1;:::;rn:Sotheonlypossiblevaluesofare=iforsomeiandthecorrespondingfisgivenbyrj=0;j6=iandri6=0.ThisisawatereddownversionversionofTheorem12.4.3.Inapplications,oneisfrequentlygivenabasisofVwhichisnotorthonormal.ThusintermsofthegivenbasisQf=XHijrirj;andSf=XijSijrirjwheref=Xrifi:TheproblemofndinganextremepointofQfsubjecttotheconstraintSf=1becomesthatofndingandr=r1;:::;rnsuchthatdQf=dSf

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12.5.THEDIRICHLETPROBLEMFORBOUNDEDDOMAINS.365i.e.0B@H11)]TJ/F11 9.963 Tf 9.963 0 Td[(S11H12)]TJ/F11 9.963 Tf 9.962 0 Td[(S12H1n)]TJ/F11 9.963 Tf 9.963 0 Td[(S1n............Hn1)]TJ/F11 9.963 Tf 9.963 0 Td[(Sn1Hn2)]TJ/F11 9.963 Tf 9.962 0 Td[(Sn2Hnn)]TJ/F11 9.963 Tf 9.963 0 Td[(Snn1CA0B@r1...rn1CA=0:Asaconditiononthisbecomesthealgebraicequationdet0B@H11)]TJ/F11 9.963 Tf 9.963 0 Td[(S11H12)]TJ/F11 9.963 Tf 9.962 0 Td[(S12H1n)]TJ/F11 9.963 Tf 9.962 0 Td[(S1n............Hn1)]TJ/F11 9.963 Tf 9.963 0 Td[(Sn1Hn2)]TJ/F11 9.963 Tf 9.962 0 Td[(Sn2Hnn)]TJ/F11 9.963 Tf 9.963 0 Td[(Snn1CA=0whichisknownasthesecularequationduetoitsprevioususeinastronomytodeterminetheperiodsoforbits.12.5TheDirichletproblemforboundeddomains.LetbeanopensubsetofRn.TheSobolevspaceW1;2isdenedasthesetofallf2L2;dxwheredxisLebesguemeasuresuchthatallrstorderpartialderivatives@ifinthesenseofgeneralizedfunctionsbelongtoL2;dx.OnthisspacewehavetheSobolevscalarproductf;g1:=Z!fx gx+rfxr gxdx:ItisnothardtocheckandwewilldosowithinthenextthreelecturesthatW1;2withthisscalarproductisaHilbertspace.WeletC10denotethespaceofsmoothfunctionsofcompactsupportwhosesupportiscontainedin,andletW1;20denotethecompletionofC10withrespecttothenormkk1comingfromthescalarproduct;1.Wewillshowthatdenesanon-negativeself-adjointoperatorwithdomainW1;20knownastheDirichletoperatorassociatedwith.IwanttopostponetheproofsofthesegeneralfactsandconcentrateonwhatRayleigh-Ritztellsuswhenisaboundedopensubsetwhichwewillassumefromnowon.WearegoingapplyRayleigh-RitztothedomainDandthequadraticformQf=Qf;fwhereQf;g:=Zrfxr gxdx:Denen:=inffLjLC10;dimL=ngwhereL=supQf;f2L;kfk=1asbefore.

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366CHAPTER12.MOREABOUTTHESPECTRALTHEOREMHereisthecrucialobservation:If0aretwoboundedopenregionsthennn0sincetheinmumforistakenoverasmallercollectionofsubspacesthanfor0.Supposethatisaninterval;aontherealline.Fourierseriestellsusthatthefunctionsfk=sinkx=aformabasisofL2;dxandareeigenvectorsofwitheigenvaluesk2a2.ByFubiniwegetacorrespondingformulaforanycubeinRnwhichshowsthatI+)]TJ/F7 6.974 Tf 6.226 0 Td[(1isacompactoperatorforthecaseofacube.Sinceanycontainsacubeandiscontainedinacube,weconcludethatthentendto1andsoHo=withtheDirichletboundaryconditionshaveemptyessentialspectraandI+H0)]TJ/F7 6.974 Tf 6.226 0 Td[(1arecompact.FurthermorethenaertheeigenvaluesofH0arrangedinincreasingorder.Proposition12.5.1Ifmisanincreasingsequenceofopensetscontainedinwith=[mmthenlimm!1nm=nforalln.Proof.Forany>0thereexistsann-dimensionalsubspaceLofC10suchthatLn+.TherewillbeacompactsubsetKsuchthatalltheelementsofLhavesupportinK.WecanthenchoosemsucientlylargesothatKm.Thennnmn+:12.6Valence.Theminimumeigenvalue1isdeterminedaccordingto.12by1=inf6=0H; ;:Unlessonehasacleverwayofcomputing1bysomeothermeans,minimizingtheexpressionontherightoverallofHisahopelesstask.Whatisdoneinpracticeistochooseanitedimensionalsubspaceandapplytheabovemini-mizationoverallinthatsubspaceandsimilarlytoapply.12tosubspacesofthatsubspaceforthehighereigenvalues.Thehopeisthatthisyieldgoodapproximationstothetrueeigenvalues.IfMisanitedimensionalsubspaceofH,andPdenotesprojectionontoM,thenapplying.12tosubspacesofMamountstondingtheeigenvalues

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12.6.VALENCE.367ofPHP,whichisanalgebraicproblemaswehaveseen.AchemicaltheorywhenHistheSchrodingeroperatorthenamountstocleverlychoosingsuchasubspace.12.6.1Twodimensionalexamples.ConsiderthecasewhereMistwodimensionalwithabasis1and2.Theideaisthatwehavesomegroundsforbelievingthatthatthetrueeigenfunctionhascharacteristicstypicalofthesetwoelementsandislikelytobesomelinearcombinationofthem.IfwesetH11:=H1;1;H12:=H1;2= H21;H22:=H2;2andS11:=S1;1;S12:=1;2= S21;S22:=2;2thenifthesequantitiesarerealwecanapplythesecularequationdetH11)]TJ/F11 9.963 Tf 9.963 0 Td[(S11H12)]TJ/F11 9.963 Tf 9.963 0 Td[(S12H21)]TJ/F11 9.963 Tf 9.963 0 Td[(S21H22)]TJ/F11 9.963 Tf 9.963 0 Td[(S22=0todetermine.SupposethatS11=S22=1,i.e.that1and2areseparatelynormalized.Alsoassumethat1and2arelinearlyindependent.Let:=S12=S21:Thisissometimescalledtheoverlapintegral"sinceifourHilbertspaceisL2R3then=RR31 2dx.NowH11=H1;1istheguessthatwewouldmakeforthelowesteigenvalue=thelowestenergylevel"ifwetookLtobetheonedimensionalspacespannedby1.SoletuscallthisvalueE1.SoE1:=H11andsimilarlydeneE2=H22.Thesecularequationbecomes)]TJ/F11 9.963 Tf 9.963 0 Td[(E1)]TJ/F11 9.963 Tf 9.962 0 Td[(E2)]TJ/F8 9.963 Tf 9.963 0 Td[(H12)]TJ/F11 9.963 Tf 9.963 0 Td[(2=0:IfwedeneF:=)]TJ/F11 9.963 Tf 9.339 0 Td[(E1)]TJ/F11 9.963 Tf 9.338 0 Td[(E2)]TJ/F8 9.963 Tf 9.339 0 Td[(H12)]TJ/F11 9.963 Tf 9.338 0 Td[(2thenFispositiveforlargevaluesofjjsincejj<1byCauchy-Schwarz.Fisnon-positiveat=E1orE2andinfactgenericallywillbestrictlynegativeatthesepoints.SothelowersolutionofthesecularequationswillgenericallyliestrictlybelowminE1;E2andtheuppersolutionwillgenericallyliestrictlyabovemaxE1;E2.Thisisknownasthenocrossingruleandisofgreatimportanceinchemistry.IhopetoexplainthehigherdimensionalversionofthisruleduetoTeller-vonNeumannandWignerlater.

