COMPLEX ANALYSIS IN LOCALLY CONVEX SPACES
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NORTH-HOLLAND MATHEMATICS STUDIES
57
Notas de Matematica (83) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester
Complex Analysis in Locally Convex Spaces
SEAN DINEEN Department of Mathematics University College Dublin Belfield, Dublin 4, Ireland
NORTH-HOLLAND PUBLISHING COMPANY
- AMSTERDAM
NEW YORK
OXFORD
Q
North-Holland Publishing Company, 1981
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: 0444863192
Publishers NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD Sole distributors forthe U.S.A.and Canada
ELSEVIER NORTH-HOLLAND, INC. 5 2 VANDERBILT AVENUE, NEW YORK, N.Y. 10017
Library of Congress Cataloging in Publication Data
Dineen, Se&, 1944Complex a n a l y s i s i n l o c a l l y convex s p a c e s . (North-Holland mathematics s t u d i e s ; 57) B i b l i o g r a p h y : p. In cl u d es index. 1. Holamorphic f u n c t i o n s . 2 . L o c a l l y convex s p a c e s . I. T i t l e . 11. S e r i e s . QA33~D637 515.713 81-16885 ISBN 0-444-86319-2(U.S.) AACB
PRINTED IN THE NETHERLANDS
To Carol, Deirdre and Stephen
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FOREWORD
The main purpose of this book, based on a course at Universidade Federal do Rio de Janeiro during the summer of 1978, was to provide an introduction to modern infinite dimensional complex analysis, or infinite dimensional holomorphy as it is commonly called, for the graduate student and research mathematician. Since we were more interested in communicating theqaturerather than the scope of infinite dimensional complex analysis and since it was clearly impossible to write a comprehensive account of the whole theory for such a short course we were obliged to limit our range and choose to develop a single theme which has made much progress in recent years and which exemplifies the intrinsic nature of the subject, namely the study of locally convex topologies on spaces of holomorphic functions in infinitely many variables. In retrospect, we feel we have provided a reasonably comprehensive view of the topoZogica2 nature of the theory, but have neglected to a large extent the geometric, aZgebraic and differential aspects. A l l of these aspects are equally important, interrelated and indeed a proper appreciation of the portion of the theory outlined in this book is not possible without an overall view of these other topics. To partially compensate for this deficiency we have written Appendix I in which we outline developments in other areas of infinite dimensional holomorphy. The main prerequisite for reading this book is a familiarity with the elements of functional analysis. An acquaintance with several complex variable theory is useful but not essential. In Appendix 11, we provide a resumk of results from these two areas for the non-specialist. On the other hand, much of the functional analysis that we use is not of the standard linear kind, but arises from the nature of infinite dimensional holomorphy and sothis text may also serve for functional analysts as a fresh view of vii
...
Foreword
Vlll
t h e i r own s u b j e c t . The p r i n c i p a l t o p i c d i s c u s s e d i n t h i s book i s t h e l o c a l l y convex s p a c e s t r u c t u r e s t h a t may b e p l a c e d on t h e set o f a l l holomorphic f u n c t i o n s d e f i n e d on a domain i n a l o c a l l y convex s p a c e .
To be s p e c i f i c , we are
p r i m a r i l y i n t e r e s t e d i n t h e p r o p e r t i e s o f , and r e l a t i o n s between, t h e T
and
T~
topologies.
T
0
i s t h e compact open t o p o l o g y ,
t o p o l o g y o f l o c a l convergence and c o u n t a b l e open c o v e r i n g s .
T&
T~
T
is a
0’
i s t h e t o p o l o g y dominated by t h e
Many o f t h e o t h e r t o p i c s d i s c u s s e d a r i s e from
avenues opened up by o u r i n v e s t i g a t i o n o f t h e s e t o p o l o g i e s . Our arrangement o f t h e m a t e r i a l i s a s f o l l o w s .
I n c h a p t e r 1, we
d i s c u s s polynomial mappings between l o c a l l y convex s p a c e s .
This, hopefully,
p r o v i d e s a g e n t l e i n t r o d u c t i o n t o i n f i n i t e dimensional complex a n a l y s i s s i n c e a holomorphic f u n c t i o n i s l o c a l l y a sequence o f homogeneous polynomi a l s which s a t i s f i e s c e r t a i n growth c o n d i t i o n s .
Furthermore, t h e t h e o r y o f
homogeneous p o l y n o m i a l s , which is e q u i v a l e n t t o t h e t h e o r y o f symmetric m u l t i l i n e a r forms, i s i n t e r m e d i a t e between t h e t h e o r i e s o f l i n e a r mappings and holomorphic mappings and t h e p r o p e r t i e s o f polynomials i n t e r v e n e a t various stages i n l a t e r chapters.
I n c h a p t e r 2 , w e d e f i n e and d i s c u s s t h e
d i f f e r e n t c o n c e p t s of holomorphic mapping between l o c a l l y convex s p a c e s . Our primary i n t e r e s t i s i n c o n t i n u o u s ( o r F r g c h e t ) holomorphic mappings, b u t w e f i n d t h a t our a n a l y s i s o f t h e c o n t i n u o u s c a s e r e q u i r e s t h e s e o t h e r concepts.
I n t h i s c h a p t e r , we a l s o d e f i n e t h e v a r i o u s t o p o l o g i e s on s p a c e s Chapter 3 i s devoted t o
of holomorphic f u n c t i o n s and g i v e some examples.
holomorphic f u n c t i o n s on b a l a n c e d s e t s .
In t h i s s i t u a t i o n , t h e Taylor
s e r i e s expansion l e a d s t o a t o p o l o g i c a l decomposition of t h e s p a c e o f holomorphic f u n c t i o n s and p r o p e r t i e s o f t h e u n d e r l y i n g s p a c e s o f homogeneous p o l y n o m i a l s are extended t o holomorphic f u n c t i o n s . holomorphic f u n c t i o n s on Banach s p a c e s .
I n chapter 4,we discuss
Here w e f i n d a n i n t e r p l a y between
t h e geometry o f t h e Banach domain, bounding s e t s and t h e t o p o l o g i e s on t h e
s e t o f holomorphic f u n c t i o n s . on n u c l e a r sequence s p a c e s .
C h a p t e r 5 d e a l s w i t h holomorphic f u n c t i o n s In t h i s chapter, w e construct a d u a l i t y theory
between t h e s e t o f holomorphic f u n c t i o n s on open p o l y d i s c s and holomorphic germs on compact p o l y d i s c s .
T h i s l e a d s t o a c l a r i f i c a t i o n o f a number o f
examples and counterexamples from p r e v i o u s c h a p t e r s and p r o v i d e s u s w i t h a holomorphic c l a s s i f i c a t i o n o f t h e s u b s p a c e s o f
s,
t h e r a p i d l y decreasing
s e q u e n c e s , w i t h i n t h e c a t e g o r y of F r g c h e t n u c l e a r s p a c e s w i t h a b a s i s .
In
ix
Foreword c h a p t e r 6 we d i s c u s s a number o f methods o f g e n e r a l i s i n g t h e p o s i t i v e r e s u l t s o f t h e p r e v i o u s c h a p t e r s t o more g e n e r a l classes o f s p a c e s , and we a l s o d e v o t e one s e c t i o n t o t h e t o p o l o g i c a l c l a s s i f i c a t i o n o f holomorphic f u n c t i o n s on power s e r i e s s p a c e s o f i n f i n i t e t y p e .
The r e s u l t s o f t h i s
c h a p t e r are g e n e r a l l y o f r e c e n t o r i g i n and, no d o u b t , w i l l b e improved i n t h e not too d i s t a n t f u t u r e . I n t h e f i f t h s e c t i o n o f each c h a p t e r , w e g i v e a s e t o f e x e r c i s e s . Some o f t h e s e might n o r m a l l y b e c o n s i d e r e d r e a s o n a b l e e x e r c i s e s , b u t o t h e r s are q u i t e d i f f i c u l t .
We i n c l u d e d t h e l a t t e r i n o r d e r t o i n t r o d u c e f u r t h e r
r e s u l t s w i t h o u t u n n e c e s s a r i l y c o m p l i c a t i n g t h e main body o f t h e t e x t , and i n Appendix I 1 1 we p r o v i d e r e f e r e n c e s and comments on t h e s e e x e r c i s e s .
A
serious attempt a t solving t h e exercises w i l l give t h e i n t e r e s t e d reader a much d e e p e r u n d e r s t a n d i n g o f t h e s u b j e c t and i n t r o d u c e him o r h e r t o many i n t u i t i o n s and s u b t l e t i e s which a r e o f t e n d i f f i c u l t t o communicate by t h e p r i n t e d word a l o n e .
Comments and r e f e r e n c e s on t h e t e x t a r e g i v e n i n t h e
f i n a l s e c t i o n o f each c h a p t e r . I t i s our o p i n i o n t h a t t o p o l o g i c a l c o n s i d e r a t i o n s w i l l e n t e r , t o a g r e a t e r o r l e s s e r e x t e n t , i n t o most problems i n i n f i n i t e d i m e n s i o n a l h o l o morphy.
On t h e o t h e r hand, we a l s o f e e l t h a t t h e t h e o r y o u t l i n e d i n t h i s
book w i l l b e more i m p o r t a n t a s a t o o l i n o t h e r b r a n c h e s o f i n f i n i t e dimens i o n a l holomorphy and a n a l y s i s r a t h e r t h a n as a n o b j e c t o f r e s e a r c h i n itself.
For t h i s r e a s o n , we f e e l it i m p o r t a n t t h a t t o p o l o g i c a l problems i n
i n f i n i t e dimensional a n a l y s i s b e motivated, i f a t a l l p o s s i b l e , e i t h e r d i r e c t l y o r i n d i r e c t l y , from o u t s i d e and t h a t t h e g e n e r a l d i r e c t i o n o f r e s e a r c h i n t o l o c a l l y convex s p a c e s s t r u c t u r e s on s p a c e s o f holomorphic f u n c t i o n s i n i n f i n i t e l y many v a r i a b l e s b e c o o r d i n a t e d and guided by d e v e l opments i n o t h e r a r e a s o f t h e s u b j e c t .
T h i s approach h a s , u n t i l now, l e d
t o t h e more i n t e r e s t i n g r e s u l t s . Most o f t h e r e s u l t s p r e s e n t e d i n t h i s t e x t have n o t p r e v i o u s l y a p p e a r ed i n book form.
The r e a d e r w i l l s e e t h a t t h e s u b j e c t i s s t i l l i n a s t a t e
o f t r a n s i t i o n and t h a t t h e r e a r e many open problems.
I t w i l l b e some time
b e f o r e t h e d e f i n i t i v e book on t h e s u b j e c t i s w r i t t e n , and t h e p r e s e n t work may b e r e g a r d e d a s a r e p o r t o f "work i n p r o g r e s s " .
The a r e a h a s , however,
been developed c o n s i d e r a b l y i n r e c e n t y e a r s , and it i s a p p r o p r i a t e t h a t t h e p r e s e n t s t a t e o f knowledge of t h e s u b j e c t be r e c o r d e d i n a r e a s o n a b l y
Foreword
X
organised fashion.
W e hope t h a t t h i s book w i l l s t i m u l a t e r e s e a r c h t o s o l v e
t h e open problems posed, and t h a t workers i n a l l i e d f i e l d s w i l l g a i n some i n s i g h t i n t o a t l e a s t one a s p e c t o f i n f i n i t e dimensional holomorphy. T h i s book would n e v e r have been w r i t t e n w i t h o u t t h e s u p p o r t , e n c o u r a g e ment and f r i e n d s h i p of J o r g e A l b e r t 0 Barroso, who a r r a n g e d my v i s i t t o Rio d e J a n e i r o i n 1978, and of Leopoldo Nachbin who f i r s t s u g g e s t e d t h e i d e a of w r i t i n g t h i s book.
To them I extend my s i n c e r e s t g r a t i t u d e .
To J . M .
Isidro
I a l s o extend my t h a n k s f o r g i v i n g me t h e o p p o r t u n i t y t o l e c t u r e on, more
o r less, t h e c o n t e n t s of c h a p t e r 5 i n S a n t i a g o d e Compostela d u r i n g J u n e of 1979.
Many o t h e r p e o p l e , by t h e i r a d v i c e , proof r e a d i n g , encouragement and
s u g g e s t i o n s enabled m e t o f i n i s h t h i s book.
I would e s p e c i a l l y l i k e t o
thank R. Aron, P . Boland, R . Ryan and R . Soraggi f o r t h e i r p a t i e n c e and i n t e r e s t and t o e n s u r e them t h a t a l l t h e e r r o r s are mine.
I thank R. Meise
and D . Vogt f o r t h e i r encouragement and f o r making a v a i l a b l e t o me t h e i r v e r y r e c e n t unpublished r e s e a r c h . J.F.
J.M. Ansemil, P . Barry, K-D.
Bierstedt,
Colombeau, C . Herves, J . Mujica, L . A . d e Moraes and Ph. Noverraz were
a l s o very helpful. The e x c e l l e n t l a y o u t and t y p i n g a r e due t o H i l a r y Hynes and Ann Lewis and I thank them v e r y s i n c e r e l y f o r a l l t h e h e l p t h e y have g i v e n m e .
/
Sean Dineen Dublin, October 3 , 1980.
CONTENTS
vii
FOREWORD CHAPTER 1.
POLYNOMIALS ON LOCALLY CONVEX TOPOLOGICAL VECTOR SPACES
1
1.1
Algebraic properties of polynomials
1
1.2
Continuous polynomials
9
1.3
Topologies on spaces of polynomials
22
1.4
Duality theory for spaces of polynomials
31
1.5
Exercises
42
1.6
Notes and Remarks
46
HOLOMOWHIC MAPPINGS BETWEEN LOCALLY CONVEX SPACES
53
2.1
Gzteaux holomorphic mappings
53
2.2
Holomorphic mappings between locally convex spaces
57
2.3
Locally convex topologies on spaces of holomorphic mappings
71
2.4
Germs of holomorphic functions
a4
2.5
Exercises
92
2.6
Notes and Remarks
99
CHAPTER 2 .
CHAPTER 3. HOLOMORPHIC FUNCTIONS ON BALANCED SETS
109
Associated topologies and generalized decompositions in locally convex spaces
110
3.2
Equi-Schauder decompositions o f (H(U;F),T)
119
3.3
Application of generalised decompositions t o the study of holomorphic functions on balanced open sets
124
3.1
xi
xii
Contents 3.4
S e m i - r e f l e x i v i t y and n u c l e a r i t y f o r s p a c e s of holomorphic f u n c t i o n s
141
3.5
Exercises
146
3.6
Notes and Remarks
153
HOLOMORPHIC FUNCTIONS ON BANACH SPACES
159
4.1
A n a l y t i c e q u a l i t i e s and i n e q u a l i t i e s
160
4.2
Bounding s u b s e t s o f a Banach s p a c e
172
4.3
Holomorphic f u n c t i o n s on Banach s p a c e s w i t h an unconditional bas i s
183
4.4
F u r t h e r r e s u l t s and examples concerning holomorphic f u n c t i o n s on Banach s p a c e s
196
4.5
Exercises
204
4.6
Notes and Remarks
210
HOLOMORPHIC FUNCTIONS ON NUCLEAR SPACES WITH A BASIS
217
5.1
Nuclear s p a c e s w i t h a b a s i s
218
5.2
Holomorphic f u n c t i o n s on f u l l y n u c l e a r s p a c e s w i t h a basis
236
5.3
Holomorphic f u n c t i o n s on DN s p a c e s w i t h a b a s i s
262
5.4
T o p o l o g i c a l p r o p e r t i e s i n h e r i t e d by s t r i c t i n d u c t i v e l i m i t s and subspaces
277
5.5
Exercises
288
5.6
Notes and Remarks
293
CHAPTER 4 .
CHAPTER 5 .
CHAPTER 6 .
GERMS, SURJECTIVE LIMITS, SPACES
1 -PRODUCTS
AND POWER SERIES
297
6.1
Holomorphic germs on compact s e t s
297
6.2
S u r j e c t i v e l i m i t s of l o c a l l y convex s p a c e s
316
6.3
€-Products
32 7
6.4
Power series s p a c e s of i n f i n i t e t y p e
336
6.5
Exercises
356
6.6
Notes and Remarks
360
Contents
...
Xlll
APPENDIX I
FURTHER DEVELOPMENTS I N I N F I N I T E DIMENSIONAL HOLOMORPHY
365
APPENDIX I1
D E F I N I T I O N S AND RESULTS FROM FUNCTIONAL ANALYSIS, SEVERAL COMPLEX VARIABLES AND TOPOLOGY
397
APPENDIX 111
NOTES ON SOME EXERCISES
41 1
Bib 1iography
433
Index
48 1
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Chapter 1
POLYNOMIALS ON LOCALLY CONVEX TOPOLOGICAL VECTOR SPACES
T h e r e a r e two s t a n d a r d methods o f i n t r o d u c i n g p o l y n o m i a l s , by u s i n g t e n s o r p r o d u c t s o r by u s i n g
n
We have found it more
l i n e a r mappings.
c o n v e n i e n t t o a d o p t t h e l a t t e r approach.
T h e . r e a d e r familiar w i t h t h e
t e n s o r p r o d u c t approach w i l l have l i t t l e d i f f i c u l t y i n c o r r e l a t i n g h i s e x p e r i e n c e w i t h o u r approach b u t w e b e l i e v e t h a t t r a n s l a t i n g r e s u l t s i n t h e o t h e r d i r e c t i o n i s more d i f f i c u l t . between v e c t o r s p a c e s o v e r
C.
I n 5 1 . 1 we d i s c u s s polynomial mappings
In 11.2 w e discuss various kinds of contin-
uous polynomial mappings between l o c a l l y convex s p a c e s .
Our main i n t e r e s t
i s i n polynomials c o n t i n u o u s w i t h r e s p e c t t o t h e g i v e n t o p o l o g y on a
l o c a l l y convex s p a c e b u t we f i n d t h a t hypocontinuous and Mackey ( o r S i l v a ) c o n t i n u o u s polynomials a l s o p l a y a n i m p o r t a n t r o l e i n o u r s t u d y .
51.3 i s
d e v o t e d t o endowing s p a c e s o f p o l y n o m i a l s w i t h l o c a l l y convex t o p o l o g i e s For t h e n o n - s p e c i a l i s t we p r o v -
and i n 11.4 a d u a l i t y t h e o r y i s d e v e l o p e d .
i d e an i n t r o d u c t i o n t o f u n c t i o n a l a n a l y s i s i n Appendix 11. 51.1
ALGEBRAIC C,R,N
PROPERTIES
and
POLYNOMIALS
OF
d e n o t e r e s p e c t i v e l y t h e complex numbers, t h e r e a l num-
Z
b e r s , t h e n a t u r a l numbers and t h e i n t e g e r s . and
msN
and
m
then
copies of
In t h i s section,
and
B
E
If
A
and
and
xnyn F
w i l l d e n o t e t h e element
n
are sets,
B
w i l l denote t h e C a r t e s i a n product o f
AnBm
n
copies of
A
"FX' ' C Y I.
n times m times
w i l l d e n o t e v e c t o r s p a c e s o v e r t h e complex
numbers.
