Introduction to holomorphy

INTRODUCTION TO HOLOMORPHY

NORTH-HOLIAND MATHEMATICS STUDIES Notas de Matematica (98)

Editor: Leopoldo Nachbin Centro Brasiteiro de Pesquisas Fisicas, Rio de Janeiro and University of Rochester

NORTH-HOLLAND -AMSTERDAM

0

NEW YORK

0

OXFORD

106

INTRODUCTION TO HOLOMORPHY

Jorge Albert0 BARROSO UniversidadeFederal do Rio de Janeiro Rio de Janeiro Brasil

1985

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD

Elsevier Science Publishers B.V., 1985

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0444 87666 9

Publishers: ELSEVIER SCIENCE PUBLISHERS B.V. P.O. Box 1991 1000 BZ Amsterdam The Netherlands

Sole distributors for the U.S.A.and Canada: ELESEVIER SCIENCE PUBLISHING COMPANY, INC. 52 Van d e r b i It Avenue NewYork, N.Y. 10017 U.S.A.

Library of Congress Cataloging in Publication Data

Barroio, dorge Alberto. Tntroduction to holomorphy. (North-Holland asthmatic studies ; 106) (Notas dc M t m t i C a ; 98) Bibliography: p. Includes index. 1. Named linear epaccs. 2. Damins of holoaorphy. I . Title. I T . Series. 111. Series: Notas dc materdtica (hterdnm, Netherlmdr) ; 98. QAl.N86 no.98 rQA322.21 510 (I r515.7'31 84-22283 ISBN 0-444-87666-9

PRINTED IN THE NETHERLANDS

T o Anna Amalia

with love.

This page intentionally left blank

FOREWORD

This book presents,

011

the one hand,

a set

of

basic

properties of holomorphic mappings between complex normed spaces and between complex locally convex spaces.

These

properties

have already achieved an almost definitive form and should

be

known to all those interested in the study of infinite dimen-

sional Holomorphy and its applications.

On the other hand, for

reasons of personal taste but also (and especially) because of the importance of the matter, some

incursions have been

made

into the study of the topological properties of the spaces

of

holomorphic mappings between spaces of infinite dimension.

An

attempt is then made to show some of the several topologies that can naturally be considered in these spaces. There has been

no concern to establish priorities

relatively few authors are quoted in the text. facts should be pointed out here.

The study of

and

Some historical differential

mapping and holomorphic mapping between spaces of infinite dimension apparently begins with V. Volterra [142],

[143], [ 1441,

[ 1451 , [ 1461 around .L887. Then D. Hilbert, in his work

[ 561 ,

outlines a theory of holomorphic mappings in an infinity of variables, in which the concept of polynomial in such a context already clearly appears.

At the same time (1909), M. Fr6chet

publishes his first work [40] on the abstract theory nomials in an infinity of variables.

vii

of poly-

Later on, the development

viii

F 0R E WO RD

,[ 423)

of the theory of normed spaces led Fre'chet to defirie([41]

real polynomial in a more general situation.

Mention must be

made of R. Ggteauxls works [ 431, [ 441 , in which he proposes definition for complex polynomials.

In the period

mid-6Os, several other names are worthy of note.

a

until the

A historical

vision of the development of the notions of polynomial and holomorphic mapping in this period can be obtained through works of A.E. Taylor, [ 1353, [136].

the

The mid-60s witnessed

a

rekindling of interest and a quickening of the development the study of questions that originate in the notion of

of

holo-

morphic mapping between complex normed spaces and between complex locally convex spaces.

Holo-

Thus, infinite dimensional

morphy appears as a theory rich in fascinating problems

and

rich in applications to other branches of Mathematics andMathematical Physics.

Once again without any desire to establish

priorities, we would like to quote the names of H. Cartan

P. Lelong in France, for their influence and work, as well the French team composed of G. Coeure',J.-F.

Ramis and J.P. V i g d .

as

Colombeau, A. Douady,

M. H e r d , A . Hirschowitz, P. K r g e , P. Mazet, P. Noverraz, Raboin, J.P.

and

P.

Still, we should especial-

ly like to stress the important role played in the development o f this theory by Leopoldo Nachbin and his doctoral students in

Brazil and the United States: Baldino, J.A.

Aragona, R.M. Aron,

Barrroso, P.D. Berner, P.J. Boland, S.B.

S. Dineen, C.P. Pombo, R.L.

A.J.

Gupta, G . I .

Soraggi, J.O.

as T. Abuabara, T.A.W.

Katz, M.C.

R.R.

Chae,

Matos, J. Mujica, D.P.

Stevenson, A.J.M.

Wanderley as well

Dwyer, J.M. Isidro, L.A.

Moraes and D.

Pisanelli, all of whom were directly influenced by him.

ix

FOREWORD

Mention should also be made of the German school, represented by K.-D.

Bierstedt, B. Kramm, R. Meise,

as

&I.

Schottenloher and D. Vogt, the Italian school as represented by E . Vesentini, and the Swedish school, as represented by C .O.

Kiselman. Let us speak a little about the contents of this book.

We begin with a study of algebraic and topological properties of m-linear mappings, m-homogeneous polynomials and power

se-

ries, and then introdlice the concept o f holomorphic mapping between complex normed spaces and complex locally convex spaces. We endorse Weierstrass's point of view, that is,that holomorphic mappings are, in a sense, locally represented Taylor series.

Several

expressions are then

by

their

derived

for

Cauchy's integral formula and for Cauchy' s inequalities; then a study is presented of the convergence of Taylor series of a holomorphic mapping.

The differences with the case of infinite

dimension are stressed, thus leading naturally to a consideration of holomorphic mappings of bounded type.

The relation-

ships are shown between the notions of holomorphic weakly holomorphic mapping and finitely (holomorphic in GGteaux's sense).

mapping,

holomorphic

mapping

We then present the infin-

ite-dimensional versions of the theorem of maximum module and uniqueness

of holomorphic

continuation.

3-bounding sets o f locally convex spaces, Josefson-Nissenzweig theorem:

"If E

In studying we apply

I((pmll = 1

in the weak topology

f o r every o(E',E)".

m E N

the

is a normed space

infinite dimension, there exists a sequence such that

the

and

'

(pm' mE

(pm

-t

0

in

as

m

of

E' -t

This theorem resolves a famous

FOREWORD

X

problem proposed by Banach, and thus enjoys an a p p l i c a t i o n i n an a r e a d i f f e r e n t from i t s i n i t i a l context. i s made of t h e p r o p e r t i e s o f topologies P a r t I and P a r t 11.

T

A d e t a i l e d study

both i n

~ T, ~ ' r,6

The spaces of thebounded type holomorphic

mappings a r e d e a l t with i n d e t a i l and i n t h e spaces we prove t h e Cartan-Thullen base space b e i n g separable,

context of such

theory i n t h e case o f t h e

t h i s bringing Part I t o a close.

P a r t I1 ends with t h e study of bornological p r o p e r t i e s of t h e spaces o f holomorphic mappings. Incomparably more could be s a i d about t h e have been l e f t out.

topics that

P a r t i c u l a r mention should be made o f the

f a c t that no p r o p e r t y of a n u c l e a r n a t u r e i s r e f e r r e d the t e x t , although q u e s t i o n s r e l a t i n g t o n u c l e a r i t y

to

in

are

be-

coming more and more i m p o r t a n t i n t h e study of Holomorphy. P r o f i t a b l e ilse of t h i s book w i l l r e q u i r e some familari t y with t h e b a s i c theorems o f Functional A n a l y s i s ,

in

c o n t e x t o f normed spaces and l o c a l l y convex spaces.

The read-

i n g o f P a r t I1 does not presume knowledge of P a r t I.

the

Theoret-

i c a l l y , i t could be s a i d t h a t t h e study of both p a r t s r e q u i r e s no previous knowledge of s e v e r a l

complex v a r i a b l e s .

however, i n the a u t h o r ' s opinion,

i s a marvellous

This, case

of

w i s h f u l thinking. T h i s work owes much t o t h e experience acquired

during

t h e courses administered a t t h e F e d e r a l U n i v e r s i t y o f R i o de J a n e i r o , a t t h e U n i v e r s i t y of Santiago de Compostela v i t a t i o n o f P r o f e s s o r J.M.

Isidro)

(on i n -

and a t t h e U n i v e r s i t y of

Valencia ( o n i n v i t a t i o n of P r o f e s s o r M .

Valdivia).

But

it

FORE WORD

xi

owes most to what was learned in the classes given by Profess o r Leopoldo Nachbin, to whom we are deeply grateful.

A special word of thanks to Dr. Raymond Ryan

of

the

University o f Galway, Ireland, f o r his English translation of a preliminary version of this book. fessors

s.

Many thanks also to Pro-

Dineen and J.P. Ansemil for their suggestions. O u r

thanks also go to the Mathematics Institute of

the

Federal

University of Rio de Janeiro f o r their financial support, and to Wilson Gdes for his excellent typing services.

Jorge Albert0 Barroso

Federal University of Rio de Janeiro August 1984

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TABLE O F CONTENTS

FOREWORD

............................................... PART I.

THE NORMED CASE

...... CHAPTER 2 . POWER S E R I E S ............................... CHAPTER 3. HOLOMORPHIC MAPPI NGS ....................... CHAPTER 4. T H E CAUCHY I NTEGRAL FORMULAS ............... CONVERGENCE OF T H E TAYLOR S E R I E S ........... CHAPTER 5. CHAPTER 6. WEAK HOLOMORPHY ............................. FINITE HOLOMORPHY AND GATEAUX HOLOMORPHY ... CHAPTER 7. T O P O L O G I E S ON S P A C E S O F HOLOMORPHIC CHAPTER 8. M A PPI NGS ................................... CHAPTER 9. U N I QUENESS OF ANALYTIC CONTINUATION ........ CHAPTER 10. T H E MAXIMUM P R I N C I P L E ...................... CHAPTER 11. HOLOMORPHIC MAPPI NGS O F BOUNDED T Y P E ....... ................... CHAPTER 1 2 . DOMAINS O F #,,-HOLOMORPHY CHAPTER 1.

CHAPTER

NOTATION AND TERNINOLOGY.

13. THE CARTAN-THULLEN O F #b-HOLOMORPHY

P A R T 11.

THEOREM

POLYNOMIALS

CHAPTER

17.

CHAPTER 18. CHAPTER 19. CHAPTER 20.

CHAPTER 2 1 . CHAPTER 22.

1

17 25 31 45 57 69 81 111

115 119 127

FOR DOMAINS

...........................

139

THE LOCALLY CONVEX CASE

1 4 . NOTATION AND M U L T I L I N E A R MAPPINGS........... CHAPTER 15. POLYNOMIALS CHAPTER 1 6 . T O P O L O G J E S ON S P A C E S O F M U L T I L I N E A R

CHAPTER

vii

................................ M A PPI NGS AND HOMOGENEOUS POLYNOMIALS ....... FORMAL POWER S E R I E S ........................ HOLOMORPHIC MAPPI NGS ....................... S E P A R A T I O N AND PASSAGE T O T H E QUOTIENT ..... #-HOLOMORPHY H-HOLOMORPHY .............. E N T I R E MAPPI NGS ........................... . SOME ELEMENTARY P R O P E R T I E S O F HOLOMORPHIC M A PPI NGS ................................... AND

xiii

155 159 167

173 177 183 185

187 191

TABLE OF CONTENTS

xiv

CHAPTER 2 3

. HOLOMORPHY.

. 25 .

................................ SETS .............................. THE ............................... ....................... LOCAL CONVERGENCE OF THE .............................. C O N T I N U I T Y AND AMPLE

BOUNDEDNESS

CHAPTER 24

BOUNDING

CHAPTER

THE CAUCHY INTEGRAL INEQUALITIES

CHAPTER 26

. THE TAYLOR

CHAPTER

. 28 . 29 . 30 . 31. 32 . 33 . 34 . 35 .

CHAPTER

36

CHAPTER CHAPTER

CHAPTER

CHAPTER CHAPTER

CHAPTER CHAPTER CHAPTER

27

.

AND

CAUCHY

REMAINDER

COMPACT AND TAYLOR S E R I E S

........................ ............... .............

THE M U L T I P L E CAUCHY I N T E G R A L AND THE CAUCHY I N E Q U A L I T I E S DIFFERENTIALLY LIMITS

STABLE

OF HOLOMORPHIC

UNIQUENESS

SPACES

MAPPINGS

OF HOLOMORPHIC

CONTINUATION

.....

........... ............... LIMITS ................................. ....................... OF ..................

HOLOMORBHY AND FINITE

HOLOMORPHY

THE MAXIMUM SEMINORE1 T H E O m M P R O J E C T I V E AND I N D U C T I V E HOLOMORPHY

195 197 209

215 221

229

233 237 241

245 249

AND

253

T O P O L O G I E S ON # ( U ; F )

263

BOUNDED S U B S E T S

273

#(U;F)

279

................................ ...........................................

AN I N D E X O F D E F I N I T I O N S

297

AUTHOR I N D E X

301

PART I THE NORMED CASE

This page intentionally left blank

CHAPTER 1

NOTATION AND TERMINOLOGY.

We denote by

IN, R

and

POLYNOMIALS

the systems of non-negative

0:

integers, real numbers and complex numbers respectively. Throughout this book, all vector spaces considered will

have

CC

as their field of scalars unless explicitly stated otherwise.

E

and

F

will denote complex normed spaces, and

empty open subset of

If 5

U

a non-

E.

is a point in a normed space and

p

a positive

real number, the open ball (respectively, the closed ball) in this space with centre B~

(respectively

(5 )

DEFINITION 1.1

Let

f

and radius

will be denoted by

p

Cp ( 5 1 ) .

...,Em

(m E N,

E1,E2,

m > 0)

be a finite

sequence of normed spaces.

...,E,;F) denotes the vector space of m-linear mappings fi Ei = El x . . . ~ Em into F, where addition and multii=1

La(E1, ~~

of

plication by scalars are defined pointwise.

s. (El,...,Em;F)

denotes the subspace of

Xa(E1,

?

...,Em;F) m

of

Ei into F. ( Ei being i=l i d In the case m = 0 we endowed with the product topology). c ontinuous m-linear mappings of

identify

S(E1,

S,(E~,

...,E,;F)

...,E,;F) with

F

with

F

as vector spaces, and

as normed spaces.

1

CHAPTER 1

2

inf{M z 0: 1IA(x1

,...,xm )I/

L

MIlx11I ...I( x , )

for all xl€E1

,...,xmEEm],

and 2)

denoting by

IIAII

the common value of the expressions

which appear in l), the mapping

is a norm. REMARK 1.1:

a)

We commit an abuse of notation, using the

same symbol to represent the norms on spaces which m a y be distinct b)

IIA(xl, c)

.

...,E,;F),

If A E S(E1,

-

,xm)ll

5

IIAll

If A E Ca(E1

it is easy to see that

.I1 Xmll

lIxlll

,...,Em;F)

for every

then

xlEE1,.

A E C(E1

..

, xf

Em

,...,Em;F)

if

and only if

...,

s UP

xl+o, d)

xdo

In the case in which

the space

C(E1,

...,E,;F),

In the case in which

IIA(X1’

YXm

Ill

11 ~111. 11 Xmll F

< =.

is a Banach space, s o too is

with the norm above. El = E2 =...=

Em = E

the space

of m-linear mappings (respectively, of continuous m-linear m a p

NOTATION AND TERMINOLOGY.

3

POLYNOMIALS

m

7

Em = G...xE

pings) of

(respectively,

order

Sa(%;F)

(m > 0 )

Sas(%;F)

the subspace of

In other words, if

m,

spaces and

Sm

is the symmetric gro-Jp of

then

In the case

m = 0,

'as (%;F)

= Cm(E;F) = F

S,(%;F)

We denote by

Ss(%;F)

=

s~("'E;F) =

F

as vector

as normed spaces.

the subspace of

S(%;F)

sisting of the continuous symmetric m-linear mappings of into

Sas(%;F)

for

Em

Ss(%;F)

is a closed subspace

S(%;F). With each

of

con-

F. Thus,

It is easy to see that

of

Em

consisting of the symmetric m-linear mappings of

F.

into

will be denoted by

x (%;F)).

We denote by

Sa(%;F)

F

into

A,

xl,

A E Sa(%;F)

which we denote by

is associated an element of As,

and call the symmetrization

defined by:

...,xm E

E.

The mapping

A

E Sa(%;F)i--.As

is linear and surjective, and is a projection of the subspace

gas (%;F);

thus

= As

(As) S

E Las(%;F) Xa(%;F)

onto

for every AEX~(~E;@

4

CHAPTER 1

By restriction we obtain a projection of

C(%;F)

onto

Cs(%;F)

s IIAll

for every

IIA,II

which is continuous since

A 5 C(%;F).

If A(x,.T.,x)’ x

m > 0, and

m E N,

m Ax ;

by

if

A E Sa(%;F),

m = 0

we denote

Axo = A

we write

for every

E E.

DEFINITION 1.2

m E N.

Let

an m-homogeneous polynomial from

A E Ca(%;F)

such that

P: E + F

A mapping

P(x) = A x

E m

F

into

is called

if there exists

for every

x E E.

When

A

P

and

A

are related in this way we write

REMARK 1.2:

If A E Ca(%;F)

restriction of

A

and

to the diagonal of

-.L = A 6 x). If Am: E + Em

,...,

Am(x)

r-7

=

lent to

P =

i,

P = A. then

Em, since

P

is the

P(x) =

is the diagonal mapping, x E

for every

E,

then

P =

is equiva-

P = AoAm.

We denote by nomials from

E

ba(%;F)

into

F.

the set of m-homogeneous polyThis set forms a vector space,

addition and multiplication by scalars being defined pointwise. DEFINITION 1 . 3

Let

m E N.

A mapping

P: E + F

continuous m-homogeneous polynomial from exists

A E S(%;F) We denote by

such that

P(%;F)

eous polynomials from

E

P(x) = Ax

E rn

into

is called a

F

for every

if there

x

E E.

the set of continuous m-homogen-

into

F.

This set forms a vector

space, addition and multiplication by scalars being defined pointwise, and is a subspace of

Pa(%;F).

NOTATION AND TERMINOLOGY.

P E r(~;F)

It is easy to see that if IIp(x)l

sup xEE,x~O

=

II xIIm

II p(x)1I

inf{N ;:: 0

5

then

sup II p(x)1I xEE,1I xll!':l

!': Nllxll

P E P(~;F)

and that the mapping

POLYNOMIALS

m

for every

~ Ilpli

x E E}
,

is a norm, where IIpli

denotes the common value of the expressions above. REMARK 1.3:

a)

In the case in which

r(~;F)

so too is the space b)

If

F

is a Banach space,

with the norm above.

