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- https://ufdc.ufl.edu/UF00099379/00001
## Material Information- Title:
- Domains of Greek letter tau-holomorphy on a Banach space
- Creator:
- Livadas, Panos E. ( Panos Evange ), 1944- (
*Dissertant*) Su-shing Chen (*Thesis advisor*) Cenzer, Douglas (*Reviewer*) Popov, Vasile (*Reviewer*) Block, Louis (*Reviewer*) Su, Stanley Y. W. (*Reviewer*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 1980
- Copyright Date:
- 1980
- Language:
- English
- Physical Description:
- vi, 63 leaves ; 28 cm.
## Subjects- Subjects / Keywords:
- Academic degrees ( jstor )
Analytics ( jstor ) Banach space ( jstor ) Cauchy Schwarz inequality ( jstor ) Distance functions ( jstor ) Graduates ( jstor ) Mathematical domains ( jstor ) Mathematics ( jstor ) Separable spaces ( jstor ) Topological theorems ( jstor ) Banach spaces ( lcsh ) Dissertations, Academic -- Mathematics -- UF Domains of holomorphy ( lcsh ) Mathematics thesis Ph. D - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Abstract:
- Let E be a complex separable Banach space, U be a nonempty open subset of E, x be a strictly positive lower semicontinuous function on U such that d(x,9U) > t(x) for every x e U. Let 8 (U) denote the collection of all finite unions of closed balls B (x) with center x e U and p < x (x) and let H _(U) be the algebra of the complex holomorphic functions in U which are bounded in every set of 8 (U) furnished with the Frechet topology of the uniform convergence over the elements of B T (U) . The main results are: if A e B (U) , then A is bounded, if and only if, its x-holomorphy hull is bounded; if A e B (U) n _ and A = u B (x.) and z n is some element of the T-holomorphy i=l p i hull of A, then, every f e H (U) is holomorphic on B (z ) and for every p < r, [| f|L . . < || f[|, . , where r = m i n p l "0 j {p> A i=l,2,...,n {t(x.) - p.}, < p. < x(x.), x. e U; if U is bounded the following are equivalent: (i) U is a domain of x-holomorphy . (ii) U is x-holoniorphically convex, (iii) Every boundary point of U is x-essential. (iv) For every sequence {£ } N of points of U with d(£ ,3U) + as n * °°, there exists f e H (U) such that S U p{|f (? ) |} = =o. n > °° If U is bounded and is approximated from the inside by a principal sequence of domains {D } â€ž, then if each domain D n is x-holomorphically convex relative to U, then U is a domain of T-holomorphy; if in addition U is T-semicomplete and if each of the domains D is T-holomorphically convex relative to D , , , then U is domain of T-holomorphy; if each D is a n+1' tr j i n domain of T-holomorphy and U is T-complete, then U is a domain of T-holomorphy. Suppose now that E is a locally convex Hausdorff space and an open subset U of E is approximated from the inside by a principal sequence of domains {D } â€ž. If each of the domains D is pseudoconvex, so is U; if each D is polynomially convex n c n L and if E has the approximation property, then U is pseudoconvex and Runge. Finally it is shown that if M is a complex analytic Banach mainifold, then the Kobayashi pseudodistance is the largest for which every holomorphic mapping from the unit disk of the complex plane into a complex analytic Banach manifold is distance decreasing while the Caratheodory pseudodistance is the smallest pseudodistance for which every holomorphic mapping from the complex analytic manifold to the unit disk of the complex plane is distance-decreasing.
- Thesis:
- Thesis (Ph. D.)--University of Florida, 1980.
- Bibliography:
- Includes bibliographic references (leaves 61-62).
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Panos E. Livadas.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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- 023437508 ( AlephBibNum )
07288598 ( OCLC ) AAL5549 ( NOTIS )
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DOMAINS OF T -HOLOMORPHY ON A BANACH SPACE BY PANS E LIVADAS A DISSERITATIONJ PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1980 Copyright 1980 by Panos E. Livadas ACKNOWLEDGEMENT The author is deeply grateful to his advisor, Dr. Su Shing Chen, who gave generously of his time, and made many helpful suggestions. His guidance and patience are much appreciated. Thanks are also due to Dr. Dong S. Kim who was responsible for generating his interest in the area of Several Complex Variables and to Dr. William Caldwell for his support and understanding. The author would like, also, to thank the many others in the Department of Mathematics at the University of Florida who contributed in their several ways to the completion of this dissertation. BIBLIOGRAPHY. ............................... BIOGRAPHICAL SKETCH................................... TABLE OF CONTENTS ACKNOWLEDGEMENTS.............................. ABSTRACT....................................... INTRODUCTION................................... PRELIMINARIES ................... ................... ... CHAPTER Page ill v 1 4 I. SEQUENCES OF DOMAINS OF T-HOLOMORPHY IN BANACH SPACES................................ 11 5l. Domains of T-Holomorphy in a Complex Separable Banach Space.................. 11 52. Sequences of Domains of T-Holomorphy On a Complex and Separable Banach Space.... 28 53. Sequences of Certain Domains on a Locally Convex Hausdorff Space.......... 40 II. KOBAYASHI AND CARATHEODORY DISTANCES FOR COMPLEX BANACH MANIFOLDS..................... 46 l.Complex Analytic Banach Manifolds Over a Complex Banach Space.................... 46 62. The Kobayashi Pseudodistance On a Complex Analytic Banach Manifold................. 50 53. The Caratheodory Pseudodistance On a Complex Analytic Banach Manifold......... 57 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DOMAINS OF T-HOLOMORPHY ON A BANACH SPACE By Panos E. Livadas December 1980 Chairman: Dr. Su-Shing Chen Major Department: Mathematics Let E be a complex separable Banach space, U be a non- empty open subset of E, -c be a strictly positive lower semi- continuous function on U such that d(x,aU) > T(x) for every x E U. Let 8 (U) denote the collection of all finite unions of closed balls B (x) with center x E U and p < T(x) and let H?(U) be the algebra of the complex holomorphic functions in U which are bounded in every set of 8 (U) furnished with the Fr~chet topology of the uniform convergence over the elements of 8 (U). The main results are: if As E B(U), then A is bounded, if and only if, its T-holomorphy hull is bounded; if As E 6(U) and A = v E (x ) and z0 is some element of the T-holomorphy i=1 Pi hull of A, then, every fe @ (U) is holomorphic on Br(z0) and for every p < r, I/ f/f f ,whrr mi {T(x ) p }, O < pi < T(xi), xi E 07 if U iS bounded the following are equivalent: (i) U is a domain of r-holomorphy. (ii) U is T-holomorphically convex. (iii) Every boundary point of U is T-essential. (iv) For every sequence CEn neN of points of U with d(5,laU) + 0 as n + m, there exists f EH (U) such that s u p{ f(En) } = m If U is bounded and is approximated from the inside by a principal sequence of domains {D }neN, then if each domain Dn is T-holomorphically convex relative to U, then U is a domain of T-holomorphy; if in addition U is r-semicomplete and if each of the domains Dn is T-holomorphically convex relative tDn+1, then U is domain of r-holomorphy; if each Dn is a domain of r-holomorphy and U is T-complete, then U is a domain of T-holomorphy. Suppose now that E is a locally convex Hausdorff space and an open subset U of E is approximated from the inside by a principal sequence of domains (Dn nEN. If each of the domains Dn is pseudoconvex, so is U; if each Dn is polynomially convex and if E has the approximation property, then U is pseudoconvex and Runge. Finally it is shown that if M is a complex analytic Banach mainifold, then the Kobayashi pseudodistance is the largest for which every holomorphic mapping from the unit disk of the com- plex plane into a complex analytic Banach manifold is distance- decreasing while the Carathdodory pseudodistance is the smallest pseudodistance for which every holomorphic mapping from the complex analytic manifold to the unit disk of the complex plane is distance-decreasing. INTRODUCTION The study of holomorphic mappings defined on Banach spaces has received considerable attention in recent years. This dissertation contains new results relating to certain kinds of holomorphy on a complex Banach space. The notion of a domain of holomorphy arises naturally with the study of holomorphic functions on a non-empty open subset of Cn. In particular, an open subset U of Cn is said to be a domain of holomorphy, if and only if, there exists a holomorphic function defined on it which cannot be extended analytically beyond any point of its boundary alU. It is known that in this case U is a domain of holomorphy, if and only if, U is holomorphically convex ([ 2 3, [101, [15], [17]). The situation is different on complex Banach spaces. As a matter of fact, if U is a holomorphically convex domain in a complex Banach space, then it need not be a domain of holomorphy [ll]. Moreover, the situation is different when one moves from one complex Banach space to another. The main reason is the behavior of the bounding sets on a complex Banach space. A closed bounded subset, A, of a complex Banach space, E, is said to be bounding if every complex valued holomorphic function on E is bounded on A. For certain complex Banach spaces, in particular, for separable or reflexive spaces, the bounding subsets are precisely the compact subsets, while there are examples of non-compact bounding subsets of other complex Banach spaces ([4], [5]). In Sections I and 2 of Chapter I of this dissertation, we primarily consider a separable complex Banach space E and a kind of holomorphy, the T-holomorphy, which is due to M. Matos. In particular, M. Miatos has proved (see Preliminaries and Section I.1 for the corresponding notations and definitions): THEOREM (MATOS). Let E be a complex nepahable Banach pace and let a be a non-empty open nubnet og E. Then, the 6ollowing (a) U La a domain od T-holomohphy. (6) o v1 e8[) in bounded and d(A ,au) > 0. (C) Thate La d in Hq(0) ouch that it La kmponnible to dind two open connected nubnets il and 012 06 E datidgging the baoLloing conditions: (i) U1n UI 012, U2 f 4, UI Q U. (ii) Theh n La E H(UI) nuch that 6 = 61 on U2. In Section I.1, the primary result is a local continuation theorem, namely: THEOREM I.1.10. Let E be a complex oepata~ble Banach pace. Let U be a non-empty open nubnet og E. Let As 8 -(UJ) add Let z0 be an element oj A Then i6 d E ~(~ hn6i holomoxphic on 8 [zg). And moneovea, goA evehy p < h and 6ot Section I.2 is concerned with the problem of convergence of a principal sequence of domains of r-holomorphy. (See Section I.2 for the corresponding notations and definitions.) We are showing that: THEOREM I.2.9. Let E be a complex nepahable Banach space and Let a be a bounded and T-compLete domain in E. Suppose 04 domains o{ T-holomohphy {9 vevN. Then U in a domain od T-holomohphy. Section I.3 deals with the problem of convergence of certain kinds of domains on locally convex Hausdorff spaces. In Chapter II, we introduce complex analytic Banach manifolds, and we furnish them with two pseudometrics, the Carathdodory and Kobayashi pseudodistances. We are also proving that the Kobayashi pseudodistance is the largest for which every holomorphic mapping from the unit disk of the complex plane into a complex analytic Banach manifold is distance-decreasing while the Carathdodory pseudodistance is the smallest pseudodistance for which every holomorphic mapping from the complex