For each ings from
nsN
into
E
we l e t b a ( n E ; F ) F.
denote t h e space of n
The s u b s c r i p t
a
n o t assume any c o n t i n u i t y p r o p e r t i e s . d e f i n e d on
En
with values i n
o t h e r s remain f i x e d .
J!a(nE;F)
F
linear mapp-
refers t o algebraic s i n c e we do
Hence i f
L E ~ ~ ( ~ E ;t hFe n)
L
is
and i s l i n e a r i n each v a r i a b l e when t h e i s a v e c t o r s p a c e o v e r t h e f i e l d of complex 1
2
Chapter 1 1 l i n e a r mappings a r e j u s t l i n e a r mappings and i n t h i s c a s e w e
numbers.
use t h e n o t a t i o n da(E;F). 2 l i n e a r mappings are a l s o c a l l e d b i l i n e a r 2 i s sometimes used i n p l a c e o f "ea( E ; F ) . mappings and t h e n o t a t i o n @,(E;F) n When F=C w e w r i t e dea( E) i n p l a c e o f &a(nE;C) and E* i n p l a c e o f La(E;C).
E*
dLa('E;F)
as t h e s e t o f a l l c o n s t a n t mappings from
E.
i s called the algebraic dual o f
space can be i d e n t i f i e d with
When
n=O w e d e f i n e
F
into
E
and t h i s
i n a natural fashion.
F
The f o l l o w i n g a l g e b r a i c i d e n t i t y i s e a s i l y e s t a b l i s h e d and i s u s e f u l i n p r o v i n g r e s u l t s by i n d u c t i o n . P r o p o s i t i o n 1.1
then
E
If
and
F
are vector spaces over
and
C
nEN
z ~ ( ~ E ; F )I- L ~ ( E ; x ~ ( ~ - ' E ; F ) : ) B,(~-'E;~~(E;F)).
Proof
The mapping
T
T(xl'.-.Xn)
=
-f
T1
+
T2,
T
Xa(nE;F),
E
[Tl(xl)3(x2,...Jn)
=
g i v e n by
~ ~ , ~ ~ l ~ . . . , ~ n ~
e s t a b l i s h e s t h e r e q u i r e d correspondence. Definition 1 . 2
xl,
f o r any numbers.
L
An n-lineor mapping
symmetric i f
W l ' . . * ,xn)
..., xn E
E
=
.
L(xql)>
* *
and any permutation
We l e t ZS("E;F) a mappings from E i n t o
E
from
F
into
is said t o be
'Xo(n))
u
of t h e f i r s t
n
denote t h e vector space o f a l l symmetric F.
natural n
Zinear
By averaging over a l l permutations of t h e f i r s t
natural numbers we can associate, i n a canonical fashion, a symmetric n-linear mapping w i t h each n l i n e a r mapping and i n t h i s way we obtain a
n
st$(nE;F)
projection from f o l l o w i n g way.
where
L
Formally we a c h i e v e t h i s i n t h e
we s e t
E
s n denotes t h e s e t of permutations of the f i r s t We c a l l
s
IS
onto z:(nE;F).
s
t h e symmetrization o p e r a t o r .
if
L
E
%a(nE;F)
then
(b)
s(L) = L
i f and o n l y i f
(c)
s ( s ( ~ ) )=
S(L)
for all
s(L) L
L
Zz(nE;F)
E
E
bz(nE;F)
in
natural numbers.
The f o l l o w i n g p r o p e r t i e s of
are easily verified: (a)
n
x~(~E;F)
3
Polynomials on locally convex topological vector spaces (d) If
s
is a linear operator. i s a v e c t o r space we l e t
E
ing from
into
E
En
into
x
An)
denote t h e diagonal mapp-
onto t h e p o i n t
x".
A mapping from a ZocaZZy convex space
Definition 1 . 3
convex space
which maps
En
(or
A
E
t o a locally
which is the composition of the diagonal mapping from E
F
n
and an
E
Zinear mapping from
F
into
n
is called an
hom-
ogeneous poZynomia2. denote t h e vector space of a l l
We Zet QaCnE;F) nomials from
E
homogeneous poly-
F.
into
Thus we s e e t h a t i f t h e r e e x i s t s an
n
i s an
P:E+F
n
n
l i n e a r mapping
homogeneous polynomial i f and only from
L
F
into
E
such t h a t t h e
following diagram commutes E
~ ( x )= x
where
n
f o r every
A polynomial from
iaZs from
E
E
into
E
L
If
En
x
in
into
F
We l e t
F.
into
polynomials from Example 1 . 4
A ___f
E.
is a f i n i t e s m of homogeneous poZynom-
@,(E;F)
is a
l i n e a r (C-valued mapping on
2
it i s well known t h a t t h e r e e x i s t s an zAwt
for a l l If
z = (zl
A = (a. .) ij
,..., zn)
16idn 16 j6n
denote t h e v e c t o r space of a l l
F.
Cn
6
then
nxn
L(z,w)
=
P(Z) =
L(Z,Z) =
1_s i s_ n 1s j s n
1
1s i s n
L B,
B = - A + tA 2
zAzt = zBzt
f o r every
tA
z
L(z,w) =
C".
aijziwj. on
C"
'J
has t h e f a m i l i a r
J
s(L),
we o b t a i n a 2 - l i n e a r form
i s t h e symmetric matrix a s s o c i a t e d with where
E
then
a..z.z..
by i t s symmetrization,
I f we r e p l a c e whose matrix,
Since
1
such t h a t
w = (wl ,..., wn)
1s j s n Hence any C-valued 2-homogeneous polynomial, P , form
A
matrix
and a l l
ncN,
(c",
A,
i.e.
i s t h e transpose o f t h e matrix
in
Cn
d e f i n e t h e same 2-homogeneous polynomial.
it follows t h a t
L
and
A.
s(L)
Chapter I
4
More generally if L in E
dQa(nE;F)
E
then L(xn)
=
s(L)(xn)
for every x
and hence we do not, in general, have a one to one correspondence
between n-homogeneous polynomials and n-linear mappings. However, if we restrict ourselves to symmetric n-linear mappings we do obtain a unique correspondence. By the definition o f n-homogeneous polynomials and the symmetrization operator, the following diagram commutes
where the non-horizontal mappings are just restrictions to the diagonal. T+T
We denote the vertical mapping
by
is easily seen to be lin-
ear and we have already noted that it is surjective. As a consequence of the polarization formula we show that is injective and thus w e have a canonical bijective linear mapping between the space of symmetric n-linear A
mappings from E
E
into F
and the space of n-homogeneous polynomials from
into F. ( P o l a r i z a t i o n Formula)
Theorem 1.5 over
Proof
(c,
L
If E
and xl, ..., xn
Ex~(~E;F)
E
E
and F are vector spaces then
By linearity and symmetry
Zm.=n -
Hence 1 n 2 .n!
1
c=il
l,
El
. . . En
A
L(
n
1 EjXj)
j=l
=
Polynomials on locally convex topological vector spaces
- - 2"
1
1
L(Xl)
O,<m.sn m 1 ! . .. mn !
m
1
...
(x,)
m
n
.I
of
L(xl,. If
c l2
... e2n
and h e n c e t h e c o e f f i c i e n t
2"
=
i n t h e above e x p a n s i o n i s
f o r some
then
i
m.=O
m +1
ml+l
c
1 E . =+1
then
l,
. . ,xn)
m.>l
n
61
e. =?I 1
i sj s n
... = m n = l
ml=m2
m +1
... enn
1
Zm.=n If
ml+l El 6.=+1
5
E.
J1
l,
1.
f o r some
1
j
and hence mj+l+l E. J+1
1-1
=+1
..,'.
mn+l
Isisn,i#j
0.
=
Hence a l l t h e o t h e r terms v a n i s h and we have proved t h e p o l a r i z a t i o n formula. Corollary 1.6 Proof
The mapping
-:%:("E;F)
By t h e p o l a r i z a t i o n formula
i
and o n l y i f
i s i d e n t i c a l l y zero.
+
L
E
Qa("E;F)
i s a linear bijection. i s i d e n t i c a l l y zero i f
%-:(nE;F)
i s a l i n e a r mapping w i t h
Hence
z e r o k e r n e l and t h u s it i s i n j e c t i v e . T h i s completes t h e p r o o f . The p o l a r i z a t i o n f o r m u l a h a s many o t h e r a p p l i c a t i o n s , some o f . w h i c h we now d e s c r i b e .
These f o l l o w from t h e s p e c i f i c n a t u r e o f t h e formula which
a l l o w s u s t o compare t h e moduli o f symmetric n - l i n e a r forms and p o l y n o m i a l s S i n c e i t i s u s u a l l y e a s i e r t o compute t h e modulus o f an n
on c e r t a i n sets.
l i n e a r form, t h e n t h e modulus o f a p o l y n o m i a l , t h i s p r o v e s a e f u l . If
and
B
f
i s a f u n c t i o n d e f i n e d on a s u b s e t A o f
i s a semi-norm on
we let
F
llfll
u n d e r s t o o d from t h e c o n t e x t , we j u s t w r i t e Theorem 1 . 7
x&.A
and
F
are vector spaces over
baZanced subset of E
and
B
i s a semi-norm on
f o r any
Remark
L
in X;("E;F)
with values i n
= supB(f(x)).
E
I f
E
F
11
If
B
is f u l l y
/]A' CC,
A
is a convex
then
.
T h i s e q u a l i t y may t a k e t h e form
c a s e we o b t a i n i n f o r m a t i o n .
+ws+ms+-
F
b u t even i n t h i s
3
6
Chapter I
Proof
The l e f t - h a n d s i d e o f t h i s i n e q u a l i t y i s t r i v i a l s i n c e By t h e p o l a r i z a t i o n formula
i ( A ) C L(An).
lii,
x.
i = l , .. . , n
A,
E
and
C. = 2 1
then
,I &ixi
1 "
E
1=1
convex and b a l a n c e d , and hence
A,
since
A
is
Hence
T h i s completes t h e p r o o f . nn - i s t h e b e s t p o s s i b l e i n g e n e r a l (example 1 . 8 ) . In n! c e r t a i n c a s e s , however, i t i s p o s s i b l e t o improve t h i s e s t i m a t e - t h i s w i l l The c o n s t a n t
depend on
n
and on t h e geometry o f t h e s e t
A.
We g i v e a few examples t o
i l l u s t r a t e t h i s p o i n t (some w i t h o u t p r o o f ) and n o t e i n p a s s i n g t h a t t h e r e are s t i l l i n t e r e s t i n g open problems i n t h i s a r e a . Example 1 . 8 w i t h norm
where
L
Let
( t h e space o f a b s o l u t e l y convergent s e r i e s
E = R1
11 ( x ~ ) ~ I= / c n l x n l ) .
i
m
(Xm)m=l
xi =
Let
for
b e d e f i n e d by
L:En+C
.
i = l, . . , n .
i s n - l i n e a r and symmetric and one e a s i l y shows t h a t
l l L l l A :=
n
where
and t h e c o n s t a n t
A
is t h e u n i t b a l l of nn n!
R1.
llLlkn =
and
Hence
.
nn i n theorem 1 . 7 i s t h e b e s t p o s s i b l e . n!
Our n e x t example c o n c e r n s r e a l polynomials o v e r an i n n e r p r o d u c t s p a c e . The analogous problem f o r complex p o l y n o m i a l s i s s t i l l open a l t h o u g h some p a r t i a l r e s u l t s have been o b t a i n e d .
I
Polynomials on locally convex topological vector spaces Example 1 . 9 let
F
d e n o t e a real inner product space with u n i t b a l l d e n o t e a r e a l Banach s p a c e and suppose L E Y:(nH;F). Then
\]LIBn = llLlb
Let
H
where
B
i s t h e u n i t b a l l of
forms and r e a l p o l y n o m i a l s , w e always have <
l l ~ l b<
Case 1 Ow'
Dim(H)
and
< m
n=2.
Dim(H) < (xl,
...,x,)
aaB
n
m,
E
Bn
it f o l l -
arbitrary. such t h a t
# 0
such t h a t
J
(yl*, . . . , y n * ) E A
W e claim
y . * = 'yy
for all
such t h a t
y * + y:
# 0 and
= YD*
+
Yq*
IIYp*+ Y,*ll
=
j = l , ..., n .
n {(yl, ...,yn) E B ; b a ,
=
and Now choose
for
1
such t h a t t h e s e t A
and
11L(x,y) = ~ ( x + y -$(x-y) )
Since
L
Now choose
= z
and w e may suppose
IlLlbn
11
Case 2
z
6
1 IIL(x>Y) b T\lillB( IIx+Y112+(1x-YI(2)
that
Choose
IILIL
m.
and hence
a>O
H.
S i n c e t h e p o l a r i z a t i o n formula h o l d s f o r r e a l symmetric n - l i n e a r
Proof
o
B,
.
Hence t h e r e e x i s t s
j = l , ...,n.
( I L ( Y 1 , e . . 9 ~ n ) I I= I I L I I p
i s non-empty.
such t h a t
i
and
y* P
-
y;
j.
I f n o t t h e r e ex s t s
# 0.
Let
The f i r s t c a s e i m p l i e s t h a t
M + ( + < a , y
L
zi
= y r
if
(z l , . . . , z n )
- 1)
p
and
q
i # p , q and E
A.
Since
8
Clzapter 1
we have a r r i v e d a t a c o n t r a d i c t i o n and completed t h e p r o o f i n t h i s c a s e . Case 3
H
and
llLl/
arbitrary.
t h e r e e x i s t s a f i n i t e d i m e n s i o n a l subspace
Given €>O
that
n
' IILIIgn- t'
(Br\E)n
lfliB
c i s a r b i t r a r y we have shown t h a t
Since
= llLll
Bn
...
such
H
and completed t h e p r o o f .
The f o l l o w i n g r e s u l t r e d u c e s t o theorem 1 . 7 on t a k i n g nl=n2
of
E
By c a s e 2 it f o l l o w s t h a t
p=l,k=m,
n =l. k
P r o p o s i t i o n 1.10
Let
denote a Banach space and l e t
E
nl,
. . . ,nk
be
n +n +...+ n = m . Suppose x l , . . . ,xk are u n i t 1 2 k k \I z i x i / \ 6 ( \ z i I p ) l / P for a l l ( z , , . . . , z k ) E Ck i=l i=l
positive integers with vectors i n
E
and
l
where
"1
/Wl
"2
'X2
k
1
if L n
,... ' X k 1 I k
1
E
C(mE) nl!
6
...
nl/P
n1
A
and
nk! mm/P nk/P
.....nk
The n e x t p r o p o s i t i o n g i v e s , on t a k i n g
P r o p o s i t i o n 1.11
E
Let 2.
denote a power of
m!
p=2,
t o complex v a l u e d polynomials o f d e g r e e
m
i s t h e u n i t ball of E
Ilill,
a n e x t e n s i o n o f example 1.9
2".
denote a compZex
AP
space,
ldp<-,
and Let
Then
m! f o r any
L
where t h e norms are taken over t h e u n i t baZ1 o f
i n if,:(mE)
E.
We conclude t h i s s e c t i o n w i t h two r a t h e r s i m p l e b u t u s e f u l r e s u l t s . The f i r s t f o l l o w s r e a d i l y from t h e b i n o m i a l theorem and t h e second f o l l o w s from a n a p p l i c a t i o n o f t h e maximum modulus theorem f o r f u n c t i o n s o f one complex v a r i a b l e . Lemma 1 . 1 2
in
E
and
If L AEC
E
L;("E;F)
we have
and
P =
i
E
Oa("E;F)
then f o r any
x,y
9
Polynomials on locally convex topological vector spaces
P(x+y)
=
P(X)
If
E
and
i s a semi-norm on
F,
A
Lemma 1 . 1 3
Moreover, if
+
A#O,AXEA
P(Y)
c
n- 1 +
F
r =1
(;)L(x)n-r(Y)r
are vector spaces over
i s a balanced subset of
and
=
=
A
E
P
(E,
and
E
@a(nE;F), 6
XEE
then
is convex then
sup B(P(x+eiey)) YEA,BER sup B(P(eiex+y))
(since
i s balanced)
A
(by homogeneity)
YEA
0 ER
If
§1.2
AXEA
then
2
sup B(P(y)) YSA
=
1$\lb,A*
1 x+A C - A + A A
CONTINUOUS
=
( 1
+
(by t h e maximum modulus p r i n c i p l e )
x1 )A
and hence
POLYNOMIALS
Our main i n t e r e s t i n t h i s book i s i n s t u d y i n g c o n t i n u o u s polynomial and holomorphic mappings between l o c a l l y convex s p a c e s .
However, w e f i n d
i t u s e f u l (and n e c e s s a r y ) t o d e f i n e more g e n e r a l c l a s s e s o f mappings -
t h i s i s analogous t o t h e s i t u a t i o n i n f u n c t i o n a l a n a l y s i s where t h e weak t o p o l o g y i s used t o o b t a i n p r o p e r t i e s o f t h e norm topology.
10
Chapter 1 Let
and
E
b e t o p o l o g i c a l s p a c e s which are a l s o v e c t o r s p a c e s o v e r
F
U s u a l l y t h e r e w i l l b e some r e l a t i o n s h i p between t h e t o p o l o g i c a l and t h e
(E.
v e c t o r s p a c e s t r u c t u r e s b u t f o r t h e moment w e w i l l n o t make any such assumption.
We l e t
/ ; ( n E ; F ) , X(nE;F)
J ? ( ~ E ; F ) denote r e s p e c t i v e l y t h e
and
E
spaces of continuous n-homogeneous polynomials from E
tinuous n-linear mappings from E
n-linear mappings from
F
to
and the continuous symmetric
t h e above s p a c e s a r e v e c t o r s p a c e s and t h a t - ( d S ( n E ; F ) ) = S’(nE;F). spaces. a
cs(E)
E,
i s a semi-norm on
Note t h a t
Ea.
Ea
is
-
Ra
Ea
a(x) = 0
{ x e E ; a ( x ) < r ) and
B,(r)
i s the c l o s u r e of f
If
Ba(r)
Ba(r)
a r e l o c a l l y convex
F
in
II (x) = 0
E.
If
{xcE;a(x)
=
Since
E.