P E P(~;F),

then

IIp(x)ll!': Ilpllllxll

m

for every

x E E.

c)

m = 0,

In the case

Pa(OE;F)

r(~;F)

as a vector space, and

is identified with

is identified with

F

F as a

normed space. EXAMPLE 1.1

In the case in which

geneous polynomial fro~ = aA

m

E

A E~,

for every

generally, taking

E

=

=F

E

into

F

=~,

every m-homo-

is of the form

and

~

A E~,

for every

F

where

In fact, every mapping A(A l,··· ,Am) = Al ••• Amb is some element of

REMARK 1.4:

If

F,

~.

where a is some element of

b

for all

and

P(A) =

is some element of

A E £a(m~;F)

and thus

A E £a(~;F)

More

an arbitrary normed space,

then every m-homogeneous polynomial is of the form

= bAm

P(A) =

is of the form

AI' ••• ,Am E V, P =

As

.. A

F.

where

b

takes the form

is its symmetrization,

6

CHAPTER 1 A

then

-

A = AS

and taking

.

x

1

T o see this, let

= x2 =...=

PROPOSITION 1.1 Let for

xl,

X

E.

Then

yields

(The Polarization Formula):

m E IN,

...,xm E

xm = x

,.., m E

X ~ ,

m

2

1, A E eas(%;F)

and

P =

i.

Then

E,

We omit the proof of this formula which is purely algebraic in nature. a)

PROPOSITION 1.2:

The mapping

A E .Ca(%;F)

k-i E

is linear and surjective for every

Pa(%;F) m E IN.

b) The mapping A

E SaS(%;F) 1-2 E Pa(%;F)

is an isomorphism of vector spaces for every

PROOF:

a)

m E IN.

Surjectivity of this mapping is an immediate con-

sequence of the definition of an m-homogeneous polynomial;

NOTATION AND TERMINOLOGY.

7

POLYNOMIALS

the proof of linearity is trivial. b)

This mapping is certainly linear, being the restriction

of the mappinggiven in a) to the subspace

X,,(%;F)

of

.Ca(%F) The mapping is surjective since, given by part a) there exists by Remark 1.4,

is = i

Xa(%;F)

A

= P,

and

P E

P =

such that

As E

Pa(%;F),

fi,

and

Xas(%;F).

T o see that this mapping is injective, let

A E .Cas(%;F).

By the polarization formula, 1

A(x~,...,x~) =

for all A(xl,

xl,

..., m ) x

...,xm = 0

c m 2 m! ci=fl

E E.

€

1.

.

C

m

fi ( C lX1+.

.

+& , X , )

A

Thus if

for all

xl,

A = 0, it follows that

...,xm E

E,

and hence

A = 0.

Q.E.D. PROPOSITION 1.3:

a)

The mapping

A E X(%;F)+

fi

E P(%;F)

is linear, surjective and continuous. b)

The mapping A E Xs(”E;F)W

E P(%;F)

is an isomorphism of vector spaces and a homeomorphism. Furthermore,

8

CHAPTER 1

A E Cs(%;F)

for every a)

PROOF:

and

m

E

IN.

This mapping takes its values in P(%;F)

and is

surjective by the definition of a continuous m-homogeneous polynomial; subspace

it is linear since it is the restriction to the

C(%;F)

Xa(%;F).

on

I f i l l ?:

ga(?E;F)

of

of a linear mapping defined

Continuity is a consequence of the inequality

l/All, whose verification is immediate. This mapping is certainly linear, being the restriction

b)

of the mapping given in a) to the subspace

Ss(%;F)

and is continuous for the same reason.

L(%;F),

The mapping is surjective since, given by part a) there exists know that if

A E

of

is = i

X(%;F),

A E P(%;F)

P E P(%;F),

such that

P =

i. We

(Remark 1.4), and it is easy to see that

then

As E Xs(%;F).

Injectivity is a con-

sequence of part b) o f Proposition 1.2, since this mapping is the restriction of the mapping considered there to the subspace

x,(%;F)

of

x,~(~E;F).

Finally, we prove the inequality

\(ill i

m m IIAll s m! I IiII

,

from which it follows that the given mapping is a homeomorm

phism.

We show that

IlAll 5

5I l i l l - as we have already in-

dicated, the other inequality is immediate. and

xl,.. .,xm

whence

E.

Let

By the polarization formula,

A € Xs(”E;F)

NOTATION AND TERMINOLOGY.

9

POLYNOMIALS

1s i s m

1sism

1s is m

Thus, if

/Ixl\l=...=

1lxml( = 1, then

1sism

and hence

REMARK

1.5:

The mapping

in general an isometry. which satisfies:

m,

P(%;F)

is not

In fact the smallest constant

IIAIl s C l I i l I

depending only on

iE

E XS(%;F)++

A

is

m m

-.m !

independently of

E

and

C

F,

This is shown by the follow-

ing example. EXAMPLE 1.2

Let

E

x = (x1,x2,...,xm,...) m

Let

m

be

4,

,

1

the vector space of all sequences

of complex numbers for which m

be a positive integer,

m

2

1,

and let

,...,xm1,...) , x2 = ( x2 p 22 ,...,xm,...),...,xm 2 m m m (x1,x2 ,...,xm,...) be elements of E. We define

x1 = (x1,x2 1 1 =

Am:Em

=

+

d:

CHAPTER 1

10

by :

It is easy to see that from

Em

into

6.

Am

is a symmetric m-linear mapping

We have

1

m!

IIx 1IIIIx 2I1

*..IIXrnll

Thus

Furthermore, taking

,...,xm =

..., (4

(0,

.. .. ) ,

x1 = (l,O,.

0,1,0,...),

,O,.

we have

1 2 x ,x

(1) and (2) together imply that

Now let

im(x)

x E E,

which implies that

IIA,ll

,...,x

m

=

..., ,...),

E E

0

and

1

mT.

.,.,

x = (x1,x2,

?--=--= xlx2...xm. = Am(~,...,~)

x2 = (O,l,O,

xm,...). Then

Thus f o r

x f E

we have

NOTATION AND TERMINOLOGY.

POLYNOMIALS

11

m A

'1 1

ilxl! = 1

1'

x = (m,m,...,m,O,...,O,...)

Taking and

E

Z(E;F)

F

and

112m !I

= - l G .Therefore m

Iim(x)I

Thus we have shown that

If

E E,

1 = - - -m- .

m

,.

mm

we have

IjAmll = xIIAm!l.

are vector spaces we shall denote by

the set of all mappings from

E

F.

into

This set

f o r m s a vector space, addition and multiplication by scalars

being defined pointwise.

REMARK 1.6:

Let

B

be a vector space and

a se-

C Bm the set o f mE IN the finite sums which can be formed with elements chosen from

B.

quence of subspaces of

the subspaces exist integers

We denote by

. In other words, x E m€C ml,m2, ...,mk and elements

Bm

Brn N x

,...

such that ,X E Bm m E Bmk 2 k 2 C Bm is a subspace o f B Then m€ N the algebraic sum of the subspaces xm

E Bm , 1 1 x = xm + x xm l m2 k it is referred to as m

+...+ .

-

Bm.

An algebraic sum o f subspaces

Bm

a direct algebraic sum if, whenever the

m E N,

9

m

j

implies

N,

are pairwise

xm

@ Bm , and we say that the subspaces rnE [N are linearly independent.

notation m

is called

= x =...= x = 1 m2 mk In the case of a direct algebraic sum, we employ the x = 0

distinct, the condition

= 0.

if there

Now let

E

F

and

sider the sequences their algebraic sums

C

of

B,

be normed spaces, and let us con-

pa(%;F) m€N

Bm

and

Pa(%;F)

fJ(%;F), and

C

mEIN

rn € I N ,

P(%;F)

and within

12

CHAPTER 1

the vector space

1.4.

PROPOSITION

3(E;F). The subspaccs

m E N,

Pa(%;F),

of

5(E;F)

are linearly independent. PROOF:

It suffices to prove the following statement f o r

every

m E

P = P

+ P1 +...+

for

j

m = 0

P j E pa('E;F),

if

(N:

the condition

Pm

= 0,1,...,m.

P = 0

m

m-1,

2

1,

and prove it for

P

j

= 0

For

m.

According-

and suppose that

X E a!

m

c

implies

We assume the truth of the

l y , let P . € P(jE;F), j = 0,1,..., m, J m C P j = 0. Then for all x E E and j=O

(a)

and

We s h a l l prove this by induction.

there is nothing to prove.

statement for

= 0,1,..., m ,

j

we have

m

o

pj(x) =

and

(b)

j=O

Pj(Xx) = 0 .

C j=O

Multiplying equation (a) by

XIn

and subtracting the

result from equation (b) we obtain: m-1 C (hm-hj)Pj(x) = 0

(c)

f o r all

X E

d:

and

x E E.

j=O

We now choose a value o f tion of any of the

m

equations

X E CC Xm-XJ

which

i s

not a s o l u -

= 0, j = 0,1,...,m - 1 .

Then the induction hypothesis applied to the relation m-1

C

(hm-hj)Pj

= 0

j=O j

= O,l,...,m-1,

Pm = 0.

given in (c) shows that P = 0 for j rn and then, since C P . = 0 , we also have j=O

J

Q.E.D.

COROLLARY 1.1

The subspaces

P(%;F),

m E IN,

of

3(E;F)

are linearly independent. PROOF:

This is a straightforward application of the proposi-

NOTATION AND TERMINOLOGY.

tion, using the fact that

P(%;F)

POLYNOMIALS

C

13

for every m EN.

Pa(%;F)

Q.E.D.

1.7

REMARK

In the language of algebraic s u m s and direct al-

gebraic s u m s Proposition 1.4 and its Corollary state that

and =

C 63(%;F) m€ Ui

Pa(E;F)

We denote by

Pa(%;F)

63

subspace

CB P(%;F). mEN

(respectively, P(E;F))

(respectively,

m€ N

DEFINITION 1.4 E

from

into

P(%;F)).

CB

m€N

An element o f

F,

pa(E;F)

is called a polynomial

P(E;F)

and an element of

is called a con-

tinuous polynomial from

E

into

Thus to say that

P

is a polynomial from

means that either

P = 0, or if

F.

#

P

written in a unique way in the form where

Pj E Pa(JE;F),

j

P.

m

+

P = Po and

#

into

F

can be

+...+ 0.

Pm, I n the

is called the degree o f the

P = 0 is

By convention the polynomial

assigned the degree

P1

Pm

E

P

0, that

= 0,1,..., m,

latter case the natural number polynomial

the

-1: m

PROPOSITION 1.5 pj

pa( j E ; F ) ,

If P E Pa(E;F) then

P E P(E;F)

and

P =

C j=O

Pj,

if and only if

where

P . E P(jE;F) J

for every

j

= 0,1,...,m.

The p r o o f of this proposition is similar to that of Proposition

1.4 and will be left to the reader to carry out,

as will the proof of the following:

14

CHAPTER 1

PROPOSITION 1.6

E, F

Let

(a)

If

P € Pa(E;F)

(b)

If

P E p(E;F)

and

G

be normed spaces.

Q E P,(E;G),

and

then

Q E P(F;G),

and

then

E Pa(E;G).

QoP

QoP t P(E;G).

We state without p r o o f the following:

If

llPROPOSITIONA :

m E

A € Cs(%;F),

N,

the following are

equivalent : a)

A

is continuous.

b)

A

is continuous at one point of

c)

There exists a neighborhood

such that

A

is bounded in

V

Em. Em

of the origin in

V."

This is used to prove PROPOSITION 1.7

If

P E pa(E;F)

the following are equivalent:

a)

P

is continuous.

b)

P

is continuous at one point of

c)

There exists a non-empty open subset

that

is bounded in

P

PROOF:

E. U

of

E

such

U.

The implications a)

3

b) a c) are easily verified.

We prove the implication c) a a). By c), there exists a non-empty open subset

M

and a constant 1)

I\P(X)~ Since

t: E

-t

E

2

s M

U

0

of

such that

for every

x E U.

is non-empty there exists

be the translation:

1) is equivalent to

U

t(x)

= x-x

0 )

x

0

E U.

x E E.

Let Then

E

NOTATION AND TERMINOLOGY.

llPot-'(y)/I

2)

5

M

f o r every

neighborhood of the origin in

E

polynomial from sition 1.6, that

ll~(~)ll

s M

Q =

t

is a

is a continuous

y E V.

P = Qot,

and

Q.

continuity of

P

We shall prove that

is Q

is

We begin with the following assertion:

j = O,l,...,m,

and

bounded in a subset if

Since

V

and

is a polynomial, and, by 2),

equivalent to continuity of continuous.

t(U)

( o f degree 1) we have, by P r o p o -

-1

P o t

for every -1 Q = Pot

AS

E

into

y f V =

E.

15

POLYNOMIALS

Qj

Qm

#

V

of

is bounded in

V

0,

then

Q

is

if and only

E

for every

j,

j = 0,1,..., m."

If the Q

Q j , j = O,l,...,m,

is certainly bounded in

by induction.

The case

V

are bounded in

also.

m = 0

V

then

We prove the converse

is trivial.

Assuming the

truth of the statement for all natural numbers less than

m

2

1, we consider the case

m.

m,

Then:

m

3)

Q(x)

=

c

and

Qj(X)

j=O

for all

x E E

and

1 E

C.

From 3 ) and

4) we obtain

m-1

5)

im Q(~>-Q(A~) = c

(~"-x~)Q~(X)

j=O

for all

x E E

and

1 E

C.

We choose a value for

which is not a solution of any of the equations

X E

Xm-Xj =

(c

0,

16

CHAPTER 1

j = O,l,...,m-1.

The polynomial m-1

C

(h"-XJ)Qj(x)

j=O

has degree at most

Q

hypothesis,

is bounded in

V,

is bounded in

V

V

and is bounded in

m-1,

and hence

for each fixed value of

since, by

h"Q(x)

-

With

h

1.

Q(Xx)

chosen

as indicated, it now follows from the induction hypothesis is bounded in

that j'

V

for

j = O,l,...,m-1,

and then

siricc m-1

Q,

V.

is also bounded in

This proves assertion ( * ) . a

A j E Sas(jE;F)

Let

with

= A J.

j '

for

j = O,l,...,m.

Then

7)

Q(x)

=

m C

m Qj(x)

=

j=O

Now let

W

C A.xj. J j=O

be a balanced neighborhood o f the origin in

E

such that

,--. w

for every

...,m.

j = 1,

j

+...+w c

Then if

v

...,xj) E

(xl,

j

WJ

w e have

.

C cixi E V if e i = k l , f o r every j = l,.. ,m. Using i=l the polarization formula, we conclude that A is bounded in j the neighborhood Wj of the origin in Ej, for j = 1,. ,m.

..

A.

is constant, and hence is continuous. It now follows from Proposition A above that

continuous for every

7),

Q

is continuous.

j = O,l,...,m.

Q.E.D.

A

j

is

Therefore, by relation

CHAPTER 2

POWER S E R I E S

D E F I N I T I O N 2.1

A power series from

point

5 E E

where

Am E X s ( % ; F )

E

into

m = O,l,...;

about the

x E E

is a series in the variable

for

F

of

the form

if we prefer to use

polynomials the series can be written: m

72

Pm(X-%)

m=O A

where

Pm = Am

for

sponding polynomials

m = O,l,...

.

The

Am

,

o r the corre-

P m , are often referred to as the coef-

ficients of the power series in question. D E F I N I T I O N 2.2

The radius of convergence, or the radius of

uniform convergence, of a power series about the point is the supremum of the set of numbers

0 s r .s

r,

that the series is uniformly convergent in p,

0

gp(S)

5 E

E

such for every

s p < r. A power series is said to be convergent, or uniformly

convergent, when its radius of convergence is greater than z e r o , that is, when there exists

converges uniformly in

Ep (5 )

.

p > 0

such that the series

CFIAPTER 2

18

C

F o r power series from

into

(c,

the radius of con-

vergence can be calculated by means of the Cauchy-Hadamard formula.

E

When

is a normed space, and

F

a Banach space,

we have the following: PROPOSITION 2.1 (Cauchy-Hadamard).

The radius of uniform con-

W

c ~ ~ ( x - 5 from ) m= 0 is given by

vergence o f a power series

5 E E

about the point

In this expression,

REMARK 2.1:

r

into

E

F

is taken to be 0 or

if

is infinite or zero respectively.

lim SUP lIP,l/ m-tm

PROOF:

a)

Given

LE

for which each

lim sup l\Pm]ll’m = m . m-w M(L) the set of all m E N

We considcr first the case El,

L > 0 we denote by

llPml\l/m > L .

m E M(L)

\lPm(tm)/l > Lm.

Then tm E E

choose Let

m E M(L).

Then

for every

m E M(L).

p

M(L)

is an infinite set. )Itml/ = 1,

with

= 1/L, and let

xm =

such that

5 + ptm

for each

This shows that the power series in ques-

tion does not converge uniformly in

Bp(5),

since its general

term does not converge uniformly to zero in this ball.

L

For

Since

is an arbitrary positive real number, we conclude that the

series does not converge uniformly in any ball p > 0.

b)

gp(s),

p E R,

Hence the radius of convergence o f the series is zero. Suppose now that

lim sup llPm\\l’m = 0 . m-w

POWER SERIES

Given

p E R,

number of values of

t E E,

E = -

> 0, let

p

/(tll s 1,

m,

2P

m E IN,

m,

f o r all but finitely many

For all but a finite

*

we have

x = 5

and let

19

+

pt.

l\Pmlll’m s

E

.

Let

Then

rn E IN,

and hence the given

B p ( ~ ) . Since

power series converges uniformly i n

p

is an

arbitrary positive real number, we conclude that the radius of convergence is infinite. c)

If

Finally, we consider the case

p E R

and

8 E R,

exists

0

< p < l/h,

then

0 < 8 < 1,

A < l/p,

jlPm// s (f3/p)m.

Thus if

A < e/p.

such that

all but a finite number of values of t E E,

and so there

m,

Hence, for

m E N,

we have

and

x = 5 + pt,

)/tl/s 1,

then

for all but finitely many

m,

m E N,

power series converges uniformly i n radius o f uniform convergence,

r,

and hence the given

Ep(5).

satisfies

If, on the other hand, we take and so there exists a n infinite subset

/IPmlI> ( l / ~ ) for ~ I(tm(l = 1,

tm E E ,

xm = 5

+

ptm

,

m E M.