analytic manifold to the unit disk of the complex plane is distance-decreasing. The pseudodistances permit us to obtain main results on complex analytic Banach manifolds by a purely topological method. They enable us also to give geometric insight into function theoretic results. In particular, for results in Cn see [3] and [12]. PRELIMINARIES In this paper the notation and terminology used, unless otherwise stated, is that of Nachbin [16]. For the sake of completeness, however, we recall certain theorems and defini- tions needed in the sequence. Let E and F be two complex Banach spaces. If z E E by Br(z) we denote the open ball with center z and of radius r while by Br(z) we denote the closed ball of center 2 and of radius r. We reserve the letters R and C to denote the set of real and complex numbers respectively throughout this paper. Similarly, we reserve the letter N to denote the set of natural numbers. THEOREM 0.1. Let Lm(E,FI =- {T: T La a continuous m-eineah map d@cm Em into F }. Then Lm(E,F) La a Banach pace with henpect to the pointwine vectoA opetationn and nohm defined by xcZ = 7,2,...,m. x I DEFINITION 0.2. Let L :(E,F) denote the eloned vectoa/ nub- space of Lm(E,F) oJ all m-lineat mapn T: Em + F which a/e DEFINITION 0.3. Let A s Lm(E,F). Define its symmettization An E L (E,F) by whehe the summation in taken oveh the m! pehmutationn Note that |iAsl lA|| and moreover that the map A + A is a continuous projection from Lm(E,F) onto Lm(E,F). s s In the sequence we write Lm(E) and Lm(E) for Lm(E,C) and Lm(E,C) respectively. Finally, if x E E and A E Lm(E,F) we write Axm to denote A(x,x,...,x) and AxO to denote A. DEFINITION 0.4. A continuous m-homogeneous polynomial P dh0m E into F Zh a mapping P doh which theae in a map A E Lm(E,F) nuck tkat Plx) = Axm hoh evehy x E E. We white P =- A to denote that P coatenpondn to A that way. THEOREM 0.5. Ig Pm(E,F) denotes the net og Laff continuous m-homogeneoun polynomiatn dhom E into F then,PmlE,F) becomes a Banach pace with Aenpect to the pointwvine vector opu~ations and nohm dedined by Observe that if m = 0 then PO(E,F) is just the set of all constant maps from E into F. THEOREMI 0.6. The map A + A 6hom Lm(E,F) onto Pm(E,FJ La a vector inomoxphinm and homeomohphism. Moneovex lA|| 5 ||Al < (mm/m!,) Al and (mm/m!) id the best univehale constant. DEFINITION 0.7. A continuous polynomial P dhom E into F in a mapping P: E F 60on which thehe ahe me W u {0) and P eP (E,FI (0 s k m) nuch that P = PO PT .. Pm Id P t 0 then L a one and only one nuch exphednion doh P with Pk < 0 dolr ome h = 0,7,2,...,m. The degree 06 P La dedined to be the numnbeh m. Id P = 0 tken the deglree of P may be taken as either equal to -7 oh to -=. We denote by P(E,F) the vector pace od all continuous pofynomitae B/rom E into F. DEFINITION 0.8. A powet netrien Rom E to F about x E E, in a saehne in the vaalable z E Eo{6 the Bohm (0.7) Z A {z-x)m = P (z-x) m=0 m=0 whee Am E Lm(E,F) and Pm = Am 6ot evexy m = 0,7,2,.... The hadius 04 convehgence oj the above poweh neales La the La~gent 4, h E [0,m], nuch that the poweh dealen La uni6otmly convehgent on eveh B 6(x) ot0h sp < P . THEOREM 0.9. (Cauchy-Hadamard) The Laudius o{ convergence oB the poweh nealeo 06 the definition 0.8. La given by n = (ZmsuLp |PmI 1/m -1 It is easy to verify that the power series of the Defini- tion 0.8. is convergent, if and only if, the sequence { I Pm/ 1/m meN is bounded?. DEFINITION 0.10. Let E and F be two complex Banack space and eet a be a non-empty open nubnet oB E. A mapping 5 dAom a into F is said to be holomohphic ia U (in the weimasttann dense) id cohhenponding to eve~y x E U thene a/e h > 0 and a powet naetre of (0.7) dRom E to F about x nuch that (LL) 5(z) = C Pm(z-X) unidohmly doh evehU z E R (x). In the case above we write f E H(UF) and observe that H((U,F) is a vector space with respect to the pointwise vector operations. We remark that the above series of Definition 0.10. is unique at every x E U and this series is called the Taylor's series of f at x. Let Pm Pm(E,F) correspond to Am m(EF yP m We set the notations dmf(x) = mlAm and dmf~x) = mlAm so we have the differential mappings dmf: x E U + d f(x) E Lm(E,F) dmf : x E U + dmf (x) E P (E,F) and the differential operators of order m EN u {0} dm: f E H(U,F) + dmf E H(U,Lm(E,F)) THEOREM 0.11. (Cauchy integral) Let d ER H(,FJ, Z E U, x E U, and tr > 0 be nuch that (1-Xlz + Ax cu U oh evehg X E C, with /Af < h. Then 7 d((17-X~z+AxJ 6(X) d THEOREM 0.12. Let g E H(U,F), Z E U, x E E, and h > D be 6uch that z + Ax E U got eve~y X E C, wit IXI h Then 04h evehU me W u {0} we have (m!-1dm(z(x =(2nri) (6(z+xxx) (+) dL THEOREM 0.13. (Cauchy inequalities) Let E and F be two complex Banach dpacen and Let a be a non-empty open nubnet od E. Suppose that Q E H(U,FI and that dox z E l thee La name A > 0 nuch that 8 (z) La contained in U. Then Bot each me W u {0) we have mI dmd(z)I r - u up ii 6x) |I Some of the properties of the holomorphic mappings on Cn can be extended on the holomorphic mappings on a complex Banach space. We are proving here that the principle of analytic continuation is valid on complex Banach spaces. THEOREM 0.14. Let E and F be, two complex Banach npacen and suppose that d in an hoLomorrphic mapping dnom E into F. Id 6 in equal to zeho on dome non-empty and open nubnet 0 od E, then we have d = 0 on E. PROOF: Let S = {Z E E: dmf~z) = 0 for all m ENu (0}}~. Clearly S is non-empty. Moreover for fixed m EN the set Sm = {z E: dmf(z) = 0} is closed because of the continuity of dmf and then the set S is closed being the intersection of all closed sets Sm for m E N. NOw, let a s S. Since f is assumed to be holomorphic on E and since a eS then,we can find some neighborhood V of a in E such that the Taylor's series expansion S(m!)-1 dmf(a) (z-a]m m=0 converges to f(z) for every z E V according to the definition 0.10. But we have that dmf~a) = 0, because a s S, for every m E N. Hence we obtain that f(z) = 0 for every z E V and therefore V cS and we can conclude that S is open. But because ofconnectedneaswe have that E = S and then f = 0 on E. COROLLARY 0.15. Let E and F be two complex Banach npacen. Let a E H(E,F) and-aUppose that thee exists nome point SE E dulch that dmd~a) = 0 dotr evehy me W u {0}. Then a = 0 on E. COROLLARY 0.16. Let E and F be two complex Banach dpacen. Let 6,g 4 H(E,FJ and nuppose that 5 and g ag~ee on 6ome open and non-emptU subset oa E. Then 6 = g on E. DEFINITION 0.17. Let a be a non-empty and open dubset od a complex Banach pace E. A mapping a 6Aom LI into another complex Banach pace F in naid to be Sinitely holomohphic on U id the mapping Bila n S: Un0S F in holomohphie 6ot eveAU {aitife dimenslonal vector nubnpace S od E. THEOREM 0.18. (Nachbin) Let E and F be two complex Banachz npacen and let a be a non-empty open nubnet og E. A mapping 6 bpom a into F 4 6initely holomohphic on 0, id and only id, doh evehy pala oj points a and b og E tke mapping u: {k E C: a + bh E U} + F defined by u(k) = 6(a + bh) in hoLomohphic. THEOREM 0.19. (Nachbin) Let E and F be two complex Banach npacen and Let a be a non-empty and open nubnet oj E. A mapping d dhom u into F Za hoLomoxphic, i6 and onLU id, La dinitely holomohphic and eithelr Lt La continuous, oh elsen it natinblen the equivalent conditions: (i) 6 mapn eveiiy compact nubnet o6 0 onto a bounded subset od F. (il d mapn nome neighborhood in U 04 eve^U compact subset of u onto a bounded nubnet a5 F. We are closing this section by giving a new definition. DEFINITION 0.20. Let E be a complex Banackz pace and let D and 0 be ;two open and non-empty nubnetn of E. Ig D in a nubnet oj 0 we will nay tkat D La al~atively compact in Ll id the closun~ ad D in contained in 0 and is compact. In thin case we white 0 cc u. CHAPTER I SEQUENCES OF DOMAINS OF T-HOLOMORPHY IN BANACH SPACES 5l. Domains of T-Holomorphy in a Complex Separable Banach Space Let E be a complex separable Banach space and let U be a non-empty open subset of E. DEFINITION I.1.1. A mapping d haom 0 into C id naid to be holomoaphic id thehe exints a sequence oj complex continuous n=0 convetrgen unioahmly doh A in a neighborhood od zeho in E. Let aU denote the boundary of U and let I be a strictly positive lower semicontinuous function of U such that T(x) 5 d(x,aU) for every x in U. Let B (U) denote the collection of all finite unions of closed balls B (x) with center x E U and p < T (x). Let H,(U) denotethealgebra of the complex holomorphic functions in U which are bounded in every set of 8 (U) fur- nished with the Frdchet topology of the uniform convergence over the elements of 8 (U) [14]. Observe that the union of H (U), for all T, is the algebra H(U) of all the complex valued holomorphic mappings in U. 11 DEFINITION I.1.2. Let E be a complex bepahable Banach space. Let U be a non-empty open subset og E, then U is said to be a domain o{ r-holomohphy, id and on~y id, it La impobbibLe to 6lad two non-empty, open and connected nubdetd UI and U2 o{ E 6a~~dyigg the do~llwing conditions: (1) U n U1 U2, U2 0, U0 Q U lii) 6ot eve/y 6 E H (0) thehe exists F E H(Uy) nuchz that Flu2p = DEFINITION I.1.3. Let E be a complex sepahabLe Banach space and Let U be a non-empty open subdet of E. Ig A is a Mon- empty subset og06 wIe define the r-holomohphy huff of A with henpect to u to be the net AU= x E 1: |4(x) A 6ot evetry 6 EH tl(ll) woheh II (|| = n a pf O~x)|}. xeA From the Definition I.1.3. it follows that for every subset A of U we have A c AU and therefore we can deduce LEMM~A I.L.4. Let E be a complex depahable Seanach pace and Cet U be a non-empty open subnet og E. Ig A is a subset of 0~ ~ an i UZ bounded,then A in bounded. If A is a subset of U we denote by co(A) the closed con- vex hull of A. Then the following theorem indicates that AU is contained in the closed convex hull of A. THEOREM I.1.5. Let E be a complex nepauable Baeach space and Let a be a non-empty open nubnet of E. Then 6ot evetr bounded nubnet A og a we have: A'U c co(A). PROOF: Let xg co(A). Then by Banach Separation Theorem [6] there exists a continuous linear functional f on E and real numbers c and 6, 6 > 0 such that (I.1) Reff(cO(A))} < c 6 < c < Re~f(x0 '. Consider the set P = {1 E C: Re(A) < & (f(x)) ItO(A). Then in view of the inequality (I.1) we obtain (I.2) f(co(A)) c P and (I.3) f(x0) ' Using the facts that f is continuous and linear and in view of (I.2) and (I.3) we can find X0 E C and r > 0 such that (I.4) f(co(A)) c Br 0o where Br 0O) denotes the open ball with center 10 and of radius r, and r can be chosen so that (I.5) f(x0) Br 0 We define a mapping from E into C by the rule #(x) = f(x)- 10. Then in view of (I.5) (I.6) /f~x0) 0[ >O r and in view of (I.4) (I.7) r > s up { f(x) XO0n' xECO(A) Hence in view of (I.6) and (I.7) we obtain /#(x0)| = |f(x0) 0 >O r s up {|f(x) 10~ = sup {[ (x~l} > s u p{((~xl)| = ll 4 xECO(A) xEA and therefore (I.8) |(xA Claim that eH ~(U). Observe that $ is bounded on every subset of 8 (U) and therefore in order to establish the claim we must show that Let 5( E E and SF = (2 E C: 5 + nze U}. Define a map u from S into C by the rule u(u) = O((+vn) for every E S. Then u(u) = f(S) + uf(n) A0 and clearly then u E H(S). Then Q is finitely holomorphic and since it is continuous it is holomorphic, by Theorem 0.19., on U and the claim is established. Finally, since 4 E H (U) and because of inequality (I.8) we can deduce that x0 i AU and hence AU c co(A). From Lemma 1.1.4. and Theorem 1.1.5. and the fact that if A is bounded, then co(A) is bounded, we obtain THEOREM I.1.6. Let E be a complex nepahable Banach pace. Let U be a non-empty open nubnet og E. Suppose