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 THOLOMORPHY IN
BANACH SPACES................................ 11
5l. Domains of THolomorphy in a Complex
Separable Banach Space.................. 11
52. Sequences of Domains of THolomorphy 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 THOLOMORPHY ON A BANACH SPACE
By
Panos E. Livadas
December 1980
Chairman: Dr. SuShing 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 Tholomorphy hull is bounded; if As E 6(U)
and A = v E (x ) and z0 is some element of the Tholomorphy
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 rholomorphy.
(ii) U is Tholomorphically convex.
(iii) Every boundary point of U is Tessential.
(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 Tholomorphically convex relative to U, then U is a domain
of Tholomorphy; if in addition U is rsemicomplete and if
each of the domains Dn is Tholomorphically convex relative
tDn+1, then U is domain of rholomorphy; if each Dn is a
domain of rholomorphy and U is Tcomplete, then U is a domain
of Tholomorphy.
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 distancedecreasing.
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 nonempty 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 noncompact 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 Tholomorphy, 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 nonempty open nubnet og E. Then, the 6ollowing
(a) U La a domain od Tholomohphy.
(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 nonempty 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 rholomorphy. (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 TcompLete domain in E. Suppose
04 domains o{ Tholomohphy {9 vevN. Then U in a domain od
Tholomohphy.
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
distancedecreasing 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 distancedecreasing. 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 meineah
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 mlineat 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 mhomogeneous 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
mhomogeneoun 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 {zx)m = P (zx)
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. (CauchyHadamard) 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 nonempty 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(zX) 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 (1Xlz + Ax cu U oh evehg
X E C, with /Af < h. Then
7 d((17X~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 nonempty 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 nonempty 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 nonempty. 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) (za]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) andaUppose 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 nonemptU subset oa E. Then 6 = g on E.
DEFINITION 0.17. Let a be a nonempty 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 nonempty 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 nonempty 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 nonempty 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 THOLOMORPHY IN BANACH SPACES
5l. Domains of THolomorphy in a
Complex Separable Banach Space
Let E be a complex separable Banach space and let U be a
nonempty 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 nonempty open subset og E, then U is said to be a
domain o{ rholomohphy, id and on~y id, it La impobbibLe to
6lad two nonempty, 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 nonempty open subdet of E. Ig A is a Mon
empty subset og06 wIe define the rholomohphy 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 nonempty 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 nonempty 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 nonempty 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 nonempty open and bounded nubnet od E. We will
naU that U in Tholomoaphically 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 nonempty open and bounded
dubnet od a nepa/rable complex Banac pace E. Then U 4 a
domain ag Tholomo/rphy id and only Lid id ThoLomaxphically
convex.
LEMM~A I.1.9. Let U be a nonempty 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
nonempty 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 nonempty 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) (qn) 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) (zz0 "
j=0
Because of inequalities (10.4), we obtain:
(I.14) j(ji)1 d f(z0) (zz0~l Mf(1) (zz0l(n )
But, (I.14) indicates that the series (I.13) is convergent
for z2z0 < 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) (zz0)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) (zz0~j f(n) (1zz0l(~~
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 
qnl
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 nonempty 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 Tholomorphy 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
n1
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
n1
If (S ) = n + ) f (5n) and Ifn lS < 2n 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
n1 cm
i=1 j=n+1
n1 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 If ((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 > n1.
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 nonempty open nubnet o{ E. A point z o{ the
boundary o6 u is a venbential 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 rholomorphy.
THEOREM 1.1.13. Let E be a nepahable complex Banach pace.
Let a be a nonempty open and bounded nubnet oj E. Then U
id a domain od Tholomoaphy, iS and onLy i6, evehy boundary
PROOF: Observe that by the definition of a Tressential
boundary point it follows that even if U is not bounded, then
U being a domain of Tholomorphy, then every boundary point of
U is Tessential.
Conversely, suppose that every boundary point of U is
Tessential, but U is not a domain of Trholomorphy. Then by
the result of Theorem I.1.8., U is not Tholomorphically
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 Tessential 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 Tholomorphy.
THEOREM I.1.14. Let E be a complex deaspable Sanach pace.
Let a be a nonempty open and bounded nubnet of E. Then the
boLLowing ahe equivalent:
(i) U in a domain 04 rholomoaphy.
PROOF: ()+i).Since U is assumed to be a domain of
Tholomorphy, then U is Tholomorphically 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 Tholomorphy.
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 nonempty open and bounded dubdet 06 E. Then the
ofL~owing ane equivalent:
(i) U id a domain oj rholomoxphW.
(ki) U La T0emholomoapeall 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
Tholomorphy.
THEOREM I.1.16. Let E be a complex, sepatrable Senach dpace.
Let U be a Monempty, open subset od E. Ig U is a domain og
Tholomo~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 9holomoaphy.
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 Tholomorphy 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 Tholomorphy,
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 4holomorphy.
THEOREM 1.1.17. Let E be. a complex nepatrablee Beach pace.