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368CHAPTER12.MOREABOUTTHESPECTRALTHEOREM12.6.2Huckeltheoryofhydrocarbons.InthistheorythespaceMisthen-dimensionalspacewhereeachcarbonatomcontributesoneelectron.Theotherelectronsbeingoccupiedwiththehydrogenatoms.ItisassumedthattheSinthesecularequationistheidentitymatrix.Thisamountstotheassumptionthatthebasisgivenbytheelectronsassoci-atedwitheachcarbonatomisanorthonormalbasis.ItisalsoassumedthatHf;f=isthesameforeachbasiselement.Inacrudesensethismeasurestheelectron-attractingpowerofeachcarbonatomandhenceisassumedtobethesameforallbasiselements.IfHfr;fs6=0,theatomsrandsaresaidtobebonded".Itisassumedthatonlynearestneighbor"atomsarebonded,inwhichcaseitisassumedthatHfr;fs=isindependentofrands.SoPHPhastheformI+AwhereAistheadjacencymatrixofthegraphwhoseverticescorrespondtothecarbonatomsandwhoseedgescorrespondtothebondedpairsofatoms.Ifwesetx:=E)]TJ/F11 9.963 Tf 9.963 0 Td[( thenndingtheenergylevelsisthesameasndingtheeigenvaluesxoftheadjacencymatrixA.Inparticularthisissoifweassumethatthevaluesofandareindependentoftheparticularmolecule.12.7Davies'sproofofthespectraltheoremInthissectionwepresenttheproofgivenbyDaviesofthespectraltheorem,takenfromhisbookSpectralTheoryandDierentialOperators.12.7.1Symbols.Thesearefunctionswhichvanishmoreorgrowlessatinnitythemoreyoudierentiatethem.Moreprecisely,foranyrealnumberweletSdenotethespaceofsmoothfunctionsonRsuchthatforeachnon-negativeintegernthereisaconstantcndependingonfsuchthatjfnxjcn+jxj2)]TJ/F10 6.974 Tf 6.226 0 Td[(n=2:Itwillbeconvenienttointroducethefunctionhzi:=+jzj21 2:SowecanwritethedenitionofSasbeingthespaceofallsmoothfunctionsfsuchthatjfnxjcnhxi)]TJ/F10 6.974 Tf 6.227 0 Td[(n.13forsomecnandallintegersn0.Forexample,apolynomialofdegreekbelongstoSksinceeverytimeyoudierentiateityoulowerthedegreeandeventually

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12.7.DAVIES'SPROOFOFTHESPECTRALTHEOREM369getzero.Moregenerally,afunctionoftheformP=QwherePandQarepolynomialswithQnowherevanishingbelongstoSkwherek=degP)]TJ/F8 9.963 Tf 9.002 0 Td[(degQ.Thenamesymbol"comesfromthetheoryofpseudo-dierentialoperators.12.7.2Slowlydecreasingfunctions.DeneA:=[<0S:Foreachn1denethenormkknonAbykfkn:=nXr=0ZRjfrxjhxir)]TJ/F7 6.974 Tf 6.227 0 Td[(1dx:.14Iff2S;<0thenf0isintegrableandfx=Rxftdtsosupx2Rjfxjkfk1andconvergenceinthekk1normimpliesuniformconvergence.Lemma12.7.1ThespaceofsmoothfunctionsofcompactsupportisdenseinAforeachofthenormskkn+1.Proof.Chooseasmoothofcompactsupportin[)]TJ/F8 9.963 Tf 7.749 0 Td[(2;2]suchthat1on[)]TJ/F8 9.963 Tf 7.748 0 Td[(1;1].Denem:=s msoisidenticallyoneon[)]TJ/F11 9.963 Tf 7.749 0 Td[(m;m]andisofcompactsupport.Noticethatforanyk1wehavekmxKkhxi)]TJ/F10 6.974 Tf 6.227 0 Td[(k1jxjmxforsuitableconstantsKk.Weclaimthatmf!finthenormkkn+1foranyf2A.Indeed,kf)]TJ/F11 9.963 Tf 9.962 0 Td[(mfkn+1=n+1Xr=0ZRdr dxrffx)]TJ/F11 9.963 Tf 9.963 0 Td[(mxghxir)]TJ/F7 6.974 Tf 6.227 0 Td[(1dx:ByLeibnitz'sformulaandthepreviousinequalitythisisboundedbysomeconstanttimesn+1Xr=0Zjxj>mjfrxjhxir)]TJ/F7 6.974 Tf 6.227 0 Td[(1dxwhichconvergestozeroasm!1:

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370CHAPTER12.MOREABOUTTHESPECTRALTHEOREM12.7.3Stokes'formulaintheplane.Considercomplexvalueddierentiablefunctionsinthex;yplane.Denethedierentialoperators@ @ z:=1 2@ @x+i@ @y;and@ @z:=1 2@ @x)]TJ/F11 9.963 Tf 9.962 0 Td[(i@ @y:Denethecomplexvaluedlineardierentialformsdz:=dx+idy;d z:=dx)]TJ/F11 9.963 Tf 9.963 0 Td[(idysodx=1 2dz+d z;dy=1 2idz)]TJ/F11 9.963 Tf 9.963 0 Td[(d z:Sowecanwriteanycomplexvalueddierentialformadx+bdyasAdz+Bd zwhereA=1 2a)]TJ/F11 9.963 Tf 9.963 0 Td[(ib;B=1 2a+ib:Inparticular,foranydierentiablefunctionfwehavedf=@f @xdx+@f @ydy=@f @zdz+@f @ zd z:Alsodz^d z=)]TJ/F8 9.963 Tf 7.749 0 Td[(2idx^dy:SoifUisanyboundedregionwithpiecewisesmoothboundary,StokestheoremgivesZ@Ufdz=ZUdfdz=ZU@f @ zd z^dz=2iZU@f @ zdxdy:ThefunctionfcantakevaluesinanyBanachspace.WewillapplytofunctionswithvaluesinthespaceofboundedoperatorsonaHilbertspace.Afunctionfisholomorphicifandonlyif@f @ z=0.SotheaboveformulaimpliestheCauchyintegraltheorem.HereisavariantofCauchy'sintegraltheoremvalidforafunctionofcompactsupportintheplane:1 ZC@f @ z1 z)]TJ/F11 9.963 Tf 9.962 0 Td[(wdxdy=)]TJ/F11 9.963 Tf 7.749 0 Td[(fw:.15Indeed,theintegralontheleftisthelimitoftheintegraloverCnDwhereDisadiskofradiuscenteredatw.Sincefhascompactsupport,andsince@ @ z1 z)]TJ/F11 9.963 Tf 9.962 0 Td[(w=0;wemaywritetheintegralontheleftas)]TJ/F8 9.963 Tf 13.678 6.74 Td[(1 2iZ@Dfz z)]TJ/F11 9.963 Tf 9.963 0 Td[(wdz!)]TJ/F11 9.963 Tf 20.478 0 Td[(fw:

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12.7.DAVIES'SPROOFOFTHESPECTRALTHEOREM37112.7.4Almostholomorphicextensions.Letbeasabove,anddenex;y:=y=hxi:LetfbeacomplexvaluedC1functiononR.Dene~fz=~f;nz:=nXr=0frxiyr r!x;y.16forn1.Then@~f;n @ z=1 2nXr=0frxiyr r!fx+iyg+1 2fn+1xiyn n!:.17So@~f;n @ z=Ojyjn;and,inparticular,@~f;n @ zx;0=0:Wecall~f;nanalmostholomorphicextensionoff.12.7.5TheHeer-Sjostrandformula.LetHbeaself-adjointoperatoronaHilbertspace,andletf2A.DenefH:=)]TJ/F8 9.963 Tf 9.472 6.74 Td[(1 ZC@~f @ zRz;Hdxdy;.18where~f=~f;n.Oneofourtaskswillbetoshowthatthenotationisjustiedinthattherighthandsideoftheaboveexpressionisindependentofthechoiceofandn.Butrstwehavetoshowthattheintegraliswelldened.Lemma12.7.2Theintegral.18isnormconvergentandkfHkcnkfkn+1:Proof.WeknowthatRz;Hisholomorphicinzforz2CnR,inparticularisnormcontinuousthere.SotheintegraliswelldenedoveranycompactsubsetofCnR.LetU=fzjhxi
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372CHAPTER12.MOREABOUTTHESPECTRALTHEOREMThenormofRz;Hisboundedbyjyj)]TJ/F7 6.974 Tf 6.227 0 Td[(1,andissupportedontheclosureofV:=fzj0n1then~f;m)]TJ/F8 9.963 Tf 11.322 2.629 Td[(~f;n=Oy2soanotherapplicationofthelemmaprovesthat~fHisindependentofthechoiceofn.Noticethattheproofshowsthatwecanchoosetobeanysmoothfunctionwhichisidenticallyoneinaneighborhoodoftherealaxisandwhichiscompactlysupportedintheimaginarydirection.

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12.7.DAVIES'SPROOFOFTHESPECTRALTHEOREM37312.7.6Aformulafortheresolvent.Letwbeacomplexnumberwithanon-zeroimaginarypart,andconsiderthefunctionrwonRgivenbyrwx=1 w)]TJ/F11 9.963 Tf 9.962 0 Td[(xThisfunctionclearlybelongstoAandsowecanformrwH.ThepurposeofthissectionistoprovethatrwH=Rw;H:Wewillchoosetheinthedenitionof~rwsothatw62supp.Tobespecic,choose=jyj=hxiforlargeenoughsothatw62supp.Wewillchoosetheninthedenitionof~rwasn=1.Foreachrealnumbermconsidertheregionm:=x;yjjxj
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374CHAPTER12.MOREABOUTTHESPECTRALTHEOREMasbefore.TheintegralsovereachofthesecurvesismajorizedbycZy2 hxi31 jyjjdzj=cm)]TJ/F7 6.974 Tf 6.226 0 Td[(1Z1 hxi2jdzj=Om)]TJ/F7 6.974 Tf 6.226 0 Td[(1:12.7.7Thefunctionalcalculus.Wenowshowthatthemapf7!fHhasthedesiredpropertiesofafunctionalcalculus,seeTheorem12.7.2below.Firstsomelemmas:Lemma12.7.4IffisasmoothfunctionofcompactsupportwhichisdisjointfromthespectrumofHthenfH=0.Proof.WemayndanitenumberofpiecewisesmoothcurveswhicharedisjointfromthespectrumofHandwhichboundaregionUwhichcontainsthesupportof~f.ThenbyStokesfH=)]TJ/F8 9.963 Tf 9.471 6.74 Td[(1 ZU@~f @ zRz;Hdxdy=)]TJ/F11 9.963 Tf 12.737 6.74 Td[(i 2Z@U~fzRz;Hdz=0since~fvanisheson@U:Lemma12.7.5Forallf;g2AfgH=fHgH:Proof.Itisenoughtoprovethiswhenfandgaresmoothfunctionsofcompactsupport.Theproductontherightisgivenby1 2ZKL@~f @ z@~g @ wRz;HRw;HdxdydudvwhereK:=supp~fandL:=supp~garecompactsubsetsofC.ApplytheresolventidentityintheformRz;HRw;H=z)]TJ/F11 9.963 Tf 9.963 0 Td[(w)]TJ/F7 6.974 Tf 6.226 0 Td[(1Rw;H)]TJ/F8 9.963 Tf 9.963 0 Td[(z)]TJ/F11 9.963 Tf 9.963 0 Td[(w)]TJ/F7 6.974 Tf 6.226 0 Td[(1Rz;wtotheintegrandtowritetheaboveintegralasthesumoftwointegrals.Using.15thetwodouble"integralsbecomesingle"integralsandthewholeexpressionbecomesfHgH=)]TJ/F8 9.963 Tf 9.472 6.74 Td[(1 ZK[L~f@~g @ z+~g@~f @ zRz;Hdxdy=)]TJ/F8 9.963 Tf 9.472 6.739 Td[(1 ZC@~f~g @ zRz;Hdxdy:

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12.7.DAVIES'SPROOFOFTHESPECTRALTHEOREM375ButfgHisdenedas)]TJ/F8 9.963 Tf 9.472 6.74 Td[(1 ZC@fg~ @ zRz;Hdxdy:Butfg~)]TJ/F8 9.963 Tf 12.108 2.629 Td[(~f~gisofcompactsupportandisOy2soLemma12.7.3impliesourlemma.Lemma12.7.6 fH=fH:ThisfollowsfromRz;H=R z;H:Lemma12.7.7kfHkkfk1wherekfk1denotesthesupnormoff.Proof.Choosec>kfk1anddenegs:=c)]TJ/F1 9.963 Tf 9.963 9.005 Td[(p c2)-222(jfsj2:Theng2Aandg2=2cg)-222(jfj2orf f)]TJ/F11 9.963 Tf 9.962 0 Td[(cg)]TJ/F11 9.963 Tf 9.963 0 Td[(c g+g2=0:ByLemma12.7.5andtheprecedinglemmathisimpliesthatfHfH+c)]TJ/F11 9.963 Tf 9.963 0 Td[(gHc)]TJ/F11 9.963 Tf 9.962 0 Td[(gH=c2:ButthenforanyinourHilbertspace,kfHk2kfHk2+kc)]TJ/F11 9.963 Tf 9.962 0 Td[(gHk2=c2kk2provingthelemma.LetC0Rdenotethespaceofcontinuousfunctionswhichvanishat1withkk1thesupnorm.ThealgebraAisdenseinC0RbyStoneWeierstrass,andtheprecedinglemmaallowsustoextendthemapf7!fHtoallofC0R.Theorem12.7.2IfHisaself-adjointoperatoronaHilbertspaceHthenthereexistsauniquelinearmapf7!fHfromC0RtoboundedoperatorsonHsuchthat1.Themapf7!fHisanalgebrahomomorphism,2. fH=fH,