that A c Ui. Then A is bounded id and onLy id A'U oud. DEFINITION 1.1.7. Let E be a complex nepahable Banach pace and a be a non-empty open and bounded nubnet od E. We will naU that U in T-holomoaphically convex id and onig id aoh eve~y As 8 B(U) we have d(A ,u)>. Then in the case of bounded open sets, Matos' theorem which is stated in the Introduction becomes: THEOREM I.1.8. (Matos) Le~t II be a non-empty open and bounded dubnet od a nepa/rable complex Banac pace E. Then U 4 a domain ag T-holomo/rphy id and only Lid id T-hoLomaxphically convex. LEMM~A I.1.9. Let U be a non-empty and open dubdet od a complex nepaxable Banach cpace E. Let Ae 8 g(U) and nuppone that *ueh exints a constant M > 0, and a SUnCtion E H (u) bUCh that |4(x)( sM doh evehy x E A. Then the name estimate extend IloZ Au Tt z) M aok evexy z E ,U PROOF: Since (f(x)( cM for every x E A, we obtain ifl AE M. But, since z E AU, we have |f(2) I |f |A; and in view of the last inequality |f(z) < M for every z e A . We are employing the following notation: Let U be a non-empty open subset of a separable complex Banach space E. Let Ae B(U) and suppose that A = u B (X.) i=1 Pi where x. E U and 0 < pi < T (x.) for every i = 1,2,...,n. Let r = m in {T(x ) p } and let 0 < q < r. Then we denote i=1,...,n by (9A the set defined by (9A i= u Bif (X ) THEOREM I.1.10. Let E be a complex nepatable Banachz spuce. Let a be a non-empty open bubbet oj E. Let As 8 &(U) and suppose tkat A = u B ( i) and Le~t h = m n {Ti- }. Let z0 be an element oQ A ;. Then, id ge E (0), then d is holomo~pkic on 8 (z0). And moneoveh, 6oh evety p < h and dox evagy 6 E H (U), We have 8zp) (p), PROOF: We first prove that if f eH (U), then f is holomorphic on Br(Z0 " Let 0 < q < r and let n > 0 such that q n > 0. Consider the set (-IA. Observe that (-IAc (r)A n ic vr f EH (U) is bounded on (rA, then every fe E (U) is bounded on (4~A. Let (I.9) fl (qnA = M ('l) for every f eH (U). Now if z is any element of A, we clearly have that B o ~(2) c A and therefore in view of (I.9) (I.10 ||f (Z) Mf(n) for every f eH (U). Now, since H (U) c H(U), every f EH (U) belongs to H(U), and therefore we can apply the Cauchy inequalities, of Theorem 0.13, to f and obtain: (I.11) (ji)-1 d f(z)) Mf (n) (q-n)- for every j = 0,1,... and Z E A. In view of Lemma I.1.9. and that z0 EA I, the above inequality becomes (I.12) (j)1df0 f Now consider the power series of f at a neighborhood of z0. (I.13) f(z) = C (j!)- d f(z0) (z-z0 " j=0 Because of inequalities (10.4), we obtain: (I.14) j(ji)-1 d f(z0) (z-z0~l Mf(1) (|z-z0l(-n ) But, (I.14) indicates that the series (I.13) is convergent for z2-z0 < q 0. Hence, f is holomorphic on B q~(z0) which in turn implies that f E H(B (Z0)) because as n approaches zero, the series is convergent on B (z0). But q is arbitrary, q < r; and therefore, f E H(Br(20) Now, we are proving the second part of the theorem. Let 0 < p < q n. Then, if fe E (U), by the first part of the theorem we obtain that fe H (B (Z0)). Therefore, the mapping f admits a Taylor's series expansion. (I.15) f(z) = E (jl)- d f(z0) (z-z0)j for every z E B (20 ' j=0 But 0 < p < q n and therefore in view of (I.15) we obtain |f(z)| < ~ (jl)-1 d f(z0) (z-z0~j f(n) (1z-z0l(-~~ j=0 j=0 < M (n) Z p (-) ]=0 and therefore (I.16) /f(2)l I Mf(n) for every z B (z0 ' Define a function $ from R+ to the R by (I.17) ~(n) = ( (fl~pzM (z0 ) We claim that ~(n) < 1 for all small n. Suppose not. Then there exists n0 > 0 such that ~(nO) > 1. Consider the functions Ok from U into C defined for every z E U and k E N by (I.18) Ok(z) = (f(z)/M (nO))k Clearly, Ok T H(U); and moreover, (I.19) I kllpEl (z0) 0))kC because I k gp(z0 = (f(z)/M (nlOl))kpg (20 1 jp~z0 )/M (nOl)k In view of (I.19) and of the assumption that ~(nO) > 1, we can deduc tha Ok (z0is large for large k. But, (I.20) M k0 because M k~ 0 k i(q-~on0 A = (2sfZ ll)/M(1) 0 I9'gA =(M ('l0 /f 0))k = 1. Now since #k E H,17), inequality (I.16) in view of (I.20) implies p 0 1 - q-nl But, the above inequality (I.21) implies that all functions {kk=1 are uniformly bounded on B (z0) contradicting that #k is large when k is large. Therefore, the claim is estab- lished. Since Q(n) < 1 for all small n, the equation (I.17) implies ( f b(z0 )/M (n)) 5 1 i m( I f ~( o/M (n)) 5 1 IIf~ (z0) and finally from the above inequality, we obtain: IIfe(z0) I fIp A THEOREM I.1.11. Let E be a complex aepahable 8anach pace and Let a be a non-empty open aubaet o( E. Let R be a non- empty nubnetf o6 0 natindying the p~opeaty that evehU mapping d E H(U) LA bounded in R. Then, thehe exists Ae 8 B(U) nuch thatf AU PROOF: Suppose not. Let Al'A2,..,,An,.. be a sequence of elements of B (U) such that every element Ae 8 r(U) is con- tained in some Aj for some j L N. Since n 4 A; for any Le S = AL. Clearly, n 4 51 ; an therefore, we can find an element (1 of n SlU. Let pl be such that 01 < pl 1) and define S2 A2 U B (5 ). Since S2 E 8 (U), we have that R Q S2 ; and therefore, we can find a new element 52 E R S2 . Let p2 be such that 0 < p2 ( (2) and define S3 by S3 = Aj uB 15) u B 25 2 Inductively then, we have obtained a sequence of subsets of U, {Sn n=1 and a sequence I5n n=1 of points of 0 satisfying the properties: (i) {Sn n=1 is a nested increasing sequence of sets. (ii) Sn T B(U) for every n E N. (iii) If Ae 6 ~(U), then there exists some j EN such that (iv) 5n E S. for every j,n E N with n < j. (v) 5, a Sn for every n E N. Let n EN and fixed. Claim that for every given positive number 6n and every positive given number Mn we can find fn E B (U) Such that (I.22) fn Sn) n and Ifn IS nM where En and Sn are as above. Since Sn SU by the definition of the T-holomorphy hull, we can find a mapping gn r H (U) such that lg ((n, 'i SnS Then we take a mapping fn from U into C defined by f (x) = (gn(x)/gn Sn))m n for every x E U and some positive integer m. Then it is clear that fn E H (U); and moreover, for some large m the mapping fn satisfies the conditions (I.22) and the claim is established. We take 61 1 and M = Apply (I.22) to get a mapping fl E H (U) such that Ifl 1L)I = 1and IIfl SL < 1. Inductively, for n-1 in n E f (n) and Mn 2(n) for every n E N- fl) .i=1 we can find according to (I.22) mappings fn ER H(U) Satisfying n-1 If (S ) = n + ) f (5n) and Ifn lS < 2-n for n E N- 1 , i=1 n By the construction of the functions {f } ,, we can see that the series I fi determines a function f E H(U), Since the i=1 series converges uniformly on each Sn and therefore on each As 8(U).We claim that the function f is unbounded on R. We have (I.23) If(Snl n nf(, n (n(, i i=1 n-1 cm i=1 j=n+1 n-1 m~ s fn(, n Ii(, n j n 'I i=1 j=n+1 Taking into account conditions (I.22), inequalities (I.23) imply |f(Sn)| z n I|f ((n ' j=n+1 But, recalling property (v), we have that whenever j < n we have that (nr S Then, f (En, l jl S .< .Hece E |f (Sn)I < 1 and the above inequalities imply j=n+1 |f(,)I > n-1. But 5,nE n, and therefore |/ flh = m, and the claim is estab- lished. But every holomorphic map f in U was assumed to be bounded on R, and this is a contradiction. DEFINITION I.1.12. Let E be a complex nepanable Banach pace, and let U be a non-empty open nubnet o{ E. A point z o{ the boundary o6 u is a v-enbential boundahU Point o{ u id ;thenr exints a mapping d ER H(UI which in ROt the ARAZ~let/OR 0( Sholomohphic mapping in an open nubnet O o{ E containing z and natibdging U c V. The following theorem furnishes us with another necessary and sufficient condition in order for an open subset of a Banach space to be a domain of r-holomorphy. THEOREM 1.1.13. Let E be a nepahable complex Banach pace. Let a be a non-empty open and bounded nubnet oj E. Then U id a domain od T-holomoaphy, iS and onLy i6, evehy boundary PROOF: Observe that by the definition of a Tr-essential boundary point it follows that even if U is not bounded, then U being a domain of T-holomorphy, then every boundary point of U is T-essential. Conversely, suppose that every boundary point of U is T-essential, but U is not a domain of Tr-holomorphy. Then by the result of Theorem I.1.8., U is not T-holomorphically convex. Therefore, we can find A E 8 (U) such that (I.24) d(AU,3U) i 0. Suppose that A is of the form A= i= u E (x.), x. t U, O < pi < T(x.) for i = 1,2,...,n. Let 0 < r < m i n{T(x ) p }. Then in view of (I.24), we can find a point z0 E AU satisfying (I.25) d(z0,3U) < r. But if fe r (U), and since 20 a AU, by applying Theorem I.1.10. we deduce that f E H(Br(z0)). Combining this result with (I.25), we get Br(z0) n Ux 0. But every boundary point is assumed to be T-essential and this is the contradiction. The next theorem also furnishes us with another necessary and sufficient condition in order for an open subset of a complex separable Banach space to be a domain of T-holomorphy. THEOREM I.1.14. Let E be a complex deaspable Sanach pace. Let a be a non-empty open and bounded nubnet of E. Then the boLLowing ahe equivalent: (i) U in a domain 04 r-holomoaphy. PROOF: ()+i).Since U is assumed to be a domain of T-holomorphy, then U is T-holomorphically convex by Theorem 1.1.8. Therefore, for every A E B (U), We have that the set ((n) i AU. But then Theorem I.1.11. supplies us with a mapping fe E (U) such that the condition (ii) is satisfied. (ii) + (i). Suppose that U is not a domain of T-holomorphy. Then there exists, by Theorem I.1.8. again, A E 8 (U) Such that d(AU,aU) = 0. Let iHn ntN be a sequence of points of AUI such that d(n,raU) 0 as n + m. Let f H ~(U) satisfying s u p{ f(Sn) }3 = m. Then, /Ifl| = m. But fe E (U), and fro th deiniionof U, t fllos tatwe must have IIfl = if/ A < and this is a contradiction. THEOREM I.1.15. Let E be a complex 6epahable Banachz apace. Let a be a non-empty open and bounded dubdet 06 E. Then the ofL~owing ane equivalent: (i) U id a domain oj r-holomoxphW. (ki) U La T-0emholomoapeall convex. (ii1) Evehy boundary point 06 u is essentially . (ku) Foh eVehY sequenCe (Sn)E o{ points 04 u with d(S,,BU) + 0 ad n + m, thene exists 6 ER C)(U 6U~t at /5 a p{|((5, )|} =- The above Theorem is the collection of Theorems 1.1.8., I.1.13. and I.1.14. We are giving now some more properties of domains of T-holomorphy. THEOREM I.1.16. Let E be a complex, sepatrable Senach dpace. Let U be a Mon-empty, open subset od E. Ig U is a domain og T-holomo~phy, and W id a townh semicontinuous function defined on 0 and datiofying Q(x) T[x) Sox eve~y x E U, then U is a domain o6 9-holomoaphy. PROOF: Let As 8 B(U). Suppose that A = ~U1 B i(xi ), where pi < $(x ), x E U for each i = 1,2,...,n. By assumption, $(x) & T(x) for every x E U; and therefore, pi T(x ) for each i = 1,2,...,n which implies that A E B (U). Claim (I.25) A~ c AU' Let z A ; then by the definition of the T-holomorphy hull, we can find a mapping Fe E (U) such that IF(z)l > IF A. But Fe E (U), and $ < T implies that F EH (U); and then in view of the above inequality involving the mapping F, we dedce hatz 4AUand the claim is established. Now since U is assumed to be a domain of T-holomorphy, we obtain that AU is bounded and that d(AU,aU) > 0. Then in view of (I.25), we obtain that AU is bounded and that d(AUBU) > 0 which indicates that U is a domain of 4-holomorphy. THEOREM 1.1.17. Let E be. a complex nepatrablee Beach pace. Let U and i, be two non-empty open nubnetn of E with non-empty intanction.n Suppode that U in a domain o{ T-holomohphy and that V is a domain oj i-holomo~phy. Then Un~V id a domain "5 v-holomo~phy wehee y = min(T|U n V,@I Un V). PROOF: Let As E (U n V), and suppose that A = u B (X ), Y ~i=1 Pi where x. E U nV and pi < Y(x.) for each i = 1,2,...,n. But, Y(x ) min{T(x ),0(x )} for each i = 1,2,...,n, implies