For every

such that

we then have

Therefore the

llPm(tm)l/

r ;r l / A .

p > l/A,

M

of

m E M

then [N

> l/p,

such that

there exists

> ( l / ~ ) ~ .Taking

CHAPTER 2

20

for every ly in

m E M,

g,(g).

and so the series does not converge uniform-

r s l/A.

Thus

r = 1/A.

Therefore

Q.E.D. m

REMARK 2.2:

C Pm(x-5) from E into F m= 0 the radius of Convergence is unchanged

F o r a power series

5 E E,

about a point if the norm on

F

is replaced by an equivalent norm;

E

this radius can change if the norm on

however,

is replaced by an

equivalent one. One can not, in general, replace the polynomials

Pm

A

by the corresponding

Am,

Pm = A m ,

in the formula for the

radius of convergence given in the preceeding proposition. This is illustrated by the following example: EXAMPLE 2.1

E

Let

be the space

C1

of absolutely summable m

IIxII =

sequences of complex numbers, with the norm

x = (x1,x2 ,...) E L 1 ,

for

we define

where

Am: Em

j x j = (xl

3

F

F = a.

and let

,...,xm’j ...) E

E,

Am

j = 1

A.

=

m

,...,

m.

is m-linear and symmetric,

and, as in Example 1.2, we find that

am^^

Am

is continuous, and

m

inT9

= 0, and s o

m E

,...

by

It is easy to see that

that

For

C lxml m= 1 m = 1,2

(N,

IIAo/l = 0 .

m 2 1. For

m = 0, we take

POWER SERIES

21 A

The continuous rn-homogeneous polynomial sociated with

Am

x

= (xl,

as-

is given by

P,(x) for

Prn= Am

...,xrn,...) E

m

= rn xl...x

E

rn

rn = 1,2,...

and

.

If

rn = 0,

Pm =O. Again following Example 1.2 we find that \lPm/l= 1 f o r

.. .

rn=1,2,.

Thus

but

m

m

COROLLARY 2.1 series about

Let

E E.

=

C P,(x-g) rn=O

C

rn

Arn(x-5)

be a power

rn= 0

Then the following are equivalent:

a)

The series is uniformly convergent.

b)

The sequence

{ \lPmll'/"'I

m E N,

is bounded.

c)

The sequence

[~~Aml~l/rn}, rn E N,

is bounded.

PROOF:

,

Statement a), that the radius of convergence is

lim sup \lpml/ l/m E R, rn-w and this in turn is equivalent to the assertion that the se-

greater than zero, is equivalent to

{~lP,,,~l~m], m E N,

quence

is bounded.

Hence a) and b) are

equivalent. The equivalence of b) and c) i s a consequence of the relation

and the fact that the sequence

[(rnm/m!)l'm],

m E IN, which

CHAPTER 2

22

converges to

PROPOSITION

e,

Q.E.D.

is bounded.

2.2

Let

E P,(%;F),

pm

5 c

m E IN,

E,

p

7

0,

a

C Prn(x-5) = 0 m= 0 for every m E N.

and suppose that Pm = 0

Then

PROOF:

x E Bp(5)-

for every

We begin by proving the following:

c 'lm3 mELN

"If

is a sequence o f elements o f m

6 > 0, and

rn

X 'um = 0

C

F,

X E G,

for every

m=O

1x1

s 6,

then

urn = 0

u 0 = 0.

= 0, w c obtain

Taking uo = u1 = * " =

k-1

--

0

for

m E IN."

for every

k

We shall prove that

1.

2

Suppose that

uk = 0, and our claim follows by induction: m

Since

m

rn

6 um = 0, lirn 5 urn = 0, and hence

C In= 0

In+-

1)

m

L = sup I/Um116


p

Bp(5) c U

such that

0,

and the power

m

c

series

~ ~ ( x - 5converges ) ~ uniformly to

f(x)

in

1.

1 3 (~5

m=O

It is shown in Nachbin [ 9 0 ] that when f: U + F ,

space and

the set of points in

is a Banach

F

U

at which

f

is holomorphic is open.

In the prcceeding definitions of a holomorphic mapping, whether in an open subset or at a point, one can dispense with the condition that f

that

Am,

m E IN,

is continuous on

In fact, in this case there

U.

BD({),

exists an open ball

be continuous, if we assume

for

5 E

m

bounded in Pm

f.

Therefore, by Proposition 1 . 7 we have that

Bp(S).

(and hence also

is continuous for every

Am)

We note too that the set the norms o n DEFINITION 3 . 2 m

C A,(x-Z) m=O

where

and

E

is bound-

m

C Am(x-5) = C Pm(x-5) converges m= 0 m=0 This implies that each P m , m f N, is

ed and the power series uniformly to

f

where

U,

m

F

remains unchanged if

are replaced by equivalent n o r m s .

f E #(U;F),

Let

g(U;F)

m E N.

5 E U, and let

m

m

=

C

Pm(x-s)

be the Taylor series of

f

at

5 ,

m=O *

Pm = A m , m E N.

Then

a)

dmf(S)

= m!A,

b)

Z"f(5)

= m!im = m!Pm E P ( % ; F )

E 2 ("E;F)

and

are called the differential of order

m

of

f

at

g.

In a)

the differential is viewed as a continuous symmetric m-linear

ISOLOMORPHIC M A P P I N G S

27

mapping, and i n b ) a s a c o n t i n u o u s m-homogeneous polynomial.

;mf(5 )

Note t h a t

6 ' 1

i s an a b b r e v i a t e d n o t a t i o n f o r

The T a y l o r s e r i e s o f

at

f

s

d

f(5).

can n o w be w r r i t e n a s

or m

f(x)

1

c

2

m= 0

m T

m,f,5

C

m

k=0 of f

degree

If f

1

(x) =

f

o r order

zkf(S)(x-4;)

5

at

E

m

we can c o n s i d e r t h e d i f f e r e n t i a l s of

N,

d m f : 4;

a s mappings d e f i n e d o n

E

5 E

;t"f:

i s t h e T a y l o r polynomial o f

a

E 3J(U;F), m,

m! 2 " f ( S ) ( X - S ) .

UH

d m f ( 4 ; )E Ss("E;F)

U H

;zmf(g)

U:

E P(%;F).

W e can go a s t e p f u r t h e r and d e f i n e t h e d i f f e r e n t i a l

m

o p e r a t o r s of o r d e r

on

S((U;F):

dm: f E 3 J ( U ; F ) w dmf E S ( U ; S s ( % ; F ) )

;m:

f

E

m > 0,

DEFINITION 3.3

Let

ed s p a c e s , and

A E Sa(E1

and

x1

1)

E

E1,...,xh

If

h = m,

E

If

h

< m,

Eh,

A(xl,

value o f t h e mapping 2)

zmf E

#(U;F)I+

A

A(xl,

let

,...,

Z(U;P(%;F)).

El,

Em;F).

A(xl,

...,Em For

...,

xh)

...,

xh) = A(xl,

a t t h e m-tuple

...,

xh)

and h

E

N,

F

be norm0 :E h i m ,

i s defined a s follows:

...,

xm) E F

(X

l,...,~m).

i s t h e mapping f r o m

i s the

CHAPTER 3

28

X

...x

Em

F

into

given by

It is easy to see that a)

x...x

...,xh)

A(x~, Em

b)

last

F,

into

If El =...=

m-h

Em = E

A(xl,. h = 0

we set

A E Sa(%;F)

Axo

for every

= A.

h E N,

0

,..., r E hl hr Ax1 ... x r h

DEFINITION f

N,

3.4

1

L

is symmetric in the

A

Ii

x1 =.

..=

E N,

1

xh = x E E L

h

5

m,

and for m E IN,

Axh E La(m-%;F)

then

we write

is defined

h s m.

?;

1

hl

+...+

r

4

h

r

L

m,

= h

5

x1 m,

,...,xr E

E

and

we define

to be the mapping

A mapping

is holomorphic in

E.

f: E + F

#(E;F)

space of entire mappings of operations.

or, more gen-

With this convention, if

If A E ga(%;F), hl

and

x E E,

and

and

X

A(x l,...,~h)E gas (m-hE;F).

Eh = E

..,xh) = Axh

for every

Em = E

Eh+l

is continuous.

A

A E eas(%;F)

and

variables, then

If El =...=

mapping of

which is continuous if

=...=

Eh+l

erally, if

is a n (m-Ii)-linear

E

is said t o be entire if

denotes the complex vector

into

F,

with the usual

29

HOLOMORPHIC MAPPINGS

PROPOSITION

3.1

cisely, if

m E IN,

if

k E

[N,

0

if

k E

[N,

k

P(E;F) A

E .@,(%;F),

k < m,

: 8

P =

i

and

More pre-

E E,

then

and

> m. P(%;F)

PROOF:

It suffices to show that

rn E

[N.

This is obvious in the cases

rn

2,

let

2

#(E;F).

is a subset of

A E Es(%;F)

and

P =

c #(E;F)

m = 0

i.

for every m = 1.

and

Then for every

For

3 E

E

we have m C

P(x) =

m

( k ) Agmmk(x-5)

k

k=O

x E E.

for every

5

f

of

E, P

P

This shows that

is holomorphic at every

and that the radius of convergence of the Taylor series at every

5 E E

is infinite, since this series is fiP(x)

nite and hence converges uniformly to every of

P

in

g,({)

for

> 0. The expressions for the differentials

p 5 R,

p

at any

g E E

follow directly from the relation above.

Q.E.D. REMARK 3 . 2 :

In the case in which

f E #(E;F),

the radius of convergence of the Taylor series of

5

has finite dimension and

5 E E.

If the dimension of

f

at

E

is infinite, the radius of convergence of the Taylor series

of

is infinite for every

E

f E #(E;F)

is either finite at every point of

finite at every point of

E.

E

or in-

The second possibility charac-

terises the entire functions of bounded type, which we shall

CHAPTER 3

30 consider later. COROLLARY 3.1

Let

m f IN,

P E p(%;F)

and

E

k

N,

0 S lc s m.

Then

P R O P O S I T I O N 3.2

Let

E, F

non-empty open s u b s e t of t o f E S((U;G)

Then

5 E

U

and

PROOF:

If

and

E,

and f

be normed spaces,

G

E

= to(dmf(g))

dm(taf)(g)

5 E U,

Bp(Z) C U

uniformly i n

there exists

p

E IR,

uniformly in

Z ) ,

L(F;G).

for every

dmf(5)

Bp(5).

dm(tOf)(s)

m € N

t: F + G

is linear and continu-

F,

1) implies that

It is easily seen, using Proposition 1.6,

E Ss(%;G)

tof E #(U;G),

f o r every

and

B p ( % ) . Since

to(dmf(g))

we have

E

> 0 and a con-

p

o u s , and hence is uniformly continuous in

that

t

a

m E N.

tinuous symmetric m-linear mapping such that

and

S((U;F)

U

for every

m E IN.

Therefore, by

and by the uniqueness of the Taylor series

= to(dmf(5))

for every

5 E U

and

rn E N.

QeEoD.

CHAPTER

4

THE CAUCHY INTEGRAL FORMULAS

REMARK

4.1:

In this chapter we make use of some properties of

integrals o f functions defined on a subset of

IR

CC

or

with

values in a real or complex Banach space (see Dieudonne' [ 2 8 ] and Hille [ 5 7 ] )

.

PROPOSITION 4.1 (The Cauchy Integral Formula). Let

f

E

(1-X)g

such that

REMARK 4.2:

5 E U, x E U

#(U;F),

+ Ax E

Although

U

F

for every

and

p E R,

h E C,

1x1

p

s p.

> 1 Then

is not necessarily complete, the

existence of the integral in ( * ) is guaranteed, since we may consider

f

identifying

as taking its values in the completion

F

with its image in

metric inclusion Iof,

and as

of

F,

F

under the natural iso-

I. The relation ( * ) will be proved for

(Iof)(x)

= f(x)

tegral in (*) is an element of PROOF:

i

A

F.

By the preceeding remark,

sider the case in which

F

it follows that the in-

F,

it is sufficient to con-

is a Banach space.

use the following well-known result:

We shall make

32

CHAPTER

4

a) Let V be a non-empty open subset of 6, z E V , 6 E IR, 6 > O , such that 6 ( z ) c V , E 6 ( z ) being the usual closed ball of radius 6 p and center z in C, and Set g

5 , x,

With V = (h E

EP(O)

(I:

V

Z

and

+

h x E U]

E D6(z).

is an open subset of

1 E Dp(0).

If

+ Ax]

1 E V,

g E #(V;F).

T

Then

as in the statement of the proposition,

: (1-1)s

g ( h ) = f[(l-l)z

that

p

E s(V;F) and

for

g: V

Applying a) to

-t

F

C,

is defined by

then it is easy to see g,

T = 1,

with

we obtain

PROPOSITION 4.2 (The Cauchy Integral Formulas). Let

f

E

3S(U;F),

such that

5 + l x E

for every

m E IN.

PROOF:

U

g E U,

x E E

for every

p

and

h E 6,

1x1

E iR, s p.

p > 0

Then

A s in the preceeding proposition, it is sufficient to

consider the case in which

F

is a Banach space.

We shall make use of the following well-known result: a)

Let

r,R E R, c V.

V

be a non-empty open subset of

0 < r < R,

Then

a E V,

such that

C,

g E #(V;F),

E C: r s

/),-a\ s R]

THE CAUCHY INTEGRAL FORMULAS

NOW

subset of

v C,

by

64x1

If

0< 0 < p

=

[A E V

and

= f (hm+l s+xx)

#

C: 2

,

P

(0)

h E V,

-

(01.

is a non-empty open

If

g: V

-I

V

is defined

it is easy to see that g E #(V;F).

{A E C :

then

E U]

0 , s+Xx

33

0

1x1

s

I;

p} c

V

and s o , by a),

we have

f o r every

m E N.

The Taylor series o f

at

f

5 ,

C

P&(Z-5;),

con-

&=O

f(z)

verges uniformly t o With

x f 0, choose

E

in a ball

E IR,

0




0

such that

Let

gp(s) c

IItlI = 1, and let

U.

f E #(U,F), Then for

x = 5 + Pt E

Bp(5:) cu.

Then, by Proposition 4.2, f

and hence

Since this holds for every

t E E

with

IItl( = 1

we have

4

CHAPTER

36

Q.E.D. REMARK 4.5:

Taking into account the inequalities mm

1/41s /IAll valid f o r every

m! ll All 6

9

we can write the Cauchy ine-

A E dis(%;F),

qualities in the f o r m

where

f 6 #(U;F)

and

The constants

rn E IN.

1 --

and

Pm

m m . -which appear respect-

1 -pm

m!

ively in the Cauchy inequalities and in ( * ) are the least possible universal constants in these inequalities.

PROPOSITION 4.5 p

Let

> 1, such that

Then f o r every

g E

f E W(U;F),

(1-1)s + Xx E U

U,

x E U

f o r every

X 6

m E IN,

where

PROOF:

For

f o r every

1 E C,

m E IN.

#

0

and

7,

#

1, we have

and C,

p E R,

1x1

4

p.

37

THE CAUCEIY INTEGRAL FORMULAS Multiplying ( * ) by

2~1

f[ (1-X)g +Ax]

1x1

the resulting equation over the circle

fc

1

(**)

2n i

( ' 1

)5+xx1

x -1

and integrating = p,

we obtain

dX =

I 1 I'P m

'

k=0

f

f

fC-_

1 Zni j *

(1-X -

)g+xx1

xk+l

dl

fr c?I?L)g 5x1d h .

1

+

2ni

I x I=P

P+l(bl)

IxI=p

Applying Proposition 4.1 to the left hand side of (**) and Proposition 4.2, with

(x-5)

in place of

x,

to the

first term of the right hand side (noting that (1-1)s

5 + X(X-~)),

we obtain the desired relation.

COROLLARY 4.1

+

Ax = Q.E.D.

Under the hypotheses o f Proposition

4.5 we have

the following estimate f o r the norm of the Taylor remainder of order

for

m

PROOF:

f

at

5:

This follows immediately from Proposition

4.5, using

the usual inequality for the norm of an integral and the fact that

inf

Ix I = P

11-11 = p - 1 .

COROLLARY 4.2 r > 0,

Let

such that

f E #(U;F),

Q.E.D.

m E N,

5 E

B r ( 5 ) c U, and let x E

U

and

Br(5).

r E R,

Then

38

CHAPTER 4

x = 5 ,

If

PROOF:

=ma r

let

1x1

the inequality is trivial.

?;

p.

Then

p

(1-X)5 +

> 1 and

Thus, by Proposition

For

Ax E U

x

# 5,

if

X E

C,

4.5, r-

Applying the usual inequality for the n o r m of an integral, and the fact that

which implies

we obtain:

The corollary now follows when

-

II

r ~

p.

is substituted for

X-lli

Q.E.D.

REMARK

4.6:

mappings o f

C o n s i d e r the vector space U

into

F.

C(U;F)

of continuous

For each compact subset

K

of

Uy

the mapping

is a seminorm on

c(u;F).

The separated locally convex topology defined on C ( U ; F )

39

THF: CAUCHY INTEGRAL .FORMULAS

(pK: K

by the family of seminorms

compact,

K c U)

is known

as the compact-open topology. PROPOSITION 4.6

F

#(U;F)

is complete, and

then

#(U;F)

is a vector subspace of

C(U;F)

If

C(U;F).

carries the compact-open topology,

is a closed vector subspace of

C(U;F),

and

hence is complete in the induced topology.

If

PROOF:

f E #(U;F),

that the Taylor series of ly in that

Bp(5).

if

f(x)

Ilx-5ll < p

ous at

1 1 5 2"f (5 )I1

Po(x-%) = P o =

formly to

5

,

5

at

f

and

converges to

in

and

s C C

and

< 1.

Pm(x-5)

It follows that

F

is a Banach space.

complete by the following result: locally compact space, and

Y

"If X

Topologie General, Chapter 10).

m

2

1,

g E U.

we define

m

[N,

such 2

1.

converges uni-

f

is continu-

C(U;F)

is

is a metrizable or C(X;Y)

(see Bourbaki [ 1 9 ] ,

To prove that

#(U;F)

is

for the compact-open topology, we take

E #(U;F) and show that Let

uniform-

C > 0

a Banach space, then

is complete in the compact-open topology"

f

f

such

#(U;F) c C(U;F).

and therefore

C(U;F)

m E

> 0

m= 0 we have

B,(g),

p OC

for every

p

a

f(s),

Suppose now that

closed in

p E R,

By the Corollary 2.1, there exists

llPm( l/m =

Since

5 E U,

let

f E a(U;F).

We set Am: Em

-b

f(s),

A.

