Title: Domains of Greek letter tau-holomorphy on a Banach space
CITATION PDF VIEWER THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00099379/00001
 Material Information
Title: Domains of Greek letter tau-holomorphy on a Banach space
Physical Description: vi, 63 leaves : ; 28 cm.
Language: English
Creator: Livadas, Panos E ( Panos Evange ), 1944-
Publication Date: 1980
Copyright Date: 1980
 Subjects
Subject: Banach spaces   ( lcsh )
Domains of holomorphy   ( lcsh )
Mathematics thesis Ph. D
Dissertations, Academic -- Mathematics -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis (Ph. D.)--University of Florida, 1980.
Bibliography: Bibliography: leaves 61-62.
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by Panos E. Livadas.
 Record Information
Bibliographic ID: UF00099379
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000100089
oclc - 07288598
notis - AAL5549

Downloads

This item has the following downloads:

PDF ( 2 MBs ) ( PDF )


Full Text















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




University of Florida Home Page
© 2004 - 2010 University of Florida George A. Smathers Libraries.
All rights reserved.

Acceptable Use, Copyright, and Disclaimer Statement
Last updated October 10, 2010 - - mvs