that pi < T(x ) and pi < $(x ) and therefore, As 8 g(U) and Ae 6(v).Claim: (I.26) AnV ~ AU nA . Wie recall that A~nV =Z EUn" V: If(z)J 6 fA for every f eH (U n V)}. Let z a AU n AV. Then, we may suppose that either z e AU and z A P~, or that Z a AU and z AV. But, i both cases, since z AV, we can find some mapping f E H (V) such that If(z)l > If |A. But, f EH I(V) and y 5 4l on U nV implies that f EH ~(U n V) which combined with the above inequality gives us that z a A~nV. Hence, in either case above, (I.26) is established. Using the fact that U is a domain of T-holomorphy, we obtain: (I.27) d(At~nV,80) d(AU,aU) > 0, and the fact that V is a domain of 9-holomorphy, we obtain: (I.28) d(AU VV dA V > 0. Then (I.27) and (I.28) imply that (I.29) d(A ,BaUU u V) > 0 and then (I.29) with the aid of the property alU u aV - a(U u V) gives: (I.30) d(AY ,3(U n V) > 0. ^UnV' Finally, V being a domain of T-holomorphy implies that AU is bounded; and similarly, V being a domain of 9-holomorphy implies that AV is bounded. Then, beas o I26,AnV i bounded. Then the above result, together with (I.30), proves the Theorem. If S is a bounding set in a domain of -r-holomorphy V on a separable complex Banach space, then S is closed in E by Theorem I.1.11. Then, by a result of S. Dineen [5], S is compact. Hence, all bounding sets of a domain of T-holomorphy are compact. We note here that the fact E is a complex, separable Banach space is critical in this case. In fact, if E is not separable, there exist bounding sets in U which are not compact. An example is given in [5]. Specifically, let E = RE (the space of all bounded sequences furnished with the sup norm topology) and A = u un where un = (0,...,0,1,0,0,...), neN the I appears in the nth place. Then A is a bounding set in R,, but not compact. We are closing this chapter by stating two more theorems. The first of which is an immediate result of Theorem 1.1.17 and the second of which is due to M. Matos. THEOREM I.1.19. Let E be a complex nepahable Banach pace. Le Qn be a dinite collection 06 open dctd in E. Suppose that Vq 4 a domain uS ?q-hoLomohphy dot each v = 7,2,...,n. Let SZ denote the intellection oS all natta V. 74 R in non- empty, then it La a domain od v-holomohphyy cve/re = min{Ty| THEOREM I.1.20. Let E be a complex nepahablee Banach pace and Let V be a non-empty open dubnet od E. Then, the following (i1 V La a domain od ?-hoLomohphy. (il Fox evehy closed subset S od V, S in bounded and d(S,av) > 0 id eveAU E H (V) in bounded on S. 52. Sequences of Domains of T-Holomorphy on a Complex and Separable Banach Space In the sequence a domain U in a complex separable Banach space will be a non-empty, open, and connected subset U of E. Also, the meaning of the function Ir wherever it appears will be the same as in 5l. That is, T will denote a strictly pos- itive lower semicontinuous function defined on a domain U in a complex separable Banach space E; and for each x E U it satisfies d(x,8U)2T(x). Finally, if D is some domain in E, which is a subset of a domain U in E, by HT(D) we will denote the collection of all holomorphic functions from D into C which are bounded on every A E R (U) Which is contained in D. DEFINITION I.2.1. Let E be a complex Banach pace. Let 0 be a domain in E and Let {9 VEN be a dequence 0( dOmaind in E. We wiCl day that the sequence oj domain {9 }VEN apphoximaten u; and we will white 1mD, = U, id 6on evehy z Eu t hene existn an open neighbohhood Uz og z, dubnet oj u, which Lied in the inte~nection oj aLmost all domainn of the sequence {9 }veN' Observe that the above definition is equivalent to limDv = U if and only if z a U,then z lies inside the inter- section of finite many of the domains of the sequence {D VeVN' DEFINITION I.2.2. Let E be a complex Banach pace, U be a domain in E, and {0 \VE be a sequence o{ domainn in E. Id 6oh each ve H we have that DV cc D,, CC U; and id 1mqv = U, then we will day that the sequence {9 vEN i hnia sequence oj domainn apphoximating a dhom the innlde. DEFINITION I.2.3. Let E be a complex nepanable Banach pace, and Let UI and up2 be two non-empty open nubnets 04 E. Suppode that in addition Uqc 02. Let A be a non-emptU subnet of ul. We dedine the T-holomohphy huff od A ul~ative to U12 to be the set AT,2 wzeatl A ~ ~ ~ ~ ~ 1A = ze 6z)s "gn eveny 6 c H (0 )} and wehee r in derjined on U . DEFINITION I.2.4. Let E be a complex nepauable Banach pace, and Let 01l and U12 be two non-empty open nubnetn of E. Suppode that in addition UI c 02. Let As 8 B(U2) duch that A c 07. We wiLl nay that Ul in T-holomohphically convex relative to U2, 4 6oh evety nuch A we have (i) A1, La bounded. (LL)d(A,2au7) > 0. THEOREM I.2.5. Let E be a complex, nepatable BanacCk pace and Let U be a bounded domain in E. Suppose that {9,, E 19 a principal sequence 04 domainn in E approximating U 6hom the inside. Ig jot evehy ve E each domain D9 La r-holomoaphi- cally convex tielative to u; then u in a domain og T-h~olomohphzy. PROOF: Suppose that U is not a domain of T-holomorphy. Then according to the Theorem 1.1.8., U is not T-holomorphically convex. Therefore, we can find some Ae 8 ?(U) such that d(AU:,aU) i 0 where A = UB (xi), xi EU and pi < T(xi) for i=1 1 each i = 1,2,...,n. Let p and r be two positive real numbers satisfying 0 < p < r < m i n{T(x ) P 1. Since d(AU,3U) 0, 1sisn if z0 E aU, We can fn euneo lmnso U a {z } ,, such that limzk = z0. Since each z~ E AU, we obtain (I.31) If(z ) I 5 f |A, for each K = 1,2,...,n and fe E (U). But then according to the Theorem I.1.10., we have that (I.32) ( f a(z ) ( 1flp)A for each K = 1,2,...,n and Using the fact that z0 is the limit of the sequence {ZK} as < approaches infinity, inequality (I.32) yields (I.33) | f g(20) A IfI~ for each fe i (U). By our hypothesis, the given sequence {D }vEN is a principal sequence of domains approximating U from the inside. But we have that A c U, and therefore, we canfidsmnaul number 9 such that PA cD, for every v I . By our assumption again each domain D, is T-holomorphically convex with respect to U; and therefore, in view of the above remark we have that (I.34)d( Ap,3D ) > 0, for every v > where (PA~ denotes the Tr-holomorphy hull of (p)A relative to U. Moreover, from Definition I.2.3. we have that (I.35) f(z )| > f Apl for 0 E (D~ A m)~~ and feiH (U). Now, z0 E aU and limD~ = U imply that we can find some natural number X such that (I.36) aD, n B (Z0) for every v r X. Let 6 = max~u,X}. Then both inequalities (I.34) and (I.35) are valid simultaneously for v > 6. That is, (I.37) d[( A m,BD.) > 0 and aD. n B (z0) E 0 for j 6. But the above inequality implies that if z* is an element of (Dj Ag)im), then for sufficiently large j we will have that z* E B (20). Combining the above result with inequality (I.35), we obtain that for sufficiently large j we have (I.39) /f(z*) > If|l (p)A, for every fH ~(U). But then the above found inequality (I.39) contradicts the inequality (I.33). Hence U is T-holomorphically convex and therefore U is a domain of T-holomorphy. DEFINITION I.2.6. Let E be a complex nepahable Banach pace and let a be a domain in E. Suppose that D ,D2 and D3 axe t~hse domain in aI nati bUing D9 c 92 c D3. Suppose, Buhtheh- mote, that 6ot evehy As 8 R(U) nuch that A c DI, wve have that Al, in bounded and that d(A1,,9)>0 ewl yta the domain U La T-nemicompLete id dot eveAU 9/ven E > 0 and evehy ge E (D2) we can dlad a function Fe E (93) "UCh that IIF-61] 0 ' We are proving: THEOREM I.2.7. Let E be a. complex nepahable Banach pace. Let U be a bounded T-nemicomplete domain in E. Suppone that {9 }VEW is a principal sequence of domain approximating U dhom the inalde. Id each o9 in r-holomohphicaell convex AtlativeI 6o ,go each v E N, then U id a domain 06 T-hoLomohphy. PROOF: In view of the Theorem I.2.5., we have to show that each of the domains Dv is T-holomorphically convex relative to U. That is, if As 8 B(U) and A cDv then d(Appm,aD ) > 0 where App = {2 E D : If(Z)1 5f forI every,,, fH (U)}. Equivalently, since each DV is T-holomorphically convex rela- tive to D+1 we have that d(A I,,BD ) > 0. Hence, it is enough to show that A c A ,or that (I.40) (Dv A r~) c (Dv A l). Let 5 E (Dv A ,+) = (DV {z E D :fZ 5 f o f EH (DV+1) }). Then we can find a function f0 eH (D +) such that lf0() 0 .lga Let 6 be a positive real number such that (I.41) If(f0 0 A/f/~+6 We select a sequence of positive real numbers, {E }peN' such that E e < (6/2). peN Now we have that DV c DV c Dy+2, and since each D, is T-holomorphically convex relative to D ,and the latter is T-holomorphically convex relative to Dy+2, we have d(A ,BD ) > 0, and since U is T-semicomplete from v,v+2' v+1 Definition I.2.6., it follows that we can find a fl T H(Dy+2) such that (I.42) If0(z) fl(2)I < El for every z E D Inductively we obtain a sequence {f }KEN Satisfying: (i) f eH (D ++) for each K E N, (ii) |fp (z) fp (Z) < E q for each Kc > v, 1 2 X=p2+ p1 > p2 > K u, where the last inequality was obtained from |f (z) f h(z) = f (z) A 1() 11 2 Pp + pl 1 f~) f-1z A=p2+1 X=p2+1 Observe that the inequalities in the property (ii) above imply that the sequence {f (2)}v=0 is a Cauchy sequence, and therefore, it determines a function fe E (U). Moreover, the function f satisfies for every Z E D the following inequality (I.43) |f0(z) f(z)| < (6/2) because |f0(z) f(z)| = lim f0(z) f (2) and in view of the property (ii) above the latter equality yields (f0(z) fI) f (z) f (z)| < C E < (6/2). p=1 p=1 We claim that (I.44) |f(5) j > f ||A' We obtain from inequality (I.43) (I.45) |f (z)I jf(z)| < (6/2) for every z E D . The above inequality for z = 5 yields (I.46) lf0(5)1 |f(51) < (6/2) or (I.47) lf(5)j + (6/2) > lf0 '11 Combining the above inequality with inequality (I.41) we obtain (I.48) lf(5)1 + (6/2) > If0l A 6 From (I.43) again we obtain /f(z)| (6/2) < If0(z)l for every 2 E D or (I.49) fA ,1- (6/2) s 5 f0 A Combining (I.48) and (I.49) we obtain If(5)1 + (S/2) > Iif01 A d llA (6/2) + 6 = IIfIA + (6/2) and hence lf(t)| > | f A' Therefore, we have shown that if 5 E (D -A ) the SE (DV A;,m) and (I.40) is established. DEFINITION I.2.8. Let E be a complex nepatable Banach pace and Let U be a bounded T-semicomplete domain in E. Let 97, 9)2, and D3 be ~thhan domain such that DI c 92 c 93 c UI. We wibl say that U in a r-complete domain, i6 and only i6, 6ot eve A 8(0 wih c ad dAT3' 2 > 0, then we can dLad a domain od T-hoLuomophy RT duch that DI c nl c D2. THEOREM I.2.9. Let E be a complex nepahable Banach pace and Let a be a bounded and T-compLete domain in E. Suppose that U id approximated 6Kom the inside by a principal sequence o{ domain o{ T-holomo~phy {9 } .N Then U in a domain o{ ?