F
E
onto
and t h e open u n i t b a l l
i s a semi-norm on
a
from a t o p o l o g i c a l s p a c e i n t o
i s continuous for every
of
and
E
and
w i l l d e n o t e t h e normed l i n e a r s p a c e
i f and o n l y if
R a { x ~ E ; a ( x )< 1 ) .
=
nB
W e now suppose t h a t
J,’(~E;F)
=
w i l l d e n o t e t h e c a n o n i c a l s u r j e c t i o n from
Ba(r)
mapping
s(Y.(~E;F))
w i l l d e n o t e t h e s e t o f a l l c o n t i n u o u s semi-norms on
( E , ~ i ) / ~ - l ( ~ )and
of
the con-
I n a l l cases we c o n s i d e r we f i n d t h a t
F.
into
F,
into
=
we l e t
E
If
accs(E)
lim
ES (F)
then
(FB,IIB), a
i s c o n t i n u o u s i f and o n l y i f
F
B E C S ( F ) . We u s e t h i s f a c t i n e x t e n d i n g
r e s u l t s c o n c e r n i n g normed l i n e a r s p a c e v a l u e d polynomials and holomorphic f u n c t i o n s t o f u n c t i o n s w i t h v a l u e s i n a n a r b i t r a r y l o c a l l y convex s p a c e . Proposition 1.14
Let
E
be a l o c a l l y convex space over
C,
F
il
P E P ~ ( ( ” E ; F ) . The following are equivalent
normed l i n e a r space and suppose
(a)
P
i s everywhere continuous;
(bi
P
is continuous a t t h e o r i g i n ;
(el
P
i s bounded on some neighbourhood of the origin;
(dl
P
i s a l o c a l l y bounded f u n c t i o n i i . e . bounded on a neighbourhood of each p o i n t ) .
Proof 1.13
The i m p l i c a t i o n s ( c ) <=> ( d ) .
suppose
A=P.
( a ) => (b) => ( c )
W e now show
( c ) => ( a ) .
Let
By t h e p o l a r i z a t i o n formula and
b a l a n c e d neighbourhood o f z e r o be a r b i t r a r y .
are t r i v i a l and by lemma
Choose
a >O
V
such t h a t
such t h a t
A
(c)
XS(nE;F)
= M < m . Let Vn By Lemma 1 . 1 2
IlAIl
axo E V .
E
and
t h e r e e x i s t s a convex x0€ E
11
Polynomials on locally convex topological vector spaces
Hence
i s continuous a t
P
xo
and
( c ) => ( a ) .
This completes t h e proof.
Let E and F be ZocnZZy convex spaces over E and Corollary 1 . 1 5 n if and only if P i s continuous a t Zet P E Pa( E;F). Then P E @("E;F) one point. I t s u f f i c e s t o use p r o p o s i t i o n 1 . 1 4 and t h e p r o j e c t i v e l i m i t represent a t i o n of
by normed l i n e a r spaces.
F
We now look a t a very u s e f u l f a c t o r i z a t i o n lemma. l o c a l l y convex spaces, Hence
?(("E,;F)
a
E
cs(E)
and
P
!?(nE,;F)
E
may be i d e n t i f i e d with a subspace of
If
E
then
P
F
are
When
F
and 0
nLYE $fnE;F)
6'("E;F).
i s a normed l i n e a r space t h e f a c t o r i z a t i o n lemma says t h a t t h e union of a l l This i s not s u r p r i s i n g i n view of lemma such subspaces covers p("E;F). 1.13.
Lemma 1.16 F
and
(Factorization Lemma).
i s a ZocaZZy convex space
i s a nomed Zinear space then
f o r every p o s i t i v e i n t e g e r Proof
Let
P
E
B(nE;F)
n. and suppose
symmetric n - l i n e a r mapping.
Since
exists a
llpll
a ( x ) < 1, n $[
E
If
E
cs(E)
such t h a t
a(y) = 0
0EFI.
n A ( x ) ~ ( X ~ ) ~= - 1 ~]
1
R=O
polynomial from follows t h a t
g
Ax'
where
to
E
M<m.
The function
Now suppose
x , y ~E ,
g(A) = $oP(x+Ay) =
@ ( A ( X ) ~ ( ~ ) A~n - -R ~ ] . i s a
):(
of degree
i s the associated
i s a normed l i n e a r space t h e r e =
R=O
Ks(nE;F)
n.
sup a(x+Ay) = a ( x ) < 1 it AEC i s a bounded polynomial and hence has degree 0 . By t h e
C
Hahn-Banach theorem t h e form
and
F
A
C
R
Since
ObRsn-1. Since any z E E has R it follows t h a t A(z) (y)n-R = 0 i f Z E E
A(x) (y)n-R = 0 a(x')
Q
for
12
Chapter 1
and Y
a(y) = 0 , R#n.
E,
b e d e f i n e d on
P
n,(z1)
=
'TI
a
(?
) = z
Thus
2" P
u n i t b a l l of
E,
P(i,).
Hence
P(z+y) = P(z)
by
$(z)
then
71
=
P(3
if
(z2-z1) = 0 I
if
a(y) = 0 . Y
~ E Es a t i s f i e s
' T I J Z )= z.
Let If
and hence
_
i s well d e f i n e d and
z
f o r any
TO'TI=
P.
~ ( 2 =~ P) ( t l + ( ? 2 - i 1 ) ) = S i n c e n,(B,(l)) is the
w e have
T h i s completes t h e p r o o f . The f o l l o w i n g example shows t h a t t h i s r e s u l t d o e s n o t e x t e n d t o arbitrary
F.
Example 1 . 1 7
Let
d e n o t e t h e s e t o f a l l sequences o f complexnumbers
CN
i s g i v e n t h e p r o d u c t t o p o l o g y o r t h e t o p o l o g y o f c o o r d i n a t e convergence
CN
and i s a F r & c h e t s p a c e ( i . e . a c o m p l e t e m e t r i z a b l e l o c a l l y convex s p a c e ) . If
a
N
cs(C )
6
then
hence i s isomorphic t o
The i d e n t i t y L
of
E
'J
Cn
f o r some
belongs t o
I
&((CN),;CN)
a E C S (CN) C N . Hence
i s a f i n i t e dimensional normed l i n e a r s p a c e and
E
I
&![C
then
u
na.
N ;C N )
L((GN),)
and
We c l a i m
I(C N )
= C
N.
If
i s a f i n i t e dimensional subspace
a E cs (CN)
Our n e x t r e s u l t i s a f a c t o r i z a t i o n lemma f o r a r b i t r a r y p o l y n o m i a l s . The p r o o f f o l l o w s d i r e c t l y from l e m m a 1.16.
Lemma 1.18
If
G-J("E;F) =
and
E
F
are Locally convex spaces over
C
then
LJ P ( ~ E , ; F ~ ) BECS(F) a ~ c s ( E )
Before d e f i n i n g hypocontinuous and S i l v a c o n t i n u o u s polynomials we f i r s t p r o v e a r e s u l t a b o u t p o l y n o m i a l s on a Banach s p a c e .
This r e s u l t is t r u e
f o r any Baire s p a c e and i s used i n p r o v i n g Zorn's theorem i n t h e n e x t chapter. Lemma 1.19
Let
E
l o c a l l y eonvex space.
be a eompZex Banach space and
F
an arbitrary
If {Pm}i=l i s a sequence i n @("E;F)
and
13
Polynomials on locally convex topological vector spaces
We may suppose without l o s s of generality that F
Proof
linear space. Let Am E&'(~E;F)
where
im =
is a normed
Pm for every integer m.
By the polarization formula Am converges pointwise on E
to a symmetric
n-linear form A and hence P = E 2a(nE;F). We complete the proof by induction. Let B denote the unit ball of E. If n=l then Pm €;t(E;F) and by the uniform boundedness principle Hence IIPIIBs M and P is continuous by proposition 1.14. result holds for n=k. I f
Pm
E
6(k+1E;F)
for all m
= M <-.
Now suppose the
then, by induction,
Zsisk+l
By the uniform boundedness principle the collection of linear mappings
1 is bounded on the unit ball of E, (A (Y2,.*'3Yk+l) Yi'B 2fiSk+l i.e. . sup " (Y, .. .,Yn)II< YiEB ltick+l Thus P and A are continuous. This completes the proof. 9
The hypocontinuous and the Mackey continuous polynomial mappings between two locally convex spaces E and F can be defined in two different ways, either by defining new topologies on E o r by using properties of the compact and bounded subsets of E. We describe both methods.
A n element
p
of
@a(nE;F)
i s said t o be hypocontinuous i f
14
Chapter I
it i s continuous on t h e compact subsets o f
E.
pHy(nE;F)
We l e t
denote
the vector space o f a l l hypocontinuous n-homogeneous poZynomia2s from E
to
F.
Da(E;F)
An element of
i s said t o be Mackey ( o r S i l v a ) continuous F.
if it maps bounded subsets of E onto bounded subsets o f
We l e t
@M(nE;F) denote t h e vector space of a22 Mackey continuous n-homogeneous E
polynomials from
F.
into
Hypocontinuous and Mackey c o n t i n u o u s n - l i n e a r forms a r e d e f i n e d i n a n S i n c e t h e v e c t o r sum of compact ( r e s p . bounded) s u b s e t s o f
a n a l o g o u s way.
a l o c a l l y convex s p a c e i s compact ( r e s p . bounded) it f o l l o w s from t h e s y m m e t r i z a t i o n formula and t h e p o l a r i z a t i o n formula t h a t f o r any l o c a l l y convex s p a c e s
and
E
s(&,fE;F))
If
x
we have
F
giy(nE;F)
=
XHy(nE;F) n f;("E;F)
=
is a topological space ue l e t
Xk
denote the'space
with the
X
f i n e s t topology which coincides with the original topology on the compact subsets o f
If
X.
x
xk
=
we c a l l
X
a k-space.
If
is a locally
E
convex s p a c e t h e n it i s n o t t r u e i n g e n e r a l t h a t Ek i s a l s o a l o c a l l y convex space.
Any m e t r i z a b l e s p a c e i s a k - s p a c e .
A l o c a l l y convex s p a c e i s a
semi-Monte1 s p a c e if i t s c l o s e d bounded s e t s a r e compact. b a r r e l l e d semi-blontel s p a c e i s c a l l e d a Montel s p a c e .
An i n f r a -
The s t r o n g d u a l o f
a n i n f i n i t e d i m e n s i o n a l F r g c h e t Montel s p a c e ( a &I>mspace) o f a k-space which i s n o t m e t r i z a b l e .
i s a n example
The f o l l o w i n g p r o p o s i t i o n f o l l o w s
e a s i l y from t h e d e f i n i t i o n above.
Proposition 1.20
Hence i f convex s p a c e
E = Ek
If
E
=
B C ~ ( E;~ F ))
and
then
F
are ZocaZZy convex spaces then
Q H Y ( " ~ ; ~ )=
Q c ~ E ; F ) f o r any l o c a l l y
F.
W e now look a t Mackey c o n t i n u o u s p o l y n o m i a l s .
convex s p a c e w i t h t o p o l o g y
Mackey convergent t o scalars,
x,
T.
A sequence M
we w r i t e
(A~)E=~ \ i, n \ -f
01
as
xn-+ x,
n+-
Let
E
be a locally
( x ~ )i n~ E
is said t o be if there e x i s t s a sequence o f
, such t h a t
1 n(xn-x)
-f
0
in
(E,T)
15
Polynomials on locally convex topological vector spaces
-.
as
n
xn
y x as
-+
n-,
E
of
A
A subset
i s said t o be
( x ~ E) A,~
M-closed i f
The M-closed s u b s e t s o f
XEA.
implies
satisfy the
E
c l o s u r e axioms f o r a t o p o l o g y which w e c a l l t h e topology of the M-closure T ~ .W e
and d e n o t e by If
a l s o write
then t h e space
T = T~
EM
i n place of
(E,rM).
i s c a l l e d a superinductive space.
(E,T)
F r g c h e t s p a c e s and t h e s t r o n g d u a l s o f Frcchet-Schwartz s p a c e s a r e superinductive spaces. convex.
i s t h e s t r o n g d u a l o f a Frgchet-Monte1
(E,T)
For i n s t a n c e i f
(d.42 s p a c e s )
i s not necessarily l o c a l l y
The t o p o l o g y
space then t h e following a r e e q u i v a l e n t ;
is a 2 3 - 3 s p a c e ,
(a)
(E,T)
(b)
T = T~
(c)
( E , T ~ ) i s a l o c a l l y convex s p a c e .
on
E,
S i n c e c l o s e d sets are s e q u e n t i a l l y c l o s e d and Mackey c o n v e r g e n t sequences a r e convergent T~
in
T~
3
T~
on any l o c a l l y convex s p a c e
>,T
i s t h e k-topology a s s o c i a t e d w i t h M (E,rM) i f and o n l y i f xn x as
Proposition 1.21
any integer
n
F i r s t suppose
Proof
B(P(
F
and
E.
in
xn
+
x
as
be locally convex spaces.
Then f o r
P
E
@(n(EM);F) b u t
P
i s n o t bounded on t h e
B(P(xm)) > m
m 1 B(P(xm)) mn)) 1/2 X
=
>
m.
for all
1 m2 f o r a l l
m.
E,
(x,),,
This implies t h a t This contradicts t h e
n
fact t h a t
Hence
P
i s bounded on bounded s e t s and
8 Now suppose
P
n+-
(E,T).
Then we c a n f i n d a bounded sequence i n
B E C S ( F ) such t h a t
m
+
c ~ ~ ( ~ E ; =F )Q(~(E,);F).
bounded s u b s e t s of and
E
Let
n
-
Note a l s o t h a t
T.
-f
where
(E,T),
E
c~(E,);F)
c
(pM(nE;F).
IP~(~E;F). To show t h a t
P
E
@(n(EM);F)
it s u f f i c e s
16
Chapter 1 x as m - t m i m p l i e s P(xm) + P ( x ) as m b e a sequence o f scalars such t h a t 1x1, -t +
t o prove t h a t
Let hm(xm-x)
+
s u b s e t of
m+
x
as
0 E.
If
m
+.
A
ki(nE;F)
E
m
.
and
B = { A , ( X ~ - X ) }u~{XI i s a bounded
The s e t
m.
m
and
x=P
then
A
i s bounded on
Bn.
Hence
m=1,2,..
m = 1,2,.
i s a bounded s u b s e t o f
F.
I t now f o l l o w s t h a t P(xm)
- P(x)
shown t h a t
R n-R A(xm-x) (x)
P(x+xm-x)-P(x)
=
P(xm) + P ( x )
Corollary 1 . 2 2
as
=
m-+m
If E and
n
-t
1 R=l
as m + m if k & n . Since n R n-R (R) A(xm-x) (x) we have
0
and t h i s completes t h e p r o o f .
F
are ZocaZZy convex spaces then
~ , , ( " E ; F ) 3 P M ( " ~ ; ~3 )
for every
.
Q,,("E;F)
3 P ( n ~ ; ~ )
N.
Pa(nE;F)
may a l s o b e r e g a r d e d as a s p a c e o f c o n t i n u o u s polynomials i f we
p l a c e on
E
space, i s
the f i n i t e Z y open topoZogy tf
open i f V n F
tf.
A subset
V
of E
,
a vector
i s open for every f i n i t e dirnensionaZ subspace
F
of E where each f i n i t e dirnensionaZ space i s given i t s unique norm top0 zogy . When do we have e q u a l i t y i n c o r o l l a r y 1.22?
T h i s i s an i n t e r e s t i n g
q u e s t i o n and t h e answer p l a y s an i m p o r t a n t r o l e i n Chapter 5. a l r e a d y n o t e d cases i n which we do have e q u a l i t y .
We have
These r e s u l t e d from
c o n s i d e r i n g l o c a l l y convex s p a c e s as o b j e c t s i n t h e c a t e g o r y of t o p o l o g i c a l s p a c e s and c o n t i n u o u s mappings.
T h i s c a t e g o r y i s much t o o l a r g e f o r o u r
p u r p o s e s and r a r e l y p r o v i d e s n e c e s s a r y and s u f f i c i e n t c r i t e r i a .
In t h e
p r e s e n t case i t p r o v i d e s s u f f i c i e n t b u t n o t n e c e s s a r y c o n d i t i o n s f o r c e r tain equalities.
On t h e o t h e r hand, w e can a l s o look a t l o c a l l y convex
s p a c e s as o b j e c t s i n t h e c a t e g o r y of t o p o l o g i c a l v e c t o r s p a c e s .
T h i s cat-
egory i s t o o small and g e n e r a l l y o n l y g i v e s c o n d i t i o n s of t h e o p p o s i t e k i n d
-
i n t h e p r e s e n t s i t u a t i o n w e g e t c o n d i t i o n s which are n e c e s s a r y .
For example, i n o u r t e r m i n o l o g y a l o c a l l y convex s p a c e s p a c e if and o n l y if
6fM(E;F)
=
d(E;F)
E
is a bornological
f o r e v e r y l o c a l l y convex s p a c e
17
Polynomials on locally convex topological vector spaces I t i s , however, u s e f u l t o keep t h e s e t y p e
F.
o f r e s u l t s i n mind a s t h e y
p r o v i d e u s e f u l estimates and g i v e u s o u r p r e l i m i n a r y examples. The c o r r e c t s e t t i n g f o r t h e p r e s e n t c h a p t e r i s t h e "category" o f l o c a l l y convex s p a c e s and co n t i n u o u s m u l t i l i n e a r mappings and w e now g i v e some e x a m p l e s w h i c h r e f l e c t t h e
E = F x F' where F i s a l o c a l l y convex spa c e B i s i t s s t r o n g d u al ( i . e . F ' i s t h e spa c e o f a l l c ontinuous
Example 1 . 2 3 and
F' B
linear
Let
mappings on
F
bounded s u b s e t s o f Let
A
E
n a t u r e of t h i s "category".
A : E x E
and
f;(*E)
F). -+
Q: be d e f i n e d by
i ( x , x t ) = XI(%).
a normed l i n e a r s p ace. w e always have
w i t h t h e t o p o l o g y o f uniform convergence on t h e
iE
i s c o n t i n u o u s i f and o n l y i f
By t h e d e f i n i t i o n o f t h e s t r o n g topology on
PM(2E) and f r e q u e n t l y
form.
CN (C")
F'
i s hypocontinuous.
Our most u s e f u l counterexample i n t h i s book, CN x C(N), and
is
E
h a s t h e above
i s t h e s p a c e o f a l l complex s eq u en ce s w i t h t h e produc t topology i s t h e s p ace of a l l f i n i t e complex se que nc e s with t h e d i r e c t
sum to p o l o g y .