=

F

in the following manner:

and f o r each

m

E

[N,

40 if

CHAPTER

,...,xm)

(xl

E E

m

,

4 (pl

there exists an m-tuple

,...,

)p,

of strictly positive real numbers, which we shall call an m-tuple associated to

(xl,

m

c

5 +

ljxj E

u

...,xm),

xj

if

j=1

such that

Jijl

E C,

j = ~~...,m.

s pj,

W e then define

m a)

...

A ~ ( x ~ , ,xm) =

__

1 -

m

m ! (2ni)

[

f(%+ I

’Ixjl=Pj

_

c

x .x.)

j=1 J J

- =xm)

-

(11.

dX1e..dXm

IS j s m We note first that the integral in a) is defined since f

is continuous and

F

is complete.

We show next that the

value of this integral is independent o f the choice of the ( p l , ...,p,)

m-tuple clear for

associated to

f E #(U;F)

as was noted in Remark

fixing

(xl,

...,

...’xrn).

(xl,

This is

since, in this case,

4.4.

xm) E Em,

Consider the following mapping: and an associated m-tuple

1s j r m This mapping i s easily seen to be continuous, and by Remark4.4,

THE CAUCHY INTEGRAL FORMULAS

,..., (xl, ...,

if to

and

p,)

(pl

,...,

are two m-tuples associated

p&)

then

xm),

Since

( p i

41

TPl,

...,Pm;xl’...7x

T

and

m

,p;ixl,

PLY

,xm

are continuous, it follows that

Am

Therefore Am

is symmetric and m-linear. P1’

prove only that

(xl,

elements

(xl,

E E

J

(pl, ...,p

...,x

j

g E W(U;F).

and

(xl,

J

...,

xm)

of

Em,

..., .,....,xm ) X‘

J

(g)

,Pm;xl’

9

xj *

xm ( g )

Tpl

+

m

Let

,...,

..., ,...yxm

pm:x1y

..,pm;xl,. ..,x .+XI.,...,x T ...,x .,...yx T ply ..., ...,x;, ...,x T p l,.

J

p1,”’,Pm;Xl’

PmiX1,

and

=

X I

j

Therefore b)

we shall

we have

p l , ...,~m;xl,...,xj+x~,...,x

9

; YX m

Choosing an

T

T PI’

.

which is associated to each of the three

m)

...,x.,...,~~),

+ x>,

,PmiX1’

is additive in each variable.

Am

...,x.,x>, ...,xm

m-tuple

This is proved using

..

T

the continuity of the mapping

x1,x2,

m E N.

is well-defined f o r every

-

J

+

J

W(U;F)

0

(8).

42

CHAPTER

4

From b ) , and t h e c o n t i n u i t y o f t h e t h r e e mappings which appear, w e have TP

I , . . . ,p r n ; x l

,...,x

,...,xm / w ; F j -

+x;

----

j

...,

x

Am(xl,

j

,

f E #(U;F)

Therefore, as

+X’.,...,X

we have

rn ) = T

pl,.

..,

pm;xl,...,x

Analogous c o n s i d e r a t i o n s show t h a t v a r i a b l e , and symmetric.

Thus

W e prove next t h a t P

r e a l number and 3 ) sup

II t-5lIhP

>

0

such t h a t

1)

0

5:

M,

and l e t

pm)

s t r i c t l y p o s i t i v e real numbers with

g,(5) c

if

U,

(xl,

...,

x ) E Em m

m then

5 +

ljxj E U

C

< p < 1,

...,

(pl,

p1

with

(Al,

for every

j=l

lXjl

s Pj,

associated j = l,...,m.

we have

1 c j

5

m.

t o every Thus for

...,xm) ...,

(X1,

E Em

Xm)

2)

(5)

I=

U,

b an m-tuple o f

+...+ (Ixl(l

= p.

p,

E Cm (p,

f o r which

Since

= 1,

IIXmll

=.me=

...,A m )

E Em,

P

Fix a

such t h a t

Therefore the m-tuple (xl,

for every mElN.

i s continuous.

M > 0

t h e r e e x i s t s a real number Ilf(t)li

eas(%;F)

m s 1,

Am,

(f) =

i s homogeneous i n each

Am

Am E

m

j+X>,...,X

with

,...,

p,)

is

IIxjl/ = 1,

/Ixll(= a * . =

II

Xmll

=

1

9

THE CAUCHY I N T E G R A L FORMULAS

1

1 -

2np1...2np

~.

(2ny

--

rn

rn

1 2

P1...Prn

43

2

Ix J+ P j

1s j s m continuous. Now let x

E B
1; then if for every

X E

C,

We claim

U.

~-

for

E

3

E H(U;F),

-l(s)

x E B

and

rn E IN.

PO

We have already proved this for tion

4.5).

If

/Ix-511 < Po-',

f E #(U;F)

the following mappings are con-

tinuous : i)

Vx: h E C ( U ; F )

Therefore c ) holds for F r o m c ) we have

I--

Vx(h)

(Proposi-

= h(x)

f E #(U;F).

E F,

44

CHAPTER

f o r every

1x1

=

m E N.

(I

\lx-~llL pa",

Since

we have for

X E 6,

DY

and s o , by 3 ) ,

sup

I x l=u

//f[(l-h)s+hx)I/

s M.

Hence, by

(*),

m

Since to

f(x)

U

> 1,

uniformly i n

it f o l l o w s that

-'(s),

B PO

C Am(x-g)" rn= 0

and t h e r e f o r e

converges f E #(U;F).

CHAPTER 5

CONVERGENCE OF THF: TAYLOR SERIES

In this chapter we consider the following problem: Given

f E #(U;F)

sets of

U

in which the Taylor series of f;

uniformly to

at

5

E = Cn,

5

at

f

represents

converges uniformly to

DEFINITION 5.1

x

f

If E

A

converges

In the case in

and contained in

1x1

(1-X)S;

+

A t E

?,x E A

and

E r,

If A

is 5-balanced and non-empty, then

5;

A subset

A

E

of

= {x-5 : x E A]

5

C

E,

and

U.

5 E E,

f o r every

5 E A.

The

are the simplest examples

E.

is 5-balanced if and only if the i s 0-balanced.

are referred to simply as balanced sets.

If 5 E A

f

1.

of g-balanced sets in a normed space

A-5

5

in every compact subset of

5

open and closed balls with centre

set

f.

is a normed space,

is said to be 5-balanced if

E

at

it is well known that the Taylor series of

the largest open ball centred at

A

f

that is, we wish to know to what extent the

Taylor series of which

5 E U, we seek to determine the sub-

and

the set

45

The 0-balanced sets

45

5

CHAPPER

is the largest {-balanced set contained in the 5-balanced kernel of an open set then

9

If

is never empty.

A.

is

4, E A c E

If

it is called

A;

A

is

Open*

where

B c A,

is {-balanced, and

A

then r.

= r(l-x)g+Xx

B

5

B"

Thus

s 11 c

A.

is the smallest %-balanced set containing

5

PROPOSITION 5.1

E

1x1

E c,

is called the %-balanced hull of

5

f

x

: x E B,

Let

an open set

V,

K c V c U,

K C U,

K,

and a number

v, x

: x E

{(l-x)g+Xx

B.

be an open 5-balanced set and let

U

F o r every compact set

Sf(U;F).

B.

1x1

E @,

p

there exists

> 1, such that

P I c

u

and

PROOF.

Denoting by

and radius

P,

5

the closed ball in

P

is continuous, and s o Since

set of

F,

p

3

+ hx E E

is a compact set contained in

T(ElxK)

f E S((U;F) c C ( U ; F ) ,

f[T(E1xK)]

and hence is bounded.

bounded neighbourhood number

with centre

the mapping

T: (X,X) E C X E H ( 1 - X ) C

U.

(I:

A

of

that 5,XV

C

Therefore there exist a

f[T(61XK)]

> 1, and an open subset

T-'[f-l(A)]

is a compact sub-

V

in of

F,

a real

E, K c V c U , such

CONVERGENCE O F THE TAYLOR SERIES

47

which i m p l i e s t h a t

T(5

P

x V ) c f-l(A)

f-l(A) c U ,

Since

f[T(EpxV)]

and

C A.

we have

(1-1)s + Xx E U

x E V

if

1x1

1 E G,

and

s p,

and sup(/IfC(l-X)S+hx]I) : x E

v, X

1x1

E c,




0

E W(U;C)

U:

such that with

Let

f E U.

G r ( 5 ) c U.

m = 0,

Ji E Y ,

Since

u

is open,

Applying Corollary we have

CHAPTER 6

60

x E Br(5)

for every

By d)

f

Q E

and

Y.

i s l o c a l l y bounded, and s o , t a k i n g

f i c i e n t l y small, s o t h a t

)

f o r every

x E B,(f

f o r every

x E Br(5),

Q E Y.

and

x

Ilf(t)ll

11 t

# 5,

x E Br(S),

f o r every

It follows t h a t f)

f

f

5+Ax E

x E E U

# 5,

JI E Y .

c

2

Pm: E

-t

U: F

Let

E

6,

It follows by b )

such t h a t

0

1x1

5. g E U.

For each

i n t h e f o l l o w i n g way:

we choose a r e a l number

f o r every

we have

and hence

i s continuous a t

i s holomorphic i n

we define a mapping each

x

+m,

suf-

Therefore

and

t h a t t h e r e e x i s t s a r e a l number

= d


0

such t h a t

and s e t

m E IN For

61

WEAK HOLOMORPHY

This integral exists since by e)

F

and by hypothesis

is complete.

5

max[p,p'],

is continuous,

We show first that the

value of the integral is independent of positive real numbers such that

f

P.

5+Xx E U

p,

p'

be

X E 6,

1x1

Let for

5

and let r

Pm(x> = -2rri

B y hypothesis,

by Proposition

Qof

,

E #(U;C)

$ E

Y

and

m E IN.

f o r every

lJ E Y

and

m

for every

Q E Y

and

m E N.

F,

Q E 1,

and hence,

4.2 (the Cauchy integral formula),

for every

of

for every

and hence

E

IN.

Similarly,

Therefore By a)

Pm(x) = Pk(x)

Y

separates the points

for every

We show next that the mappings

= Q[Pk(x)]

$[Pm(x)]

m

Pm, m E

E

IN. N,

are con-

62

CHAPTER 6

tinuous.

If

xo E E ,

is continuous at the point such that and

g+Xx E U

IIx-xoII

for every

5

p.

(O,xo),

for all

x E Bp(xo).

E

and so there exists p > 0 and

x E E

with

N o w , using the fact that

is continuous at

We now show that

Pm E P ( % ; F )

1x1

L p

is con-

f

xo

for every

m E

for every

m E IN.

For

[N.

this is immediate, since

and applying Proposition 4.2 the functions

for every

$of

x E E

the points of

E

W(U;C),

and

$ E

F, P o ( x ) =

(the Cauchy integral formula) to $ E

Y.

f(5)

Y,

Since, by a),

Y

for every

x E E.

separates Therefore

E P('E;F). For

by

C

5 +Ax E

-B

and the usual inequalities for integrals,

Pm

it follows that

p0

1 E

CXE

Then

tinuous (part e)),

m = 0

(1 , x ) E

the mapping

m

N,

m 2 1, we define a mapping

k : Em

-B

F

63

WEAK HOLOMORPHY

L: ei=±l

&le 2 ••• e m Pm(elxl+ ••• +emXm)

1" i"m where

m E tN,

(Xl, ••• ,x ) m m :2: 1,

E

Em.

are symmetric and m-linear, and that

It is easy to see that to prove that

~

E 'l',

Am

Am

A- m

Pm .

is symmetric, and i t then suffices

is linear in the first variable.

Am(Xl+X~'X2""'Xm)

only that

We shall prove that the mappings Am'

= Am(xl

We show

, x 2, ••• ,xm) +

we have

HAm(Xl+X~,X2"",xm)J

=

1

L: el ••• em(*oPm)(el(Xl+x~)+e2X2+ 2 mm.I &i=±l

•• .+emxm) =

l"i:s:m

=

L:

&l"'&m

~!

am(*of)(O(&1(Xl+X~)+&2X2+··.+emXm)

e i=±l 1" i:s;m

= 2

1 mm!

L:

1

&1 • • • e m

Ii1T am(*of) (~)

e 1 ••• & m

Ii1T am(W o f) (~)

&i=±l

(e lX l+"

'+&mXm) +

(e lX~+"

'+&mxm)

1" i:s;m 1 2 mm.,

L:

e i=±l

1

l"i:s;m 1 =-L: &1'''& (wop )(e x m m m l 1+ . . . +& mx m ) + 2 m.I &i=±l 1" i"m

=

64

CHAPTER 6

-m 1 C 2 m! E =il

el...s m

($0

Pm) (E

r+C

lX;+

mxm)

i 1s i s m

16 i s m

14 i s ; m

Therefore

= $[Am(x19x2,.. Since

Y

$[Am(x1+x;,x2,.

.,xm )

..,xm)]

=

+ A m (x‘1 ,x2,. . . , x m )] for every

separates the points o f

F,

Jr E Y .

it f o l l o w s that

A

Next, we show t h a t that

Pm E P ( ? E ; F ) .

$Cim(X)l

Am = P m , from which i t follows

We have

,L = $[Am(%

,X)l

1si s m

for every

E Y,

x E E

and

m E N,

m

2

1,

and hence

WEAK HOLOMORPHY

im(x) = Pm(x)

f o r every

x E E

rn E N,

and

m z 1.

To complete the proof of the proposition we show that m

C Prn(x-5) m=O

of

converges uniformly to

in a neighbourhood

f

5. Since

M > 0

numbers sup

f

llf(t)ll

is locally bounded (part d)) and

L

II t-s I1 =o such that

r < ap

> 0 such that

0

M.

.

to the functions

f o r every

$[ Pk( x)]

=

1

;Ik(

$ 0

r > 0

and

C

Prn(x-5)

m= 0

5;

uof,

x E B,(g)

p > 1

We claim that the series f

p,

~ ~ ( 9 ) .

in

x E Br(Z)

1x1

E C,

for all

and

U

m

converges uniformly to

For every

gu(g) t

Choose real numbers

-1

there exist real

we have

(1-x)g

+ xx E

fi ( 5 )

C

U

and s o , applying Proposition

u

4.5

$ E Y,

and

rn E

f)( 5 ) (x)

,

Using the fact that

[N.

this yields r

for every

$ E Y,

the points of

for every

F

x E Br(s)

and

m E

[N.

it follows that

x E Br(C)

and

rn E N.

Hence

Since

Y

separates

66

CliAPTER

6

m f o r every

x E B,(g)

m E

and

It follows that

!N.

C

P,(x-~\

k=0 converges uniformly to

in

f

Q.E.D.

Br(g).

A s a corollary to Proposition

If F

PROPOSITION 6.2

6.1 we have

is a Banach space and

f: U

-t

F

a

mapping, the following are equivalent:

t

W(U;F)

1)

f

2)

$of E a ( U ; C )

topological dual of PROOF:

for every

$ E F'

F

is bounded, and so

Y = F'

PROPOSITION 6.3

U

complete, .S(F;G)

E, F

Let

satisfies the con-

and

Q.E.D. G

be normed spaces,

a non-empty open subset of

U

a mapping.

E,

F

and

and

Then the following are equivalent:

~(u;c(F;G))

a)

f E

b)

the mapping

for every

PROOF:

is the

Thus Proposition 6.2 is a par-

ticular case of Proposition 6.1.

-t

F'

F.

ditions of Proposition 6.1.

f: U

where

By the Banach-Steinhaus Theorem every weakly bounded

subset o f

G

,

For

y

E

F

y E F

x

E U

w[ f( x ) (y)]

E d:

is holomorphic in

w E G'.

and and

-t

w E G'

J I ~ , ~u: E C ( F ; G ) is linear and continuous.

I-

Let

the mapping

L U U C ~E( (C~ ) I

WEAK HOLOMORPHY

If x E U ,

y E F

w E G’,

and

then

and hence condition b) is equivalent to E H(U;C)

of

L

Y

for every

w E G‘,

y E F;

this can be written: ct)

Jiof

~ ( u ; c ) for

E

By hypothesis, complete.

G

every

$ E Y.

is complete, and so

x(F;G)

is also

Thus by Proposition 6.1 the equivalence of a) and

c1) is established once we have proved the following: B c C(F;G) for every

is bounded if and only if

is bounded

Q E Y.

To say that that

$(B)

$(B)

I

sup Iw[u(y)]


0

m E IN.

E P(%;F),

Let

U

and

and

ur

h

r

1 E Vx

for

be real numbers such that

we know that

1x1

E C,

(foT

=

Gr(5) c Ep(5)

4.5 to the points X

Y-5

)(1)

-

s

0,

m

C

k=0

U).

Let

y E

foTx E #(Vx;F).

0

and

1

of

1 -k k-! d (foT

Y-5

)(0)(1)

m E IN,

Thus

for every

y E

Er(5),

rn E N.

It follows that

m

M

k=0

the series

Urn(U-l) rn E IN.

r n l

C

m! Cmf($)(y-5)

m= 0

in

Gr(5).

Q.E

s ince

we have

gr(5 ) ,

isr($),

fjr(5).

Vx,

y E

y E

1,

Applying

for every

for every

CJ

We shall prove that the series

p.

x = y-5,

Proposition

is continuous, that is, m E IN.

f o r every

(Note that Taking

i"f(5)

Thus

O D .

Since

u > 1 we conclude that

converges uniformly to

f(y)

75

FINITE HOLOMORPHY AND GATEAUX HOLOMORPHY

DEFINITION 7 . 2

A mapping

f: U

phic (Gateaux-holomorphic) in b E E

-b

F

is said to be G-holomor-

if for every

U

a E U

and

the mapping

1 E (X E C : a+Xb E U} c C

f(a+Xb)

E F

( X E C : a+Xb E U]

is holomorphic in the open subset REMARK 7 . 2

t--

of

This definition is equivalent to the following

statement: a)

If x

and

y

are points in

A = { A E C : x+Xy E U}

E

such that the set

is not empty, then the mapping

X E A c CH

f(x+hy) E F

is holomorphic in the open set

A.

It is clear that a) implies the condition given in Definition 7.2. points in

E

Conversely, suppose that

x

and

y

are

such that the mapping

Xo

is not holomorphic at a point

of its domain.

Then

a = x+Xoy E U , and the mapping

x

E

(X

E

(I:

: a+Xy E U}

is not holomorphic at the point existed

bm E F,

m E N

and

c CH

f(a+Xy)

1 = 0. F o r , suppose there

p > 0

such that a

C.

76

CHAPTER 7

= Po.