-holomohphy. PROOF: Let for every 9 = 1,2,.. an ,12. M and m be two real numbers satisfying for each 9 and v: M = max~d (q)}, m =m i n~d ,(q)} geSD~ qaD~ where by d v(q) we denote the distance of the point q E aD~ from the boundary of D, and by d ,(q) we have denoted the distance of the point q E aD~ from the boundary of U. Now, from the sequence of domains (D \cEN we extract a subsquece o doains(D peNin the following manner: (i) Choose D =D. (ii) D 2 is so chosen so that M 20< m l 37 (iii) D is so chosen so that M < m and that V3 "2' 3 l1' 3 M 3,0 < m2,0 It is clear that the above constructed sequence {D } pN i a principal sequence of domains of T-holomorphy and in general the domain D for p > 1, has been chosen so that (I.50) Mp p+ p- + for every p = 2,3,..., and (I.51) M p10< m p0for every p = 2,3,.. We claim that if Ae E B(U) and A cD, then (I.52) d(A ,aD ) > 0. ~p-1l'p+1 p If not, then we can find a point z0 E aDV and a sequence of points {z }K=1 of A' such that limz = 0. But since Vp-1 Vp+1 K Ap-' + = {z E D p-:f(z)l 5 f ~, for every fe @(D p+1 )} c {Z D v : If(Z) < If A, for every fe E (D ) =A we oti htZeA p+1 Up+1 + (r) Let r = m ; then Ae 8 B(U) and clearly Lp-1' p+1 (r) A cD .Then appealing to the Theorem I.1.10., we VP+1 obtain that every mapping f E Hr(Du ) is holomorphic on Br(z ). But limz~ = z0, and therefore for every fe E (D ) K+m p+1 we have, f E N 20(Z)). But in view of (I.50) we have that r > M pp+.But D was assumed to be a domain of T-holomorphy, and the latter inequality leads to a contradiction. Hence (I.52) has been established. By assumption, U is T-complete and therefore in view of (I.52), we can find a domain of T-holomorphy Rp for each p = 2,3,..., such that Because of the above inclusion, we can infer that limR U where {0 pk2 is a principal sequence of domains of T-holomorphy which are approximating U from the inside. We now claim that each domain Rp is T-holomorphically convex relative to Op+ for every p = 2,3,... Let Ae 8 R(U) such that A c R We must show that (I.53) An' + cA because, then, since R0 is a domain of T-holomorphy, we will have that d(A _,aC ) > 0, and then (1.53) will imply that d(Anp a p+ ,3) > 0; and therefore this will establish that Rp is T-holomorphically convex relative to Rp+1' Le n +1 hnze and If(z)| r |I fjA for every fe E (Rp+1). If now Fe E (n ), then, since U is T-complete, we can find some G eH (Rp+1) such that (I.54) IG -Fil < (E/2). For z E ,p the above inequality implies (I.55) /F(z)| jG(z) < (E/2), (I.56) IF(z) < (E/2) + IG(z). On the other hand, we have that z E App+,G H( ) and hence (I.57) IG(z) r ||G A. Then combining (I.56) and (I.57) we obtain (I.58) IF~z) < (E/2) + IG A. Appealing one more time to (I.54) we obtain (I.59) 1 GlA < (E/2) + ||F |A. Finally, combining the last two inequalities, we obtain (I.60) |F(z) |< E + |F||A. But c is arbitrary and also F is an arbitrary element of H,(R ). Therefore (I.60) establishes (I.53). Hence, we have exhibited a principal sequence of domains {0 p>1~ approximating U from the inside; and moreover, each domain np is T-holomorphically convex relative to Rp+1 for every p = 2,3,... Then, the above conclusion together with Theorems I.2.5. and I.2.7. completes the proof. 93. Sequences of Certain Domains On a Locally Convex Hausdorff Space Let E be a locally convex Hausdorff space over C, and let cs(E) denote the set of all continuous seminorms on E. For cl E CS(E), x E E and r > 0 the "a-ball" about x of radius r is defined to be the set Ba(x) = {y E E: t(x-y) < r). The "a-boundary distance" d : U + [0,m] for an open non-empty subset U of E, is defined for all x E U by dU(x) = sup~r > 0: B (x) c U). For a subset K of U we put dU(K) = inf~dU(x): x E K). Another distance function 8U: UxE + [0,m] is given for all pairs (x,a) E UxE by 6U(x'a) = sup~r > 0: x + Aa e U for all ACEC With jX| < r). DEFINITION I.3.1. Let E be lz LocaL~g convex Harundon66 pace. An open non-empty nubse-t a oa E is naid to be pneudoconvex Let P(U) denote the set of all plurisubharmonic functions on U; let H(U) denote the vector space of all holomorphic functions on U; and let W(E) denote the space of all con- tinuous polynomials from E into C. For Q c P(U) and K c U, we define as the "Q-convex hull" of K to be the set Kg defined by KQ = Cx E U: v(x) < supy(y), v E Q, y E K}. For A c H(U) and K c U we define as the "A-holomorphy convex hull" of K to be the set KA defined by KA = {x E U: If(x)l < f ~, f e A). DEFINITION I.3.2. Let E be a Locally convex Haundcag{ npace. An open non-empty nubnet U od E is ed~Led holomotphicaL@y convex id KH(u) La pnecompact in 0 {oh evehg compact and non-empty subset K od u. A subset K of U will be called precompact here, if it is relatively compact; and moreover, if there exists a s cs(E) such that dU(K) > 0. We note here that a holomorphically convex open set U cE is pseudoconvex. The converse is true for E = Cn, for E = C(IN) [9], for CA [1], and for certain Banach spaces E which are separable and have basis [18]. It is an open question whether the converse holds in general. DEFINITION I.3.3. Let E be a Localig convex Haundohdd pace. An open non-empty dubnet a od E La naid to be polynomidl~y convex i6 KlE) in paecompael in al 6oA all compare and non- empig nubdetn K od U. DEFINITION I.3.4. Let E be a Localig convex Haundo/rdd npace. An open non-empty nubset U 04 E in naid to be Range id v(E) La denne in H(u) With suspect to the compact open topology. Then a will be ca~lld ainitely Range (Redpectively diniteff polynomially convex) ii; do/ eve/y finite dimensional vectotr adanpace F 04 E, a n F 4 Range (Acapectively polynomially convex) in F. THEOREM I.3.5. Let E be a Locally convex Hausardog pace. Let a be a non-empty open subset od E apphoximated dtom the innide by a painelpal sequence od domain { }~nenN I each od the domains 9 La Range, 6o is UI. PROOF: Suppose that U is not Runge. Then according to the Theorem I.3.4., we can find a function f E H(U), a compact subset K of U such that (I.61) If(x) p(x)l > E for every pe E (E) and x E K where E is some given positive real number. Now for each x E K, we select an open set Dx of the given sequence Dn which contains x. Then the collection of all such selected domains DX forms an open cover for K. But K being compact admits a finite subcover, say {Dx3 m=1Let j be the maximum of the numbers xl,x2,,..,xm Then since the given sequence is principal, we obtain that the domain D. covers K. But f E H(U), so its restriction F on D. is holomorphic there; and then in view of (I.61), we obtain that Dj is not Runge and this is absurd. THEOREM I.3.6. Let E be a locally convex Hausdoh66 space. Le~t a be a non-empty open and connected nubne~t U 04 E. Suppose that {9 REN id a dequence 06 pseudoconvex domain in E apphoximating U 6hom the inside. I6 Dn c D ,l aot evehy ne W then U in pbeudoconvex. PROOF: Consider the distance functions 6D : DnxE + [0, ], for each n E N, and the distance function 6 : UxE +t [0,ml Clearly, from the definition of 61? and the fact that the sequence of the domains {Dn ne is nested, it follows that if x E U, then there exists a natural number nx such that for all n r nx~ and a E we have 6D (x,a) < SD (x,a), iD (x,a) < 6 (x,a) n n+1 n or by taking logarithms we can infer that (I.62) -log6D (x,a) r -log6n (x,a) > -log6U(x,a) n n+1 where this holds for every n r nx and (x,a) E UxE. According to the Definition I.3.1. and the fact that each domain Dn is assumed to be pseudoconvex, we obtain that each function -10g6D is plurisubharmonic on DnxE for each n E N. Hence, the restrictions of the functions -logSD (x,a) on every complex line of UxE, in view of (I.62), converge to the function -log6U(x,a). But then, [81, -logd (x,a) is a plurisubharmonic function on UxE; and therefore, U is pseudoconvex. A locally convex Hausdorff space is said to have the approximation phopeaty if for every compact subset K of E, every a E cs(E) and every E > 0 there exists a continuous linear map T from E into E such that dim T(E) < m and a(x-T(x)) < E for all x E K. M. Schottenholer has shown that in such space E, every open subset U of E which is polynomially convex is pseudoconvex and finitely Runge and conversely [18]. In particular, THEOREM I.3.7. Let E be a CocaCLy convex Hausdohd{ pace with the app~oximation phapehty. Let a be a domain in E; then the doCLowing ahe equivalent: (i) a in pseudoconvex and dinitely Range. (iL) 0 in holomohphicalfU convex and Range. (11) a in polynomiaelf convex. THEOREM I.3.8. Let E be a LocaLLy convex Haundoh66 pace w~ik the approximation p~opehty. Let U be a domain in E, and Letf {9 }~ be a Mneted inneasing sequence o{ domain in E apphoximating U dhom the inside. Then, id each od the domain D9 in polYnomibtlY convex, then a in pneudoconvex and Range. PROOF: Since Dn is polynomially convex, we obtain from Theorem I.3.7. that each Dn is pseudoconvex and finitely 45 Runge. But Dn being finitely Runge implies Runge [18]. Hence, each Dn is pseudoconvex and Runge. According to the Theorems I.3.5. and I.3.6., the limit of the given sequence is pseudoconvex and Runge. CHAPTER II KOBAYASHI AND CARATHEODORY DISTANCES FOR COMPLEX BANA.CH MANIFOLDS 5l. Complex Analytic Banach Manifolds Over A Complex Banach Space In this section we intend to extend the notion of a complex analytic manifold to a complex analytic Banach manifold over a complex Banach space. We need some definitions. DEFINITION II.1.1. Let E and F be two complex gaeach spaced and let a be a non-empty open nubnet of E. A map d bAom u into F Za naid to be holomohphic in U iS doh evagy x0EU we can 6ind a 2neah map L E L(E,F) duch that 2m Blx)-b(X0)- L(X-X0) = X'X0 O Ix- xOI Observe that the Goursat's Theorem for Cn extends in this case of the complex Banach spaces and therefore the above definition is equivalent to the definition given in Definition 0.10. DEFINITION II.1.2. Let M be a locally connected Haundohbh pace and let E be a complex Sanach pace. Let U be a non- empty, open and connected nubnet od M. A map 4 which La a homeomohphinm (Aom U onto dome open nubnet od E in called a ecohdinate map and the pain (u,OJ Za called a ecohdinate DEFINITION II.1.3. Let M be a locafll connected Hlaundohdd pace and let E be a complex Banach pace. A complex analytic nsauctate on M id a collection F od coordinate nyntemd {(ua' a): a E A), A n0me index net, nat 4 y g the dO eOWing thee phopehtien: (il M = u Ut" acA -1 [LL) The mapn 060a : a(Ua nu U) B(Ua n U) ar~e biholomosrphic got all paler a,B E A. (iii) The collection La maximne with huspeCt to phopehty (iil; that LL, id (II,0) id a ecotdinate nyntem dUch that tog and m,00 ne holomo~pkie wvheneveh they Rae defined 6ot aff a E ten lu,) 6 F. DEFINITION II.1.4. A complex analytic Banach manifold ove/L a complex BeMach pace La a pala (M,F) connisting od a locally connected Haundogg6 pace M together with a complex analytic nttactate F. In the sequel a complex analytic Banach manifold (M,F) will be denoted by simply M. It is clear that if U is a non-empty open subset of a complex analytic Banach manifold M, then U itself is a complex analytic Banach manifold with complex analytic structure FU given by FU = {(Ua n U,$a Ua n U): (Ua' a) E F}. DEFINITION II.1.5. Let Mt be a complex analytic Banach manifold and let a be a non-empty open nubnet o6 M. A mapping 6 dhom u into C is naid to be holomohphic in a id the mapping d04-1i a holomohphic mapping got eve49 coordinate map $ on M. In thin cane we wlaite d E H(U,C). DEFINITION II.1.6. Let M and W be two complex analytic Banach manigoldn. A continuous mapping d Brom M into W in naid to be holomohphic id god La an holomoxphic mapping on d( {domain o{ g) son evehy g E H(W,C). In thin cane we waite 5 E H(M,N). We note here that the above definition does not require the two manifolds M and N to be manifolds over the same Banach space. We are now proving a Theorem similar to the "analytic continuation Theorem" for complex analytic Banach manifolds. THEOREM II.1.7. Let M and N be two complex analytic Banack manidoldn oveA the complex Bevach npaced E and F usnpectiveLy. Suppose that M in connected and that d and g ane two holomosphic mappiays SAom M into W. Ig thene existn some non-empty and open subset U o{ M nuch that d = g on U,then 6 =- g on M. PROOF: Let S = {x E M: f = g on some neighborhood of x in M). By hypothesis and the definition of the set S it follows that S is both an open and non-empty subset of M. We claim that the set S is closed too. Since f = g on S and since both f and g [II.1.6.] are continuous we have that f = g on S. Let z eE and let y = f(z) = g(2). Let Wk be a neighborhood of y in N such that the coordi- nate map wkc is an homeomorphism onto some open subset P of F. Let Qm0 be a neighborhood of z in M such that the coordinate map qm0 is an homeomorphism onto some open subset P1 of E, -1 -1 (II.1) QmkE -1 k n Qmo n f1 (k where here by? ,we denote the subset of Wk where both f-1\ k -1 and g (Wk) are defined. Since by itsdefinitiaao~m contains z we obtain that the setQkisnnepyndorvrits clear that is open. Define two maps tl and t2 on qm0(S n Qmk) by -1 -1 tl =wkofoqgm and t2 wkopqgm From the definitions of S and Qmk it follows that we can find a be q m(S n Q k) and then because of (II.1) we obtain tl(b) = t2(b). Since b is an arbitrary element we obtain (II.2) tl = 2 on q 0(S n Omk) But clearly qm0(S n Qmk) c q 0 Qmk) and then in view of the (II.2) and of the Theorem 0.14., we infer that t1 = 2 on qm m), r wkfog = wkgog on m ( ).But both mappings wk and q 0 are homeomorphisms and therefore the later equality in view of that fact yields (II.3) f = g on Qm But (II.3) implies that the set Qmk is a subset of the set S. But from the construction of the set Qmk we have that z E Qmk and hence we have that z e S. Therefore we have shown that Sc S and hence we obtain that the set S is closed. Finally, since M is assumed to be connected and since we have shown that S is both open and closed we obtain S = M and the Theorem is proved. 