I n l a t e r c h a p t e r s we s h a l l u s e t h e f o l l o w i n g p r o p e r t i e s o f
it i s a r e f l e x i v e n u c l e a r s p ace w i t h a n a b s o l u t e b a s i s , it i s
CN x C " ) ;
a s t r i c t i n d u c t i v e l i m i t o f FrLchet n u c l e a r s p a c e s and a n open compact s u r j e c t i v e l i m i t of
( s t r o n g d u a l of F r 6 c h e t n u c l e a r ) s p a c e s , each
compact s e t i s c o n t a i n e d i n a s e t o f t h e form subset of
CN
and
KxL
where
K
i s a compact
i s a c l o s e d bounded f i n i t e dim e nsiona l s u b s e t o f
L
e v e r y neighbourhood o f t h e o r i g i n c o n t a i n s a neighbourhood o f t h e o r i g i n which h a s t h e . form U x CN-' x V where LEN, U i s a neighbourhood
(C(N),
i s a neighbourhood o f z e r o i n
CL and V
o f zero i n
_-Example 1.24
Let
E
(C").
b e a c o u n t a b l e i n d u c t i v e l i m i t o f normed l i n e a r
s p a c e s i n t h e c a t e g o r y o f l o c a l l y convex s p a c e s and c o n t i n u o u s l i n e a r
l i m En and E i s a b o r n o l o g i c a l spa c e which con-+ n m t a i n s a fundamental sequence o f bounded s e t s , (Bn)n,l. W e may suppose t h a t
mappings. each
Bn
Thus
E
=
i s convex and b al an ced .
S i n c e a l o c a l l y convex s p a c e i s b o r n o l -
o g i c a l if and o n l y i f ev er y convex b a l a n c e d s e t which a b s o r b s evEry bounded set i s a neighbourhood o f z e r o , w e f i n d t h a t s e t s o f t h e form
~n,lXnBn
Chapter 1
18
form a b a s i s of neighbourhoods of zero i n a l l sequences o f p o s i t i v e r e a l numbers
(ln=lAnBn m
m
ranges over m = Iln=lhnbn; bnEEn, m
i s convex and balanced and
IZ=lAnBn
a r b i t r a r y } ) . This follows s i n c e
as
E
absorbs every bounded s e t and hence i s a neighbourhood of zero.
Conversely
i s a convex balanced neighbourhood of zero then f o r every ntN m an t h e r e e x i s t s a n > 0 such t h a t anBn C V and hence V 3 Bn. Zn if
V
In=1
The countable d i r e c t sum of normed l i n e a r spaces and t h e strong dual I t can a l s o be
of a Frgchet Monte1 space a r e examples of such spaces.
e a s i l y shown t h a t t h i s c l a s s of spaces coincides with t h e c l a s s of bornological
DF
spaces.
E
We now show t h a t f o r such convex space
F.
$JM(nE;F) = @("E;F)
f o r every l o c a l l y
We may assume without l o s s o f g e n e r a l i t y t h a t
normed l i n e a r space.
Let
P
E
pM(nE;F)
and suppose
is a
F
A E%("E;F)
where
A = P.
B1 i s bounded I I P / I < M < B1 have been chosen so t h a t
Since
A2,
..., Am
If
x
m
E
I
AiBi, i=l
p(X+Xy)
1
i=l
=
and
YEB,,,+~
P(x)
XiBi
n
A >O
-.
Let
A1
= 1
and suppose
then
1
( " , A ( x ) " - ~ ( ~ ) X~R . and hence
IIAlL
i s f i n i t e and we can choose
+
R=l
+
where
ll A l l m Since
A
E
-
A;(~E;F),
s u f f i c i e n t l y small s o t h a t
X
=
A m+ 1
19
Polynomials on locally convex topological vector spaces m+l
s i=l
I
1 1
By induction that
?4(1 -i=l 2 i - 1
m
we can choose a sequence of p o s i t i v e numbers
llpll
such
(Xm)m=l
s 2M. m =1
Hence
P
i s bounded on a neighbourhood of zero and i s continuous by
p r o p o s i t i o n 1.14.
We a l s o n o t e t h a t t h e above proof shows t h a t i f
i s a subset of :J(nE;F), F i s a normed l i n e a r space, and ('cr)aE~ f o r every m then the collection is a locally s;p IIPall B, < (Pa)aEA bounded family of functions. Let
Example 1 . 2 5 convex space.
and only i f each I f each
E
=
Ci=lEm
PHy(nE;F')
where each
=
?("E;F)
Em
i s a normed l i n e a r space then
Em
i s a metrizable l o c a l l y
f o r every i n t e g e r
nz 2
if
i s a normed l i n e a r space.
Em
by example 1.24. Since
E
Then
Q("E;F)
=
pHy(nE;F)
i s not a normed l i n e a r space.
El
Conversely, suppose
i s a countable i n d u c t i v e l i m i t , it has a fundamental neighbourhood
system a t t h e o r i g i n c o n s i s t i n g of s e t s of t h e form a neighbourhood of zero i n
Em
i s contained and compact i n
f o r each
lm=iE
m
($m)m=2 denote a sequence i n (Elf and l e t Using t h e n a t u r a l embedding of each Em i n P
i t follows t h a t
P
Now suppose
=
E
P
lz=2$m$:-1
where
Pa("E).
Vm
and each compact s u b s e t of k.
is E
Let
Qm # 0 E EA f o r every m32. E and t h e p o l a r i z a t i o n formula
Since
@HY(nE) E
p("E).
neighbourhood of zero i n IIPII
Vm
Then t h e r e e x i s t s a sequence
Em, such t h a t
6 1.
Cm=1
E
Lo
f o r some p o s i t i v e i n t e g e r
m
i t follows t h a t
lm=lVm
m
(Vm)m=l,
Vm
a
20
Chapter I
For each
q,m(ym) # 0 .
choose
ym Vm such t h a t n-1 P(x+ym) = @,(XI ( ( ~ ~ ( y ~ ) ) and hence m22
Hence t h e r e e x i s t s a neighbourhood o f z e r o
II@ml)
v1 <
w
f o r every
ma2.
in
V1
a sequence i n
Hence we c a n c o n s t r u c t
Ei
11 @mil!,,m
such t h a t such t h a t
P
i n any l o c a l l y convex s p a c e
P
(Wm)i,2,
0,
and
m.
f o r every i n t e g e r
=
Since
CC
c a n b e embedded
as a c l o s e d complemented s u b s p a c e t h i s
F
q(nE;F) #
example can b e m o d i f i e d t o show t h a t
i s m e t r i z a b l e and n o t normable and
pH,(nE;F)
whenever
El
1122.
We now i n t r o d u c e a f u r t h e r s p a c e o f polynomials nomials.
then
V1
Q
such t h a t
El
@(nE).
g!
x
i s m e t r i z a b l e and n o t norm-
El
c o n t a i n s a neighbourhood system a t
a b l e t h e n it (@m):=z
However i f
If
-
t h e nuclear p o l y -
In c o n t r a s t t o t h e o t h e r s p a c e s o f polynomials we have i n t r o d u c e d ,
t h e s e form a s u b s p a c e of t h e s p a c e o f c o n t i n u o u s p o l y n o m i a l s .
The c o n c e p t
as a d u a l s p a c e and t o
o f n u c l e a r polynomial a l l o w s u s t o r e g a r d 6'("E)
The c o n c e p t i s a l s o u s e f u l i n d e v e l o p i n g t h e
develop a d u a l i t y theory.
t h e o r y of holomorphic f u n c t i o n s on n u c l e a r s p a c e s . D e f i n i t i o n 1.26
(a)
E
Let
F
and
be l o c a l l y convex spaces.
n
L E $ ( ~ E ; F )i s called a nuclear F
into
U
bourhood (A k ) i = l
k
E
and
R
yk
E,
in
(Yk>i,l
and
E
a bounded subset
from E (b)
E
and sequences where + i , k E B
for a l l
k
B
of
F,
m
( @ i , k ) k = l , i = l , .* q > n f o r aZZ i and
8
such t h a t
L ( X 1 > . . . > X n ) = l i = l A k @ l , k ( X 1 )' . * (X1, ...,X,) E E n .
We Let j t ' , f ~ ; ~ )
l i n e a r mapping from
i f there e x i s t a convex balanced zero neigh-
@n,k(xn)Yk f o r
denote t h e space of alL nuclear n
l i n e a r mappings
into F. P
E
@(%;F) i s called a nuclear
n
homogeneous polynomial
i f there e x i s t a convex baZanced zero neighbourhood
a bounded subset W
( @ k ) k = l IUo
and
B
of
F,
(Xk)L=l GB
E
R1
such t h a t
U
and sequences
in
E,
21
Polynomials on locally convex topological vector spaces
1"k = l A k @"(x)yk k
~ ( x )=
x
f o r every
E.
in
we l e t P 1 , ( " ~ ; ~ ) denote the space of aZZ nucZear n-homogeneous polyE
nomiaZs from Taking
F.
into
1.26(a) we obtain the d e f i n i t i o n of a nucZear Zinear
n=l i n
A ZocaZZy convex space
mapping between ZocaZZy convex spaces. t o be nucZear i f f o r every
a
E
cs(E)
B
there e x i s t s
EB
t h a t the carwnicaZ mapping from
Ea
onto
E
cs(E),BZa,
i s nucZear.
e x i s t s a sequence of
E'
such t h a t E
convex space
(An)n=l
in
p(x)
C n = l ( a n l I @ n ( ~ ) f o r every
and
.tl
1
EA
i s said t o be duaZ nucZear i f
Theorem 1 . 2 7
If E dN("E)
p
x
in
;k("E)
and
on
E
there
PN("E)
E.
A ZocaZZy
is nucZear.
i s a nuclear ZocaZZy convex space and
=
such
(@n)n a n equicontinuous s u b s e t
m
,<
i s said
This i s equi-
v a l e n t t o t h e c o n d i t i o n t h a t f o r e v e r y c o n t i n u o u s semi-norm m
E
nEN
then
p("E).
=
Proof
The second e q u a l i t y f o l l o w s immediately from t h e f i r s t s i n c e
-[x,("E))
= PN(nE).
show t h a t i f Since IL(xl
L
L
,..., xn) I
E
By d e f i n i t i o n c??~(~E)c- J?-(nE)
X(nE)
L
then
E
$("E).
is continuous t h e r e e x i s t s Q a(x,)
...
a(xn)
f o r any
f a c t o r i z a t i o n lemma, we may look upon E
is nuclear t h e r e e x i s t s a
from
EB
onto
(Yk)k C E a ,
x
For any
xl,
Ea
i s nuclear.
( @ k ) F = lc E '
=
in
E
L
cs(E)
~ E C S ( E ) such t h a t xl
,..., xn
E
Hence, by t h e
E.
as a n element o f
%(nEa).
Hence t h e r e e x i s t
in
w e have
Ea
Since
such t h a t t h e c a n o n i c a l mapping (Ak)kE k , ,
such t h a t
lE=lAk$k(x)yk
..., xn
in
B
and so it s u f f i c e s t o
f o r every
x
in
E
where
22
Chapter I
Since
t h i s completes t h e p r o o f .
51.3
TOPOLOGIES ON
SPACES OF
POLYNOMIALS
We now look a t t o p o l o g i e s on t h e v a r i o u s s p a c e s of polynomials we have S i n c e @(lE) = E '
defined i n t h e previous section.
( a l g e b r a i c a l l y ) , we
h a v e v a r i o u s g u i d e s from t h e d u a l i t y t h e o r y o f l o c a l l y convex s p a c e s .
The
most u s e f u l t o p o l o g y from t h e l i n e a r v i e w p o i n t i s t h e s t r o n g t o p o l o g y
-
t h a t i s t h e t o p o l o g y o f uniform convergence on t h e bounded s u b s e t s o f
E.
T h i s t o p o l o g y w i l l b e d e n o t e d by B .
We s h a l l s e e t h a t i t i s not t h e most u s e f u l t o p o l o g y from t h e holomorHowever, i t does s e r v e a p u r p o s e i n o u r development
p h i c p o i n t o f view. and m o t i v a t i o n .
For t h e s a k e o f e f f i c i e n c y we s h a l l always t r y t o d e f i n e
o u r t o p o l o g y on as l a r g e a s p a c e as p o s s i b l e . De finit i on 1.28
topology or the
Let B
E
and
F
be l o c a l l y convex spaces.
topology on @ ("E;F)
M
i s defined t o be the topoZogy of
uniform convergence on t h e bounded subsets o f (PM(nE;F),B) b y t h e semi-norms
The strong
E.
i s a l o c a l l y convex s p a c e and i t s t o p o l o g y i s g e n e r a t e d
I\
(/a,3
t h e bounded s u b s e t s o f
where
c1
ranges over
cs(F)
and
B
ranges over
E.
The f o l l o w i n g a r e e a s i l y proved. P roposi t i on 1.29
space, then
I f
E
(?("E;F),B)
P roposi t i on 1.30
If
l o c a l l y convex space then
i s a normed linear space and
F
i s a Banach
i s a Banach space. E
i s a ZocalZy convex space and
F
i s a complete
23
Polynomials on locally convex topological vector spaces
(~~,,("E;FI,B)
i s a complete ZocaZZy convex space.
we naturally consider the compact open topology
On @Hy(nE;F)
Let
Definition 1.31
E and
compact subsets of
F be ZocaZZy convex spaces.
We denote t h i s topoZogy by
E.
The compact
i s the topoZogy of uniform convergence on t h e
open topoZogy on $IHy("E;F)
T ~ .
(pHy(nE;F),~o) is a locally convex space and its topology is generated by the semi-norms 11 11 where c1 ranges over cs(F) and K ranges a,K
over the compact subsets of E. Since every compact subset of a locally convex space is bounded it follows that B fj =
on FHy(%;F).
3 T~
By the Hahn-Banach theorem we have
~ if and only if the closed convex hull of each bounded subset of E
T
is compact
i.e. if and only if E
is a semi-Monte1 space.
Since the uniform limit of continuous functions on a compact set is continuous the following is true. -Proposition 1.32
E be a ZocaZZy convex space and
Let
ZocaZZy convex space.
Then
space.
(BHy(n~;~) s ~ o )
By restriction B
convex topologies on
F a complete i s a compZete ZocaZZy convex and
T~
define locally
6 ("E;F) . However, we shall need a further
topology on
!?fE;F). This topology, denoted by T ~ , is the strongest topology we It is motivated by the factorization formula, the define on p(nE;F). definition of the strong topology on @("E;F)
when E' is a normed linear
space and certain properties of analytic functionals in several complex variables theory. It is perhaps more useful than T~ or B since it has stronger topological properties but it is more difficult to characterize in a concrete fashion. We first consider polynomials with values in a normed linear space. Let
E be a ZocaZZy convex space and l e t F be a T~ topology on Q("E;F) i s defined a s t h e i n d u c t i v e l i m i t topoZogy i n t h e category of ZocaZly convex spaces and conDefinition 1.33
normed Zinear space.
The
tinuous l i n e a r mappings
(B("E;F)
, T ~ ) =
o ~ P (;F), ~ Ea E CS(E) , t h a t i s lim a Zcs(E)
(P(~E~;F), B ) .
24
Chapter 1 p
Hence a semi-norm V
every neighbourhood P(P) ,<
on
d:
("E;F)
o f zero in C(V) I/pIIV
E
is .ru-continuous if and only i f f o r there e x i s t s
f o r every
P
in
c(V) 2 0
such t h a t
F("E;F).
W e w i l l subsequently s e e t h a t t h i s amounts t o saying t h a t a semi-norm on
i s .ru-continuous i f and only i f it i s ported by t h e o r i g i n and
,>("E)
f o r t h i s reason we c a l l
( 3 (nEa;F) ,B)
Since
t h e ported topology.
Tu
i s always a normed l i n e a r space
i s a bornological space when Banach space
F
i s a normed l i n e a r space.
i s a Banach space and
?(nEa;F)
(
( ?(nE;F)
When
, T ~ )
t i v e l i m i t of Banach spaces, i . e . an u l t r a b o r n o l o g i c a l space. ular,
(
? (nE;F) ,
(nE;F) ,.rW) F
is a
i s an inducIn p a r t i c -
i s then a b a r r e l l e d l o c a l l y convex space, t h a t i s
T ~ )
every closed convex balanced absorbing subset i s a neighbourhood of zero. For a r b i t r a r y
we use t h e weak form of t h e f a c t o r i z a t i o n lemma
F
(lemma 1.18) and d e f i n i t i o n 1.33 t o d e f i n e Definition 1.34
E
Let
Q("E;F)
defined on
and
F
.ru
on
P("E;F).
be l o c a l l y convez spaces.
Then
T*
as
The following elementary r e s u l t shows t h e r e l a t i o n s h i p between t h e topologies we have defined. Proposition 1.35
any p o s i t i v e i n t e g e r
(a) .ru Ibi
B
3
B
.ro
2,
and
For arbitrary locaZ7.y n we have
.ro
on
convex spaces
.
and
?("E;F).
d e f i n e t h e same bounded subsets o f
and hence have t h e same associated bornological topology
E
6(nE;F)
F
and
is
25
Polynomials on locally convex topological vector spaces We now g i v e a number o f e l e m e n t a r y examples r e l a t i n g t h e above t o p o l -
-
ogies
Afterwards, we d e f i n e a
f u r t h e r examples appear i n l a t e r c h a p t e r s .
topology on t h e s p a c e o f n u c l e a r p o l y n o m i a l s . Example 1 . 3 6
i s a n i n f i n i t e dimensional Banach s p a c e and
E
If
a l o c a l l y convex s p a c e , t h e n T
=
T~
2
6
have t h e same bounded sets and hence
a s s o c i a t e d with
T
Example 1 . 3 7
~
T
g'(lE)
If
bounded s u b s e t s o f
T~
i f and o n l y i f
= E'
Example 1 . 3 8
T~
and
Let
Example 1.39
Moreover, i f
If
subsets of
compact s u b s e t o f n
all
u
n
v
n
norm on P(*E) P(P) a
C")
E
=
cs(E)
-
..., 1
E
=
( P ,..., 4.0
d e f i n e d by
do n o t
and l e t
t h e n t h e r e e x i s t compact
such t h a t
K2,
K c K1 x K 2 .
Now e v e r y
IIPnIIK
i s a bounded s u b s e t o f
B
T~
= 0
for
(8( 2 E ) , T o ) .
0 . E and l e t C ntil p o s i t i o n
(0,
then
and
T~
P n ( ( x n ) n , ( y n ) n ) = xnyn,
Thus
=
I",=,
<'("E;F).
i s f i n i t e dimensional and hence
sufficiently large.
Let
?(2E)
and
K1
C"),
There-
and hence are n o t e q u a l .
i s a compact s u b s e t o f
K
and
EN
on
We show t h a t
E =
b e d e f i n e d by
Pn:E + E
B = (Pn):=l.