B (0)

uniformly in

P

(*)

We may write

E C

as: co

= f(x+>" o y) +

f(x+>.. o Y+AY)

L: m=l

co

whence

f(x+>" 0 y) +

B (0).

formly in

p

L: b mAm converges to m=l Thus the function x+>..y E U} c ..E"0'

f(x+(A o +A)Y)

uni-

'-,> f(x+>"y) E F

contradicting our hypothesis.

It

follows that Definition 7.2 implies condition a).

Let p ~ 1,

F

U

be a non-empty open subset of

a complex normed space, and

CPo

nonical basis for

We denote by

Zk = (Zl""'Zk_l,o,zk+l'."'Zp) E CPo k = 1,2, ••• ,p

J

z,k

pEN,

{el, ••• ,e p} Jk

1 :s;

CP,

the ca-

the mapping

k :s;

p,

Then for

>.. E C

and

we have

denotes the continuous affine linear mapping

xE

.. e k E C P •

z, k (>.. )

be a non-empty open subset of

is separately holomorphic in

satisfies the following condition: Z = (zl""'zp) E CP,

Uz,k = J-lk(U) ~ ¢ z,

and

we have

U

For every

k = 1,2, ••• ,p foJ z" k E

~(uz

such that kIF).

CPo if it

77

FINITE HOLOMORPHY AND GATEAUX HOLOMORPHY

We remark that this condition is equivalent to the

..,

z = ( zl,.

requirement that for every k = 1,2,...,p,

the restriction of

determined by

zl,. . . , Z

zp) E Cp

f

to the sectaon of

z

is holomorphic.

* P

k-1,Zk+l’

PROPOSITION 7.3 (Theorem of Hartogs): open subset of

U

ping of

F

(Cp,

F.

into

U

be a non-empty

a complex normed space, and

f

a map-

The following are equivalent:

U.

f

is holomorphic in

b)

f

is separately holomorphic in

7.3

U

Let

a)

REMARK

and

U.

The proof of the theorem of Hartogs may be found

in [ 6 O ] .

7.4 Let U be a non-empty open subset

PROPOSITION f

a mapping of

F.

into

f

is G-holomorphic in

2)

f

is finitely holomorphic in

1)

=

such that

for

S.

Let

2).

V = S

n

S

U

E and

The following are equivalent:

1)

PROOF:

E

U

of

U. U.

be a finite dimensional subspace of

#

and let

@

[al,

...,ap]

be a basis

Since the mapping cp:

(al,

...,zp) E

(C

P

PH

c

zkak E

s

k=1

is a homeomorphism and an isomorphism between the vector spaces (Cp (Cp.

and

S

the set

W = cp -‘(V)

is a non-empty open subset of

Consider the mapping

,..,.,zP)

g: (zl

We have

f = gorp S

-1

E

w

P

t--

f(

c

zkak) E F.

k=1

and s o , by Proposition

7.1, to show that

78

CHAPTER

fS E g(UnS;F)

7

it suffices to prove that

position 7 . 3 this is equivalent to

g

t H(W;F).

By P r o -

being separately h o l o -

g

morphic. For

...,

z = (zl,

zp) E Cp

k E N,

and

1 s k s p,

such that

we must show that the mapping

is holomorphic in

Wz,k

k

We have

k

= ( f o c p ) ( z +Xek) = f[p(z

g ( z +Xek)

k )+lak],

and

from the definition of a G-holomorphic mapping and Remark 7 . 2 , we k n o w that the mapping

is holomorphic in 2)

1).

3

Let

E [A E

and

b b

E

If

E.

then

a

E

is the subspace of

S

U

n

S

E

and by 2),

Applying Proposition 7 . 1 to the composition

fS E #(UnS;F).

X

a E U,

a

generated by

Wz,k

(I:

: a+Xb E U } H

this mapping is holomorphic.

a+Xb E U Thus

n

f

SH

f(a+Xb) E F,

is G-holomorphic. Q.E.D.

COROLLARY 7.1

Let

f

lowing are equivalent:

be a mapping of

U

into

F.

The fol-

79

F I N I T E HOLOMORPHY A N D GATEAUX HOLOMORPHY

a)

f

is holomorphic in

b)

f

is G-holomorphic and continuous in

c)

f

is G-holomorphic and locally bounded i n

PROOF:

U.

U. U.

T h i s corollary is an immediate consequence of P r o p o -

sitions 7.2 and

7.4.

Q.E.D.

This page intentionally left blank

CHAPTER 8

TOPOLOGIES O N SPACES OF HOLOMORPHIC MAPPINGS

DEFINITION 8.1

A non-empty subset

to the boundary of

px: f E

into

F.

is a U-bounded subset of

E,

Wb(U;F).

U

sup{Ilf(x)IJ

t h e mapping

: x E X] E

The family of seminorms

ranges over all the U-bounded subsets of rated locally convex topology on ~

U.

will denote the vector space o f holomorphic

w~(u;F)I+ pX(f) =

norm on

X

is said t o 5 e of b o z n d e d type if

mappings of bounded type from

If X

is said to be

is positive.

U

is bounded on every U-bounded subset of

UL,(U;F)

T

E

of

is bounded and the distance from

f E J$(U;F)

DEFINITIOW 8.2 f

X

X t U,

U-bounded if

X

Hb(U;F),

where

px

E,

IR is a semiX

defines a sepawhich we denote by

This topology is known as the natural to2ology o n Wb(U;F).

.

PROPOSITION 8.1 topology PROOF:

F

is complete then

Sb(U;F),

with the

is a Fre'chet space.

Tb

n = 1,2,...,

For

xn

If

= Ex E

There exists

u

no E N

let

. )/xIIs *

n

and

such that for

81

dist(x;aU) n

5 no ,

5.

Xn

1

1.

is non-empty

CHAPTER 8

82

For

and hence U-bounded. m

m

u

u =

n=1

where

1

2

Vm = { x E U : dist(x;hU)

and

there exist positive integers

Vmo

unO

which

m= 1

Un = Ex E U : IIxI/ s n]

3

n

n'

#

0.

#

X

u vm

u =

and

m,,

for

n

such that

I t follows that there is an index

a 1

n

for

0. E

Moreover, every U-bounded subset of Xn

n0

and

5

sufficiently l a r g e .

n = 1,2,...

is contained in

c pxn 3

Thus the sequence

determines the topology

r

.

Therefore

'rb

is

a rnetrizable topology.

be a ~,,-Cauchy sequence i n

Let

is U-bounded,

Bb(U;F). Erm3mcN

Since every compact subset of

U

a ~ - C a u c h ysequence, where

is the topology induced on

T

Ub(U;F) by the compact-open topology of tion

4.6,

#(U;F)

is closed i n

topology, and hence for this topology. every

x E U.

then since m

0

C(U;F)

C(U;F).

B y Proposi-

for the compact-open

{fm]mE(N converges to some

f E B(U;F)

f(x) = lim fm(x) m-tm is a U-bounded set, and

for

I t follows that

N o w if

X c U

is

E

> 0,

ifm]mt'N is a ' ~ ~ - C a u c hsequence, y there exists

E IN* = [ 1 , 2 , . ..]

such that

llfm(x)-fn(x)l/ < E Therefore

which implies that

for every

x E X,

m,n

2

m0

.

TOPOLOGIES ON SPACES O F HOLOMORPHIC MAPPINGS

sup /lf(x)ll xE x

f

f o r the topology

DEFINITION 8.3

rb.

(x)ll

U

+ c


0, there exists

n0 E IN

(SIB(U;F),ll

such that f o r

n0 ’

A s in the proof of Proposition 8.1, it follows that

f E a(U;F)

converges to s o m e

compact-open topology on f E

11).

w~(u;F).

Since

‘

for the topology induced by the

C(U;F).

W e m u s t show that

lim fm(x) = f(x)

for every

x E U,

m-tm

follows from (*) that there exists

for every

n

2

n

0

.

Hence

fm’ mEN

no E IN

such that

it

84

CHAPTER 8

Therefore in

f

E

WB(U;F);

( q m ,I1

11).

and we conclude from ( * ) that fm

-t

f

Q.E.D.

We now consider the varioas ways of topologizing SI(U;F). DEFINITION

8.4

will denote the topology on

T~

duced by the compact-open topology of separated locally convex topology o n

C(U;F). 3I(U;F),

#(U;F) T o

in-

is then a

defined by the

seminorms

PK: as

K

E

f

#(U;F)M

ranges over the compact subsets of

DEFINITION

8.5

If

m E N,

cally convex topology on

as

K

#(U;F)

logy on

U.

will denote the separated lodefined by the seminorms

U,

and

n

over

.. .

[0,1,2,.

DEFINITIOlV

K

Tm

ranges over the compact subsets of

the set

as

PK(~) = SUP Ilf(x)li xE K

8.6

#(U;F)

,m}

r m denotes the separated locally convex topo-

defined by the seminorms

ranges over the compact subsets of

REMARK 8.1:

a)

and

n

over

IN.

It follows from Proposition 1.3 that the to-

pologies defined in

is replaced by

U,

d.

8.5 and 8.6 are unchanged if the symbol

;

TOPOLOGIES ON SPACES OF HOLOMORPHIC MAPPINGS

b)

It is obvious that

T~

Tm

S

S

s rm

Tm+l

85

for every

m E N.

We introduce next the topology DEFINITION 8.7

A seminorm

L3

T

on

W

is said to be port-

#(U;F)

3n

U(U;F).

if f o r every neighbourhood

ed by a compact subset

K

of

V

U

there is a real number

of

K

contained in

U

C(V) > 0

such that p(f)

c(v)

5

sup Ilf(x)ll

f o r every

E #(u;F).

f

xE v T

W

denotes the separated locally convex topology on

defined by the seminorms subset

K

(K

U

of

p

[9l],

which are ported by some compact

may vary with

REMARK 8.2: The topology

T

#(U;F)

p).

was introduced by Nachbin in

UI

having as a motivation Martineauts concept of a linear

analytic functional ported by a compact subset.

See Martineau

1761 REMARK 8.3: If the seminorm

p

compact

is another compact set with

KIC U

K1 c K2 c U,

and if

then

p

K2

on

is ported by a

#(U;F)

is also ported by

K2.

On the other hand, if the seminorm each of two compact subsets of in general, true that

p

U,

K1

is ported by

and

K1

is ported by

p

n

K2,

it is not,

K2.

In other words, a seminorrn which is ported by a compact set does not, in general, possess a minimal compact porting set. EXAMPLE 8.1

P(f) =

If(%)

Let

,

U = E =

f E w(C,C).

F = C,

and for

5 E

It is clear that

C, p

let is a semi

86

CHAPTER 8

norm o n

la-51 < r,

and

for every

f

Therefore

p

r'

tlien

= l r ( 5 ) 1 s s u p ( If(x)l

p(f)

E )$(C,C),

where

c

= (x E

~,(a)

p

then

n

K2

since

n

K~

K2. K~

sr(a)}

= Sr(a).

/%-a1 < r' < r ,

is also ported by K1

not be ported by

: x E

: [ x - a \ = r].

K1

is ported by the compact set

is a real number such that

= Srl(a),

a E C

By the maximum modulus theorem, if

)$(C,C).

If K2 =

and

However,

= 6 , and

can-

p

p

is not

the zero seminorm. We note that the unique seminorin ported by the a m p t y set is the seminorm which is identically z e r o . REMARK

8.4:

say that

3

Let

3

for every real number

x E

f E SI(U;F)

converges to

> 0

g

g E M

such that if

)$(U;F)

be a filter i n

thcn

uniformly on

if,

M = M(c) 6 F

therc exists

s E

(jf(x)-g(x)(l

X

We

for every

x.

P R O P O S I T I O N 8.3

p

Let

a)

p

b)

If

V

K

of

PROOF:

a)

x

a b).

2

03 Let

W(U;F)

V,

U

U.

be a filter i n

The family

b'

p(3)

3

conin

0.

#(U;F)

V

which conof

K

con-

o f sets of the form

W

= [O&],

verges uniformly to zero on a neighhourhood tained in

such that

the filter base

converges to

3

for which there exists

contained i n

verges to zero u n i f o r m l y o n IR+ = {x E [R :

K

and let

The following are eqnivaleat:

is a filter i n

a neighhourhood

W(U;F)

K.

is ported by

3

be a seminorm on

U.

be a compact subset of

E

X c U.

and

E

ranging over the set of positive real numbers, is a base

87

TOPOLOGIES ON SPACES OF HOLONORPIIIC MAPPINGS

for the filter C(V)

of neighbourhoods of

b

3

M E 3

such that if

x E V.

converges to zero unif'ormly on

E M,

g

g E M.

Therefore a).

p(3)

converges to

Suppose that

K

of

V

V

which

is finer than

b.

0.

3

while

#(U;F)

in

U

contained in

zero uniformly in

=

s C(v)*c/C(V)

satisfies b) but is not ported by K.

p

We shall construct a filter hood

p(5)

G

for every

= [ p ( g - ) : g E M} c W E ,

p(M)

Hence

< €/C(V)

8.7, p(g)

Therefore, using Definition

for

a

Let

there exists

V

lig(x)ll

then

shows that the filter generated by

b)

R+.

> 0 be the number given by Definition 8.7, and let

Since

= c

in

0

and a neighbour-

such that

3

converges to

does not converge to z e r o ,

p(3)

contradicting b).

If

V

K

of

p

fk

E #(U;F)

gk = fk/p(fk)

k 2 1.

U

contained in

there is an

Then

3

Let

for every

.n *

2

k

N-{O},

k]

k E IN-CO]

,

E

k

2

for every k

E

k

bJ,

1,

for which

and

k E

E

= 1

p(g,)

#(U;F)

be the filter in

Fr6che-t filter in

there is a neighboarhood

such that for every

t SI(U;F)

image under the mapping

Nk = [n E N

K,

is not ported by

N-[O}

H

gk

E

N,

generated by the #(U;F)

of the

which has as base the sets [N,

k

and every

converges to zero uniformly on

2

1.

x

E V.

V.

From ( * ) we have

(**) shows that

However,

p(3)

3

does not

CHAPTER 8

88

0

converge to k

; I

1.

in

R+,

since

This contradicts b),

= 1

p(g,)

and so

p

for every

is p o r t e d by

k E N,

K. Q.E.D.

EXAMPLE 8 . 2

K

Let

be a compact subset o f

U,

a n d let

p

be defined by p: f E H ( U ; F ) H

p(f)

= sup l l f ( x ) l l . xE K

I t is easy to see that the seminorin EXAMPLE 8.3

a =

Let

b m l m C Nbe

khat

K

p

is ported by

U

be a compact subset of

K.

and let

a sequence of non-negative real numbers such

lim

=

Then the mapping

0.

m-t-

is a seminorm on let

V

H(U;F)

be a neighbourhood of

is a real number

r,

K.

which is ported by

K

contained i n

r > 0, such that

Vr

=

T o see this,

U.

Then there

'J Er(x) c V. xE K

Applying the Cauchy inequalities, Proposition

and s o

Then

4.4,

we have

89

TOPOLOGIES ON SPACES OF HOLOMORPHIC MAPPINGS

I t follows that if

K.

is ported by f E #(U;F), mediate.

is a seminorm on

We show that

K' ,a

(f)
0, there exists a real

c,

> 0, such that

F o r every real number

neighbourhood number

of

f E #(u;F).

for every

c)

K

V

K

of

c(c,V) > 0

E

>

and for every open

0,

contained in

U

there exists a real

such that

f E #(U;F).

for every

The proof of this proposition requires the following two lemmas. LEMMA 8.1

such that

Let

B

2P

f E 3((U;F)

(g) c U

then

and

5 E

U.

If

p E R,

p

>

0

is

91

TOPOLOGIES ON SPACES OF HOLOMORPHIC MAPPINGS

f o r every

x E

Bp(5),

k E IN.

,...

By Proposition

PROOF OF LEMMA 8.1: f o r every

k = 0,1,2

Since

5.4,

’-Akf k!

E W(U;P(kE;F))

g2p(g) is 5-balanced, the Corollary

5.1 shows that

for

,...,

k = 0,1,2

x E B2p(5),

wise with respect to the n o r m o f

the convergence being point-

P(kE;F),

where

1

For

defined by

m € N,

m

2 k,

consider the mapping

Pk(y) = Pm(y-5).

Pk E W ( U ; F )

Pk: U

-P

F

and so, applying

the Cauchy inequalities,

for

x E

i p (k{= ) 0,1,... ,

and

x E

g,(g)

and

m

5

k.

But if

then

Therefore

for

k

= 0,1,...,

rn E IN,

m

2 k

and

x E ip(5).

/Ix-yll = p

CHAPTER 8

92

From 1) and 2) we conclude that

for

= 0,1,...

k

LEMMA 8.2

Let

is such that

and f

x E gp(5).

E W(U;F)

fi,p(X)

c U,

and

Q.E.D.

X c U.

If

p

E

R,

p > 0

then for every real number

E

> 0,

we have

Let

PROOF O F LEMMA 8.2:

such that

y E

B'

(x),

P

for

k

= 0,1,...

.

for

k

= 0,1,...

.

Let

and

y E

gp(X);

then there exists x E X

and s o b y Lemma 8 . 1 ,

Therefore

93

T O P O L O G I E S ON S P A C E S O F HOLOMORPHIC M A P P I N G S

Then

c zm pm-k

'

,

Mk

Mk myk k E N we have

and

m

C

k 6

and hence for every

m

oa

C ( C e k=O m=k

M k s

k=O

k

2

m

p

m-k

MA)

which is the statement of the Lemma 8.2. PROOF O F P R O P O S I T I O N

a) E

c).

3

U.

in

p

* >

Let

b). 0,

p

E

< e/4.

c BZp(K) c U,

E R,

Then and


0 such that

< dist(K,aU),

p < 2P

Applying Lemma 8.2

m= 0

C(V)

c ) follows f r o m 1) with

K.

C

K.

But since

ing

2)

is ported by

be an open neighbourhood of

f E #(U;F).

f E #(U;F),

for

p

There is a real number

for every

c)

V

Q.E.D.

8.5:

Suppose that

> 0, and let

E R, e > 0

6

sup

x€Bp(K)

c(e,V)

= c(V).

and let

< e / 2 < dist(K,aU).

is an open subset of to

e/4 1 llz

f € #(U;F),

we have ^k

d f(x)ll

p E R, Bp(K) c

U

contain-

to the compact

CHAPTER 8

94

c/4 >

N o w , by h y p o t h e s i s , g i v e n t h e r e a l number

V = B (K) P

t h e open s u b s e t

a r e a l number (since by

depends on

p

f E #(U;F).

f o r every

f

*

let

a).

p

E

containing

t h e r e exists

E

and which we may a c c o r d i n g l y denote

F r o m 2 ) and 3) we o b t a i n :

E H(u;F).