92. The Kobaya~shi Pseudodistance O~n a Complex Analytic Banach Manifold In the sequel by a we will denote the open unit disk in the complex plane and by Ba its boundary. If M is a complex analytic Banach manifold by Hl(n,M), we will denote the set of all holomorphic mappings from n into M. We will denote by a the PoincarB-Bergman distance on the unit disk a. Recall that p is defined for every pair of points z and z' of A by tanh(4p(z,z')) = /z z'l I zz' DEFINITION II.2.1. Let M be a complex analytic manifold. Let x and il be two points od M. Chzoode n + I points p0'O', p2" '**n o06 M, duch that p0 = x and pn = y. Choose 2n points (11.4) di(ai) = pil and ~i(b~) = P, Foh each choice od -the above dets oO points a~~ {q1, { j {P } =0 and mappingn Cd 1 ,l thun made, eondeth u (II.5) C p(a ,b .] We denote by d (x,g) the in imum, taken ovetr af po66ible choices og the points a { _,Ci2 },{Q} ,{p j and mappings {Q}2= datiagging (II.4), og the numbers obtained in (II.5). The function obtained from the above definition has the following properties. THEOREM II.2.2. Let M be a complex analy-tic Banack manidold. The SLunCtion d : MXMl R dedined by (x,y) + d l(x,y) is a pneudodintance on M. PROOF: Consider two points x and y in M. If x = y, we may choose points pO and pl such that pO = x and pl = y. Let f E H(aM) and points al'b1 E n such that f(al) PO and f(b1) 1.p Since p0 = 1, we can take al = bl. But then p(a ,b ) = 0 for that choice of al'b1 0 '1 and f. Since dM is always greater than or equal to zero and since d (x,x) is the infimum of the numbers (II.5) of the Definition II.2.1., that infimum in this case must be equal to zero. Therefore, we have shown that for each x in M we have d (x,x) = 0. The triangle inequality is satisfied trivially and hence the Theorem is proved. THEOREM II.2.3. Let M be a complex analytic Banach manidold. The, function d : MxM - R dedined bW (x,y) +t d (x,g) La con- ~tinuoun. PROOF: Let r be any positive real number. It suffices to show that the sets A = ((x,y): d (x,y) < r) B = ((x,y): d (x,y) > r) are open in lxul. We first prove that A is open. Let (a,B) be any point of A. Then d (a,B) = q < r. Take p = M(r-q) and consider the open set U in MxM defined by U = B (a) x B (B). If (x,y) E U, then we have that dM(x,y) E d (x,a) + dM(a,B) + d (O,y) < p+q+r = q+2p = r. The last inequality implies that (ad,) E Uc A; and therefore, A is open. Now, let (a,B) be any point of B. Then we have that d (a,B) > r. Take p = Jr(q-r) and consider the open set V in MxM defined by V = B (a) x B (a). If (x,y) E V then we have that d (a,a) Ed a(a,x) + d (x,y) + d (y,B). The above inequality implies that d (x,y) 2 d (a,B) d (a,x) d (B,y) > q-p-p = q-2p = r. The above inequality implies that (a,B) E V c B; and there- fore, the set B is open. The pseudodistance dr defined in the Definition II.2.1. on a complex analytic Banach manifold is called the Kobayashi pseudodistance on M. This pseudodistance has the property, as we will see below, that it is the largest pseudodistance defined on M for which every holomorphic mapping from D into M is distance decreasing. Moreover, if M and N are two com- plex analytic Banach manifolds furnished with the Kobayashi pseudodistances dM and dN respectively, then every biholomor- phic mapping from M onto N is an isometry. THEOREM II.2.4. Let M and W be two complex analytic Banach manifoldn. Ig d in a hoLomohpkie mapping dhom M into W, then 6ot each pala x,g EM we have PROOF: It is enough to observe that each choice of points {a =1{b =1 FPi =0 and holomorphic mappings {f @1 ad for defining d yields a choice of points {a @n=1'{b @n=1' {f(pi =0, and holomorphic mappings {fo } i=1 ede o generating d . THEOREM II.2.5. Let M and N be two complex analytic Banach manidoldn. Ig d in a bikofomohphiC mapping 6hom M onto W, d~(xy] = dN(d(x),d(y)). The proof of the above theorem follows immediately from the Theorem II.2.4. We are proving THEOREMI II.2.6. Le~t Mi be a complex analy~tie Banach manidoled and Let d' be anU pneudodintance on iM such that Bo/r evetry Then we have that dq(x,y) d'(x,y) goh x,y E M. PROOF: Let {a ) =,{b ~r1 ,{p =0 and {f @2=1 be as in Definition II.2.1. Then n n d'(x,y)~ C E 'p-1i d'-(f (ai ),f (b 0 i=1 i=1 i=1 But, the infimum of the right hand side of the above inequal- ity taken over all possible choices of the points {a }ni=1' {b~ @= 0admapns{ = is by definition equal to d (x,y). Hence, the above inequality proves the Theorem. THEOREMII .2.7. Let M and N be two complex analytic Banach manidoldo. Then dot evety paih x,g E M and doh eve~y paik (II.6) d ~(x,Y) + d (XL',g) r dhlxN((X~X')),(YI')1 z max(d (x,y),d,(x',y')). PROOF: Define a mapping f from Ml into MxN by f(x) = (x,x'), and a mapping g from NJ into MxN by g(x') = (y,x'). Theorem II.2.4. implies that both mappings f and g are distance-de- creasing, and hence we have that d (x,y) + d (xI'y') d~xN~f(x),f.(y)) + d~xN(g(x'),g~y')) = = d~xN((xIx'),r(yrx')) + dglxM((yrx') ,(y,y')) > d~xN((xrx'),r(yIy')). From the above inequalities we obtain (II.7) d (x,y) + d (x',y') > dgxN((x,x'),(y,y')). Define a new mapping K from MxN into M by the rule K(x,y) = x, and a mapping k from MxN into N by the rule k(x,y) = y. Appealing to the Theorem II.2.4;., once again we obtain (II.8) d~xN((x'y),(x',y')) d (K(x,y),K(x',y')) = dM(x,x') and (II.9) dgxy((x,y),(x',y')) d (k(x,y),k(x',y')) = d (y,y'). Finally, combining inequalities (II.7), (II.8) and (II.9) we obtain (II.6). The inequality d~xN((x,y),(x',y')) > max(dM(x,y),d (x',y')) can actually become an equality. In particular, we obtain such an example by taking M = N = a. Also the equality holds if a is substituted by Am = axax...xa. h ateult also serves as an example to the fact that the Kobayashi pseudodistance does not coincide with the Bergman metric on Am unless m = 1. DEFINITION II.2.8. Let M be a complex analytic Banach mani- goLd and Let X be a connected and Loca&Ly pathwine connected topological pace. We wift nay that X in a covering manifold od M, id and ond& id, thhe4 in a continuous mapping n dRom X onto M with the p~opedty that each point y oj M hao a Meigh- bohhood V whose invehne image undeh x in a disjoint union od open oetn in X each homeomotrphie with V undeh v. THEOREM II.2.9. Let M be a complex analy~tic Banach maniSold and Leet X be. a covering manigold oA M with covering projection v.Let p,q E M and x,y E X be nuch that O(x) = p and x(y) = q. Then dplp,q) = i n 5 {d [%,y): My) = q}. y EX PROOF: Appealing to the Theorem II.2.4., we obtain d (p,q) < i n f d (x,y). yEX Assuming that strict inequality holds, we can find some positive real number E such that (II.10) d (p,q) + E < i n f d (x,y). yeX But by the definition of d it follows that there exist points al,a2,...,ak,bl,b2,...,bk of the unit disk a, points pO l1".' k of M, and holomorphic mappings fl f2""'f k a into M such that P = fl(al) f2(bl) =2(a2 '"'" k-1(bk-l) =k(ak) Ik(bk)= and d (p,q) + E > p(a ,b ). i=1 Now, we can lift the holomorphic mappings flf2'"'"k to holomorphic mappings F ,F2,...,Fk of a into X [19] in such a way that x = Fl(al) F (b ) = Fi+1(ai+1) for each i = 1,2,...,k-1 ieF = f. for each i = 1,2,...,k. By letting y = Fk(bk), then Ti(y) = q and dX(x,y) < E p(a ,bi ' i=1 Hence, dX(x,y) < d ?(p,q) + E; and then the above found inequality contradicts (II.10). It is not known whether the infimum is attained for some y in X even for the case of the complex manifolds. 53. The Caratheodory Pseudodistance On a Complex Analytic Banach Manifold In the sequel by H(M) we will denote the set of all holomorphic mappings from a complex analytic Banach manifold M into the unit disk D of the complex plane. Also, by p we will denote, as previously, the Poincard-Bergman metric on a. DEFINITION II.3.1. Let M be a complex analytic Banach mani- tjold and Let x and y be two points 06 M. We will denote by eq(XIY) the heal numbah dedined an the naphemum o{ the numbusi p~dlx),6(g)) taken with asnpect to the SamiLy od the mapping belonging to H(M). In otheh wohnd, we define c,(x,y) = n u p {plg(x),Siyg)). It is easy to see that THEOREM II.3.2. Let M be a complex analy~tic Banach manidold. The function eg: MxM +t R dedined by (GUIl cglx,y) is con- tinuoun and a pneudodistance on M. The above defined pseudodistance is called the Caratheodory pseudodistance on M. From what it follows, we can see that the Caratheodory pseudodistance shares many properties with the Kobayashi pseudodistance, and in particular, the dg is greater than or equal to cM. Moreover, the Caratheodory pseudodistance may also be considered as a generalization of.. the Poincare-Bergman metric for a. We return now to the properties of c . THEORIEM~ II.3.3. Let Ml be a complex anafytic Banach manidold. Fox evehy paiA 06 points x and y oj M we have PROOF: Choose points p0'91'"'"'P nof M, points al'a2,...,an' bl'b2,...,b of a, and mappings fl f2'"'" no ,)sc that for each i = 1,2,...,n we have fi(a ) = pi-1 and fi(bi) Pi' Let F be a holomorphic mapping from MI into n. Then the mappings Fofi are holomorphic mappings from a into a for each i = 1,2,...,n. Schwarz's lemma implies that all these mappings are distance decreasing with respect to the Poincare- Bergman metric, and hence, for each i = 1,2,...,n,we have p(a ,b ) p (Fof (a ),Fof (b 0 . Therefore, n n E p(a ,b ) Z p(Fof (a ),Fof (bi)) P(Forl(al 'Fogn(bn) i=1 i=1 = p(F(x),F(y)). Finally, in view of the above inequality, we obtain d (x,y) = inf C p~a ,b ) '- sup p(F(x),F(y)) = cM(x,y). i=1 The proofs of the following Theorems are similar to the ones in the previous section and therefore will be omitted. THEOREM II.3.4. Let M and W be two complex analytic Banadh manidolds. Let d be a holomoaphic mapping 6hom Ml into N, Then 6ox aCL paihs o6 pointa x and g o M. THEOREM II.3.5. Let Ml and N be two compLex analytic Banach manidolds. Then eve~y bihotomohphic mapping d drom M onto W The following Theorem indicates that the Caratheodory pseudodistance cM is the smallest pseudodistance defined on a complex analytic Banach manifold for which every holomorphic mapping f from M into a is distance decreasing. THEOREM II.3.6. Let M be a complex analytic Banach manidold. Let d be any pneudodistance defined on M nuch that 5ot evehy paih od pointa x and y 04 M we have Soh evehy holomoxphie