Using t h e method o f example
i s semi-Monte1 and hence a
E
= B = T~
T~
d e f i n e t h e same bounded s u b s e t s o f
Let
is a
E
E'.
d e f i n e t h e same bounded s u b s e t s of
T
s p a c e , it f o l l o w s t h a t
g 3 1'1
T~
on
= B
.?w
(j' ("E) ,@) i s m e t r i z a b l e and hence b o r n o l o g i c a l .
on ?(("E;F).
= @
R
~
c o n t a i n s a c o u n t a b l e fundamental system o f bounded s e t s
E
it follows t h a t
T~
T
Consequently i f
on
is a
be a c o u n t a b l e i n d u c t i v e l i m i t o f normed l i n e a r
T~,@
Since
> >
i s distinguished.
E
F
a r e l o c a l l y bounded.
,?(nE;F)
b e a normed l i n e a r s p a c e .
F
1.24 w e s e e t h a t 'j, ("E;F).
E
Let
s p a c e s and l e t
and
T
i s t h e b o r n o l o g i c a l topology
T~
i s t h e bornological topology a s s o c i a t e d with
w
is
F
Moreover,
i s a m e t r i z a b l e l o c a l l y convex s p a c e and
E
non-distinguished Frgchet space then
fore
?("E;F).
.
Banach s p a c e t h e n t h e Hence
If
on
To
I P(nun,vn)
,...
)
E
)
E
Let
C").
p
d e n o t e t h e semi-
nth position
- P(o,~,)
a(un,O) = 0
for all
I
for all n
P
E
~ ( 2 ~ 1 .
sufficiently large.
Hence
26
Chapter 1
P(nun,vn) = P(O,vn) P
i n P (2E).
n
and hence
T o
it f o l l o w s t h a t
for all
s u f f i c i e n t l y l a r g e and
S i n c e t h e semi-norm which maps
is
P((O,vn)(
n
for all
p
is a T
P(2E)
E
(Q( ~ E ) , T)
c o n t i n u o u s and B
( P (2E)
is not a
,T~]
T ~ , @
and
E
convex subset of a ZocalZy convex space
and
L
and
P
B
If
.
II
we l e t
$ k , i ~El}
8 N ( n E ) we l e t
E
where t h e i n f i m a are taken over a l l possible representations of IIB(L) IIB(L) hood
and
and
( r e s p . P) V
These
i s a balanced
$,("E)
E
and each and if
is
P(nE).
p N n( E ) , no, li B r e s p e c t i v e l y o n 8 ("E) .
T~
!?(2E). T~
on Y ( ~ E ) .
T~
We now d e f i n e t h r e e t o p o l o g i e s on correspond t o
p(P n ) = n
bounded s u b s e t o f
T~
-
is barrelled Since
i s a b o r n o l o g i c a l s p a c e and
t h e b a r r e l l e d topology a s s o c i a t e d with This r e s u l t extends e a s i l y t o
f o r every
m
I P(nun,vn)
to
c o n t i n u o u s semi-norm on :?(2E).
w e h a v e shown t h a t
We s h a l l s e e l a t e r t h a t
P
p(P) <
IIB(P)
may b e i n f i n i t e .
ItB(P)
are always f i n i t e .
&:,(%)
(respectively
E
of z e r o such t h a t
ITv(L)
However, i f
L
and
P.
i s bounded t h e n
B
Moreover, by d e f i n i t i o n , i f
L
t h e n t h e r e e x i s t s a neighbour-
Q,(nE)) <
and
TIv(€')
<
m.
These a l l o w us t o
give t h e following d e f i n i t i o n . Definition 1.40
(a)
The
IIo
topology on
t h e l o c a l l y eonvex topology generated by subsets of
AN(nE)
flK
PN(nE)I is K ranges over a l l compact
as
iresp.
E.
( b ) the
IIB
topology on J,("E)
( r e s p e c t i v e l y @,(nE)
locaZ2y convex topo2ogy generated by
nB
as
3
)
i s the
ranges over
all bounded subsets of E. ( c ) A semi-norm be
p
on J'",lnE)
IIwcontinuous
(respectiveZy ? N ( n E ) )
if for every neighbourhood V
is said t o in
of zero
21
Polynomials on locally convex topological vector spaces E
c(v) > 0
there e x i s t s p(L) s c(v)nV(L)
nu
11 LII B n 6
Since
of
B
E
it f o l l o w s t h a t for all
d N ( n E ) and '$"(nE) Proposition 1.41 TIo = IIR = B =
K1
ced subset
E
of
Since
~
,
n.
for every
%
3 T
w
on
Then
c K >0
I\L/I
(L) i c:
and a convex balan-
(Kl)n
for every
L
in
n.
i s q u a s i - c o m p l e t e and d u a l n u c l e a r i t s c l o s e d bounded
E
B =
and
T
II 0
= ItR.
E,
+Ei
such t h a t t h e c a n o n i c a l mapping from E' m a, K1 (xk)k,l E E l , ( a k ) k = l a sequence i n
i v e ) and a bounded sequence
@
Now l e t A
nu
Moreover, for each convex
exist
~1 E
and
E ' i s n u c l e a r w e c a n , g i v e n K a convex b a l a n c e d compact subB choose a convex b a l a n c e d compact s u b s e t K1 o f E c o n t a i n i n g K
Since
f o r every
f o r e v e r y convex b a l a n c e d
II R > B
there e x i s t
such t h a t
s e t s a r e compact and hence
set i n
;J~(~E))
be a quasi-complete dual nuclear space.
and for every non-negative
Proof
IIB(P) T
n.
K of E
balanced compact subset $N(nE)
E
Let
on d,("E)
T~
Xo 3
E
continuous serni-norms.
11 Pllgi
and
IIB(L)
P
irespectively
is t h e topology generated by a l l Xu
subset
p ( P ) 6 c(V)IIv(P))
(respectively
i n A.,("E)
L
f o r every
such t h a t
i n E'
K1
i n L ~ ( ~ E )NOW . suppose K.
E
Hence t h e r e i s semi-reflex-
such t h a t
E' K1
K.
There e x i s t s , by t h e Hahn-Banach theorem,
such t h a t
v e r g e s u n i f o r m l y on
(@k)k=l i n
(since
where t h e s e r i e s converges u n i f o r m l y on
L €AN("E).
(XN(nE),no)'
m
i s nuclear. K~
a ( L ) = IIK(L) L
=
I",l
We t h e n have
and
$m,l..
Ia(A)I
.
Q IIK(A)
f o r every
J,m,n where t h e series con-
28
...,kn =1xk 1
=
*
* . 'kn
1
...
@k 1 L(a , . . . , a 1 n kl kn
T h i s completes t h e p r o o f . Combining theorem 1 . 2 7 and p r o p o s i t i o n 1 . 4 1 w e o b t a i n t h e f o l l o w i n g result. C o r o l l a r y 1.42
then
X("E)
=
If
gN(nE)
If
Theorem 1 . 4 3
E
E
is a quasicomplete nuclear and dual nuclear space and
TIo =
T~
for every
n.
is a quasi-complete dual nuclear space then
( L ( ~ E ) , T ~is ) a nuclear space.
K
Proof
If
sequence
(Xm)m=l
f o r every
m
@
in
i s a compact s u b s e t of in
E'.
E and
Hence i f
m
n
E
E
R1
t h e n t h e r e e x i s t s a compact such t h a t
i s any p o s i t i v e i n t e g e r and
then
lrisn- 1
(and by induction)
L
E
&(("E)
29
Polynomials on locally convex topological vector spaces
a compact sequence
t h i s completes t h e proof.
The form o f t h e above i n e q u a l i t y w i l l b e u s e d i n c h a p t e r 3 Proposition 1.44 T
E
Let
on &("E)
= i7 w w
be a quasi-complete nuclear space.
Proof
nu
II~(L)c cn
such t h a t
C>O
Since
= T
IIw
2 T
w
W
of zero, contained i n V
Vn f o r every
L
in
, and
~ ( n ~ ) , n = 1 , ... 2,
it s u f f i c e s t o p r o v e t h e above i n e q u a l i t y t o show
w'
By t h e n u c l e a r i t y o f bourhood
V
i c a l mapping m
($k)k=l in
IILI/
of zero, there
V
Moreover, f o r any convex balanced neighbourhood e x i s t a convex balanced neighbourhood
Then
n.
f o r any p o s i t i v e i n t e g e r
W"
E
E we can choose, g i v e n a convex b a l a n c e d n e i g h -
o f z e r o , a n e i g h b o u r h o o d of z e r o EW + EV
and
m
is nuclear.
L ~ d ( % ) and
Now suppose If
Y1,.
Hence t h e r e e x i s t
. . ,Yn
V
E
L1,
E
co
( x ~ ) c~ V= ~such t h a t
where t h e convergence i s i n
E
s u c h t h a t t h e canon-
WCV
x = rk=lXkQk(x)xk
for all
EV'
IILII
= 1.
V"
then
and h e n c e
=
Now
Ikl,
lL(xk , 1
...,kn . .. , x
kn
'kl )
1
... x
kn
,< 1
.
L(Xk ,. .,Xkn) bkl 1
f o r any c h o i c e o f Hence
x
. .. $k .
,. . . , x
kl
n
kn
in
V
since
x
Chapter 1
30
.,knIhkl ."
kn
I
=
then
Thus nw(L) if 11 LII vn
6 =
cn IlLll vn and since this inequality is trivially satisfied we have completed the proof. m
The preceeding results can be transferred to n homogeneous polynomials by using the inequality II*(L)
for any L B
of
E.
6 IIB(L)
in XS("E)
nn
c n! nB(L)
n A~("E)
=
~ki(~E) and any convex balanced subset (X ZPE) ,no)
In particular, we find
(A i(nE> ,nu)
=
($?N(nE)
=
WNN(nE),no)
and
,nu).
We now summarize results obtained in this way. Proposition 1.45 Let E be a quasi-complete Locally convex space and l e t n be a non-negative i n t e g e r . ( a ) If E i s dual nuclear then
no
= B =
and
('J(~E),T~)i s a nucZear space. ( b ) IS E i s nuclear then 2bN(n~) = 'J'("E and
= T
w
on J"(nE)
.
Moreover, the estimates given in propositions 1.41 and 1.44 are still valid, with minor modifications, for spaces of homogeneous polynomials on the appropriate locally convex spaces.
31
Polynomials on locally convex topological vector spaces 51.4
DUALITY THEORY FOR SPACES OF POLYNOMIALS I n t h i s s e c t i o n w e c o n s i d e r l i n e a r f u n c t i o n a l s on t h e l o c a l l y convex This t o p i c is currently t h e subject
s p a c e s o f polynomials d e f i n e d i n 5 1 . 3 .
o f r e s e a r c h and s h o u l d p l a y an i m p o r t a n t r o l e i n t h e development o f t h e Our p r e s e n t a t i o n of r e s u l t s i s n o t f u l l y com-
subject i n t h e near future.
p r e h e n s i v e b u t h o p e f u l l y o u t l i n e s t h e main developments and p r o v i d e s a g l i m p s e o f f u t u r e developments. We show t h a t c o n t i n u o u s l i n e a r f u n c t i o n a l s on s p a c e s o f polynomials Our main t o o l i n o b t a i n i n g
can themselves b e r e p r e s e n t e d by polynomials. t h i s r e p r e s e n t a t i o n i s t h e Borel t r a n s f o r m . D e f i n i t i o n 1.46
Let
A be a vector space of
.
form on h
The Bore2 transform of
T,BT,
vaZued n-homogeneous
C
E
poZynomiaZs defined on a ZocaZZy convex space
and Zet
i s defined on
T
be a Zinear
{$IEE*I+"EA >
by the formuZa BT($I) If BT
to
then
=
i s a subspace o f
F
T(99
EX
and
ms(Fn)c&
i s an n-homogeneous polynomial.
F
'4
if BT E $ ~ ( ~ E * and )
=
?(nE)
then t h e r e s t r i c t i o n of
For example i f
o r 3N(nE)
then
A
= qa("E)
BT E ? a ( n E 7 ) .
A i s a l o c a l l y convex s p a c e and T i s c o n t i n The Borel t r a n s f o r m w i l l o n l y be u s e f u l i f it i s i n j e c t i v e . T h i s
In t h e c a s e s w e consider uous.
w i l l always b e t h e case i f
p r o p e r t y and
fi
= PN(nE)
or if
E
h a s t h e approximation
'J'("E).
=
P r o p o s i t i o n 1.47
The Bore2 transform i s a vector space isomorphism from
( i ) ( 2 , ( n ~ ),n6)
1
onto D ( ~ E ; )
and
(ii)( @ N ( n ~,no) )
Under t h i s isomorphism the equicontinuous subsets o f espond t o the locaZZy bounded subsets o f (PN(%) ,no)
subsets of P(n(E,.ro) Proof
p("E;)
onto
P(~(E:T~)
(PN(("E),ITB)
corr-
and the equicontinuous
correspond t o the Zocally bounded subsets of
'1. S i n c e b o t h c a s e s a r e proved i n a s i m i l a r f a s h i o n , we o n l y c o n s i d -
er t h e case There e x i s t
(PN(E),i16).
B
i s l i n e a r and i n j e c t i v e .
and
B
an a b s o l u t e l y convex bounded s u b s e t o f
c>O
Let
such t h a t IT(P) I f
c nB(p)
f o r every
P
i n P,("E).
T E (@N(nE),TIbgll E
32
Chapter 1 In p a r t i c u l a r i f
c 11$"1/, 6 c .
$
Hence
tinuous subset of Now suppose
1 then
h
T h i s a l s o shows t h a t t h e image by
PI
and
P("Egt)
E
of
B
E.
If
/I PillBo P
E
T(P)
li=lP1($i).
=
ear o p e r a t o r on 3'N(nE), t h e representation of
c
6
of z e r o i n
V
ll$ill
Moreover, i f
P.
and
?(nEgt).
E,
then w e
T
is a well defined l i n -
T(P)
i s independent o f
One e a s i l y shows t h a t i.e. the definition of
Eb
o f an equicon-
f o r some c l o s e d a b s o l u t e l y m n and P = l i = l $ i , $ i E~l
PN(nE)
f o r some neighbourhood m
B
i s a l o c a l l y bounded s u b s e t o f
(?,("E),TIB)'
convex bounded s u b s e t
let
/ B T ( $ ) / = IT($")[ h
i s bounded on a neighbourhood o f z e r o i n
BT
so it i s continuous.
I/$ //
and
E E I
= 0
then
Pt(Oi) = 0
and
hence
=
Hence
T
cnB(P).
( ? N(nE) ,")
E
and
BT($)
=
T($n) = P I ( $ ) .
i s s u r j e c t i v e and a v e c t o r s p a c e isomorphism.
{P
=
E
v("Ei) ;
)I P \ \ Bo 8
C}
T h i s shows t h a t
B
The above a l s o shows t h a t
f o r any
c>O
and t h i s completes
t h e proof. Proposition 1.48 (?N(nE)
E'
,nu)
The Borel transform i s a vector space isomorphism from
onto
,FE ("El)
f t h e space of n
homogeneous poZynomiaZs on
which are bounded on the equicontinuous subsets of
El).
Under t h i s
isomorphism t h e equicontinuous subsets of subsets of
? ("El)
subsets of
El.
Proof T
E
5
6
(?N(nE) ,II) I correspond t o which are uniformly bounded on :he equicontinuous
The p r o o f i s v e r y similar t o t h e proof of p r o p o s i t i o n 1 . 4 7 .
(?N(nE) ,nu)
and l e t
V
Let
d e n o t e a n a b s o l u t e l y convex neighbourhood of
33
Polynomials on locally convex topological vector spaces 0
in
and
E.
BT
E
c(V)> 0
There e x i s t s
? ("El). 5
uous subset of
such t h a t
This a l s o shows t h a t t h e image by
i s a subset of
(yN("E),na)'
5
B
("El)
of an equicontin-
c o n s i s t i n g of fun-
E'.
c t i o n s which a r e uniformly bounded on t h e equicontinuous s u b s e t s of Now suppose
PI
(nE').
E
We d e f i n e
i s a neighbourhood of zero i n
Moreover, s i n c e isomorphism.
E
and
BT($) = PI($)
as i n proposition 1.47.
T P
E
?N(nE)
If
V
then
t h i s shows t h a t
B
i s a v e c t o r space
The r e s u l t about equicontinuous s e t s a l s o follows from t h e
above. We g i v e
Quite a number of c o r o l l a r i e s can be deduced from t h e above. The f i r s t i s perhaps t h e most i n t e r e s t i n g .
j u s t a few examples.
Since
t h e c o l l e c t i o n of spaces which occur i n t h i s c o r o l l a r y i s r a t h e r i n t e r e s t -
s),
ing, (see chapters 3and Definition 1.49 EL;
we g i v e them a s p e c i a l name.
A locally convex space
E
E
i s fuzzy nuclear i f
and
are both complete infrabarrelled nuclear spaces. A f u l l y n u c l e a r space i s a r e f l e x i v e nuclear space and t h e s t r o n g dual
of f u l l y n u c l e a r space i s f u l l y n u c l e a r .
Every FrGchet n u c l e a r space i s
f u l l y nuclear. Corollary 1.50 @("E),
n
I f
E
is a f u l l y nuclear space then
T~
= T*
a p o s i t i v e integer, i f and only i f F'M(nEA) = ?(nE;j)
B -bounded subsets of p:d(nEL)
are locally bounded.
on
and the
Chapter 1
34 Proof
Since
i s an i n f r a b a r r e l l e d l o c a l l y convex s p a c e t h e equicon-
E
c o i n c i d e w i t h t h e bounded sets and hence
tinuous subsets of
Ek
PM("Eb) = Pt(("E').
I t now s u f f i c e s t o a p p l y theorem 1 . 2 7 , and p r o p o s i t i o n s
1 . 4 5 , 1 . 4 7 and 1 . 4 8 t o complete t h e p r o o f . In particular w e note t h a t
o = Also t h i s shows t h a t
nuclear space.
T
T
w
T~
on
#
?(nE) on
T~
if
i s a Frgchet
E
if
F(nC(N) x CN)
1122,
a r e s u l t which w e h a v e a l r e a d y proved d i r e c t l y (example 1 . 3 9 ) . Corollary 1.51
If E (8(nE)
If E
Corollary 1.53
If E
ng
Q,(nE;)
I
C o r o l l a r y 1.52
FreTchet space then
is a r e f l e x i v e nuclear space then
=
i s an infrabarrelled l o c a l l y convex space then
i s an infrabarreZled
nw
DF s p a c e o r a distinguished
on P N ( n E )
We now look a t some examples i n which t h e Bore1 t r a n s f o r m g i v e s a t o p o l o g i c a l isomorphism.