Let U and i, be two nonempty open nubnetn of E with nonempty
intanction.n Suppode that U in a domain o{ Tholomohphy and
that V is a domain oj iholomo~phy. Then Un~V id a domain
"5 vholomo~phy wehee y = min(TU 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 Tholomorphy, we
obtain:
(I.27) d(At~nV,80) d(AU,aU) > 0,
and the fact that V is a domain of 9holomorphy, 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 Tholomorphy implies that
AU is bounded; and similarly, V being a domain of 9holomorphy
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 rholomorphy 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 Tholomorphy
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 ?qhoLomohphy 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 vholomohphyy cve/re = min{Ty
THEOREM I.1.20. Let E be a complex nepahablee Banach pace and
Let V be a nonempty 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 THolomorphy on a
Complex and Separable Banach Space
In the sequence a domain U in a complex separable Banach
space will be a nonempty, 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 nonempty open nubnets 04 E. Suppode
that in addition Uqc 02. Let A be a nonemptU subnet of ul.
We dedine the Tholomohphy 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 nonempty 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 Tholomohphically 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 rholomoaphi
cally convex tielative to u; then u in a domain og Th~olomohphzy.
PROOF: Suppose that U is not a domain of Tholomorphy. Then
according to the Theorem 1.1.8., U is not Tholomorphically
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 Tholomorphically
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 Trholomorphy 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*) > Ifl
(p)A, for every fH ~(U).
But then the above found inequality (I.39) contradicts the
inequality (I.33). Hence U is Tholomorphically convex and
therefore U is a domain of Tholomorphy.
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 TnemicompLete id dot eveAU 9/ven E > 0 and
evehy ge E (D2) we can dlad a function Fe E (93) "UCh that
IIF61] 0 '
We are proving:
THEOREM I.2.7. Let E be a. complex nepahable Banach pace.
Let U be a bounded Tnemicomplete domain in E. Suppone that
{9 }VEW is a principal sequence of domain approximating U
dhom the inalde. Id each o9 in rholomohphicaell convex
AtlativeI 6o ,go each v E N, then U id a domain 06
ThoLomohphy.
PROOF: In view of the Theorem I.2.5., we have to show that
each of the domains Dv is Tholomorphically 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 Tholomorphically 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
Tholomorphically convex relative to D ,and the latter is
Tholomorphically convex relative to Dy+2, we have
d(A ,BD ) > 0, and since U is Tsemicomplete 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~) f1z
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 Tsemicomplete 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 rcomplete domain, i6 and only i6, 6ot
eve A 8(0 wih c ad dAT3' 2 > 0, then we can
dLad a domain od ThoLuomophy 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 TcompLete domain in E. Suppose that
U id approximated 6Kom the inside by a principal sequence o{
domain o{ Tholomo~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 Tholomorphy 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.
~p1l'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
Vp1 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
Lp1' 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
Tholomorphy, and the latter inequality leads to a contradiction.
Hence (I.52) has been established.
By assumption, U is Tcomplete and therefore in view of
(I.52), we can find a domain of Tholomorphy 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
Tholomorphy which are approximating U from the inside.
We now claim that each domain Rp is Tholomorphically
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 Tholomorphy, 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 Tholomorphically 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
Tcomplete, 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 + FA.
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 Tholomorphically 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 "aball" about x of radius
r is defined to be the set
Ba(x) = {y E E: t(xy) < r).
The "aboundary distance" d : U + [0,m] for an open nonempty
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 nonempty nubset 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 "Qconvex 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 "Aholomorphy
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 nonempty nubnet U od E is ed~Led holomotphicaL@y
convex id KH(u) La pnecompact in 0 {oh evehg compact and
nonempty 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 nonempty 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 nonempty 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 nonempty 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 nonempty 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(xT(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 nonempty 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(XX0) =
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
nonempty 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 nonempty open nubnet o6 M. A mapping 6 dhom u
into C is naid to be holomohphic in a id the mapping d041i
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 nonempty 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 nonempty 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 f1\ 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 PoincarBBergman 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 analytic 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(rq) 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(qr) 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) > qpp = q2p = 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 'p1i 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 distancede
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 '"'" k1(bkl) =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,...,k1
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 PoincardBergman 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 PoincareBergman 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 ) = pi1 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 kdimensional 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. 305307, 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. 2026, 1974.
[10] Gunning, R. C., and Rossi, H., Analytic Functionn o{
SeventL Complex Vahiablen, PhenticeHabll EngLewood
Cli5dg, NJ., 7965.
[ll] Josefson, B., A Countehexample to the Levi Pr~obLem,
"Phcceedings on Inbinite 7imenaionaL HoLomoaphy",
SphlageA Leefute Wotes 264, pp. 1681777, 1974.
[12] Kim, D. S., Canathiodohy Distance and Bounded HoLomoaphic
Functions, Dake Mathematical Jou~nal, V. 47, pp. 333338,
1974,
[13] Kobayashi, S., Hype~boLie Manidolds and Holomohpkie
Mapping, MaxceL Dekheh, N.Y., 7970.
[141 Matos, M., Domai~n od rHolomo~phy in a Sepahable
Banach Space, Math. Ann. 195, pp. 273277, 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
Applicatto~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.
SuShing 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