Let

be an open s u b s e t o f

V

with

R,

0

< p < dist(K,aV).

U

containing

f o r every

E g(U;F),

f

m E

[N.

t h e r e i s a r e a l number

f o r every

f E #(U;F).

f o r every

f E #(U;F).

depends on

V,

p o r t e d by

K.

Choose .(€)

Thus, f r o m

As

we may t a k e

Q.E.D.

E

>

gp(K)

Then

s o , by t h e Cauchy i n e q u a l i t i e s ( P r o p o s i t i o n

thesis,

K,

which i n f a c t depends o n l y on

0,

c)

U

such t h a t

c(a),

f o r every

b)

>

c(c,V)

of

and

0

4.4),

c = p/2. 0

K,

and and

c V

we have

By hypo-

such t h a t

4),

depends on

C(V) = 2c(g).

p,

which i n t u r n

Therefore

p

is

95

TOPOLOGIES ON SPACES OF HOLOMORPHIC MAPPINGS

PROPOSITION 8.6 lanced subset o f be p o r t e d by V

number

>

c(V)

f o r every PROOF:

f

U,

and l e t

K,

be a compact 2-ba-

I n o r d e r t h a t a seminorm

U.

of 0

which c o n t a i n s

U

p

on

U(U;F)

K,

there i s a r e a l

such t h a t

E U(U;F).

The c o n d i t i o n i s n e c e s s a r y :

ed by

K

i t i s n e c e s s a r y and s u f f i c i e n t t h a t f o r every

K

open s u b s e t

5 E

Let

and t h a t

Suppose t h a t

i s an open s u b s e t o f

V

p

i s port-

c o n t a i n i n g K.

U

Let

w

=

K C W.

v

: (1-h)S

x

+ xx E v ,

E c,

1x1

.s

i s t h e l a r g e s t open 1 - b a l a n c e d s u b s e t o f

W

Then

[ X E

13. V,

and

B y C o r o l l a r y 5.1 we have m

f(x) =

c

Pm(x-l)

m= 0

for e v e r y m E IN. C(W)

>

x E W

Since 0

p

and

f E #(U;F),

i s p o r t e d by

K,

where

pm =

1

m! :"f(5)

,

t h e r e i s a r e a l number

such t h a t

f or e v e r y

f E #(U;F).

Using

f o r every

f E #(U;F).

Since

(*), we have

W

depends o n l y on

V,

we may

96

CHAPTER 8

for every

f

E x(u;F).

The c o n d i t i o n i s s u f f i c i e n t : of set

containing

U W

of

K.

By Remark

containing

V

(1-1)s + Ax E V

that

K,

Let

5.1

V

be an open s u b s e t

t h e r e e x i s t s an open subp

and a r e a l number

for e v e r y

x

E

W

> 1, such

E C,

and

Applying t h e Cauchy i n t e g r a l f o r m u l a , P r o p o s i t i o n 4.2,

for

f

1x1

s p.

we have,

E ~(u;F),

f o r every

x E W,

a r e a l number

C(W)

m = 0,1,...

>

0

.

By h y p o t h e s i s , t h e r e e x i s t s

such t h a t for e v e r y

f

E #(U;F),

and s o

Taking

C(V)

for every

f

-c(w)p 7 we have '

E

#(U;F).

Hence

p

i s p o r t e d by

K.

Q.E.D.

TOPOLOGIES ON SPACES O F HOLOMORPHIC MAPPINGS

PROPOSITION 8.7

5

PROOF:

is 5-balanced, the Taylor series about

f E S(U;F)

every

of

If U

converges to

f E g(U;F),

Let

and let

W = { g E kl(U;F)

for the topology subset of

Vf

C(Vf)

U

of

5

about

f

T

: p(g)

UJ

.


0

E

the

is a neighbourhood of zero

E}

By Proposition 5.2

K

containing

w

be a seminorm on S f ( U ; F )

p

which is ported by the compact set set

97

there is an open

such that the Taylor series

converges uniformly to

f

in

Vf

.

Let

> 0 be such that

Now there exists

for every

m

2

m o E IN

mo

and

E W

if

such that

Applying (*) to

x E Vf.

-

f

T

m,f&

we have

thus m

-t

m

f-T

m,f&

in the topology

m

2

TUJ

mo

.

Therefore f E

We now introduce another topology on DEFINITION 8.8

Let

empty open subsets.

#(u;F).

We denote by U

f

as

HI(U;F)

into

F

Q.E.D.

S((U;F).

I be a countable cover of

of holomorphic mappings of each open set of

-t

m,f,S

for every

'

T

U

by non-

the vector space

which are bounded on

I.

The natural topology of

gI(U;F)

is the separated local-

ly convex topology defined by the seminorms

CHAPTER 8

98

where

V

ranges over

If

PROPOSITION 8.8

I. is a Banach space then

F

with

aI(U;F)

the natural topology is a Fre‘chet space. PROOF:

Since

I

is countable, and the natural topology is

clearly separated,

F

whether or not

WI(U;F)

with this topology is metrizable,

is complete.

be a Cauchy sequence in

e > 0 and

given

m,n E N,

m

I

mo

V E I and

n

there exists

Then

WI(U;F).

mo E N

such that if

m o t then

B

(*> If K

is a compact subset of

argument shows that f o r every

U,

a classical compactness

e > 0 there exists

mo E

if

mo.

[N

such that

(**) Thus

sup l\fm(x)-fn(x)II x€K

‘

E

2

mo

and

n

2

is a Cauchy sequence for the topology gI(U;F)

by the compact-open topology on

the sequence to a function and that

‘

‘

fm’ mEN f

r

in-

C(U;F).

is complete, it follows from Proposition 4.6

F

that

converges in the compact-open topology

E W(U;F).

We must show that

converges to fn’mEN It follows from (**) that

x E V,

m

fm’ mEN

duced on Since


0 , there exists

f(x)

V E I

f E WI(U;F),

TOPOLOGIES ON SPACES OF HOLOMORPHIC MAPPINGS

m

E

such that

N

Therefore

llf(x)ll

sup

XE v

sup

5

a)

I1 and

Let

by non-empty open subsets. if f o r every c

v 1.

If

then given

+

(X)II

G


0

p

bounded, there exists

x

and so, since

#(U;F)

is a

s 13

: pK(f)

T 0

is

-

0-

px c pKl([O,l]),

such that

that is,

X

Hence

E

: x

sup{jlf(x)ll

Ill =

SUP Ilf(x)ll XE

au

We note first that the equation sue Ilf(x)li xc u

= SUP llf(x)ll

XE u

follows f r o m the continuity of

f

in

-

U.

And since

we have

SUP

XE au

Ilf(x)ll

g

SUP llf(x)ll XE 0

Thus we must prove that

We shall divide the proof into three parts. a)

U

E = F = C:

we assume this case as known.

a U c

c,

THE MAXIMUM PRINCIPLE

E = 6, F

b)

117

#

F

an arbitrary normed space,

{O]

.

We

can reduce this to case a) by using the Hahn-Banach theorem. x E U,

Fixing

I/.

Since

x

$ E F'

this theorem yields a

such that

Now, by Proposition 3 . 2 ,

U

was an arbitrary element o f

this completes the

proof in case b). c)

E

F

and

are normed spaces,

F

#

[O]

.

We leave it

to the reader to show that this can be reduced to case b). Finally, the case REMARK 10.3

F =

E

f: 6

f E C(f?;C),

Let 6

-I

is essential, even when the di-

U

is finite.

EXAMPLE 10.2 let

Q.E.D.

is trivial.

The following simple example shows that the hypo-

thesis o f boundedness o f mension of

[O]

be

E = F = 6, U = f(z)

= eZ

.

and

f E #(U;C).

sup

If(z)l = 1

{ z E

Then

C : Re z > 01,

f E g(C;G),

However, since

we have

zeau

< sup I f ( z ) l = +". zE u

and

and s o

a U = [ix:x€!R},

This page intentionally left blank

CHAPTER 11

HOLOMORPHIC MAPPINGS OF BOUNDED TYPE

DEFINITION 11.1 type if gb(E;F)

f

A mapping

f: E

-t

F

is said to be of bounded

is bounded on every bounded subset of

#(E;F)

denotes the subspace of

E.

consisting of the

entire mappings of bounded type. When

E

is finite dimensional, or

entire mapping of

E

into

F

F =

(03,

is of bounded type.

every Examples

5.1, 5.2 and 5 . 3 s h o w that there exist entire mappings which are not of bounded type. PROPOSITION 11.1

F

Let

be complete.

For

f

E

#(E;F),

the

following are equivalent: a)

f

b)

There exists

is of bounded type.

5 E E

such that

lim

I\&

zmf(g)II l/m

= 0.

m-rm

ll&

c)

lim m-tm

d)

There exists

e)

lim

Zmf(x)\l

I/-&

l/m

=

5 E E

dmf(x)l/

l/m

o

for every

such that

= 0

x E E.

lim m-tm

for every

x

1 1 1 dmf(5))I ~

l/m

= 0.

E E.

m-tm

f)

F o r every

Taylor series o f PROOF:

of

f

For

at

x E E

the radius of convergence of the

f

x

x E E, x,

at let

and let

is infinite.

rb(x)

R(x)

be the radius of boundedness

be the radius of convergence of

CHAPTER 11

120

the Taylor series of rb(x) =

states that

at

Then condition a), which

X.

x E E,

for every

+m

is equivalent

5.3, to the statement R(x) =

by Proposition

x E E.

f

Therefore a)

o

f o r every

f m

f).

The Cauchy-Hadamard formula (Proposition 2.1) shows that f) is equivalent to c), and so a) e f)

Q

c).

By Proposition 1.3,

x E E

for every

and

m

Since

N.

mm l / m lim (-)m! = e,

we

m-w

have

e) and b)

c ) e

e d).

Clearly, c )

a

d), and so the proof

is complete if we show that b) * a).

The Cauchy-Hadarnard formula (Proposition 2.1) applied to b) shows that

R(g) =

every bounded subset therefore

f(X)

B M A R K 11.1:

and

g E 6,

X

9 ,

of

and thus

E

r,(S)

=

+=.

Now

is contained in a ball Bp(S);

is bounded for every bounded

X c E.

Q.E.D.

is a sequence of complex numbers,

If

m

C

the power series

a,(z-g)

m

is the Taylor

rn= 0 series at

5

o f a function

lim lamll/m = 0 .

f E #(C;C)

if and only if

We shall see that in the general case the

m+

situation is not quite the same. PROPOSITION 11.2 quence with

Let

F

be a Banach space,

Am E C s ( % ; F ) ,

and

4; E E.

EAm’mEN

a se-

Then the power

m

series

c

A , ( X - ~ ) ~

about

g

is the Taylor series of a

m=O

holomorphic mapping of bounded type f r o m

E

into

F

if and

HOLOMORPHIC MAPPINGS OF BOUNDED TYPE m

PROOF:

C

Suppose that

Am(x-g)

m

is the Taylor series at

m= 0 of an entire mapping of bounded type.

tion a)

121

Then the implica-

e) of Proposition 11.1 and the uniqueness of the

Taylor series shows that

= 0.

lim llAml\ m+w

Conversely, suppose that

Given p > 0,

lim llAml\l’m = 0. m-rm

mo

there exists

m > mO.

mo

N,

5.

x E Bp ( 5 )

Then if

1,

such that

c

llAm(x-5)mll

m

c

h

m= 0

IIA,I//IX-511

+

m0

I/AmI/pm+

C

F

C

2-m

m>m

m=0 Therefore, since

for

l l ~ m l l l l ~ - ~ l ml

c

m>m 0

m=O

=

r: 1/2p

,

m0

W

llAm\ll’m


0

p

f

by

*

CPX(f) Since fm

= c SUP xE x Wb(U)

(m f N)

If(4I

is an algebra over

in ( * ) :

C

THE CARTAN-THULLEN THEOFG3M FOR DOMAINS OF Wb-HOLOMORPHY

for every

f E W,(U),

m E N.

f E Hb(U),

m

149

Therefore

(**I for every

E

m

Letting

!N.

tend to

we

m,

obtain

PO) f E Wb(U).

for every

PJf)

It follows from the definition of

p

that

for every the

f

E Wb(U),

Wb(U)-hull,

gn

hypothesis,

n

E

and s o , by the definition of

N,

5, E

Wb(u) -+ 5 E a U as

n

A

,aU) = 0. #b(’> which contradicts c).

Therefore

dist(X

d)

a).

3

such that for every fIW

which

r:

-

[O,l] -+ V

glw.

f

V

?

E Ub(U)

Choose

be the first point on

aWo

Then there exist non-

W,

and

r(0) =

‘5 n’nEN

image of as

‘

n

-+

tn’nEm

n -+

60,

m.

r,

W”C

U

r(l) =

Then

$ U,

V, V

g E W,(V)

b E V\U, a,

n

for

and let b.

Let

5 E V

n

aU

5

r

which lies in the image o f

(see the proof of Proposition 12.3). N o w let

with

there exists

a E W

be a path with

is not U-bounded,

Wb(’)

and

But by

IN.

which implies that

-t m ,

Suppose that a) is false.

empty connected open sets

n E

for every

n

aWo

be a sequence of points belonging to the

and strictly preceding

5,

such that

5,

-+

5

This can be accomplished by choosing a sequence

c [O,t),

where

and taking We then have

gn

r(t)

= 5,

such that

tn -+ T

as

= r(tn).

In E

Wo c U

for every

n E lN

and so by

.

150

CHAPTER 13

SUP I f({,)/ = m e HOWn€IN ever, by o u r initial assumption, there exists g E Wb(V) such

f E gb(U)

d) there exists

that

fIW

5 E aWo

,

-

such that

g I w . This implies that

,

f

and since

we have

Ig(5)I

which is absurd, since

is finite, and

lim f(5,)=-. n-tm

Q.E.D. DEFINITION 13.3

Let

f E ab(U),

g E aU.

and

be a non-empty connected open set,

U

5

is said to be Sfb-regular at

f

if there exists a pair of non-empty connected open sets V, W,

n

W c U

such that

and there exists

5

Conversely,

5 E V

V,

Ub(V)

g

(which implies that such that

au

if every point o f

= f(W*

f

W c U

which

g

n

= f

Sb(U)

V in

and

V

#

is a gb-singular point of

f.

V, W

of

U,

E

for

W.

f E #,(U)

will denote the set of all

domain of existence if

aU.

Sb(U)

f

separable, and let

U

which are

is said to be a gb-

U

@.

THE CARTAN-THULLEN THEOFLEM (part 11):

E.

g E Wb(V)

there is no

Wb-singular at every point of

of

if

f

is said to be gb-singular o n

This means that for all non-empty open subsets with

U),

is said to be a gb-singular point for

no such pair of sets exist. aU

g(w

#

V

Suppose that

E

is

be a non-empty connected open subset

Then the following are equivalent:

a)

U

is a domain of

b)

U

is a Wb-domain of existence.

#b-holomorphy.

THE CARTAN-THULLEN THEOREM FOR DOMAINS OF kib-HOLOMORPHY

c)

The complement

CSb(U)

of

Sb(U)

in

#,(U)

151

is of

#,(u).

first category in

In order to prove this theorem, we need the following propositions: PROPOSITION 13.3 (Montelts Theorem).

Ef,lnEN

and

is a sequence in

sup Ifn(x)I < xEX,n€[N cular, if sup

-

Wb(U)

If E

is separable,

such that

for every U-bounded set

< -),

Ifn(x)I

X

(in parti-

then there is a subsequence

x€U,nEN

'

which converges uniformly on every compact subfn' nEN set of u to a function f E w,(u).

of

LEMMA 13.4 (Ascoli). and

Ef,3"

Let

M

be a separable metric spaces

a sequence of complex functions on

that the sequence

M.

Suppose

is equicontinuous, and that

sup Ifn(x)( < = for every x E M. Then there is a subsenEN quence of [fnInEN which converges uniformly on every compact subset of

M

to a continuous function on

PROOF OF PROPOSITION 13.3: separable metric space. is equicontinuous in 0

< r < dist(5 ,aU).

U.

Since

E

M.

is separable,

We claim that the sequence

T o see this, let

g E U

By Corollary 4.2 we have

U

'

is a

fn' ng" r E R,

and

CHAPTER 13

152

II x-5 ll r-llx-5 I1

I fn( x) -fn(s ) I

Therefore shows that

is equicontinuous.

Efn’nEN
0 be the distance

Let

c V

r

is contained in

V,

V‘

sup I g l

U.

Br(C)

and

sufficiently close to

be such that

E CSb(U)

From the proof of Proposition 1 2 . 2 ,

n

aU

f

be the connected component

Wo

sufficiently close to

= Bs(q)

Let

W.

containing

there exists of

W,

V, W.

If

E.

be a countable dense subset of

$ i ! !

g

Now choose

and a rational

so that the ball

#

U

sup I g ( < V‘

and

s m,

and let

t

a.

be a suf-

V

ficiently small positive rational number so that Bt(q) c Wo. Let so

W’ = B t ( q ) f

E

and

g‘

= glw’.

f = g‘

Then

Since the family of sets H

# b , m (U,V’ yW’).

defined in this way is countable,

REMARK

13.5:

the above.

w,(u).

W’,

and

(U,V’ ,W‘)

b,m is the union of a

CSb(U)

countable family of nowhere dense sets, o f first category in

in

Therefore

@Sb(U)

is

Q.E.D.

There are various open questions relating to F o r example, what are the complex Banach spaces

f o r which the Cartan-Thullen theorem holds?

same question with

H(U)

in place of

Hb(U).

Also open is the The answer to

the last question is y e s when the Levi problem has solution, f o r example, when

See Dineen [ L

3.

E

is a Banach space with a Schauder basis.

PART I1

THE LOCALLY CONVEX CASE

This page intentionally left blank

CHAPTER 14 NOTATION AND MULTILINEAR MAPPINGS

Unless stated otherwise, U

locally convex spaces, and

IN, IR

E.

subset of

G

and

E

and

F

will denote complex

will denote a non-empty open denote respectively the sets of

natural numbers, of real numbers and of complex numbers. IN*

{1,2,3, ...).

denotes the set and

SC(E)

SC(F)

denote respectively the sets of conand

tinuous seminorms on

E

DEFINITION 14.1

m E N*.