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376CHAPTER12.MOREABOUTTHESPECTRALTHEOREM3.kfHkkfk1,4.Ifwisacomplexnumberwithnon-zeroimaginarypartandrwx=w)]TJ/F11 9.963 Tf -311.056 -11.955 Td[(x)]TJ/F7 6.974 Tf 6.227 0 Td[(1thenrwH=Rw;H5.IfthesupportoffisdisjointfromthespectrumofHthenfH=0.Wehaveprovedeverythingexcepttheuniqueness.Butitem4determinesthemaponthefunctionsrwandthealgebrageneratedbythesefunctionsisdensebyStoneWeierstrass.InordertogetthefullspectraltheoremwewillhavetoextendthisfunctionalcalculusfromC0Rtoalargerclassoffunctions,forexampletotheclassofboundedmeasurablefunctions.Infact,Daviesproceedsbyusingthetheoremwehavealreadyprovedtogetthespectraltheoremintheformthatsaysthataself-adjointoperatorisunitarilyequivalenttoamultiplicationoperatoronanL2spaceandthentheextendedfunctionalcalculusbecomesevident.Firstsomedenitions:12.7.8Resolventinvariantsubspaces.LetHbeaself-adjointoperatoronaHilbertspaceH,andletLHbeaclosedsubspace.WesaythatLisresolventinvariantifforallnon-realzwehaveRz;HLL.IfHisaboundedoperatorandLisinvariantintheusualsense,i.e.HLL,thenforjzj>kHktheNeumannexpansionRz;H=zI)]TJ/F11 9.963 Tf 9.963 0 Td[(H)]TJ/F7 6.974 Tf 6.227 0 Td[(1=z)]TJ/F7 6.974 Tf 6.227 0 Td[(1I)]TJ/F11 9.963 Tf 9.963 0 Td[(z)]TJ/F7 6.974 Tf 6.227 0 Td[(1H)]TJ/F7 6.974 Tf 6.227 0 Td[(1=1Xn=0z)]TJ/F10 6.974 Tf 6.226 0 Td[(n)]TJ/F7 6.974 Tf 6.227 0 Td[(1HnshowsthatRz;HLL.Byanalyticcontinuationthisholdsforallnon-realz.SoifHisaboundedoperator,ifLisinvariantintheusualsenseitisresolventinvariant.Weshallseeshortlythatconversely,ifLisaresolventinvariantsubspaceforaboundedself-adjointoperatorthenitisinvariantintheusualsense.Lemma12.7.8IfLisaresolventinvariantsubspaceforapossiblyunboundedself-adjointoperatorthensoisitsorthogonalcomplement.Proof.If2L?thenforany2LwehaveRz;H;=;R z;H=0ifImz6=0soRz;H2L?:NowsupposethatHisaboundedself-adjointoperatorandthatLisaresolventinvariantsubspace.Forf2LdecomposeHfasHf=g+hwhereg2Landh2L?.TheLemmasaysthatRz;Hh2L?.ButRz;Hh=Rz;HHf)]TJ/F11 9.963 Tf 9.963 0 Td[(g=Rz;HHf)]TJ/F11 9.963 Tf 9.963 0 Td[(zf+zf)]TJ/F11 9.963 Tf 9.963 0 Td[(g

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12.7.DAVIES'SPROOFOFTHESPECTRALTHEOREM377=)]TJ/F11 9.963 Tf 7.749 0 Td[(f+Rz;Hzf)]TJ/F11 9.963 Tf 9.963 0 Td[(g2Lsincef2L,zf)]TJ/F11 9.963 Tf 10.338 0 Td[(g2LandLisinvariantunderRz;H.ThusRz;Hh2LL?soRz;Hh=0.ButRz;Hisinjectivesoh=0.WehaveshownthatifHisaboundedself-adjointoperatorandLisaresolventinvariantsubspacethenitisinvariantunderH.Sofromnowonwecandropthewordresolvent".Wewilltakethewordin-variant"tomeanresolventinvariant"whendealingwithpossiblyunboundedoperators.Onboundedoperatorsthiscoincideswiththeoldnotionofinvari-ance.12.7.9Cyclicsubspaces.Wearegoingtoperformasimilarmanipulationwiththewordcyclic".Letv2HandHapossiblyunboundedself-adjointoperatoronH.WedenethecyclicsubspaceLgeneratedbyvtobetheclosureofthesetofalllinearcombinationsofRz;Hv:Lemma12.7.9vbelongstothecyclicsubspaceLthatitgenerates.Proof.FromtheHellfer-SjostrandformulaitfollowsthatfHv2Lforallf2Aandhenceforallf2C0R.Choosefn2C0Rsuchthat0fn1andsuchthatfn!1point-wiseanduniformlyonanycompactsubsetasn!1.Weclaimthatlimn!1fnHv=v;whichwouldprovethelemma.Toprovethis,chooseasequencevm2DomHwithvm!vandchoosesomenon-realnumberz.Setwm:=zI)]TJ/F11 9.963 Tf 9.963 0 Td[(Hvm;sothatvn=Rz;Hwn.Letrzbethefunctionrzs=z)]TJ/F11 9.963 Tf 8.646 0 Td[(s)]TJ/F7 6.974 Tf 6.227 0 Td[(1onRasabove,sovm=rzHwm.

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378CHAPTER12.MOREABOUTTHESPECTRALTHEOREMThenfnHv=fnHvm+fnHv)]TJ/F11 9.963 Tf 9.962 0 Td[(vm=fnHrzHwm+fnHv)]TJ/F11 9.963 Tf 9.962 0 Td[(vm=fnrzHwn+fnHv)]TJ/F11 9.963 Tf 9.963 0 Td[(vm:SofnHv)]TJ/F11 9.963 Tf 9.963 0 Td[(v=[fnrzH)]TJ/F11 9.963 Tf 9.963 0 Td[(rzH]wm+vm)]TJ/F11 9.963 Tf 9.962 0 Td[(v+fnHv)]TJ/F11 9.963 Tf 9.962 0 Td[(vm:Nowfnrz!rzinthesupnormonC0R.Sogiven>0wecanrstchoosemsolargethatkvm)]TJ/F11 9.963 Tf 10.552 0 Td[(vk<1 3.Sincethesupnormoffnis1,thethirdsummandaboveisalsolessinnormthat1 3.Wecanthenchoosensucientlylargethatthersttermisalso1 3:Clearly,Listhesmallestresolventinvariantsubspacewhichcontainsv.Hence,ifHisaboundedself-adjointoperator,ListhesmallestclosedsubspacecontainingalltheHnv.Soforboundedself-adjointoperators,wehavenotchangedthedenitionofcyclicsubspace.Proposition12.7.1LetHbeapossiblyunboundedself-adjointoperatoronaseparableHilbertspaceH.ThenthereexistaniteorcountablefamilyoforthogonalcyclicsubspacesLnsuchthatHistheclosureofMnLn:Proof.LetfnbeacountabledensesubsetofHandletL1bethecyclicsubspacegeneratedbyf1.IfL1=Hwearedone.Ifnot,theremustbesomemforwhichfm62L1.Choosethesmallestsuchmandletg2betheorthogonalprojectionoffmontoL?1.LetL2bethecyclicsubspacegeneratedbyg2.IfL1L2=Hwearedone.Ifnot,thereissomemforwhichfm62L1L2.choosethesmallestsuchmandletg3betheprojectionoffmontotheorthogonalcomplementofL1L2.Proceedinductively.EitherthiscomestoanendwithHanitedirectsumofcyclicsubspacesoritgoesonindenitely.IneithercaseallthefibelongtothealgebraicdirectsuminthepropositionandhencetheclosureofthissumisallofH:

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12.7.DAVIES'SPROOFOFTHESPECTRALTHEOREM379IntermsofthedecompositiongivenbytheProposition,letDn:=DomHLnLetHndenotetherestrictionofHtoDn.WeclaimthatDnisdenseinLn.HnmapsDnintoLnandhencedenesanoperatorontheHilbertspaceLn.TheoperatorHnontheHilbertspaceLnwithdomainDnisself-adjoint.Proofs.Fortherstitem:letvbesuchthatLnistheclosureofthespanofRz;Hv;z62R.SoRz;Hv2LnandRz;H2DomH.SothevectorsRz;HvbelongtoDnandsoDnisdenseinLn.Fortheseconditem,supposethatw2Dn.Letu=zI)]TJ/F11 9.963 Tf 10.407 0 Td[(Hwforsomez62Rsothatw=Rz;Hu.InparticularRz;Hu2Ln.Ifweshowthatu2LnthenitfollowsthatHw2Ln.Letu0betheprojectionofuontoL?n.Wewanttoshowthatu0=0.SinceLnisinvariantandu)]TJ/F11 9.963 Tf 10.12 0 Td[(u02LnweknowthatRz;Hu)]TJ/F11 9.963 Tf 9.676 0 Td[(u02LnandhencethatRz;Hu02Ln.ButL?nisinvariantbythelemma,andsoRz;Hu02LnL?nsoRz;Hu0=0andsou0=0.Toprovethethirditemwerstprovethatifw2DomHthentheorthog-onalprojectionofwontoLnalsobelongstoDomHandhencetoDn.Asintheproofoftheseconditem,letu=zI)]TJ/F11 9.963 Tf 9.354 0 Td[(Hwandu0theorthogonalprojectionofuontotheorthogonalcomplementofLn.Sow=Rz;Hu=Rz;Hu0+Rz;Hu)]TJ/F11 9.963 Tf 8.535 0 Td[(u0.ButRz;Hu0isorthogonaltoLnandRz;Hu)]TJ/F11 9.963 Tf -335.963 -11.955 Td[(u02Ln.SotheorthogonalprojectionofwontoLnisRz;Hu)]TJ/F11 9.963 Tf 10.142 0 Td[(u0whichbelongstoDomH.Ifw0denotestheorthogonalprojectionofwontotheorthogonalcomplementofLnthenHw0=HRz;Hu0=)]TJ/F11 9.963 Tf 7.749 0 Td[(u0+zw0andsoHw0isorthogonaltoLn.Nowsupposethatx2LnisinthedomainofHn.ThismeansthatHnx2LnandHnx;y=x;Hyforally2Dn.Wewanttoshowthatx2DomH=DomHforthenwewouldconcludethatx2DnandsoHnisself-adjoint.Butforanyw2DomHwehavex;Hw=x;Hw0+Hw)]TJ/F11 9.963 Tf 9.963 0 Td[(w0=x;Hw)]TJ/F11 9.963 Tf 9.963 0 Td[(w0=x;Hnw)]TJ/F11 9.963 Tf 9.962 0 Td[(w0=Hnx;wsince,byassumption,Hnx2Ln.Thisshowsthatx2DomHandHx=Hnx:NowletusgobackthethedecompositionofHasgivenbyPropostion12.7.1.Thismeansthateveryf2Hcanbewrittenuniquelyasthepossiblyinnitesumf=f1+f2+withfi2Li.

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380CHAPTER12.MOREABOUTTHESPECTRALTHEOREMProposition12.7.2f2DomHifandonlyiffi2DomHiforalliandXikHifik2<1:IfthishappensthenthedecompostionofHfisgivenbyHf=H1f1+H2f2+:Proof.Supposethatf2DomH.Weknowthatfn2DomHnandalsothattheprojectionofHfontoLnisHnfn.HenceHf=H1f1+h2f2+and,inparticular,PkHnfnk2<1:Conversely,supposethatfi2DomHiandPkHnfnk2<1:Thenfisthelimitofthenitesumsf1++fNandthelimitoftheHf1++fN=H1f1+HNfNexist.SinceHisclosedthisimpliesthatf2DomHandHfisgivenbytheinnitesumHf=H1f1+H2+:12.7.10Thespectralrepresentation.LetHbeaself-adjointoperatoronaseperableHilbertspaceH,andletSdenotethespectrumofH.Wewillleth:S!Rdenotethefunctiongivenbyhs=s:WerstformulateandprovethespectralrepresentationifHiscyclic.

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12.7.DAVIES'SPROOFOFTHESPECTRALTHEOREM381Proposition12.7.3SupposethatHiscyclicwithgeneratingvectorv.ThenthereexistsanitemeasureonSandaunitaryisomorphismU:H!L2S;suchthat2HbelongstoDomHifandonlyifhU2L2S;.Ifso,thenwithf=UwehaveUHU)]TJ/F7 6.974 Tf 6.227 0 Td[(1f=hf:Proof.DenethelinearfunctionalonC0Rasf:=fHv;v:Wehave f= f.Ifisnon-negativeandwesetg:=f1 2thenf=kgHvk20.SobytheRieszrepresentationtheoremonlinearfunctionsonC0weknowthatthereexistsanitenon-negativecountablyadditivemeasureonRsuchthatfHv;v=ZRfdforallf2C0R.IfthesupportoffisdisjointfromSthenfH=0.thisshowsthatissupportedonS,thespectrumofH.WehaveZf gd=gHfHv;v=fHv;gHvforf;g2C0R.LetMdenotethelinearsubspaceofHconsistingofallfHv;f2C0R.TheaboveequalitysaysthatthereisanisometryUfromMtoC0RrelativetotheL2normsuchthatUfH=f:SinceMisdenseinHbyhypothesis,thisextendstoaunitaryoperatorUformHtoL2S;.Letf1;f2;f2C0Randset2=f1Hv;2=f2Hv.Sofi=Ui;i=1;2.ThenfH1;2=ZSff1 f2d:Takingf=rwsothatfH=Rw;HweseethatURw;HU)]TJ/F7 6.974 Tf 6.227 0 Td[(1g=rwgforallg2L2S;andallnon-realw.Inparticular,UmapstherangeofRw;HwhichisDomHontotherangeoftheoperatorofmultiplicationbyrxwhichisthesetofallgsuchthatxg2L2.Iff2L2S;theng=rwf2DomhandeveryelementofDomhisofthisform.Wenowuseouroldfriend,theformulaHRw;H=wRw;H)]TJ/F11 9.963 Tf 9.962 0 Td[(I:

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382CHAPTER12.MOREABOUTTHESPECTRALTHEOREMAppliedtoU)]TJ/F7 6.974 Tf 6.227 0 Td[(1gthisgivesHU)]TJ/F7 6.974 Tf 6.227 0 Td[(1g=HRw;HU)]TJ/F7 6.974 Tf 6.227 0 Td[(1f=)]TJ/F11 9.963 Tf 7.749 0 Td[(U)]TJ/F7 6.974 Tf 6.226 0 Td[(1f+wRw;HU)]TJ/F7 6.974 Tf 6.227 0 Td[(1f=)]TJ/F11 9.963 Tf 7.749 0 Td[(U)]TJ/F7 6.974 Tf 6.227 0 Td[(1f+U)]TJ/F7 6.974 Tf 6.227 0 Td[(1wrwf=U)]TJ/F7 6.974 Tf 6.226 0 Td[(1w w)]TJ/F11 9.963 Tf 9.962 0 Td[(x)]TJ/F8 9.963 Tf 9.962 0 Td[(1f=U)]TJ/F7 6.974 Tf 6.226 0 Td[(1hrwf=U)]TJ/F7 6.974 Tf 6.226 0 Td[(1hg:Wecannowstateandprovethegeneralformofthespectralrepresentationtheorem.UsingthedecompositionofHintocyclicsubspacesandtakingthegeneratingvectorstohavenorm2)]TJ/F10 6.974 Tf 6.227 0 Td[(nwecanproceedaswedidlastsemester.Wecangettheextensionofthehomomorphismf7!fHtoboundedmeasurablefunctionsbyprovingitintheL2representationviathedominatedconvergencetheorem.Thisthengivesprojectionvaluedmeasuresetc.

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Chapter13Scatteringtheory.ThepurposeofthischapteristogiveanintroductiontotheideasofLaxandPhillipswhichiscontainedintheirbeautifulbookScatteringtheory.ThroughoutthischapterKwilldenoteaHilbertspaceandt7!St;t0astronglycontinuoussemi-groupofcontractionsdenedonKwhichtendsstronglyto0ast!1inthesensethatlimt!1kStkk=0foreachk2K:.113.1Examples.13.1.1Translation-truncation.LetNbesomeHilbertspaceandconsidertheHilbertspaceL2R;N:LetTtdenotetheoneparameterunitarygroupofrighttranslations:[Ttf]x=fx)]TJ/F11 9.963 Tf 9.962 0 Td[(tandletPdenotetheoperatorofmultiplicationby1;0]soPisprojectionontothesubspaceGconsistingofthefwhicharesupportedon;0].Weclaimthatt7!PTtisasemi-groupactingonGsatisfyingourcondition.1:TheoperatorPTtisastronglycontinuouscontractionsinceitisunitaryoperatoronL2R;Nfollowedbyaprojection.AlsokPTtfk2=Z)]TJ/F10 6.974 Tf 6.226 0 Td[(tjfxj2dxtendsstronglytozero.383

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384CHAPTER13.SCATTERINGTHEORY.Wemustcheckthesemi-groupproperty.ClearlyPT0=IdonG.WehavePTsPTtf=PTs[Ttf+g]=PTs+tf+PTsgwhereg=PTtf)]TJ/F11 9.963 Tf 9.963 0 Td[(Ttfsog?G:ButT)]TJ/F10 6.974 Tf 6.226 0 Td[(s:G!Gfors0.Henceg?GTsg=T)]TJ/F10 6.974 Tf 6.227 0 Td[(sg?GsinceT)]TJ/F10 6.974 Tf 6.227 0 Td[(sg;=g;T)]TJ/F10 6.974 Tf 6.227 0 Td[(s=082G:13.1.2Incomingrepresentations.Thelastargumentisquiteformal.Wecanaxiomatizeitasfollows:LetHbeaHilbertspace,andt7!UtastronglycontinuousgrouponeparametergroupofunitaryoperatorsonH.AclosedsubspaceDHiscalledincomingwithrespecttoUifUtDDfort0.2tUtD=0.3 [tUtD=H:.4LetPD:H!Ddenoteorthogonalprojection.TheprecedingargumentgoesoverunchangedtoshowthatSdenedbySt:=PDUtisastronglycontinuoussemi-group.Werepeattheargument:TheoperatorStisclearlyboundedanddependsstronglycontinuouslyont.Forsandt0wehavePDUsPDUtf=PDUs[Utf+g]=PDUt+sf+PDgwhereg:=PDUtf)]TJ/F11 9.963 Tf 9.963 0 Td[(Utf2D?:Butg2D?Usg2D?fors0sinceUsg=U)]TJ/F11 9.963 Tf 7.749 0 Td[(sgandU)]TJ/F11 9.963 Tf 7.749 0 Td[(sg;=g;U)]TJ/F11 9.963 Tf 7.749 0 Td[(s=082Dby.2.

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13.1.EXAMPLES.385Wemustprovethatitconvergesstronglytozeroast!1,i.e.that.1holds.Firstobservethat.2impliesthatU)]TJ/F11 9.963 Tf 7.749 0 Td[(sDU)]TJ/F11 9.963 Tf 7.749 0 Td[(tDifs0U)]TJ/F11 9.963 Tf 7.749 0 Td[(tD?isdenseinH.Ifnot,thereisan06=h2Hsuchthath2[U)]TJ/F11 9.963 Tf 7.749 0 Td[(tD?]?forallt>0whichsaysthatUth?D?forallt>0orUth2Dforallt>0orh2U)]TJ/F11 9.963 Tf 7.748 0 Td[(tDforallt>0contradicting.3.Therefore,iff2Dand>0wecanndg?Dandans>0sothatkf)]TJ/F11 9.963 Tf 9.963 0 Td[(U)]TJ/F11 9.963 Tf 7.748 0 Td[(sgk
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386CHAPTER13.SCATTERINGTHEORY.Wecanallowforthepossibilityofmoregeneralconceptsofinformation",forexampleinmartingaletheorywherethespacesUtDrepresentthespaceofrandomvariablesavailablebasedonknowledgeattimet.Inthesecondexample,itismorenaturaltoallowttorangeovertheintegers,ratherthanovertherealnumbers.Butinthislecturewewilldealwiththecontinuouscaseratherthanthediscretecase.Inthethirdexample,wemightwanttodispensewithUtaltogether,andjustdealwithanincreasingfamilyofsubspaces.13.1.3Scatteringresidue.Inthescatteringtheoryexample,wewanttobelievethatatlargefuturetimestheobstacle"haslittleeectandsothereshouldbebothanincomingspace"describingthesituationlongbeforetheinteractionwiththeobstacle,andalsoanoutgoingspace"reectingbehaviorlongaftertheinteractionwiththeobstacle.Theresidualbehavior-i.e.theeectoftheobstacle-iswhatisofinterest.Forexample,inelementaryparticlephysics,thismightbeobservedasablipinthescatteringcross-sectiondescribingaparticleofaveryshortlife-time.SeetheveryelementarydiscussionofthebliparisingintheBreit-Wignerformulabelow.Solett7!UtbeastronglycontinuousoneparameterunitarygrouponaHilbertspaceH,letD)]TJ/F8 9.963 Tf 10.754 1.494 Td[(beanincomingsubspaceforUandletD+beanoutgoingsubspacei.e.incomingfort7!U)]TJ/F11 9.963 Tf 7.749 0 Td[(t.SupposethatD)]TJ/F14 9.963 Tf 9.492 1.495 Td[(?D+andletK:=[D)]TJ/F14 9.963 Tf 8.939 1.494 Td[(D+]?=D?)]TJ/F14 9.963 Tf 8.939 2.463 Td[(D?+:LetP:=orthogonalprojectionontoD?:LetZt:=P+UtP)]TJ/F11 9.963 Tf 6.725 1.494 Td[(;t0:Claim:Zt:K!K:Proof.SinceP+occursastheleftmostfactorinthedenitionofZ,theimageofZtiscontainedinD?+.Wemustshowthatx2D?)]TJ/F14 9.963 Tf 9.492 2.463 Td[(!P+Utx2D?)]TJ/F8 9.963 Tf -222.384 -19.973 Td[(sinceZtx=P+UtxasP)]TJ/F11 9.963 Tf 6.724 1.494 Td[(x=xifx2D?)]TJ/F8 9.963 Tf 6.725 2.463 Td[(.NowU)]TJ/F11 9.963 Tf 7.748 0 Td[(t:D)]TJ/F14 9.963 Tf 10.58 1.494 Td[(!D)]TJ/F8 9.963 Tf 10.698 1.494 Td[(fort0isoneoftheconditionsforincoming,andsoUt:D?)]TJ/F14 9.963 Tf 9.492 2.463 Td[(!D?)]TJ/F11 9.963 Tf 6.724 2.463 Td[(:

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13.2.BREIT-WIGNER.387SoUtx2D?)]TJ/F11 9.963 Tf 6.725 2.463 Td[(:SinceD)]TJ/F14 9.963 Tf 9.492 1.494 Td[(D?+theprojectionP+istheidentityonD)]TJ/F8 9.963 Tf 6.725 1.494 Td[(,inparticularP+:D)]TJ/F14 9.963 Tf 9.493 1.494 Td[(!D)]TJ/F8 9.963 Tf -199.047 -20.782 Td[(andhence,sinceP+isself-adjoint,P+:D?)]TJ/F14 9.963 Tf 9.492 2.463 Td[(7!D?)]TJ/F11 9.963 Tf 6.725 2.463 Td[(:ThusP+Utx2D?)]TJ/F8 9.963 Tf 10.045 2.463 Td[(asrequired.QEDByabuseoflanguage,wewillnowuseZttodenotetherestrictionofZttoK.Weclaimthatt7!Ztisasemi-group.Indeed,wehaveP+UtP+x=P+Utx+P+Ut[P+x)]TJ/F11 9.963 Tf 9.963 0 Td[(x]=P+Utxsince[P+x)]TJ/F11 9.963 Tf 10.076 0 Td[(x]2D+andUt:D+!D+fort0.AlsoZt=P+UtonKsinceP)]TJ/F8 9.963 Tf 10.406 1.494 Td[(istheidentityonK.ThereforewemaydroptheP)]TJ/F8 9.963 Tf 10.406 1.494 Td[(ontherightwhenrestrictingtoKandwehaveZsZt=P+UsP+Ut=P+UsUt=P+Us+t=Zs+tprovingthatZisasemigroup.WenowshowthatZisstronglycontracting.Foranyx2Handany>0wecanndaT>0anday2D+suchthatkx)]TJ/F11 9.963 Tf 9.962 0 Td[(U)]TJ/F11 9.963 Tf 7.749 0 Td[(TykTUtU)]TJ/F11 9.963 Tf 7.748 0 Td[(Ty=Ut)]TJ/F11 9.963 Tf 9.963 0 Td[(Ty2D+soP+UtU)]TJ/F11 9.963 Tf 7.749 0 Td[(Ty=0andhencekZtxk<:WehaveprovedthatZisastronglycontractivesemi-grouponKwhichtendsstronglytozero,i.e.that13.1holds.13.2Breit-Wigner.TheexampleinthissectionwillbeofprimaryimportancetousincomputationsandwillalsomotivatetheLax-Phillipsrepresentationtheoremtobestatedandprovedinthenextsection.

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388CHAPTER13.SCATTERINGTHEORY.SupposethatKisonedimensional,andthatZtd=e)]TJ/F10 6.974 Tf 6.226 0 Td[(tdford2Kwhere<>0:Thisisobviouslyastronglycontractivesemi-groupinoursense.ConsiderthespaceL2R;NwhereNisacopyofKbutwiththescalarproductwhosenormiskdk2N=20:Thenkfdk2=Z0e2
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13.3.THEREPRESENTATIONTHEOREMFORSTRONGLYCONTRACTIVESEMI-GROUPS.389Theorem13.3.1[Lax-Phillips.]ThereexistsaHilbertspaceNandanisometricmapRofKontoasubspaceofPL2R;NsuchthatSt=R)]TJ/F7 6.974 Tf 6.227 0 Td[(1PTtRforallt0.Proof.LetBbetheinnitesimalgeneratorofS,andletDBdenotethedomainofB.Thesesquilinearformf;g7!)]TJ/F8 9.963 Tf 7.749 0 Td[(Bf;g)]TJ/F8 9.963 Tf 9.962 0 Td[(f;Bgisnon-negativedenitesinceBsatisesReBf;f0:DividingoutbythenullvectorsandcompletinggivesusaHilbertspaceNwhosescalarproductwewilldenoteby;N.Ifk2DBsoisStkforeveryt0.Letusdenefk)]TJ/F11 9.963 Tf 7.749 0 Td[(t=[Stk]where[Stk]denotestheelementofNcorrespondingtoStk.Forsimplicityofnotationwewilldropthebracketsandsimplywritefk)]TJ/F11 9.963 Tf 7.748 0 Td[(t=Stkandthinkoffasamapfrom;0]toN.Wehavekf)]TJ/F11 9.963 Tf 7.749 0 Td[(tk2N=kStkk2N=)]TJ/F8 9.963 Tf 7.749 0 Td[(2ReBStk;StkN=)]TJ/F11 9.963 Tf 10.743 6.74 Td[(d dtkStkk2:Integratingthisfrom0torgivesZ0)]TJ/F10 6.974 Tf 6.227 0 Td[(rkfsk2N=kkk2)-222(kSrkk2:Byhypothesis,thesecondtermontherighttendstozeroasr!1.ThisshowsthatthemapR:k7!fkisanisometryofDBintoL2;0];N,andsinceDBisdenseinK,weconcludethatitextendstoanisometryofDwithasubspaceofPL2R;Nbyextensionbyzero,say.AlsoRStk=fStkisgivenbyfStks=S)]TJ/F11 9.963 Tf 7.749 0 Td[(sStk=St)]TJ/F11 9.963 Tf 9.963 0 Td[(sk=S)]TJ/F8 9.963 Tf 7.749 0 Td[(s)]TJ/F11 9.963 Tf 9.963 0 Td[(tk=fks)]TJ/F11 9.963 Tf 9.963 0 Td[(tfors<0,andt>0soRStk=PTtRk:ThusRKisaninvariantsubspaceofPL2R;Nandtheintertwiningequationofthetheoremholds.QEDWecanstrengthentheconclusionofthetheoremforelementsofDB:

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390CHAPTER13.SCATTERINGTHEORY.Proposition13.3.1Ifk2DBthenfkiscontinuousintheNnormfort0.Proof.Fors;t>0wehavekfk)]TJ/F11 9.963 Tf 7.749 0 Td[(s)]TJ/F11 9.963 Tf 9.963 0 Td[(fk)]TJ/F11 9.963 Tf 7.749 0 Td[(tk2N=)]TJ/F8 9.963 Tf 7.749 0 Td[(2ReB[St)]TJ/F11 9.963 Tf 9.962 0 Td[(Ss]k;[St)]TJ/F11 9.963 Tf 9.963 0 Td[(Ss]k2k[St)]TJ/F11 9.963 Tf 9.963 0 Td[(Ss]Bkkk[St)]TJ/F11 9.963 Tf 9.962 0 Td[(Ss]kkbytheCauchy-Schwarzinequality.SinceSisstronglycontinuoustheresultfollows.QEDLetusapplythisconstructiontothesemi-groupassociatedtoanincomingspaceDforaunitarygroupUonaHilbertspaceH.Letd2Dandfd=Rdasabove.WeknowthatU)]TJ/F11 9.963 Tf 7.749 0 Td[(rd2Dforr>0by.2.NoticealsothatSrU)]TJ/F11 9.963 Tf 7.749 0 Td[(rd=PUrU)]TJ/F11 9.963 Tf 7.749 0 Td[(rd=Pd=dford2D.Thenfort)]TJ/F11 9.963 Tf 18.264 0 Td[(rwehave,bydenition,fU)]TJ/F10 6.974 Tf 6.227 0 Td[(rdt=S)]TJ/F11 9.963 Tf 7.749 0 Td[(tU)]TJ/F11 9.963 Tf 7.749 0 Td[(rd=PU)]TJ/F11 9.963 Tf 7.749 0 Td[(tU)]TJ/F11 9.963 Tf 7.749 0 Td[(rd=S)]TJ/F11 9.963 Tf 7.748 0 Td[(t)]TJ/F11 9.963 Tf 9.963 0 Td[(rd=fdt+randsobytheLax-Phillipstheorem,kU)]TJ/F11 9.963 Tf 7.749 0 Td[(rdk2D=Z0kfU)]TJ/F10 6.974 Tf 6.226 0 Td[(rdk2NdxZ)]TJ/F10 6.974 Tf 6.227 0 Td[(rkfU)]TJ/F10 6.974 Tf 6.227 0 Td[(rdxk2Ndx=Z0jfdxj2Ndx=kdk2:SinceU)]TJ/F11 9.963 Tf 7.749 0 Td[(risunitary,wehaveequalitythroughoutwhichimpliesthatfU)]TJ/F10 6.974 Tf 6.227 0 Td[(rdt=0ift>)]TJ/F11 9.963 Tf 7.749 0 Td[(r:Wehavethusprovedthatifr>0thenfU)]TJ/F10 6.974 Tf 6.226 0 Td[(rdt=fdt+rift)]TJ/F11 9.963 Tf 18.265 0 Td[(r0if)]TJ/F11 9.963 Tf 7.749 0 Td[(r
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13.4.THESINAIREPRESENTATIONTHEOREM.391Theorem13.4.1IfDisanincomingsubspaceforaunitaryoneparametergroup,t7!UtactingonaHilbertspaceHthenthereisaHilbertspaceN,aunitaryisomorphismR:H!L2R;NsuchthatRUtR)]TJ/F7 6.974 Tf 6.227 0 Td[(1=TtandRD=PL2R;N;wherePisprojectionontothesubspaceconsistingoffunctionswhichvanishon;1]a.e.Proof.Weapplytheresultsofthelastsection.Foreachd2Dwehaveobtainedafunctionfd2L2;0];NandweextendfdtoallofRbysettingfds=0fors>0.WethusdenedanisometricmapRfromDontoasubspaceofL2R;N.NowextendRtothespaceUrDbysettingRUrdt=fdt)]TJ/F11 9.963 Tf 9.963 0 Td[(r:Equation.5assuresusthatthisdenitionisconsistentinthatifdissuchthatUrd2Dthenthisnewdenitionagreeswiththeoldone.WehavethusextendedthemapRtoSUtDasanisometrysatisfyingRUt=TtR:SinceSUtDisdenseinHthemapRextendstoallofH.AlsobyconstructionRPD=PRwherePisprojectionontothespaceoffunctionssupportedin;0]asinthestatementofthetheorem.WemuststillshowthatRissurjective.ForthisitisenoughtoshowthatwecanapproximateanysimplefunctionwithvaluesinNbyanelementoftheimageofR.RecallthattheelementsofthedomainofB,theinnitesimalgeneratorofPDUt,aredenseinN,andford2DBthefunctionfdiscontinuous,satisesft=0fort>0,andf=nwherenistheimageofdinN.HenceI)]TJ/F11 9.963 Tf 9.963 0 Td[(PUdismappedbyRintoafunctionwhichisapproximatelyequaltonon[0;]andzeroelsewhere.SincetheimageofRistranslationinvariant,weseethatwecanapproximateanysimplefunctionbyanelementoftheimageofR,andsinceRisanisometry,theimageofRmustbeallofL2R;N.

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392CHAPTER13.SCATTERINGTHEORY.13.5TheStone-vonNeumanntheorem.LetusshowthattheSinairepresentationtheoremimpliesaversionforn=1oftheStone-vonNeumanntheorem:Theorem13.5.1LetfUtgbeaoneparametergroupofunitaryoperators,andletBbeaself-adjointoperatoronaHilbertspaceH.SupposethatUtBU)]TJ/F11 9.963 Tf 7.749 0 Td[(t=B)]TJ/F11 9.963 Tf 9.963 0 Td[(tI:.6ThenwecanndaunitaryisomorphismRofHwithL2R;NsuchthatRUtR)]TJ/F7 6.974 Tf 6.227 0 Td[(1=TtandRBR)]TJ/F7 6.974 Tf 6.226 0 Td[(1=mx;wheremxismultiplicationbytheindependentvariablex.Remark.IfiAdenotestheinnitesimalgeneratorofU,thendierentiating.6withrespecttotandsettingt=0gives[A;B]=iIwhichisaversionoftheHeisenbergcommutationrelations.So.6isapartiallyintegrated"versionofthesecommutationrelations,andthetheoremassertsthat.6determinestheformofUandBuptothepossiblemulti-plicity"givenbythedimensionofN.Proof.Bythespectraltheorem,writeB=ZdEwherefEgisthespectralresolutionofB,andsoweobtainthespectralreso-lutionsUtBU)]TJ/F11 9.963 Tf 7.749 0 Td[(t=Zd[UtEU)]TJ/F11 9.963 Tf 7.749 0 Td[(t]andB)]TJ/F11 9.963 Tf 9.963 0 Td[(tI=Z)]TJ/F11 9.963 Tf 9.963 0 Td[(tdE=ZdE+tbyachangeofvariables.WethusobtainUtEU)]TJ/F11 9.963 Tf 7.749 0 Td[(t=E+t:RememberthatEisorthogonalprojectionontothesubspaceassociatedto;]bythespectralmeasureassociatedtoB.LetDdenotetheimageofE0.ThentheprecedingequationsaysthatUtDistheimageoftheprojectionEt.Thestandardpropertiesofthespectralmeasure-thattheimageofEtincreasewitht,tendtothewholespaceast!1andtendtof0gast!)1(1areexactlytheconditionsthatDbeincomingforUt.HencetheSinairepresentation

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13.5.THESTONE-VONNEUMANNTHEOREM.393theoremisequivalenttotheStone-von-Neumanntheoremintheaboveform.QEDHistorically,SinaiprovedhisrepresentationtheoremfromtheStone-vonNeumanntheorem.Here,followingLaxandPhillips,weareproceedinginthereversedirection.