mapping B dhom M into a. Then doh eve^U paik of points x and y o{ M we have THEOREM II.3.7. Let M and N be two analy~tic Banaclz manngocdn. We are closing this section with the remark that in the case of the k-dimensional polydisk ak, we have that the Kobayashi and Carathdodory pseudodistances agree [13]. BIBLIOGRAPHY [1] Aurich, V., Chahactakization 06 Domainn od HoLomohphy Oveh an A~bit~akU P~aduct of Complex Lined, Diplomahbeit, M~achen,1973. [2] Bers, L., Inthoduction to SeventL Complex Untiabled, Lectaxeo, Couhant Inntitute od Mathematical Sceancen, W.v.U., N.Y., 1964. [3] Chen, S. S., CRathof~doxy Dintance and Convexity With Respect to Bounded Hoalomohphic FUnCtions, P~aceedingo od A.M.S., V. 39, pp. 305-307, 1973. [4] Coeurd, G., Analytic Functio~n and Manidoltdn in Indinite Dimendional Spaces, Wotan de Matembtica (52), Nohth Holland, 1977. [5] Dineen, S., Bounding Subsetn oj a Banack Space, Math. Ann. 192, 1977. [6] Dunford, N. and Schwartz, J., Lineah Opehatoth, Paht I, GenehaL Theohy, Inteuselace, W.Y., 1957. [7] Fuks, B. A., ARR~ytic Function4 of Seve~aL Complex VahiabLed, Thandkations o{ Mathematical monogtaphn, A.M.S., V. 13, 1965. [8] Fuks, B. A., Analytic FUnctiond od Seve~aL Complex Vahiablen, T~anslations 05 MathematicaL monogtlaph6, A.M.S., V. 14, 1965. [93 Gruman, L., The LeuZ P~obLem La Cehtain Inf~inite Dimen- slonaL Vectok Spacen, 121. J. Math. 18, pp. 20-26, 1974. [10] Gunning, R. C., and Rossi, H., Analytic Functionn o{ SeventL Complex Vahiablen, Phentice-Habll EngLewood Cli5dg, NJ., 7965. [ll] Josefson, B., A Countehexample to the Levi Pr~obLem, "Phcceedings on Inbinite 7imenaionaL HoLomoaphy", SphlageA Leefute Wotes 264, pp. 168-1777, 1974. [12] Kim, D. S., Canathiodohy Distance and Bounded HoLomoaphic Functions, Dake Mathematical Jou~nal, V. 47, pp. 333-338, 1974, [13] Kobayashi, S., Hype~boLie Manidolds and Holomohpkie Mapping, MaxceL Dekheh, N.Y., 7970. [141 Matos, M., Domai~n od r-Holomo~phy in a Sepahable Banach Space, Math. Ann. 195, pp. 273-277, 1972. [15] Nachbin, L., HoLomokphic Functioon& Domainn ag Holomoaphy, Local Plop~ehtien, Wohth HoLland, 1970. [16] Nachbin, L., Conce/ning Spacen o{ Holomoapkie Mappingn, Seminar Lectaten Rutgets Unive~nity, NJ~., 1970. [17] Narashimhan, R., SevehaL CompLex Vahiables, Univehnity og Chicago Passn, Chicago, 1977. [18] Schottenhloher, M., PoLynomial Apptoximationn on Compact Seto, Inginite 9Zmennlonal HoLomohphy and Applicationn, Noath Ho~Land, 1977. [19] Warner, W. F., Foundationns o Didgehentiable ManigoLdn and Lie Ghaups, Scott and Fotresman and Co., Glenuiew, ILL., 1977. [20] Iniinite Dimenalonae Holomonphy and Appli-catto~ns, Itehnational Sympoalum, Notas d Matemitica, oahth Holland, 1977. BIOGRAPHICAL SKETCH Panos E. Livadas was born on February 22, 1944, in Athens, Greece, to Evangelos and Mary Livadas. He attended private schools until he received his high school diploma from "Parthenon" high school of Athens. He attended Aristotle University of Thessaloniki in Greece where he graduated Magna Cum Laude from the school of Arts and Sciences and received his Bachelor of Science in mathematics with a minor in physics in February of 1970. He began his graduate studies in March of 1970 at Georgia Southern College, in Stateboro, Georgia; and he received his Master of Science degree in mathematics in August of 1971. In December of 1971, he married the former Debra Anne Waters; and in September of 1972, he entered the University of Florida where he had been awarded a teaching assistantship and would further his studies toward the Ph.D. degree. He was introduced to the Theory of Several Complex Variables by Dr. D. S. Kim who served as his advisor until the year 1976 when Dr. Kim's association with the University of Florida ended. Since that time his new advisor has been Dr. S. Chen. The past two years the author was employed by the University of North Florida where he served the first year as an instructor and the second as instructor and academic advisor during which time Dr. S. Chen was on sabattical. 63 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Su-Shing Chen, Chairman Associate Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Douglas/ nzer Associati Professor of M~athema tic s I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Vasile Popov Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Louis Block Associate Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope anduaiy as a dissertation for the degree of Doctor of Ph osophy. Professor of Electrical Engineering This dissertation was submitted to the Graduate Faculty of the Department of Mathematics in the College of Liberal Arts and Sciences and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 1980 Dean, Graduate School |

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REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID E5I4QH1C3_MJ4ZAW INGEST_TIME 2017-07-20T21:29:56Z PACKAGE UF00099379_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES PAGE 1 DOMAINS OF T-HOLOMORPHY ON A BANACK SPACE BY PANOS E. LIVADAS A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1980 PAGE 2 Copyright 1980 by Panos E. Livadas PAGE 3 ACKNOWLE DGEMENT The author is deeply grateful to his advisor, Dr. Su Shing Chen, who gave generously of his time, and made many helpful suggestions. His guidance and patience are much appreciated. Thanks are also due to Dr. Dong S. Kim who was responsible for generating his interest in the area of Several Complex Variables and to Dr. William Caldwell for his support and understanding. The author would like, also, to thank the many others in the Department of Mathematics at the University of Florida who contributed in their several ways to the completion of this dissertation. 111 PAGE 4 TABLE OF CONTENTS Page ACKNOWLEDGEMENTS iii ABSTPACT v INTRODUCTION 1 PRELIMINARIES . 4 CHAPTER I. SEQUENCES OF DOMAINS OF T-HOLOMORPHY IN BANACH SPACES 11 Â§1. Domains of x-Holomorphy in a Complex Separable Banach Space 11 Â§2. Sequences of Domains of x-Holomorphy On a Complex and Separable Banach Space.... 28 Â§3. Sequences of Certain Domains on a Locally Convex Hausdorff Space 40 II. KOBAYASHI AND CARATHEODORY DISTANCES FOR COMPLEX BANACH MANIFOLDS 46 Â§1. Complex Analytic Banach Manifolds Over a Complex Banach Space 46 Â§2. The Kobayashi Pseudodistance On a Complex Analytic Banach Manifold 50 Â§3. The Caratheodory Pseudodistance On a Complex Analytic Banach Manifold 57 BIBLIOGRAPHY 61 BIOGRAPHICAL SKETCH 6 3 PAGE 5 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DOMAINS OF X-H0L0MORPHY ON A BANACH SPACE By Panos E. Livadas December 1980 Chairman: Dr. Su-Shing Chen Major Department: Mathematics Let E be a complex separable Banach space, U be a nonempty open subset of E, x be a strictly positive lower semicontinuous function on U such that d(x,9U) > t(x) for every x e U. Let 8 (U) denote the collection of all finite unions of closed balls B (x) with center x e U and p < x (x) and let H _(U) be the algebra of the complex holomorphic functions in U which are bounded in every set of 8 (U) furnished with the Frechet topology of the uniform convergence over the elements of B T (U) . The main results are: if A e B (U) , then A is bounded, if and only if, its x-holomorphy hull is bounded; if A e B (U) n _ and A = u B (x.) and z n is some element of the T-holomorphy i=l p i hull of A, then, every f e H (U) is holomorphic on B (z ) and for every p < r, [| f|L . . < || f[|, . , where r = m i n p l "0 j {p> A i=l,2,...,n {t(x.) p.}, < p. < x(x.), x. e U; if U is bounded the following are equivalent: PAGE 6 (i) U is a domain of x-holomorphy . (ii) U is x-holoniorphically convex, (iii) Every boundary point of U is x-essential. (iv) For every sequence {Â£ } N of points of U with d(Â£ ,3U) + as n * Â°Â°, there exists f e H (U) such that S U p{|f (? ) |} = =o. n > Â°Â° If U is bounded and is approximated from the inside by a principal sequence of domains {D } Â„, then if each domain D n is x-holomorphically convex relative to U, then U is a domain of T-holomorphy; if in addition U is T-semicomplete and if each of the domains D is T-holomorphically convex relative to D , , , then U is domain of T-holomorphy; if each D is a n+1' tr j i n domain of T-holomorphy and U is T-complete, then U is a domain of T-holomorphy. Suppose now that E is a locally convex Hausdorff space and an open subset U of E is approximated from the inside by a principal sequence of domains {D } Â„. If each of the domains D is pseudoconvex, so is U; if each D is polynomially convex n c n L and if E has the approximation property, then U is pseudoconvex and Runge . Finally it is shown that if M is a complex analytic Banach mainifold, then the Kobayashi pseudodistance is the largest for which every holomorphic mapping from the unit disk of the complex plane into a complex analytic Banach manifold is distancedecreasing while the Caratheodory pseudodistance is the smallest pseudodistance for which every holomorphic mapping from the complex analytic manifold to the unit disk of the complex plane is distance-decreasing. vi PAGE 7 INTRODUCTION The study of holomorphic mappings defined on Banach spaces has received considerable attention in recent years. This dissertation contains new results relating to certain kinds of holomorphy on a complex Banach space. The notion of a domain of holomorphy arises naturally with the study of holomorphic functions on a non-empty open subset of C . In particular, an open subset U of C n is said to be a domain of holomorphy, if and only if, there exists a holomorphic function defined on it which cannot be extended analytically beyond any point of its boundary 9U. It is known that in this case U is a domain of holomorphy, if and only if, U is holomorphically convex ([ 2], [10], [15], [17]). The situation is different on complex Banach spaces. As a matter of fact, if U is a holomorphically convex domain in a complex Banach space, then it need not be a domain of holomorphy [11]. Moreover, the situation is different when one moves from one complex Banach space to another. The main reason is the behavior of the bounding sets on a complex Banach space. A closed bounded subset, A, of a complex Banach space, E, is said to be bounding if every complex valued holomorphic function on E is bounded on A. For certain complex Banach spaces, in particular, for separable or reflexive spaces, PAGE 8 the bounding subsets are precisely the compact subsets, while there are examples of non-compact bounding subsets of other complex Banach spaces ([4], [5]). In Sections 1 and 2 of Chapter I of this dissertation, we primarily consider a separable complex Banach space E and a kind of holomorphy, the x-holomorphy , which is due to M. Matos. In particular, M. Matos has proved (see Preliminaries and Section 1.1 for the corresponding notations and definitions) ; THEOREM (MATOS). Let E be a complex i>epafiable Banach Apace and let U be a non-empty open tablet ofa E. Thin, the faollowLna ane equivalent: (a) U A~i> a domain ofa i-holomoftphy . (b) Fo/i evety A e B ill), Aj. li> bounded and d(A u ,3U) > 0. (c) Ihene li> fa -in H (U) 6uch that It i-6 impoAA-ible to faJLnd two open connected AubAet-b U-. and LiÂ„ o fa E A,at PAGE 9 Section 1.2 is concerned with the problem of convergence of a principal sequence of domains of i-holomorphy . (See Section 1.2 for the corresponding notations and definitions.) We are showing that: THEOREM 1.2.9. Let E be a complzx &epan.able. Banach Apace, and let U be a bounded and T-compZete. domain In E. Suppose. that U i.h approximated fatiom the PAGE 10 PRELIMINARIES In this paper the notation and terminology used, unless otherwise stated, is that of Nachbin [16]. For the sake of completeness, however, we recall certain theorems and definitions needed in the sequence. Let E and F be two complex Banach spaces. If z e E by B (z) we denote the open ball with center z and of radius r while by B (z) we denote the closed ball of center z and of radius r. We reserve the letters R and C to denote the set of real and complex numbers respectively throughout this paper. Similarly, we reserve the letter N to denote the set of natural numbers. THEOREM 0.1. Let L m (E,F) = {T: T is a continuous m-LlnQ.dK. map faJiom E Into F }. Then L (E,F) is a Banach space with nespect to thz pointwise vectot operations and noim defined by I ' 1 X -I , Xn , . Â• Â• , X^ J || \T\\ = s u p {.on. II x-ll * and II II ll II x, r m x t x XeE ii j ii ii m ii i = 1 , 2 , . . . ,m . DEFINITION 0.2. Let LÂ™[E,F) denote the closed vector subspace 0^ L (E,F) o (5 all m-linean maps T: E m -* F which an.e s ymmetnic. PAGE 11 DEFINITION 0.3. Let A e L m (E,F). V z.