We f i r s t need some p r e l i m i n a r y r e s u l t s .
E, a Locally convex space, has property E
of
K
subset
E K
such t h a t
EB.
is contained and compact i n
(EB i s t h e v e c t o r s u b s p a c e o f t h e norm whose u n i t b a l l i s
E
If
B).
g e n e r a t e d by B
EB
i s a Banach
S t r i c t i n d u c t i v e l i m i t s o f FrGchet s p a c e s and s t r o n g d u a l s o f
space.
f u l l y nuclear space has property Lemma 1 . 5 4 f,,(nE;F)
9M ("E;F)
If in
B
K E
i s compact i n such t h a t
continuous.
Lemma 1 . 5 5
If E
(EJ)
K
Hence
E, If
K. E
F
(s)
then
nEN.
and any
t h e n t h e r e e x i s t s an a b s o l u t e l y convex
i s compact i n P
In p a r t i c u l a r , every
has property
for any l o c a l l y convex space
i n d u c e t h e same t o p o l o g y on T
(5).
(s).
If the locally convex space =
Proof
hence
and endowed w i t h
B
i s complete t h e n
i n f r a b a r r e l l e d Schwartz s p a c e s have p r o p e r t y
set
if for each cornpact B of
(5)
there e x i s t s an absolutely convex bounded subset
P
E
EB.
Hence
PM(("E;F) t h e n
QHy(nE;F)
and
T , T ~
PIK
and
/I I / B
is
T~
and
$h(nE;F) = PH,,("E;F).
is a f u l l y nuclear space then @Hy(nE) is equal t o
35
Polynomials on locally convex topological vector spaces
t h e cornpZetion of Proof
The completion o f
convex s p a c e
.
~ )
( L 1 ( n E ) , ~ o ) l i e s i n j?Hy(nE) (j'("E)
and q u a s i c o m p l e t e w e c a n c h o o s e
KIC-K,
m
m
(yn)n=l C K 1
(Xn)n=l
E
(Qn)zX1
R1,
such t h a t f o r e v e r y x
x
A(xl,.
. .,xn)
. .,xn
xl,.
1
n
IA(yi
1
. . . . .yi
n
choose a f i n i t e s e t o f i n d i c e s
in
Kl
EK
E;<
with
.
Since
-+
E
i s nuclear.
K1
llQnll
C
1
and
m
m
6 (li=llxil)n
)
I
m.
<
m
and
Hence, f o r any 6 ' 0 ,
w e can
such t h a t
F
1 A . ... X i
<
i s c o n t i n u o u s on
A
. . ,&=lXi+ii(xn)Yi)
A(1.1=1X .1 + . 1( x l ) y i , .
1
-
E
K
E
m
=
A. , . . . , X i
IIA
and l e t
m
=
Now
?UP l , . . . ,in
LiY("E)
E
K
in
where t h e series converges u n i f o r m l y i n
we have f o r any
E
A
E. Since E i s dual nuclear a n a b s o l u t e l y convex compact
K1, such t h a t t h e c a n o n i c a l i n j e c t i o n
E
Hence t h e r e e x i s t
K:
Let
, T ~ ) .
b e a n a b s o l u t e l y convex compact s u b s e t o f
s u b s e t of
f o r any l o c a l l y
and t h u s t o p r o v e t h i s r e s u l t i t s u f f i c e s t o show
E
l i e s i n t h e completion o f
S'HY(nE)
K
,T
(t ("E)
A(yi ,..., yi ) $ i . . . $i ( 1 < 6/2. S i n c e E l i s d e n s e n 1 k 1 k K w e c a n choose a sequence of c o n t i n u o u s l i n e a r forms on E , ( $ i ) i = l , l1
ElK
such t h a t
Ill
F
Q
il
... Q
in
-
1 JI F
il
... 1L i II
< 6/2
n K
Combining t h e s e two i n e q u a l i t i e s we o b t a i n t h e d e s i r e d r e s u l t . If
(E,T)
iated with
i s a l o c a l l y convex s p a c e t h e n t h e Mackey t o p o l o g y a s s o c ( n o t t o b e c o n f u s e d w i t h t h e t o p o l o g y o f t h e M-closure
i s t h e f i n e s t l o c a l l y convex t o p o l o g y on
d u a l as T
(E,T).
E
If t h e Mackey t o p o l o g y a s s o c i a t e d w i t h
t h e n we s a y t h a t
(E,r)
T ~ )
which h a s t h e same c o n t i n u o u s
i s a Mackey s p a c e .
T
coincides with
An i n f r a b a r r e l l e d l o c a l l y
Chapter I
36
convex space is a !.lackey space.
If E i s a f u l l y nuclear space then
Proposition 1.56
is the completion o f a nuclear space and hence it @Hy(nE),~o) is a complete nuclear space. Let s s be the strong topology on
Proof
(nEb)
'
CJ?,,("E)
(proposition 1.47).
, T ~ )
($("E;)
, T ~ )
is the strong
dual of a semi-reflexive space and hence is a barrelled Mackey space. (jJ(nE;),~u) is also a barrelled space and hence a Mackey space. To complete the proof we need only show
'
(?(nEb) Since
' .
, T ~ )
(P(~E;) ,TB)
is semi-reflexive,
, T ~ )
osition 1.48,
(j)("EA)
=
, T ~ )
'
(5'(nEb)
, T ~ )
2
$'M(nE)
FHy(nE),
=
1
=
S;iY(n~).
BY prop-
and this completes the
proof. i s a fully nuclear space then
If E
Proposition 1.57
(!j"nE) ,TJ i f and only i f the If
Proof ((P("E)
,T,);)
(PHy ("E;)
, @(%)
su bounded subsets o f
(P (nE), g
;
T
~
@("E)
;a )
(PHY("E;),
and hence
T ~ )
then by proposition 1.56
(Q("E),T~)
space and thus it is reflexive. Hence
are Locally bounded.
is a barrelled semireflexive
(gHy(nEb),~o)
is also reflexive
and the equicontinuous subsets of the dual coincide with the strongly bounded subsets. The strong topology on (BHy(nE ; ~ ) , T ~ ) ' is the T topology by proposition 1.56 and the equicontinuous subsets are the locally bounded sets by proposition 1.47. Conversely if the su bounded subsets of
p("E) are locally bounded then the bounded subsets of (QHy("E;) , T ~ ) are equcontinuous (propositions 1.47 and 1.56). Hence (PHy(nEk),~o) is infrabarrelled and thus reflexive. By proposition 1.56, ((3(nE) , T ~ ) = (!!Hy(nE;),~o)~ is also a reflexive space and (OYEI,TJ;
=
((pHy("~;),
T ~ I ; ) ~
=
(pHY(n~t),~o~.
This completes the proof. We now look at linear functionals on spaces of homogeneous polynomials
37
Polynomials on locally convex topological vector spaces d e f i n e d on F r e c h e t s p a c e s w i t h t h e a p p r o x i m a t i o n p r o p e r t y . nomials s t i l l a p p e a r and p l a y a n i m p o r t a n t r o l e . uous forms as t h e c o r r e s p o n d i n g t h e o r y f o r developed and indeed t h e g e n e r a l t h e o r y f o r an u n t i l l e d f i e l d .
T~ T~
Nuclear p o l y -
We s t u d y o n l y
T~
contin-
c o n t i n u o u s forms i s n o t y e t c o n t i n u o u s forms i s a l m o s t
One c a n e a s i l y show t h a t t h e v e c t o r s p a c e isomorphisms
o f p r o p o s i t i o n 1.1 y i e l d a t o p o l o g i c a l isomorphism when t h e a p p r o p r i a t e s p a c e s are endowed w i t h t h e compact open t o p o l o g y .
This implies t h e f o l l -
owing r e s u l t .
Lemma 1.58
If E
i s a Frgchet space and F
If
P r o p o s i t i o n 1.59
T
convex space and
E
'
IT(L)I
for all L i n .Y.(E;F) then there e x i s t
E
i s a FrLchet spaee,
(x (E;F) ,T
'
~ )
F
i s an arbitrary locally
satisfies
CIILIly,K ~ E C S ( F ) and
where
( a ) a null sequence ($n)n
( b ) a sequence
i s a locally convex space
n
then f o r any p o s i t i v e integer
K
i s a compact subset of E
( x ~ )i n~ E ,
in
F'
such that
f o r all y i n F and all n and m (c) (hn)n=l E k1 where l n l h n l < l
I$,(y)
I S cY(y)
such that
m
T(L)
=
Cn,lXn+n(L(Xn))
for every Proof
L
i n X(E;F)
The compact s u b s e t
K
. of
E
i s c o n t a i n e d i n t h e a b s o l u t e l y con-
vex h u l l o f a n u l l s e q u e n c e , i . e . t h e r e e x i s t s
as n*
Hence
such t h a t
m
(Xn)n=lCE
where
Xn+o
Chapter 1
38
Thus
I
IT&) Since
c SUP Y(LX,). n
L
E
( L ( x ~ ) ) i~s a n u l l sequence i n
X(E;F),
{ ( Y ~ ) : = ~ ; y,
E
Fy and
y(yn)
-to
as
n -1.
F.
Let
co(Fy)
=
i s a normed l i n e a r
co(Fy)
s p a c e w i t n norm
II (Y,),"=, II
SUP Y(Yn). n
=
By t h e Hahn-Banach theorem w e can e x t e n d co(Fy)
s c supy(fn)
I?'(ifn}n)l Since
m
($n)n=l
T
t o a l i n e a r functional
'? on
such t h a t
n
co(Fy)l
=
R1((F
such t h a t %rfnIn) for all
(fn)n
Hence € o r any
in L
t h e r e exist
) I )
Y where
(Fy)l C F '
E
=
f o r every
$,(y)l
s cy(y)
{fnIn
in
m
E
co(FY). R1,
y
for all
lz=llh,,\ in
F
,< 1,
and a l l
n
In= m
co(Fy).
in
Z(E;F)
we h a v e
and t h i s c o m p l e t e s the p r o o f . Proposition 1.60
space.
If
in 2("E;F) B
E
cs(F)
T
E
Let
(d("E;F)
where
K
E
F
be a Frechet space and
'
,T~)
satisfies
I
I
C
a ZocaZZy convex
IIL//B,Kn f o r every
i s an absolutely convex compact subset o f
then there e x i s t an absolutely convez compact s e t
4
K
E
and
and:
L
39
Polynomials on locally convex topological vector spaces
(a)
for every
K
T(L)
such that
Proof
Nn
a sequence i n
L
, {x
lk
,. . .,xn
k k=l
,
1"
=
A~$~(L(X ,..., x ) ) k=l lk nk
%("E;F).
in
We proceed by induction on n .
Proposition 1.59 covers the case
n=l. Assume the proposition is true f o r the positive integer n . T
E
it a ,
x(n+lE;F).
4
Define T
of lemma 1.58.
? E
on
%("E;
f(E;F))
Let
by the isomorphism, call
( % ( n E ; ( ~ ( E ; F ) , ~ O ) ) , ~ O ) and '
Hence, by induction, there exist
-n
a sequence in K
n
By proposition 1.59, there exists a null sequence in K, (yj)y=l each k there exists a sequence of scalars
and f o r
Chapter I
40
Hence
in Q ~ + ' E ; F ) .
for every L
we may reorder the above to obtain a sequence with the required properties This completes the proof. Proposition 1.61
If
E
i s a FrLehet space w i t h t h e approximation prop-
e r t y , then t h e Bore1 transform,
Proof
T
E
( p ("E)
B , i s a l i n e a r isomorphism from
Since E has the approximation property B is injective. Let
'
, T ~ )
IT(P)
and suppose c/lpIIK
for all
P
E
P ("E)
where K is an absolutely convex compact subset of E . By the polarization formula and proposition 1.60 there exist a relatively compact sequence
such that
41
Polynomials on locally convex topological vector spaces A further application of proposition 1.60 shows that
that the equicontinuous subsets of the form {P
B
is surjective and
(~("E),T~)' correspond to sets of Since (E',T~)'= E and d cK}.
@N(n(Ef,~o));IIPI/KO 8 (nE) = BE(nE) it follows, by proposition 1.48, that the closed convex hull of sets of the form U(@~(~(E~,T~)), @(("E)) E
K compact are a fundamental neighbourhood system at 0 in ( @N(n(Eq,~o];flu) c ranges over all possible sets of positive numbers.
as
K
A fundamental neighbourhood system at
0 in
( 6'(nE),~o)h
by the polars of bounded sets. Since
K compact =
U((~("E),T~)',@
U { P E P PE); KCE
("E))
closed convex hull of 0
I I P Ic IcK} ~
K compact
=
U(((P(~E),T~)',
K compact This completes the proof.
@("E))
closed convex hull of
is given
42
Chapter I
51.5
EXERCISES
The f o l l o w i n g e x e r c i s e s d e v e l o p t o p i c s which w e s h a l l e n c o u n t e r i n
l a t e r c h a p t e r s and a l s o c e r t a i n material which we d i d n o t f i n d c o n v e n i e n t t o include i n t h e t e x t . difficult.
Consequently, some o f t h e s e e x e r c i s e s a r e r a t h e r
A s e r i o u s a t t e m p t a t s o l v i n g them, w i l l , however, p r o v i d e a
good d e a l o f i n s i g h t i n t o t h e t h e o r y f y i n g n o n t r i v i a l problems.
-
even i f o n l y as a means o f i d e n t i -
For t h e r e s e a r c h worker t h e y c o u l d e a s i l y l e a d
t o new t e c h n i q u e s and worthwhile r e s e a r c h p r o j e c t s .
Starred exercises are
commented o n i n Appendix 111. 1.62
Show t h a t
Dim(E) 6 1 o r 1.63* -
If
F =
If
E,F
for all and
show t h a t
1.65
E
If P
i f and o n l y i f e i t h e r
and
i f and o n l y i f
n
E
2
are v e c t o r s p a c e s o v e r
G
Q €!fa(F;G)
that
= X:(mE;F)
m=l,m=O,
i s a n i n f i n i t e d i m e n s i o n a l l o c a l l y convex s p a c e , show t h a t
E
Fa(nE) = @("E) 1.64 -
fa(mE;F)
to}.
a"). P
E,
E
pa(E;F)
and
Q o P E(~).(E;G).
a r e l o c a l l y convex s p a c e s and
F
i s c o n t i n u o u s a t one p o i n t i f and o n l y i f
P
P
E
Pa(E;F)
show
i s everywhere
continuous. 1.66
Replace c o n t i n u o u s by hypocontinuous ( r e s p . Mackey c o n t i n u o u s ) i n
e x e r c i s e 1 . 6 5. 1.67 I f E i s a m e t r i z a b l e l o c a l l y convex s p a c e , F i s a l o c a l l y convex s p a c e and P E p a ( E ; F ) show t h a t P E. @(E;F) i f and o n l y i f I$o P
E
1.68*
Let
E
and
mapping from
E
into
and d e f i n e
Ayl by2
nomial o f d e g r e e Y1'.
. . ,yn+l
+
f o r every
@(E)
and
F
F'.
b e r e a l Banach s p a c e s and l e t F.
...
Let
Aynf(x)
Ayf(x) = f ( x + y ) - f ( x ) inductively.
i f and o n l y i f
in x
in
in
E.
Ayl
...
f
be a continuous
for all
Show t h a t
f
x,y
in
i s a poly-
A ~ ~ + ~ f =( x0 ) f o r a l l
Show t h a t t h i s r e s u l t d o e s n o t e x t e n d t o
Banach s p a c e s o v e r t h e complex f i e l d .
E
43
Polynomials on locally convex topological vector spaces Let
1.69*
be a Frgchet space and suppose
E
P
E
Pa(nE).
Show t h a t
i s continuous i f i t s r e s t r i c t i o n t o a 2nd category subset of
P
is
E
continuous.
3
endow
0=
E =
Let
1.70*
m
,&
Space of
be t h e Dirac d e l t a function a t t h e point E
(
6('a)
=*
I:=,
. €in
(ansa)
Show t h a t t h e bounded s u b s e t s of
bounded and hence deduce t h a t
on @(nJl)
Show t h a t
for a l l
n
and
@Hy(nJl I ) =
#
T~
("a
I ) .
on @(rial)
T~
Let
spaces.
m Show t h a t t h e following a r e equivalent:
+
m
(b)
PPnEl;)
(c)
!?("El;) = PHy(nEl;)
E
E;
In=, $n$n
Em
E
E
a neighbourhood that
= !?Hy(nEb)
=
lm=o Em m
and l e t
m
$1 .
Conclude t h a t for all
T~ = T~
n22.
Ei
is a basis for
admits a continuous norm,
each
Let
('
be a s t r i c t inductive l i m i t of FrGchet Monte1
(a)
($n)n=l P =
E = lim E
?.!
(!?(n$jt),~O a r)e l o c a l l y
1.73* -
m
(ansa) .6n = f m
HY ( 2 B )\
E
1.72" I f E i s a Frgchet nuclear space and ($n)n show t h a t (9n@m)n>m=l i s a b a s i s f o r ( @('E),T,).
__ 1.74
We
st
E
m
Show t h a t
T~).
a.
Let
(fm)m=2 i s a Cauchy sequence i n
and t h a t
!?(*a) ,
R.
-functions of compact support i n
with i t s usual s t r i c t i n d u c t i v e l i m i t topology.
0
8H,(2E)
f o r some
n?2,
for a l l
nEN.
where each
Em
# J,n
E
n.
for a l l
EA
and t h a t
i s a l o c a l l y convex space.
P
'6 ( 2 E )
E
V
of zero such t h a t
(6 (2E) , T ~ )
i s not complete i f
I\$nIIV<
Let
Show t h a t
i f and only i f t h e r e e x i s t s f o r each
n.
Hence show
i s a non-normed metrizable
Eo
l o c a l l y convex space.
1.75
Let
E
=
lm=lEm m
where each
Em
i s a Banach space.
P E 8("E) and f o r each p o s i t i v e i n t e g e r m l e t m m E E j and y E Ej. Show t h a t Pm E
X
lj=l
as
m
+ m
lj=m+l
Let
Pm(x+y) = P(x) ("E)
uniformly on a neighbourhood of each point of
and t h a t E.
where Pm
+
P
Chapter I
44
1.76*
If
i s a m e t r i z a b l e l o c a l l y convex s p a c e and
E
i n t e g e r , show t h a t t h e compact open t o p o l o g y on l o c a l l y convex t o p o l o g y on
If
is a positive
is t h e finest
?! ("E)
which c o i n c i d e s w i t h t h e t o p o l o g y o f
("E)
p o i n t w i s e convergence on e v e r y e q u i c o n t i n u o u s s u b s e t o f 1.77 -
n
!?