Let

F.

ra(%;F)

E~ = EXE

all m-linear mappings o f

denotes the set of

x...~ E

times) into F,

(m

the operations of addition and scalar multiplication being defined pointwise. Ca(%;F)

Ca,(%;F)

denotes the subspace of

of all symmetric m-linear mappings. (%;F)

'as

for every

means that

xl,...,x

m E E

and every

set of all permutations of

If A E Sa(%;F), element

As

Thus

of

u E Sm, Sm

{1,2,...,m].

the symmetrization of

.Cas(%;F)

being the

defined by

155

A

is the

14

CHAPTER

156 for

x1,x2

,...,

xm E E .

We d e n o t e by

C(%;F)

v e c t o r subspaces of

and

ga(%;F)

Xs(%;F)

respectively the

gas (%;F)

and

c o n s i s t i n g of

c o n t i n u o u s mappings.

m = 0,

For

we d e f i n e

La(OE;F)

a s v e c t o r s p a c e s , and we s e t

:= L(OE;F) := Ss(OE;F) := F

= A

A

for

E

:= SaS(OE;F) :=

~,(OE;F).

Ak+As

i s a pro-

which maps

S(%;F)

I t i s e a s y t o s e e t h a t t h e mapping j e c t i o n of onto

ea(%;F)

Ss(%;F)

onto

= S(%)

write

Sa(’E;F)

t h e spaces

m

for e v e r y

E

= ga(E;F)

gas

ga(”’E),

= Cs(%).

gs(%;C)

and

and

X(’E;F)

,...,1,

E

(c,

s o t h e mapping

then

A(X1

and

f i n e d as follows:

m = 0,

if

Ax

0

we

If E = C,

Ls(mC;F) a r e

A(1,

...,I),

and

i s an isomorphism.

A E Sa(%;F)

Let

= Sas(%),

For if

= X1...l,

.,.,l) E F

A+-A(l,

DEFINITION 1 4 . 2

,..,X m )

we

m = 1,

= L(E;F).

a l l n a t u r a l l y isomorphic w i t h one a n o t h e r .

hl

(%el

For

gas (mC;F), S(%;F)

Ca(mG;F),

F = C

I n the case

[N.

s ~ ( ? E ; c )=

w r i t e for s i m p l i c i t y ,

L(%;C)

Sas(%;F)

and

x E E.

= A E F.

Axm

m E

If

i s deIN”,

m times

...,x ) . Sa(%;F), x1 ,...,xk E 7----L_\

Axm More g e n e r a l l y , l e t

E N,

m,nl,n2,,..’nk n Axl’.

If

.

n .xkk

A

E

and

= A(x,x,

n = n +n2 1

i s d e f i n e d as f o l l o w s :

m = n > 0,

+...+ If

n

E,

s m.

m = 0,

k E IN”,

Then

nl Axl

.,

n .xkk = A.

NOTATION AND MULTILINEAR MAPPINGS

where

each

xi

is repeated

by

where

~

1

Then

, Y ~ ,E ~E

9

n l Axl

times if n m > n, Axl

...x2

5

n n 1 k (AX1 * * * X k)(Y1,***tYm-,)

...

A

is defined

nk times ,. - J

nl in each casey and Ax1

E Xa(m-%;F)

is symmetric if

and

0,

'(X~Y...YX~Y***,~,*~. ,X~,Y~Y***YY,-,)

nk

xk

times

,-A

=

ni >

ni

ni = 0. And if

omitted if

157

is symmetic, and continuous if

...xn

k

k

is

A

continuous. LEMMA 14.1 (Newtonfs Formula). x1

,...,xk E

E,

k E N*,

m,n E IN

A(x 1+X 2+...+x~)~ =

PROOF: A(xl+

A E Las(%;F),

and

n c m.

' ...

the sum being taken over all n = n,

Let

nl!

Then

n

n!

nk!

nl,...,nk E

ax^ l [N

nk k '

f o r which

+...+nk The case

...+xk)

n

then f o r each

n = 0

is trivial.

If

...+x~,...~x1+...+x (y, ,...,ymen) E Em-n,

= A(xl+

m = n > 0, then

k).

If m > n >

0,

In each case, the given expression may be expanded, using the fact that

A

is multilinear and symmetric, and it is easy to

see that the number of occurrences of.:xA the number of permutations of

xl,.. .,xk,

.

.xF

where

is equal to x1

is re-

158

peated

CHAPTER

nl

times,

this number is

...,xk

... n!

nl!

!nk!

14

i s repeated

nk

times.

t h e lemma is proved.

Since

Q.E.D.

CHAPTER 15

POLYNOMIALS

DEFINITION 15.1

m E IN,

polynomial, where

-+ F

P: E

A mapping

is an m-homogeneous

if there exists

A E

la(%;F)

such

that m P(x) = AX

f o r every

P

T o express this relationship between A

P = A.

A: E

each

A

Em

i s the diagonal mapping,

x E E,

P = AoA.

then

we write

If m z 1, and

...,

= (x,

A(x)

If P

A,

and

A

As = A .

It is easy to see that

-I

x E E.

X)

for

is an m-homogeneous poly-

nomial, we have ~ ( x x )= xmp(x)

We denote by

for

pa(%;F)

E

being defined pointwise.

P(%;F)

case

E E,

E

(c.

the vector space of all m-homo-

geneous polynomials from

Pa(%;F)

x

into

F, the vector operations denotes the subspace of

of continuous m-homogeneous polynomials.

m = 0,

we take

= p(OE;F)

pa(OE;F)

space of constant mappings of

E

into

isomorphic with the vector space P a (%;c)

= pa(%)

Pa(m@;F)

and

and 6'("C;F)

as vector spaces.

p(%;C)

F.

to be the vector

F, which is naturally

When

= p(%).

In the

F =

When

(c,

we write

E =

(c,

are both naturally isomorphic with

F

CHAPTER 15

160

REMARK 15.1:

...,Em

If El,

m E IN*,

convex spaces,

Xa(E1,.

..

,Em;F)

F

and

are complex locally

then the vector spaces

..

L' (El,.

and

,Em;F) of m-linear mappings

Em into and of continuous m-linear mappings of El x...x m Em);F) and respectively, are subspaces of Pa( (El x...x

P(m(E1 x...x

Em);F)

To see this, let m 1 If X1 = (xl,.-.,Xl), x2 = (x2,

respectively.

1

A E ga(%l,...,Em;F).

,..., Xm =

1

(xm

,...,

m xm) E El x . . . x

Em )" + F

B: (El x . . . ~

X1 = X2 =...=

E m , define

B E Ca(m(E1

x...~

Em);F),

and that

...,xm), then = BXm = A(x1,x2, ...,xm)

Xm = X = (xl, 6(X)

for every

...,xrn)

by

It is easy to see that if

F,

...,xm) E

El x...x

(xl,

Em. Thus statements con-

cerning polynomials may also be applied to multilinear mappings. LEMMA 15. 1 (The Polarization Formula). m

E

N*,

and

,...,xm E

x1

1 A(x~,x~,...,x~) =- m! 2

c

E,

If

A E Xas(%;F),

then

C1C2...E

m ;i(e 1x 1+E 2x 2+...+

the summation extending over all possible values of

c1 = r t l , . . . , e m = PROPOSITION 15.1

fl.

The mapping

A E Sa(%;F),-

E Pa(%;F)

EmXm),

2

161

POLYNOMIALS

is linear, and induces an isomorphism between the vector spaces

‘as

(%;F)

and

ba(%;F),

spaces

LS(%;F)

and

PROOF:

The case

m = 0

ping

A €

ea(%;F)

I--*

and between the vector

ra(%;F),

fi

for every

is trivial.

Let

m

m

E

2

1.

Since

As = A ,

the mapping induces a surjective mapping of

onto

Pa(%;F).

The polarization formula shows

Ls(%;F)

that this mapping is injective and maps P (%;F).

Let

aa(E;F)

and

3(E;F)

ly the vector space of all mappings of

E

vector space o f all continuous mappings of

denote respective-

Za(E;F),

m E

[N,

Thus, a mapping

E

and

is called a polynomial from

P: E -+ F

P = P pa(E;F)

P(E;(C) =

into

F.

When

An

Pa(%;F) E

into

of

F.

+

Pl

+...+

such that

’ m

denotes the vector space of polynomials from

F, and P(E;F) nomials.

and the

is a polynomial if there exist k = 0,1,...,m,

Pk E ba(%;F),

F

into

element o f the algebraic sum of the subspaces

E into

denotes the subspace of continuous poly-

F = C,

we write

Pa(E;C) = P,(E)

and

P(E).

PROPOSITION 15.2 of the families

1Y.

onto

Q.E.D.

DEFINITION 15.2

E D

Pa(”E;F).

A

.Cas(%;F)

m

The map-

is easily seen to be

€ ea(%;F)

linear and surjective, from the definition of A

N.

Pa(E;F)

and

(Pa(%;F)jmElN

P(E;F)

are the direct sums respec tive-

162

CHAPTER

PROOF:

We show first that the family of subspaces

;Pa(E;F), m E

of Pk E

15

Pa(%;F),

is linearly independent.

N,

k = 0,1,...,m,

we must show that induction.

Po = P1 =...=

m-1,

m

2

1.

0,

Pm = 0. We prove this by m = 0; we assume

The assertion is trivial if

its truth for

Thus if

and

N,

+...+Pm =

Po + P1

(1)

m E

Pa(%;F)

Now condition (1) implies that

m

c

hm

and

= 0

P,(x)

k=0

m

c

(3)

h

k

Pk(X) = 0

k=0

x E E,

f o r every

1 E

Subtracting ( 3 ) from ( 2 ) , we have

(c.

(hrn-1)Po(X) +...+

(4)

x E E,

f o r every

Am # Xk

for

applied to

X E C.

k = O,l,...,m-1,

To show that m E N,

Pk E Pa(%;F),

so that

then our induction hypothesis

P(E;F)

0.

is the direct sum of the family

P E b(E;F).

k = 0,1,..., m,

We must show that

Pm,l =

Pm = 0.

let

P = p0 + p1

trivial f o r

C

0

(4) yields

And from (1) we have

(a)

1 E

I f we choose

Po = P1 =...=

P(”E;F),

=

( A m -1 m-1 )Pm,1(X)

+...+

Then there exist

m E N

such that

pm.

Pk E P ( % ; F ) ,

k

0,1,..., m.

I

This is

m = 0; we assume the truth of this assertion

POLYNOMIALS

for

F r o m (a) we obtain

m-1.

(b)

x E E,

f o r every

k = O,l,...,m-1.

?,

?, E G

Fix

x,

m-1

)Pm-,(x)

such that

Xm

#

h k for

it follows by the induction hypo-

Po,P1,...,Pm,l

thesis that

( X -1

Since the left hand side of (b) is a con-

tinuous function of

Pm

E C.

rn

+. ..+

= (Xm-l)Po(x)

X"P(x)-P(?,x)

are continuous.

is also continuous.

Hence, from (a),

Q.E.D.

REMARK 15.2:

The preceeding proof shows that the following

is true:

m t N,

for

if

k = O,l,...,m,

Pk E P ( % ; F )

unique

rn E N

The polynomial

REMARK

-1

15.3:

P

#

0

and

P = or

#

and

15.4

a neighbourhood

SUP eCf(x)3 xE v


1.

Therefore

IIqmII

Ml/m r

5

rn

r > 1.

s M/rm

E N",

m

m E IN";

Then

E W(E)

Ilcpmll

hence

r;

but this implies that

m E IN",

which is absurd.

have shown that for every closed ball f

is unbound-

Suppose this were not so.

for every

for every

there exists

cp

We claim that

Using the Cauchy inequalities we have

for every

j m l < r s M'

205

Thus we r > 1,

with

Er(0)

gr(0).

which is unbounded on

It

follows easily that the same is true of the closed balls

-

(0) where r1 > 0, and hence, by translation, it follows 1 that for every 5 E E and every r > 0 there exists

Br

f

E W(E)

which is unbounded on

LEMMA 24.2 let

X

Let

E

be a complex locally convex space, and

be a subset of

E

which is bounded and not precompad.

Then there exist a sequence

6 > 0 x0

such that, if

,...,xm '

PROOF:

Since in

Fix

6,

then

X

X,

Q.E.D.

Er(f).

Sm

U(U-X,+~)

EXmlmEN

in

x, u

is the subspace of 2

6

for every

€ CS(E),

E

and

generated by

u E Srn

,

m E N.

is not precompact, there is a sequence

a E CS(E)

0 < 6 < y/2.

mensional subspace of

and

y > 0, such that

We claim that if

S

E,

p 6 IN

there exists

is a finite disuch that

206

C I U P T E R 24

a(u-t )

b

2

P

u E S.

f o r every p E N,

then f o r every a(up-tp) < 6 .

Since

Suppose this were not so;

there exists (a(tp))

u

P

E S

such that

is bounded,

[U(up)]

P€I N also bounded, and s o , since po < p1 1

v, x

such that

E 0,

1x1

h 03

= €4
1 such that the set

p

U,

is contained in

and

f(T)

is bounded in

F.

T o see this, consider the continuous mapping G:

(X,t) E CxE If

( A ( i 11

(l-x)%+kt E E.

I---

x E U

5:’

the compact set

is contained in

there is a neighbourhood is bounded in

F.

Since

U

4;

U‘ G-’(U’

.

M = {(l-X)s+),x :

Since

M

of

)

f

in

X

E CC,

is locally bounded,

U

such that

f(U’)

is a neighbourhood of

223

COMPACT AND LOCAL CONVERGENCE O F THE TAYLOR SERIES

gl(0) x {x]

in

U

1x1

5

in p

and p

and

(I:

> 1

x E

there exist a neighbourhood (l-X)S+kx E U'

such that

t E W.

if

and

Since

V

series of

REMARK

t E W,

27.1:

series of

f

p E CS(F),

at

5

If

f E P(E;F)

8,

it follows that the Taylor

converges to

at any point o f

and E

f(t)

E,

whether or not

example shows that

f

f

is separated, the Taylor

F

contains only a finite

is locally bounded.

is not seminormable then the identity mapping

REMARK 27.2: every

f

uniform-

A simple

need not be locally bounded:

not locally bounded, but

t E W.

uniformly in

number of non-zero terms, and hence converges to ly in

(I:,

then

is independent of f

X E

x

of

This establishes o u r claim.

It follows from Corollary 26.1 that if m E LN

W

If E

I: E -+ E

is

I E P(E;E).

There are two interesting situations in which

f E #(U;F)

is locally bounded;

finite dimensional, the other where

F

one is where

E

is

is seminormed.

We have the following partial converse to Proposition

27.3: PROPOSITION 27.4 f E #(U;F).

converges t o f

Let

E

be seminormable,

If the Taylor series of f

f

F separated, and

at a point

uniformly in some neighbourhood o f

is locally bounded at

5.

< 5,

of

U

then

CHAPTER 27

224

PROOF:

E

Since of

V

bourhood

is seminormable, there is a bounded neigh-

5

U

in

such that

m

uniformly in

x E V.

Continuous polynomials are bounded on

bounded sets, and hence

p E CS(F).

Therefore, since

bounded on V.

27.3:

REMARK

is bounded on

B o f

V

for every

is independent o f

8,

f

is

Q.E.D.

E

The requirement that

Proposition 27.4 is essential, for if the identity

V

I: E

-I

E

be seminormable in

E

is not seminormable,

is not locally bounded.

We now present two examples of a holomorphic mapping

g

whose Taylor series at a point

% .In Example

ly in any neighbourhood of

are Fdchet-Monte1 spaces if

E

27.2

EXAMPLE

g E W(C),

Let

If I

I

is a normed space and

27.1 Let

f

5.

c > 0

E E

j)

F

a Fsechet-Monte1 space.

E = F = CI

by

does not converge uniform-

Suppose this were not s o . there exists a finite subset

j€I

J

of

5

converges uniformly in

and

F

f E E

at any point

5 = (5

and

is not a polynomial, the Taylor

g

ly in any neighbourhood of

I

E

is countable, and i n Example

f E #(E;F)

and define

Then, for some

27.1

I be a non-empty set, and let

is infinite, and

series of

does not converge uniform-

such that the Taylor series of

v =

(x

I

f

at

.

COMPACT AND LOCAL CONVERGENCE OF THE TAYLOR SERIES

for

j E J].

p(Y) = l Y k l

k E I\J

Let

Y = (Yj)j~~ E F.

9

Taylor series of

5 ,

at

f

dimension.

Let

am = 0

for

m

F = CR,

xk E C .

n,

m

2

5

n,

which means that

contrary to our hypothesis.

E

be a complex normed space of infinite

Then, by Proposition 24.4, there exists g E #(E)

which is unbounded on some bounded subset of gn E # ( E ) ,

n E N,

and let

by

gn(x)

f E #(E;F)

Suppose that for some

= g(nx),

E.

x E E.

We define Now let

be given by

r > 0

the Taylor series of

the origin converges uniformly in the open ball, radius

r.

to the

a = m m!

where

C,

for every

is a polynomial,

EXAMPLE 27.2

E

1

But this implies that g

xk

such that

5

8

Then, applying

n E N

lam(xk-5k)ml

be defined by

we find that the series

is uniformly convergent for Hence there exists

p E SC(F)

and let

225

f

V,

at of

Then

converges uniformly to

gn

in

V

for every

n E N.

This

nV

for every

n E

Since

implies that

converges uniformly to

g

in

[N.

CHAPTER 27

226

every continuous polynomial is bounded onthe bounded set n 6 N,

for every

n E N.

every set of

E,

it follows that

Therefore

is bounded on

g

nV

nV

for

is bounded on every bounded sub-

g

which contradicts o u r choice o f

Therefore

g.

f

is not locally bounded at the origin. REMARK 27.4:

U,

in

Given

5 E U, a neighbourhood

E,

and a separated space f E #(U;F)

exists

F =

mension o f

E

If E

V

#

F

r~ E

(Ea ) ‘

f

there exists

V.

a l E CS(E)

such that

such that

cp

B

(0) c V.

Since E

Ul,l

a 2 E CS(E)

c E IR+.

is false f o r all

2

g

E

whose Taylor series at the origin does not c o n -

Choose

cUl

S

is separated, then

{O]

o f the origin in

is not seminormable, there exists a2

does not con-

i s not seminormable (hence the di-

is infinite) and

verge uniformly in PROOF:

4,

A solution to this problem when U = E ,

for every neighbourhood f E #(E;F)

g

implies a general solution.

(r:

PROPOSITION 27.5

V.

of

one can ask whether there

whose Taylor series at

verge uniformly in

C = 0 and

F,

V


0, such

P 1 , ...,pk

for every

..,xk E

be

lhjl

E 6,

P j

Then n

..%!