{lnz Its s ymme.tn,lzatlo n A 4 e lJ(E,F] by A [x u x 2 , .. . ,x ] = (.I/mi] Z A(x. , x, . . . , x . ] u)ken.e the. s ummatlo n Is taken 0v2.fi tke ml premutations Note that || A || ^ || a|| and moreover that the map A -* A is a continuous projection from L (E,F) onto L (E,F) . In the sequence we write L m (E) and L m (E) for L m (E,C) and L m (E,C) respectively. Finally, if x e E and A e L m (E,F) we write Ax to denote A(x,x,...,x) and Ax to denote A. DEFINITION 0.4. A continuous mkomoge.ne.ouA polynomial V {n.om E Into f Is a mapping P {on. wklck tke.ne. Is a map A e L (E,F) suck tkat P(x] = Ax {on, even.y x e E. We uiilte. P = A to denote tkat P con.n.esponds to A tkat way. THEOREM 0.5. I { P m ( E , F ) denote* tke. set o{ all continuous m-komoge.neous polynomials {n.om E Into F tken, P (E,F) become.* a Eanack space wltk nespzct to tke. polntwlse. vecton. opzn.atlons and nonm defined by ||P || .6up J|P(*)| | x*0 || x || Observe that if m = then P (E,F) is just the set of all constant maps from E into F. THEOREM 0.6. Tke map A * A {/torn L^(E,F) onto P m (E,F) Is a vecton. Isomonpklsm and komcomonpklsm. Mon.e.oven. PAGE 12 A|| < || All < (m m /m!) || A| and [m /m! ) i.& thz bzi>t univ zn.A al constant. DEFINITION 0.7. A continuous polynomial P fan.om E Into F PAGE 13 It is easy to verify that the power series of the Definition 0.8. is convergent, if and only if, the sequence CII P II 1/m > m is bounded. 11 m" meN DEFINITION 0.10. Lzt E and F b PAGE 14 THEOREM 0.11. (Cauchy integral) Let i e H{U, F) , z e (J, x e (J, and A > be iucfe ifiat [7-A)z + Ax e (J ^oa. every A e C, w^-tfr | A | < /l. T/ien 7 tf( (7-A)z + Ax) rf{x) = / dX 2-nl \\\Â»JL A7 THEOREM 0.12. Let $ e H(U,F], z e U, x e E, and H > be iacfi ^ha-t z + Ax e U Â£oa. every A e C, wu.;tn |A| < a.. Then ^oa every m e M u {0} we have (m!)" J d m ^(z)(x) = [It\1)] i (^(z + Ax)A" (m+n )dA. THEOREM 0.13. (Cauchy inequalities) Let E and F be Â£wo complex Banach ApaceA and let U be a non-empty open &u.bi>et o j5 E. Suppose Â£kaÂ£ (J e H(U,F) and tfiat ^oa z e U there li> iome r > tuck that S (z) li> contained In U. Then ^or each m e N u {0} we have Wjpd. m iiz)\\ * -L4 a p || i(x)\\ r ii n Some of the properties of the holomorphic mappings on C can be extended on the holomorphic mappings on a complex Banach space. We are proving here that the principle of analytic continuation is valid on complex Banach spaces. THEOREM 0.14. Let E and F be two complex Banack t>pacei> and AuppoAe that fi t-i> an holomorphic mapping farom E tnto F. I jj (J x.4 equal to zero on &ome non-empty and open AubAet ii o Â£ E, then we have $ = on E. PAGE 15 PROOF: Let S = {z e E: d m f(z) = for all m e N u {0}}. Clearly S is non-empty. Moreover for fixed m e N the set S = {z e E: d f (z) = 0} is closed because of the continuity of d f and then the set S is closed being the intersection of all closed sets S for m e N. Now, let a Â€ S. Since f is m assumed to be holomorphic on E and since a e S then, we can find some neighborhood V of a in E such that the Taylor's series expansion CO I (ml)" 1 d m f(a) (z-a) m m=0 converges to f(z) for every z e V according to the definition 0.10. But we have that d f (a) = 0, because a e S, for every m e N. Hence we obtain that f (z) = for every z e V and therefore V c S and we can conclude that S is open. But because of connectedness we have that E = S and then f = on E. COROLLARY 0.15. Lzt E and F be two complex. Ba.na.ch ApaczA . Lzt fa e H(E,F) and-AuppoAZ that thznz zx-lt>ti> 4ome potent a e E 6uch that d ft [a] = ion. zvzuy m e H u {0}. Then i = on E. COROLLARY 0.16. Let E and F be two complex Banach i>pacz& . lzt i,g e H(E,F) and Auppo-bz that i and g agKzz on i>omz opzn and nonempty AubAzt o I E. Thzn i = g on E . DEFINITION 0.17. Lzt U bz a non-zmpty and opzn AubAzt o & a compLzx Banach i>pacz E. A mapping i in.om U -into anothzn. PAGE 16 10 complex Banach *pace F I* *ald to be finitely holomorphlc on U Ifi the ma.ppj.ng Â£\U n S: U n S + F ii holomorphlc ^or every finite dlmen*lonal vector *ub*pace S o ^ E. THEOREM 0.18. (Nachbin) let E and F be -two complex Banach *pace* and let U be a non-empty open *ub*et o Â£ E. A mapping I ^rom U Into F I* ltni.te.lij holomorphlc on U, Ifi and only l{, ^or every pair oÂ£ point* a and b o jj E the. mapping u: {k e C: a + bfe e 11} -* F de^lntd by a[k) = ^[a + bk) I* holomorphlc. THEOREM 0.19. (Nachbin) let E and F be two complex Banach *pace* and let U be a non-empty and open *ub*et o ^ E. A mapping fa ^rom U Into F it holomorphlc, l^ and only ifi, it I* finitely holomorphlc and either tt I* contlnuou* , on. el*e It *atl*{ i le* the equivalent condition* Â•Â• [l] (5 map* every compact *ub*et o {, U onto a bounded *ub*et o {, F. {11} & map* *ome neighborhood In U o ^ every compact *ub*et o (} U onto a bounded *ub*et o ^ F. We are closing this section by giving a new definition. DEFINITION 0.20. Let E be a complex Banach *pace and let V and U be two open and non-empty *ub*et* o ^ E. I { V I* a *ub*et o{ U we will *ay that V I* relatively compact In U II the clo*ure o & V I* contained In U and I* compact. In thl* ca*e we write V cc u. PAGE 17 CHAPTER I SEQUENCES OF DOMAINS OF T-HOLOMORPHY IN BANACH SPACES Â§1. Domains of x-Holomorphy in a Complex Separable Banach Space Let E be a complex separable Banach space and let U be a non-empty open subset of E. DEFINITION 1. 1.1. A mapping (J Â£n.om U Into C Li -i>a-id to be. koLomon.ph.-iQ. -ifi thzxz e.x-i-t>ti> a ie.qu.znce. oÂ£ compLzx cont-inuou* n-h.omoge.ne.ou-!> poLynom.la.Li> -in E: [nl] d j$(x) ion. e.ve.n.y n e H v{0} -buck that oo b[x+k) = I [n\)~ ] cl n i(x) [h] n = convzn.gz& un-i{on.mLy ^on. k -in a. ne.-igkbon.kood o^ ze.n.0 -in E. Let 3U denote the boundary of U and let x be a strictly positive lower semicontinuous function of U such that t (x) ^ d(x,3U) for every x in U. Let B (U) denote the collection of all finite unions of closed balls B (x) with center x e U and p < x (x) . Let H (u) denote thealgebra of the complex holomorphic functions in U which are bounded in every set of 8 (U) furnished with the Fr^chet topology of the uniform convergence over the elements of 8 (U) [14], Observe that the union of H (U) , for all T, is the algebra H (U) of all the complex valued holomorphic mappings in U. 11 PAGE 18 12 DEFINITION 1.1.2. Let E be a complex 6Q.paA.able. Banach Apace. Let U be a non-empty open AubAet o fi E, then U Ia Aald to be. a domain o fi T-holomofiphy, PAGE 19 13 THEOREM 1.1.5. Lit E be a complex i,zpafiabl2. Banack -6po.ee and Izt U be a non-empty open i,Uibi>e.t o { E. Then {on. zvefiy bounded Â•6ub4eÂ£ A Of) U we have: A,, <= cF(.A). PROOF: Let xÂ„ \ CO (A) . Then by Banach Separation Theorem [6] there exists a continuous linear functional f on E and real numbers c and 5 , 5 > such that (1.1) Re{f(co(A))} < c 6 < c < Re{f(x Q )} Consider the set P = {A Â€ C: Re(A) < || (f(x)) |^ (A) } Then in view of the inequality (1.1) we obtain (1.2) f (co(A) ) c p and (1.3) f(x Q ) k P Â• Using the facts that f is continuous and linear and in view of (1.2) and (1.3) we can find A n e C and r > such that (1.4) f(co(A)) c B r (A Q ) where B (A n ) denotes the open ball with center AÂ„ and of radius r, and r can be chosen so that (1.5) f (x Q ) k B r (A Q ) . We define a mapping <$> from E into C by the rule $ (x) = f (x) AÂ». Then in view of (1.5) PAGE 20 14 (1.6) |f(x Q ) A Q | > r and in view of (1.4) (1.7) r > s u_p {|f (x) X | }. xeco(A) Hence in view of (1.6) and (1.7) we obtain (x Q ) | = |f (x Q ) X-l > r > s u_p { | f (x) A Q | } XeCO(A) = s u_p { | <|)(X) | } >sup{| PAGE 21 15 THEOREM 1.1.6. Let E be a complex AQ.pa.fiab I e Banach Apace. Let U be a non-empty open AubAet o Â£ E. Suppose, that A c LI. Then A iM bounded -l^ and only L{ A., La bounded. DEFINITION 1.1.7. Let E be a complex Aepafiable Banach Apace and U be a non-empty open and bounded AubAet o {, E. We will Aay that U La 1-holomoh.phLcally convex LI and only L{ fcofi evety A e B^iil) we have rf[A^,3U) > 0. Then in the case of bounded open sets, Matos ' theorem which is stated in the Introduction becomes: THEOREM 1.1.8. (Matos) Let U be a non-empty open and bounded AubAet ol a Aepafiable complex Banach Apace E. Then U La a domain o{ x-holomon.phy L{ and only L& U Ia tkoto mo n.p hLc ally convex. LEMMA 1.1.9. Let U be a non-empty and open AubAet o{ a complex Aepafiable Banach Apace E. Let A e 8 (U) and AuppoAe that thefie T exlAtA a conAtant M > 0, and a function (J e H (U) Auch that |jj(x]| < M fiosi eve^iy x e A. Then the Aame eAtLmate extendA to Ay. That LA, | jj ( z } | < M ioft eveny z e aJ.. PROOF: Since |f(x) | < M for every x e A, we obtain ||f|L ^ M. But, since z e A^, we have |f (z) | < || f || ; and in view of the last inequality |f (z) | < M for every z e a1. We are employing the following notation: Let U be a non-empty open subset of a separable complex Banach space E. Let A e 8 T (U) and suppose that PAGE 22 16 A = u B n (x.) 1=1 1 where x. e U and < p. < t(x.) for every i = l,2,...,n. Let r = m i n (x(x.) p.} and let < q < r. Then we denote i=l, . . . ,n by q A the set defined by (q) A = u B , (x. ) . p.+q l i=l K i THEOREM 1. 1.10. Le.t E be. a. complzx bepatiable. Banack Apace.. Let U be. a non-empty opzn i>ubi>et o{ E. Let A e 8 (U) and n buppoiz that A = u 8 (x.) and Itt K = m I n {t(x-)-p-} ' ' -1P--C -in A, A, A.= 7 W A. A.= 7 , 2, . . . , n Let z Q be an ele.me.nt o & A... Thtn, a.1 & e H (U), tke.n & ti> kolomoh.ph.te. on 8 (z fl ). And motie.o\)e.h., ^on. evesiy p < n. and {,on. evtfiy fa & H (U), we have. PROOF: We first prove that if f e H (U) , then f is holomorphic on B r (z Q ) . Let < q < r and let n > such that q n > . Consider the set (q-n) A. Observe that (q_n) A c (r) A and since every (r) f e H (U) is bounded on A, then every f e H (U) is bounded on (q ~ } A. Let (1.9) || f|| . . = M f (n) for every f e H T (U) . Now if z is any element of A, we clearly have that B (q-n) (z) c A and therefore in view of (1.9) PAGE 23 17 (1. 10) || flip r,^ s M-(n) for every f Â« H(D). B (q-nr ' Now, since H (U) c H(U), every f e H (U) belongs to H(U), and therefore we can apply the Cauchy inequalities, of Theorem 0.13, to f and obtain: (1. 11) | (jl)" 1 d j f (z) | PAGE 24 (1.15) f(z) = I (jl) 1 d j f(z Q ) (z-z ) : for every z e B (z Q ) . j=0 But < p < q n and therefore in view of (1.15) we obtain CO ,00 , |f(z)| < I | (j'.) _1 d j f (z ) (z-z ) j | < S M. (n) (|z-z |/(q-n)) D j=0 u j=o 00 < M (n) 2 (p/(q-n)) 3 j-0 and therefore (1.16) |f(z)| < M f (n) Â— ~rfor every z e B (z ) . q-n Define a function H*klt (Â«Â„) " <* I U n U) (U , |