(nE).
i s a l o c a l l y convex s p a c e i n which e v e r y n u l l sequence i s a
E
i s t h e s p a c e o f F-valued sequen-
Mackey n u l l sequence, show t h a t P M ( E ; F ) t i a l l y c o n t i n u o u s polynomials from
into
E
F.
denote an uncountable s e t .
1.78 __
Let
A
k-space,
but t h a t P ( ~ ( c ~ )=
p H yn(cA
=
CA
Show t h a t
8 ("cA)
for all
M
i s not a
n.
1.79 Show t h a t t h e f o l l o w i n g two c o n d i t i o n s on a l o c a l l y convex s p a c e E a r e equivalent: (a)
e v e r y compact s u b s e t o f
(b)
(i)
E
i s s t r i c t l y compact;
e v e r y n u l l sequence i n
i s a Mackey n u l l
E
sequence ; (ii)
e v e r y compact s u b s e t of
E
is contained i n t h e
a b s o l u t e l y convex h u l l o f a n u l l sequence. If
1.80 -
i s a l o c a l l y convex s p a c e , show t h a t
E
k-space a s s o c i a t e d w i t h 1.81 If B C !? ("E;F) x
in
E.
P
E
If
E
P,("(ExF))
1.83*
F
i s a Banach s p a c e , show t h a t
s u p I P(x) I < f o r every PEB C o n s t r u c t a counterexample which shows t h a t t h i s r e s u l t i s n o t is
Let X
bounded i f and o n l y i f
T~
true for arbitrary 1.82*
(E',u(E',E)).
i s a F r e c h e t s p a c e and
E
and
E. F
are b o t h F r 6 c h e t s p a c e s o r b o t h
i s s e p a r a t e l y c o n t i n u o u s , show t h a t
P
ayF/L
Show t h a t f o r each
n
s p a c e s and
i s continuous.
b e a c o m p l e t e l y r e g u l a r H a u s d o r f f s p a c e and l e t
t h e s p a c e o f E-valued c o n t i n u o u s f u n c t i o n s on topology.
( E ' , T ~ ) is the
h(X)
be
X with t h e compact open
45
Polynomials on locally convex topological vector spaces (a)
fM ("A (XI)
(b)
PHY(*,&(X))
=
p ( n J , (X))
E = Co(T),
r
uncountable,
1.84"
If
P E !?("E;F) -,
P
from
Co(r)
P F = II ( r l ) ,
r2
Co(r2).)
r
in
rl
u n c o u n t a b l e and
and
is t h e n a t u r a l p r o j e c t i o n
(Lr2
= Lr2.
=
i s Lindelbf.
X
.r/
such t h a t
onto
{ $ J n l $E JC O ( T ) ' } s p a n s a d e n s e
Show t h a t
( C? (nCo(r) 1 , 6).
subspace of
1.85
if
i s paracompact;
show t h a t t h e r e i s a c o u n t a b l e
P(nCo(r2 ) ; F )
E
x
B ( n a ( ~ ) ) if
=
K
Let
2 L ([O,lln+l)
E
i s symmetric w i t h r e s p e c t
K
and suppose
t o its coordinates. [P(x)l(t)
Let
j' ... jol
K tl,.
=
..,tn,t)x(tl).
.. x ( t n ) d t l . .
.dtn
0
f o r every
x P
1.86"
L2([0,1]).
E
If
6
8 (nL2[o,l];L2[o,ll) i s a s e p a r a b l e H i l b e r t s p a c e and
E
t h e r e e x i s t s an such t h a t
1.87" __
x
hP(x) If
Show t h a t
in
a
P
E
@ (nE;E)
C,
a r e Banach s p a c e s , we s a y t h a t
F
weakly compact i f i t maps t h e u n i t b a l l o f compact s u b s e t o f
in
show t h a t
1x1
= 1,
T
6
llpllx.
=
and
E
I / x ( I = 1 and
E,
F.
If
T
E
6 (E;F)
E
E
(E;F)
is
o n t o a r e l a t i v e l y weakly
show t h a t t h e f o l l o w i n g a r e
equivalent (i)
T
i s weakly compact,
(ii)
T"
E
(iii) 1.88" __
8
(F';E')
(the adjoint of
A Banach s p a c e
into
that
E
i s weakly compact,
T**(E") C F . E
i s s a i d t o have t h e polynomial D u n f o r d - P e t t i s
p r o p e r t y i f f o r e v e r y Banach s p a c e E
T)
F
F
t h e weakly compact polynomials from
map weak Cauchy s e q u e n c e s o n t o s t r o n g Cauchy s e q u e n c e s .
Show
h a s t h e polynomial D u n f o r d - P e t t i s p r o p e r t y i f and o n l y i f e v e r y
Banach v a l u e d weakly compact l i n e a r mapping maps weak Cauchy s e q u e n c e s o n t o s t r o n g Cauchy sequences.
Chapter 1
46
1.89* P
E
If
P(F)
i s a n u c l e a r subspace o f a l o c a l l y convex s p a c e
F
show t h a t t h e r e e x i s t s If
P c PN(E)
(v
such t h a t
i s a Banach s p a c e , show t h a t
E
=
E
and
P.
is also a
(@,("E),IIw)
Banach s p a c e .
Let
1.91*
d e n o t e t h e t o p o l o g y on
T~
8 a (nE)
o f uniform convergence
on t h e f i n i t e dimensional compact s u b s e t s of t h e v e c t o r s p a c e E:
phism from 1.92* let
( @ a ( n E ) , ~ f ) ' onto
m
6(nE*o)
( x ~ ) ~b e= an ~ orthonormal s u b s e t o f a H i l b e r t space
be a p o s i t i v e i n t e g e r .
only if
In=1I A n I <
__ 1.93*
If
03
E;
1.94
If
= F;iEAwhere
m
and
n
1.95*
(
8f
~ ) -c0) ,
(p (",&
1.96*
(X))
X
,B)
Show t h a t
xm
E
E
and
@N(mE) i f and
.
Ei = (E',T~) i f and
which c o n t a i n s
a s a subspace
E
{@EF';$I~=O}. msn,
and
E
is a locally
( @ ( m E ) , ~ o ) i s isomorphic t o a complemented sub-
A compact Hausdorff s p a c e
closed subset of that
EL =
F
are p o s i t i v e i n t e g e r s ,
convex s p a c e , show t h a t space of
In=,A
=
and a l s o t h a t
m
o n l y i f f o r each l o c a l l y convex s p a c e
w e have
P
Show t h a t
i s a l o c a l l y convex s p a c e , show t h a t
E
and l e t
.
m
Let
E
Show t h a t t h e Bore1 t r a n s f o r m i s an a l g e b r a i c isomor-
(E*,a(E*,E)).
=
-
i s s a i d t o be d i s p e r s e d i f e v e r y
X
contains an isolated point.
If
X
i s d i s p e r s e d , show
h a s t h e approximation p r o p e r t y . ( i f , ( " ~ ~ ) ,B) =. El
/-
QE
El
A
. . . 8,E l .
n times
51.6
NOTES AND REMARKS Mathematicians began e x p l o r i n g t h e c o n c e p t s o f polynomial and holomor-
p h i c mapping in i n f i n i t e dimensions a t a t i m e when i d e a s and t h e o r i e s such a s t h e t o t a l d e r i v a t i v e , p o i n t s e t topology and normed l i n e a r s p a c e , e t c .
Polynomials on locally convex topological vector spaces
47
were either still in their infancy o r not yet discovered. Moreover, it appears that the search for fundamental concepts in infinite dimensional differential calculus stimulated much of the work which resulted in the satisfactory linear theory that we now know as functional analysis. These pioneers were motivated by many different considerations, and at times were not aware of one another's work.
We provide here a brief outline of the
early development of polynomials, a similar treatment of holomorphic functions is given in 52.6, and refer to the historical survey of A . E . Taylor [680] for further details.
It is generally recognised that the definitive step in the creation of infinite dimensional analysis was taken by V. Volterra in 1887. I n a series of notes [705,706,707,708,709], which later evolved into the book [710], he developed a theory of scalar valued differentiable functions on ,&[a,b] and obtained the following Taylor expansion [705,p.105] for the real-valued analytic function y on &[a,b]
where
@,$
E
&[a,b].
The nth term in the above expansion is an n on ,& [a,b]. mappings.
homogeneous polynomial
Volterra did not, however, specifically discuss polynomial
The next step was taken by I). Hilbert who outlined a theory of holomorphic functions in infinitely many variables at the international congress in Rome in 1908 and published his results the following year, [332]. To Hilbert, each variable was a coordinate evaluation and he used a monomial expansion with absolute convergence on polydiscs as we do in chapter 5. Each holomorphic function had, in his notation, the following Taylor series expansion
48
Chapter 1
=
cc
n l . . .nk
n 1 x1
...
x
"k k
1 I IE
\x31 i / c 3 \ ,
IC 21
I
a b s o l u t e l y on x1 $ 1 , I x2 5 , I t i s c l e a r from t h e above t h a t H i l b e r t had a d e f i n i t e
t h e s e r i e s converging
. .. .
c o n c e p t o f polynomial i n i n f i n i t e l y many v a r i a b l e s . During t h e same y e a r , 1909, M. F r g c h e t p u b l i s h e d h i s f i r s t c o n t r i b u t i o n [240] t o t h e a b s t r a c t t h e o r y o f polynomials i n i n f i n i t e l y many variables.
Motivated by Cauchy's o b s e r v a t i o n t h a t any c o n t i n u o u s r e a l
valued function
o f a r e a l v a r i a b l e which s a t i s f i e d t h e ' e q u a t i o n
f
f(x+y) - f(x) had t o have t h e form
-
f(y) = 0
for a l l
x,y
in
R
f ( x ) = Ax, h e gave a n a b s t r a c t " d i f f e r e n c e " c h a r a c t -
e r i z a t i o n o f r e a l polynomials o f one o r s e v e r a l r e a l v a r i a b l e s ( s e e e x e r c i s e 1.68).
H e t h e n used t h i s c h a r a c t e r i z a t i o n t o d e f i n e r e a l polynomials
depending on a c o u n t a b l y i n f i n i t e number o f v a r i a b l e s . was
RN
H i s domain s p a c e
and on i t h e d e f i n e d , f o r c o n t i n u i t y p u r p o s e s , a m e t r i c which
g i v e s t h e u s u a l c o o r d i n a t e w i s e convergence t o p o l o g y .
The.following year
h e used t h e same method i n [241] t o d e f i n e r e a l polynomials on , & [ a , b ] and showed t h a t a r e a l n-homogeneous polynomial
U
on t h i s s p a c e c o u l d be
r e p r e s e n t e d as (xl,
"f
where
11;~)
. . ., x n ) f (x,) . . . f (xn)dxl. . .dxn, f c &,
[a,b] ,
..
i s a s e q u e n c e o f n-homogeneous polynomials i n
independent o f
f
xl,. 'n and t h e l i m i t i s uniform o v e r t h e compact s u b s e t s of
& [ a , b ] . He a l s o showed t h a t any polynomial c o u l d be r e p r e s e n t e d a s a f i n i t e sum o f homogeneous p o l y n o m i a l s . I n a subsequent p a p e r , [243], h e o b t a i n e d a R i e s z r e p r e s e n t a t i o n theorem f o r b i l i n e a r forms on The n e x t s t e p i s due t o R . GGteaux.
,&
[a,b]
.
H e made v e r y fundamental c o n t r i b -
u t i o n s t o t h e t h e o r y o f i n f i n i t e d i m e n s i o n a l c a l c u l u s ( s e e § 2 . 6 ) , and h i s s i m p l e e l e g a n t s t y l e makes f o r v e r y p l e a s a n t r e a d i n g .
G s t e a u x ' s work
c o n s i s t s e s s e n t i a l l y o f two p a p e r s [252,253], which he w r o t e d u r i n g t h e p e r i o d 1912-1914.
H e d i e d i n 1914, and h i s r e s u l t s were e d i t e d by P . L'evy
and p u b l i s h e d i n 1919 and 1922. KN
( K = IR
or
C),
R2
Gzteaux worked o n l y on t h e s p a c e s
and g [ a , b ] .
H e noted t h a t F r e c h e t ' s d e f i n i t i o n
o f polynomial was i n a d e q u a t e f o r f u n c t i o n s d e f i n e d on v e c t o r s p a c e s o v e r
49
Polynomials on locally convex topological vector spaces t h e f i e l d o f complex numbers and proposed i n s t e a d t h a t a c o n t i n u o u s function p
P
such t h a t
f o r any v e c t o r s
degree
n.
z
P(Xz+ut) and
t
i s a polynomial o f d e g r e e
n
in
and
X
i n t h e domain b e c a l l e d a polynomial o f
H e showed t h a t h i s d e f i n i t i o n c o i n c i d e s w i t h F r & h e t ' s f o r
real
v a l u e d f u n c t i o n s of r e a l v a r i a b l e s and went on t o prove v a r i o u s r e s u l t s such as t h e r e l a t i o n s h i p between t h e homogeneous p a r t s and t h e "G$teauxq' d e r i v a t i v e s o f a polynomial - w i t h h i s d e f i n i t i o n .
The development o f t h e
concept of normed l i n e a r s p a c e and a s s o c i a t e d i d e a s between 1910 and 1925 allowed F r g c h e t t o extend h i s d e f i n i t i o n of r e a l polynomial t o a r a t h e r g e n e r a l s e t t i n g i n [244] and [246]. I n 1931-1932, A . D . Michal, a s t u d e n t o f F r g c h e t , gave a s e r i e s o f l e c t u r e s a t t h e C a l i f o r n i a I n s t i t u t e o f Technology i n which h e o u t l i n e d t h e r e l a t i o n s h i p between symmetric n - l i n e a r forms and homogeneous polynomials. T h i s r e l a t i o n s h i p had been n o t i c e d e a r l i e r f o r b i l i n e a r forms and 2-homogeneous polynomials by M . F r g c h e t [243] and R . Ggteaux [252]. F u r t h e r work on t h e d e f i n i t i o n o f polynomial between Banach s p a c e s was c a r r i e d o u t by A . D . Michal and h i s s t u d e n t s A.H.
C l i f f o r d , R.S. M a r t i n , I . G . Highberg
and A . E . T a y l o r [331,492,493,449,677].
R.S. M a r t i n , i n h i s t h e s i s [449]
proved t h e p o l a r i z a t i o n formula and I . G . Highberg [331] c l a r i f i e d t h e r e l a t i o n s h i p between t h e d i f f e r e n t d e f i n i t i o n s and showed t h a t F r 6 c h e t ' s d i f f e r e n c e method could b e extended t o t h e complex case i f one added t h e h y p o t h e s i s o f G3teaux d i f f e r e n t i a b i l i t y . W . O r l i c z [481,482],
I n d e p e n d e n t l y , S. Mazur and
e s t a b l i s h e d t h e c o n n e c t i o n between t h e n - l i n e a r
approach and t h e now c l a s s i c a l approach of F r g c h e t and G2teaux f o r real Banach s p a c e s , and proved t h e p o l a r i z a t i o n formula.
T h i s ends o u r b r i e f
s k e t c h o f t h e development o f t h e concept o f a b s t r a c t polynomial.
For t h o s e
i n t e r e s t e d , we s t r o n g l y recommend t h e o r i g i n a l s o u r c e s as i n t e r e s t i n g reading
.
We r e t u r n now t o commenting on t h e t e x t and w i l l t r y t o a t t r i b u t e
r e s u l t s t o t h e i r o r i g i n a l sources.
There a r e b a s i c a l l y t h r e e approaches
t o s t u d y i n g polynomials, by c o n s i d e r i n g r e s t r i c t i o n s t o f i n i t e dimensional spaces,by means of t e n s o r p r o d u c t s
and by u s i n g m u l t i l i n e a r mappings.
of t h e s e methods a r e u s e f u l and none should be n e g l e c t e d .
All
The r e s t r i c t i o n
method, a s a l r e a d y n o t e d , was one o f t h e o r i g i n a l methods used and r e a p p e a r s i n o u r work e v e r y so o f t e n .
The t e n s o r p r o d u c t approach i s due
t o R . S c h a t t e n [626] and A . Grothendieck [287].
I n [620], R.A.
Ryan shows
50
Polynomials on locally convex topological vector spaces
t h a t most of t h e r e s u l t s we p r e s e n t can b e o b t a i n e d by t h i s method and t h e
same approach i s a l s o t o be found i n C . P . Gupta [295,296], [214,215,217,218,220,223],
T . A . Dwyer
A.Colojoar?i [138,139] and P . Krze [402].
We
f o l l o w t h e m u l t i l i n e a r approach i n t h i s book. Theorem 1 . 7 i s due t o R.S. M a r t i n [449] and an a l t e r n a t i v e proof i s g i v e n i n L.A. Harris [310].
Example 1 . 8 i s due t o L . Nachbin [SO91 and
example 1 . 9 was d i s c o v e r e d i n d e p e n d e n t l y by O . D .
Kellogg [379], J . G . van
d e r Corput and G . Schaake [169] and S . Banach [ 4 5 ] .
The proof g i v e n h e r e
i s due t o S . k o j a s i e w i c z and can be found i n [ 7 3 ] .
Prop.ositions 1.10 and
1.11 a r e due t o L.A.
found i n t h a t a r t i c l e
Harris [310] and f u r t h e r s i m i l a r r e s u l t s may a l s o b e
and i n [316].
The f a c t o r i z a t i o n lemma (and t h e c o r r e s p o n d i n g r e s u l t f o r holomorphic f u n c t i o n s ) h a s been i m p l i c i t i n t h e works o f many a u t h o r s , e . g . A . Hirschowitz [335], C . E . R i c k a r t [605] and L . Nachbin [ 5 1 4 ] .
A system-
a t i c s t u d y o f t h i s i d e a and i t s consequences i s u n d e r t a k e n i n S. Dineen [190] and E . Ligocka [443].
A l l r e s u l t s using s u r j e c t i v e l i m i t s (see
c h a p t e r 6) depend i n some way on a f a c t o r i z a t i o n p r o p e r t y . due t o S. Mazur and W. O r l i c z [482].
Lemma 1 . 1 9 i s
I t i s a l s o a consequence o f t h e
uniform boundedness p r i n c i p l e f o r polynomial mappings on a Banach s p a c e and more g e n e r a l r e s u l t s on t h e same t o p i c can be found i n J . Bochnak and J . S i c i a k [73], P . Lelong [431] and P . Turpin [687].
J . Bochnak and
J . S i c i a k [73,74] make e x t e n s i v e u s e of t h e "Polynomial lemma o f Leja"
[424] i n p r o v i n g t h e i r r e s u l t s . P r o p o s i t i o n 1 . 2 1 i s w e l l known, and p r o b a b l y due t o J . S a b a s t i a g e Silva.