1

nl!.

-

N",

E

U,

F

Let

dmf(g)xl

f(5+hlX1+.

1

(2ni)

k

...xkn .

l

x1

-

-

.

.+hkXk)

nl+l

*/hjl"Pj

k

nk+ 1

dh l...dXk

"'h,

1sj s k REMARK 28.1: The proof of Proposition 28.1 is similar to the

proof of Proposition 25.2, the single integral in the latter case being replaced by a multiple integral.

Alternatively,

Proposition 28.1 can be obtained by repeated application of Proposition 25.2.

Proposition 25.2

Proposition 28.1 in which where

nl =...=

k = m,

COROLLARY 28.1 m E N*,

x1

,...,xm E

g + xlxl +...+ j = l,...,m.

Let

xmxmE Then

k

is the extreme case of

= 1. The other extreme case,

nm = 1

is as follows:

F

be separated,

E

and

u

p1

,...,pm

for every

xj

5

f E a(U;F),

E U,

> 0 such that E C,

IxjI

4

p j

9

9

2 30

CHAPTER 2 8

Xx = XIXl and if

+...+

IX,I

...,nk) E

n = (nl,

If we write

REMARK 2 8 . 2 :

'kXk

= pl,

Xn+l

'

nl+l

= X,

...,I

hkl

= pk

...Iknk+l ,

i s written

k

N

,

d ( n ) = k,

dX = dX,.

1x1

= p ,

..dx,

,

then

integral formula given in Proposition 2 8 . 1 becomes f

a form similar to Proposition 2 5 . 2 . REMARK 2 8 . 3 :

sition 2 8 . 1 ,

Cauchy inequalities can be derived from Propoo r from Remark 2 8 . 2 ,

in the same way as the

Cauchy inequalities of Chapter 2 5 were derived from Proposition 2 2 . 2 . REMARK 2 8 . 4 : A E Ss(%;F),

If we apply Corollary 28.1 to

5 =

with

and

0

U = E,

f =

i,

where

we obtain a new po-

larization formula:

A(xl,

...,xm) =

f1

1

m!(2ni)m

m xm )

2(X1Xl+...+h

.$ jl=l i

(XI*

-__ 2

dX1. .dX,

*Xm)

1s j 4 m

where we have taken

p1

=...=

P,

= 1;

in this case, it can

be shown easily by a change o f variable that any choice of

t

THE MULTIPLE CAUCHY INTEGRAL AND THE CAUCHY INEQUALITIES

> 0 gives the same value for the integral.

p l , ...,pm

231

Like

the original polarization formula (Lemma 15.1), we can use this formula to obtain an estimate f o r a(xj)

r;

1

REMARK 28.5:

to

f =

i,

j = l,...,m,

for

BIA(xl,

...,xm)];

then

More generally, if we apply Proposition 28.1 where

A E Xs(%;F),

m E N",

5

= 0

U = E,

and

we obtain another new polarization formula:

n

k k -

. -

if

1 k

m! ( m i )

!Ixjl=l 1sjsk

n +1 1

1,

nk+ 1

"'1,

dX l...dXk

.

This page intentionally left blank

CHAPTER 29

DIFFERENTIALLY STABLE SPACES

DEFINITION 29.1

F

F

Let

be a complex locally convex space.

is said to be differentially stable if for every

every non-empty open subset recall that

F

H(U;F) = # ( U ; F )

U

of

n Fu.

E

and

E, #(U;F) = H(U;F). Thus if

F

We

is separated,

is differentially stable if and only if, for every

f E H(U;F),

we have

5 E

xl,

and

U,

dmf(g)(xl,

...,xm E

E.

...,xm) E

F

for every

m E N*,

Bearing in mind the proof of the

Cauchy integral formula (Proposition 2 5 . 2 ) , it suffices to

E = C,

consider the case f(m)(0)

E F

for every

U = B1(0),

and to show that

f E #(U;F).

It is easy to see that a complete space is differentially stable.

The following proposition describes two more ge-

neral conditions, either of which guarantees differential stability. PROPOSITION 29.1

F

is differentially stable in each of the

following cases: 1)

F

2)

The closed convex balanced hull of every compact sub-

is sequentially complete.

set of

F

PROOF:

It suffices to consider the case where

is compact.

233

F

is separated.

CHAPTER 29

234

Let

f

that

E H(U;F),

5 E

(l-X)g+Xx E U

x E E.

and

U,

1x1

X E C,

for

s p.

> 0

p

Choose

such

Then, by Propo-

sition 25.2, --1

f = % 12 \

dmf({)xm

for every

d?,

~

m!

Xm+1

m E N.

-IXI=P

Since

f(C+Xx)

E F

5

for every

,

A , x,

conditions 1) and 2)

each imply that the integral which appears in this equation takes its values in f

Hence, by the polarization formula,

F.

E SI(U;F).

Q.E.D.

PROPOSITION 29.2 the space

WF

F

is differentially stable if and only if

= (F, a(F,F'))

is differentially stable.

In order to prove this proposition we need the following result from the theory of weakly holomorphic mappings. LEMMA 29.1

Let

and only if PROOF:

@of

f: U + F.

E = C,

and

E H(U)

f o r every

f

E

H(U;F)

suppose that f

F

@of E H ( U )

is separated, and let

is continuous at

then f o r

z E Bp (g )

5.

If

for every

5 E U.

g E z(U)

r

-

1

I t-i: I =P r

and hence

t-z

dt,

T o prove

JI E F',

We show first

and

we have

g(z)

if

JI E F'.

The necessity of this condition is obvious.

its sufficiency, suppose that

that

Then

gp(5) c

U,

235

DIFFERENTIALLY STABLE SPACES

Theref ore

g = $of,

Applying t h i s t o

$ E F',

Since t h i s holds f o r every

f o r every

p E CS(F).

f

,

we have

i t follows that

It i s e a s y t o s e e t h a t

i s bounded

f

@ [ f ( z ) - f ( r ) ] -+

on e v e r y compact s e t , and hence Therefore

(I E F'

where

0

as

z-5.

i s continuous.

We have shown t h a t

r

f o r every

$ E F'.

Since

is s e p a r a t e d , i t f o l l o w s t h a t

F f

where

for

z E Bp(5).

I t-g 1

= p

i s uniform f o r

We have

and

It-51

z

E = p

Bp (

5).

and

Furthermore, z

E

Er(g),

t h e convergence

where

0

< r < p.

23 6

CHAPTER 2 9

I t follows that

,

uniformly on every compact subset of

W(U;i),

f E

Therefore

follows that

and since

f E H(U,WF).

$ o f

F

Suppose that

it

is differential-

Therefore

d:

and let

Jr E F’, and

f o r every

H(U)

f E H(U;V) = W(U;F).

f E w(u,WF).

F,

into

Q.E.D.

so,

Since the topology

F, #(U;F) c s J ( U , W F ) ,

is weaker than the topology of

and hence

U

be a non-empty open subset of

Then

by Lemma 29.1,

u(F,F’)

U

Let

maps

f

f E H(U;F).

PROOF OF PROPOSITION 29.2:

ly stable.

Bp(

E = C,

Let

and

However, f o r

i s constant in aneighborhood

of the o r i g i n , but I/fll i s not constant i n E ; Ilf(0)ll i s the minimum of

11 fll

i n E. Example 33.1 shows t h a t , i n c o n t r a s t t o t h e case F = C,

we cannot, i n g e n e r a l , conclude i n P r o p o s i t i o n 3 0 . 2 i s constant i n a neighbourhood i s constant i n

another c o n d i t i o n on

nected,

such t h a t

Y E F,

Y

F

Let

lowing property:

5,

but only t h a t I ( f ( (

F: be a normed space, l e t

f € W(U;F).

and l e t

of

f

This conclusion i s v a l i d if we impose

V.

PROPOSITION 33.3

V

that

q E F,

if

II$I/

= 1,

f 7,

))YII

$(q) = L

s 1.

(Iqll = 1, and

Then, if

has a l o c a l maximum a t a p o i n t

F

Suppose t h a t

g

be con-

has t h e f o l -

there e x i s t s

IQ(y)I < 1

IIflI: in

U

for every

x E U -llf(x)/j U,

f

Q E F’

E IR

i s constant i n

U .

PROOF:

If

[If(g)ll =

0,

the r e s u l t i s t r i v i a l .

Suppose t h a t

CHAPTER 33

252

IIf(g)II

>

0.

assume that

# f(5).

Ilf(x)ll

f

//f(s)\l= 1.

Let

= 1,

$CfG)I y

Multiplying

5

V

x E V,

Therefore

I$ofl

be such that

Y E F,

when

x E V.

f o r every

has a local maximum in

x E V

Thus f o r

Ilf(x)ll

r~

tinuation,

33.1:

f

V,

f(x) =

U

in $of

= 1,

such that

E U(U)

and,

U,

and it follows

$of

is constant in

= $ff(g)]

+[f(x)]

1, which implies that

is constant in

REMARK

we have

//$I/

IlYII s 1,

Then

f r o m the proof of Proposition 33.2 that U.

5

be a neighbourhood of

l\f(g)/I = 1

for every

$ E F’

I$(Y)I < 1

and

Let

by a suitable constant, we may

f(g).

= 1, while Therefore

f

and so, by uniqueness o f holomorphic con-

must be constant in

U.

This proposition applies when

space, and in particular when

F = CC.

Q.E.D. F

is a Hilbert

CHAPTER 34

PROJECTIVE AND INDUCTIVE LIMITS AND HOLOMORPHY

PROPOSITION 3 4 . 1

Let

{F.] 1

ly convex s p a c e s , l e t i E I,

each

let

F

pi:

F

be a f a m i l y o f complex l o c a l -

i€I

be a complex v e c t o r s p a c e , and f o r -I Fi

be a l i n e a r mapping.

Let

F

be g i v e n t h e p r o j e c t i v e l i m i t t o p o l o g y d e f i n e d by t h e map-

i €

Pi,

pings

I.

If

i s a non-empty

U

E,

complex l o c a l l y convex s p a c e into

F,

then

f E H(U;F),

If

C o n v e r s e l y , suppose t h a t let

ii

i s a mapping o f

f

i f and o n l y i f

U

E H(U;Fi)

piof

i E I.

f o r every PROOF:

f E H(U;F)

and

open s u b s e t of a

i t i s obvious t h a t piof

be a c o m p l e t i o n o f

E H(U;Fi).

Fi.

Pi

E H(U;Fi).

For each

G =

If

o f

Fi

i E I,

and

i€ I

Fi,

= i€I g : U -t G

Then

then

6

i s a completion o f

Now d e f i n e

p:

F + G

p : Y E F+.r ( p i ( Y ) ) S = p(F).

Then

Since t h e topology o f

logy o f

Define

by

g E H(U;G).

and l e t

G.

S

F

g(U) c S ,

i EI

by

E G,

and hence

g E H(U;S).

i s t h e i n v e r s e image o f t h e topo-

u n d e r t h e mapping

p,

253

and

g = pof,

it follows

254

CHAPTER 34

that

f 6 H(U;F).

Q.E.D.

In the proof of Proposition 34.1 we have made

REMARK 34.1:

if

use o f the following fact:

p:

F

surjective, and the topology of

F

P

pof

of the topology of

then

G,

-t

G

is linear and

is the inverse image under

E

H(U;F)

if and only if

f 6 B(U;F).

COROLLARY

34.1

If F

is a complex locally convex space,

COROLLARY

34.2

If F

is a complex locally convex space,

and if

denotes the space

WF

,

u (F,F‘ )

F

with the weak topology

then

H ( u ; W ) = {f:

u + F

:

$ o f

E W(u)

for every

$ E F’].

Corollary 34.1 follows immediately from Proposition 34.1, and Corollary 34.2 follows from Corollary 34.1. O u r next example shows that it is not, in general,

possible t o replace the symbol Thus Proposition valid if EXAMPLE

H

by

a

in Corollary 34.2.

34.1 and Corollary 34.1 are not, in general,

H

i s replaced by

34.1

Suppose that

Then, by Proposition 29.2,

W. F

is not differentially stable.

WF

is not differentially stable.

It follows that there exists a non-empty open subset such that

H(U;WF)

be false for

#(U;WF).

C

PROPOSITION 34.2

Let

#

a(U;WF).

‘Em’ me”

locally convex spaces, let

E

U

of

Hence Corollary 34.2 must

be a sequence of complex be a complex vector space,

25 5

PROJECTIVE AND INDUCTIVE LIMITS AND HOLOMORPHY

m E IN

f o r each

om:

let Pm

--

Em

-i

Pm+loum

E

and let

let

Ern+ 1

p,:

Em

E

-t

be a compact linear mapping such that

f o r every

E

m

Uo

and

E,

let

REMARK

Urn =

If

a mapping of

only if

fop,

34.2:

m E N,

-l(U), Pm

If U

is an open

and suppose that

is a complex locally convex space,

F

U

m E IN.

p,,

into

F,

f E #(U;F)

then

E H ( u ~ ; F ) for every

if and

rn E N.

T o say that the linear mapping

Om: Em

is compact means that there exists a neighbourhood the origin in in

Em+l.

Pm(Em),

me N be given the locally convex inductive limit to-

is non-empty. f

u

Suppose that E =

IN.

p o l o g y defined by the mappings

subset of

be a linear mapping, and

Em

such that

Gm(Vm)

+

Vm

Em+l of

is relatively compact

The compatibility condition in the statement of

the proposition concerning the mappings

Urn

and

p,

states

that the diagram

commutes f o r every implies that

Urn

m E N.

The condition

#

= p,l(U)

for every

Q

m E N.

This fol-

lows by induction from the relation:

To prove (*),

let

xm = p

-1

(x),

= x E U, which implies that and hence

um(xm) E P,,~

(u).

where

x E U.

Then

P~+~[U~(X~) = ]pm(xm)

pm(xm) =

= x E U,

25 6

CHAPTER 34

PROOF OF PROPOSITION 34.2:

#(u;F),

f E

If

it is clear

that

fop,

E B(Um;F)

for every

m E IN. Conversely, suppose

that

fop,

E B(U,,,;F)

for every

m E IN.

is finitely holomorphic.

f

E

Suppose that

Em

for which

E.

contains a subspace

and

S.

Sm

such that

i

zi = um-l~um-2

E Em

for every

yi = pm(zi). nerated by

{zl,

of

S Sm

is separated. with

...,nk].

Choose

# 0

m E N

we have

,...

is linearly independent, and

Sm

denotes the subspace o f

then

S.

p,

Furthermore,

p,

Therefore

if

U flS

#

Em

ge-

is an isomorphism of the

p,

is continuous

is also a homeomorphism

S.

pm(UmnSs,)

N

Then

Furthermore, we have

and s o

ni E

be

i = 1 ,k. Then i i and { zl,., , z k ] is linearly inde-

zk],

and

[ nl ,

is a

(yl,...,yk]

there exists

un (xn ) ,

...,

Sm

Let

p,

.

i,

Hence, if

vector spaces and

0 . . . 0

{yl,...,yk]

pendent, since

S.

with

i = l,...,k

and similarly, for each

such that

yi = pni(xni).

greater than the maximum of

Let

Sm

i = l,.,.,k

For each

xni E E ni

be a finite

We claim that there exists m E N

topological isomorphism of a basis for

S

is separated, and let

dimensional subspace of

z

We show first that

@.

Now, since

257

PROJECTIVE AND INDUCTIVE LIMITS AND HOLOMORPHY

fop,

E #(U,;F),

U

(fopm)/UmnSm

(foPm)/UmlSm = (f/UlS)o (pm/Um1Sm),

But 3

we have that

n

S

is bijective.

and it follows that

W(umnsm;~).

E

and

pm/UmlSm: UmlSm

Therefore

f

is holomorphic in

U

n

S.

Hence

f

is finitely holomorphic. We complete the proof by showing that bounded in where

may assume

0 6 U,

It is not difficult to show that we

and that it suffices to prove that

locally bounded at

m E N.

Then

Em+l.

such that

om(Vm)

-t

o,(V,)

is relatively compact

is bounded, and so, if

neighbourhood of the origin in

Em+l,

Wm+l

there exists

is a

1 > 0

such that

that i s ,

-1

is a neighbourhood of the origin in

om (Wm)

Therefore

om

0.

Em.

is continuous.

We now assume bounded at

is

Em+l is continuous for Vm be a neighbourhood of

om: Em

T o see this, let

the origin in Em in

f

0.

We show first that every

is amply

F o r this, it suffices to consider the case

U.

is semnormed.

F

f

Let

the origin in

E,

-1 U h = p, (U’);

each

0 E U,

U‘ c U

and prove that

is locally

be a closed neighbourhood o f

and consider f o r each Uh

f

m E

(N

the set

is a closed neighbourhood of the

CHAPTER 34

25 8

origin in

E m , and

U k c Urn= p,l(U).

,

Since

fm = f o p

Let

m

m f IN.

zero, there is a neighbourhood and real numbers

Since

uo

Let

Vo = V b

and

ao(Vo)

_

oo(Vo)

_

Eo

Vb

of zero contained in

c

n vh

I

Uo(V’A)

. Then

c ao(Ub)

_

E

so

~ , +a,(~,) ~

fm+laam

,

is compact in

m f IN

such that -1

P,+~(U’),

am(Vm) and

f Vm} = M m < M.

we then have

suPEllfm+l(Y)ll

= sup{llf,(x)ll

: y

E

El

we have a convex neigh-

Em

u,’,,+~= : x

Vo c Tfo,

Also, since

oo(Vo)

c

sup[l)fm(x))/

fm =

El ; hence

_

of the origin in

Vm

compact in

and hence

9

is compact in E1’

which is closed in

U;,

C

~,(v0),and

bourhood

u;

such that

such that

NOW suppose that f o r

Since

is locally bounded at

c a o ( U b ) c U; c U1 = p;’(U).

_oo(Vo)

M

fo

is compact, there exists a convex neighbourhood

of the origin in

V’L

_

and

Mo

Also,

Om(Vm)3

: x E Vm}

= Mm
0 such that Mm+l =

E

< M.

M,+E

Since

compact, there exists a convex neighbourhood origin in Since

Em+l

Em+l

-

and

of the

is compact in E

m+2 '

there exists a closed neighbourhood

origin in

-

(0))

um+l(v;,+l

Vk+l(0)

is uniformly continuous on the compact set

fm+l

GmTm),

such that

is

'm+l

=

V>+l(0)

of the

such that

_U __

_

(Y+{VL+~(O)O)

c

u;+~

9

where

~ ~ m ( V m )

[V;+,(O)] Let

0

is the interior of

Vm+l(0) = [Vk,,(O)

..