The r e s u l t of example 1 . 2 4 i s proved f o r a c o u n t a b l e d i r e c t sum of
Banach s p a c e s i n S . Dineen [185] and f o r
33 N s p a c e s
i n S . Dineen [ 1 9 4 ] .
The p r o o f g i v e n h e r e i s modeled on t h o s e g i v e n i n [185] and [194] and r e l a t e d r e s u l t s are t o b e found i n [SO].
Example 1 . 2 5 i s due t o P . J .
Boland and S. Dineen [92] and t h e p r o o f i s s i m i l a r t o t h a t o f t h e p a r t i c N u l a r case C XE") which a p p e a r s i n S. Dineen [185]. The method o f p r o o f ha5 been f u r t h e r developed by L . A . de Moraes [498] i n h e r s t u d y of holomorphic f u n c t i o n s on s t r i c t i n d u c t i v e l i m i t s . Nuclear polynomials on Banach s p a c e s were i n t r o d u c e d by C . P . Gupta
Chapter 1
51
[295,296,297] i n o r d e r t o prove e x i s t e n c e and approximation p r o p e r t i e s of c o n v o l u t i o n o p e r a t o r s on Banach s p a c e s and m o t i v a t e d L . Nachbin [508,509] t o i n t r o d u c e t h e concept o f holomorphy t y p e .
T h i s a l l o w s one t o d i s c u s s
compact, i n t e g r a l , n u c l e a r , H i l b e r t Schmidt, e t c . polynomial and holomorp h i c mappings between l o c a l l y convex s p a c e s - c o n c e p t s which have proved u s e f u l i n l i n e a r f u n c t i o n a l a n a l y s i s and i n p r o b a b i l i t y t h e o r y on l o c a l l y convex s p a c e s ( s e e appendix I ) . P.J.
Boland [79,82,83,84]
i n i t i a t e d and developed t h e t h e o r y o f
holomorphic f u n c t i o n s on n u c l e a r s p a c e s and f o r such s p a c e s n u c l e a r p o l y nomials p l a y a fundamental r o l e ( s e e c h a p t e r s 3 and 5 ) . R.A.
Recent work by
Ryan [620] h a s shown t h e i r importance f o r holomorphic f u n c t i o n s on
F r g c h e t s p a c e s w i t h t h e approximation p r o p e r t y . The s t r o n g t o p o l o g y and t h e compact open t o p o l o g y are d e r i v e d from f u n c t i o n a l a n a l y s i s and p o i n t s e t topology r e s p e c t i v e l y .
The
T,,,
topol-
ogy i s more o r less s p e c i a l t o t h e t h e o r y of holomorphic f u n c t i o n s on l o c a l l y convex s p a c e s , a l t h o u g h some l i n e a r p r o p e r t i e s o f t h i s t o p o l o g y a r e d i s c u s s e d i n R . E . Edwards [229, p.511-5131, [397, p.4001 and J . A . B e r e z a n s k i i [ 5 7 ] .
K . F l o r e t [237], G . Kb'the
This t o p o l o g y was i n t r o d u c e d by
L . Nachbin [509] and was m o t i v a t e d by r e s u l t s of A. Martineau [450,453] on
a n a l y t i c f u n c t i o n a l s ( o f s e v e r a l complex v a r i a b l e s ) s u p p o r t e d by e v e r y neighbourhood o f a compact s e t b u t n o t by t h e compact s e t i t s e l f .
I t may
a l s o b e d e s c r i b e d as t h e t o p o l o g y o f l o c a l convergence. Example 1 . 3 8 i s g i v e n i n S. Dineen [185] f o r a c o u n t a b l e d i r e c t sum o f Banach s p a c e s . s e t s of a
$ 3 2 space i s 83% s p a c e s
[50] and f o r S . Dineen,
The same r e s u l t f o r holomorphic f u n c t i o n s on open subproved by J . A .
Barroso, M . C . Matos and L . Nachbin
by S. Dineen [ 1 9 4 ] .
Example 1.39 a p p e a r s i n
[185].
I n d e a l i n g w i t h n u c l e a r polynomials and t h e Bore1 t r a n s f o r m w e have a t t e m p t e d t o c o r r e l a t e v a r i o u s r e s u l t s on n u c l e a r and d u a l n u c l e a r s p a c e s by P . J .
Boland, [79,82,83,85],
r e s u l t s on f u l l y n u c l e a r s p a c e s by P . J .
Boland and S. Dineen [go] and r e s u l t s on F r g c h e t s p a c e s w i t h t h e approxi m a t i o n p r o p e r t y due t o R . A .
Ryan [620].
T h i s l e a d s t o a more compact
t r e a t m e n t b u t e s s e n t i a l l y t h e o n l y new r e s u l t s i n t h e f i n a l two s e c t i o n s
a r e p r o p o s i t i o n s 1.44 and 1 . 4 8 .
D e f i n i t i o n 1 . 4 0 ( a ) i s due t o P . J .
Boland
52
Polynomials on locally convex topological vector spaces
[ 7 9 ] . The fundamental i n e q u a l i t y needed i n d e f i n i t i o n 1 . 4 0 ( b ) i s a l s o due t o Boland [87]. The d e f i n i t i o n i s , however, new a l t h o u g h s p e c i a l c a s e s have p r e v i o u s l y been c o n s i d e r e d by C . P . Gupta [295] and R . A .
Ryan [620].
P r o p o s i t i o n 1 . 4 1 , c o r o l l a r y 1 . 4 2 and theorem 1 . 4 3 are due t o P . J . Boland [79,83].
P r o p o s i t i o n 1.44 i s proved f o r f u l l y n u c l e a r s p a c e s i n [ 8 7 ] .
The Bore1 ( o r F o u r i e r - B o r e l ) t r a n s f o r m was f i r s t used i n i n f i n i t e dimensional holomorphy by C . P . Gupta [ 2 9 5 ] .
Subsequently, i t h a s been
a p p l i e d by v a r i o u s a u t h o r s , see c h a p t e r s 3 , 5 , 6 and appendix I , i n t h e s t u d y o f c o n v o l u t i o n o p e r a t o r s and d u a l i t y t h e o r y .
Proposition 1.47 i s
due t o C . P . Gupta [295] f o r Banach s p a c e s and t o P . J . r e f l e x i v e n u c l e a r and d u a l n u c l e a r s p a c e s .
Boland [79] f o r semi-
Other p a r t i c u l a r cases o f
p r o p o s i t i o n 1 . 4 7 and 1.48 can b e found i n R . A . Ryan [ 6 2 0 ] .
Fully nuclear
s p a c e s ( d e f i n i t i o n 1 . 4 9 ) were d e f i n e d by P . J . Boland and S. Dineen [go] and f i g u r e p r o m i n e n t l y i n c h a p t e r s 5 and 6 .
C o r o l l a r y 1 . 5 0 and lemma 1.55
a r e proved f o r f u l l y n u c l e a r s p a c e s w i t h a b a s i s i n P . J . Boland and S. Dineen [go] and t h e l a t t e r r e s u l t h a s r e c e n t l y been extended t o q u a s i complete d u a l n u c l e a r s p a c e s by J . F . Colombeau, R . Meise and B. P e r r o t
Lemma 1.54 i s due t o S. Dineen [190].
[153].
P r o p o s i t i o n s 1 . 5 6 and 1.57 a r e due t o P . J . Boland and S . Dineen [go] when
E
h a s a b a s i s , and t o P . J .
Boland [87] i n t h e g e n e r a l c a s e .
p r o o f g i v e n h e r e o f p r o p o s i t i o n 1.57 i s new. 1 . 5 9 , 1.60 and 1 . 6 1 are a l l due t o R.A.
Ryan.
The
Lemma 1.58 and p r o p o s i t i o n s They appeared i n a p r e l i m -
i n a r y d r a f t o f h i s t h e s i s [620], b u t were r e p l a c e d by more e l e g a n t and p e r h a p s s l i g h t l y less g e n e r a l r e s u l t s i n t h e f i n a l v e r s i o n .
Chapter 2 HOLOMORPHIC MAPPINGS BETWEEN LOCALLY CONVEX SPACES
I n t h i s c h a p t e r , we g i v e t h e v a r i o u s d e f i n i t i o n s o f holomorphic mappi n g s between l o c a l l y convex s p a c e s , which we s h a l l u s e as well as t h e d i f f e r e n t t o p o l o g i e s on t h e s e s p a c e s o f mappings.
I n many c a s e s , i t i s o n l y
n e c e s s a r y t o c o n s i d e r Banach s p a c e v al u ed mappings and t h e r e s u l t s may be extended t o a r b i t r a r y v e c t o r v al u ed mappings q u i t e e a s i l y . The compact open t o p o l o g y and t h e
T
w
t opology e a s i l y e xte nd from
polynomials t o holomorphic f u n c t i o n s on l o c a l l y convex s p a c e s , b u t t h e s trong topology does n o t g e n e r a l i s e i n a s u i t a b l e fashion.
While t h e
T~
t o p o l o g y p l a y s a n i m p o r t an t r o l e i n o u r s t u d y , it doe s n o t , i n g e n e r a l , have good t o p o l o g i c a l p r o p e r t i e s .
We i n t r o d u c e t h e
-r6
topology which may b e
d e s c r i b e d as t h e t o po l o g y s u p p o r t ed by t h e c o u n t a b l e open c o v e r s .
We prove
e le m e n t a r y p r o p e r t i e s o f t h e s e t o p o l o g i e s and g i v e some sim ple examples.. The s i g n i f i c a n c e o f t h e s e examples and counterexamples i s c l a r i f i e d i n l a t e r chapters.
The r emai n i n g c h a p t e r s a r e d evote d t o a d e e p e r s t u d y o f
t h e s e t o p o l o g i e s on c e r t a i n classes of l o c a l l y convex s p a c e s and on s p e c i a l domains.
W e a l s o d e f i n e germs o f holomorphic f u n c t i o n s .
These are o f
i n t r i n s i c i n t e r e s t and a l s o p l a y a r o l e i n d u a l i t y t h e o r y . 52.1
G ~ T E A U X HOLOMORPHIC MAPPINGS
De finit i on 2.1 A subset U of a vector space E i s said t o be f i n i t e l y open i f UnF i s an open subset of t h e Euclidean space F f o r each f i n i t e dimensional subspace F of E. The f i n i t e l y open s u b s e t s o f ogy
tf.
The b a l a n c ed
neighbourhoods o f z e r o .
tf
E
define a translation invariant topol-
neighbourhoods o f z e r o form a b a s i s f o r t h e
On a l o c a l l y convex s p a c e
(E,T),
t h a n any o f t h e t o p o l o g i e s we have p r e v i o u s l y c o n s i d e r e d on Tf
3 TM 3 Tk 3 T.
53
T~
E,
is f i n e r i.e.
tf
Chapter 2
54 Pefinition 2.2
A function E
a vector space Gateaux or
f
defined on a f i n i t e l y open subset F
with values i n a l o c a l l y convex space
-
b EE
aE U,
G-holomorphic i f for each
of
i s said t o be
Q, E F '
and
U
t h e complex
valued f u n c t i o n of one complex variable A
$of(a+Xb)
i s holornorphic i n some neighbourhood of zero. s e t of a l l
G-holomorphic mappings from
U
HG(U;F)
We l e t
into
denote t h e
F.
Hartog's theorem i n f i n i t e dimensions says t h a t s e p a r a t e l y holomorphic f u n c t i o n s on
UxV
(UCK", V C C m )
a r e holomorphic.
Hence
f:UCEjF
i s G-holomorphic i f and only i f @ o f ) U n G i s a holomorphic function of
Q, i n
s e v e r a l complex v a r i a b l e s f o r each subspace
G
of
F'
and each f i n i t e dimensional
Consequently, one can u s e any of t h e equivalent
E.
f i n i t e dimensional conditions (e.g. Taylor s e r i e s expansions, Cauchy Riemann equations, e x i s t e n c e of t h e t o t a l d e r i v a t i v e ) i n t h e d e f i n i t i o n of G-holomorphicity. A t t h i s s t a g e , one may wonder why we demanded a l o c a l l y convex range
space i n d e f i n i t i o n 2 . 2 .
a:
v e c t o r space over
We could a l s o g i v e an analagous d e f i n i t i o n with a However, w e would then run i n t o
a s t h e range space.
d i f f i c u l t i e s i n showing t h a t a G2teaux holomorphic function has a Taylor s e r i e s expansion about each p o i n t s i n c e t h i s r e q u i r e s a convergence s t r u c t u r e on t h e range space and t h e f i n i t e open topology on
i s so f i n e t h a t
Even with a l o c a l l y
very few f u n c t i o n s would have t h e d e s i r e d expansion. convex range space w e s t i l l have t o be c a r e f u l .
F
We f i r s t show t h a t G-holo-
morphic functions a r e continuous f o r t h e f i n i t e open topology. Lemma 2 . 3
I f
E
i s a v e c t o r space,
i s a l o c a l l y convex space and
i s given t h e f i n i t e open topology. I t i s e a s i l y seen t h a t
i s a f i n i t e l y open subset of
f &HG(U;F)
F U
Proof
U
tf
then
f
i s continuous when
i s t h e inductive l i m i t topology, i n
t h e category of t o p o l o g i c a l spaces, given by t h e i n c l u s i o n mappings where
G
function
ranges over a l l f i n i t e dimensional subspaces of f
defined on a
tf
open subset
U
of
E
E.
G-E
Hence a
i s continuous i f and
only i f i t s r e s t r i c t i o n t o t h e f i n i t e dimensional s e c t i o n s of uous.
E,
U
a r e contin-
Since an a n a l y t i c f u n c t i o n of s e v e r a l complex v a r i a b l e s i s continuous
t h i s completes t h e proof. A
We now look a t Taylor s e r i e s expansions of Gateaux holomorphicfunctions.
Holomorphic mappings between locally convex spaces
55
If U i s a f i n i t e l y open subset of a vector space E, F i s a ZocaZly convex space and fEHG(U;F) then f o r each 5 E U there Proposition 2.4
e x i s t s a unique sequence of homogeneous polynomiak from
E into
A
F,
such t h a t
f o r all y i n some
Proof
where
Let
p,
P (x) m,S,@
neighbourhood of zero.
tf
E E U be fixed.
For each positive integer m,
@
EF' and
is chosen so that is independent of
By lemma 2.3, f is continuous and we may use Riemann's definition of the integral to define Pm,E(X)
=
dh .
The limit (of the Riemann sums) may not exist in F
but will always exist
A
in F , the completion of F. It will lie in F if F is sequentially complete o r if the closed convex hull of each compact subset of F is compact. F o r this reason, we sometimes place a completeness condition on the range space. Since f is continuous, Pm,S,@ (XI all 6 and x.
=
@(Pm,S(x))
for
Since the restriction of @of to any finite dimensional section of U is a holomorphic function of several complex variables, it follows that the function x -+ P (x) lies in @a(m~> f o r every positive integer m m,S,@ and each @ in F'. Let L be the associated symmetric m-linear m,S,@ mapping on E.
56 If
Chapter 2 xl,
...,xm
L
: Em
Let
m, 5
t h e n by t h e p o l a r i z a t i o n formula (theorem 1 . 5 )
E
E
-
h
F
b e d e f i n e d by 1
E
By t h e Hahn-Banach theorem Let
1
Lm, 5
i z (mE;F)
El...€
and
P n m,{
(1"i = lE.X.) 1 1
= A Lm,< E $,("E;?).
Pm,<
b e a t f - b a l a n c e d neighbourhood o f z e r o such t h a t
V
example, t a k e
V
{ x E E ; S + A X E U , I A ~ & ~ II)f.
=
a compact s u b s e t o f
S+{Ax;IA/&p} C U .
such t h a t
B ECS(F) let
If
Y E F ~ . By l e m m a 2 . 3
p>l
B = { $ E F ' ; l$(y)(sB(y) B i s c o n t i n u o u s and w e have
f
sup B(f(S+Ax))
sup
=
I$of(S+Ax)
I A I&P,OEBB
I A l W
I
(for
S + { X X ; I A ~ S ~i}s
then
XEV
and hence t h e r e e x i s t s
U
<+VCU
<
=
m
for all
.
Hence
f o r every p o s i t i v e integer
Hence f(S+x)
=
LmZo m
m.
T h i s shows t h a t
P
(x)
m, 5
f o r every
x
in
V.
Using t h e u n i q u e n e s s o f T a y l o r series expansions i n one complex v a r i a b l e we
57
Holomorphic mappings between locally convex spaces
see t h a t t h e sequence
m
i s u n i q u e l y d e t e r m i n e d by
('m, S)m=O
The f i n i t e d i m e n s i o n a l t h e o r y a l s o shows t h a t partial derivative of
f
at 5
i n the direction
f.
is the
Pm,s (x)
m
th
x, -and following
c l a s s i c a l terminology, P
The c o r r e s p o n d i n g
m, 5 m
m! l i n e a r form
Our expansion now becomes
L
i s d e n o t e d by
=
'mf(S) ___ (x) m!
m, 5
f(S+x)
proof.
.
dmf(S) ~
m!
T h i s completes t h e
I n p r o v i n g p r o p o s i t i o n 2 . 4 , w e h a v e a l s o shown t h e f o l l o w i n g : P r o p o s i t i o n 2 . 5 (Cauchy i n e q u a l i t i e s )
i s a balanced subset of
B
and
cs(F)
in
If f
such t h a t
and every non-negatiue integer
HOLOMORPHIC
12.2
E
Definition 2.6
MAPPINGS
Let
E
BETWEEN
and
a f i n i t e l y open subset of E.
F
HG(U;F),
E
c+pBCU
LEU;
p
LOCALLY
A function
5
in
~A { O }
m
CONVEX
SPACES
be locally convex spaces and l e t
if it i s G-hoZomorphic and f o r each
E
then for every
f:U U
+
F
U
be
i s calZed hoZornorphic
t h e function
converges and defines a continuous function on some r-neighbourhood of zero. We Zet U
into
H(U;F)
denote t h e vector space of a12 hoZomorphic functions from
F.
We u s u a l l y c o n s i d e r f u n c t i o n s d e f i n e d on open s u b s e t s o f
E
and i n
t h i s case, b e c a u s e o f t h e u n i q u e n e s s o f t h e T a y l o r series expansion and t h e
f a c t t h a t t h e f i n i t e open neighbourhoods of z e r o a r e a b s o r b i n g , a G-holomor-
Chapter 2
58
p h i c f u n c t i o n i s holomorphic i f and only i f it i s c o n t i n u o u s . The f o l l o w i n g o b s e r v a t i o n i s e a s i l y proved and f r e q u e n t l y a p p l i e d . Lemma 2 . 7
U
If
is an open subset of a ZocalZy convex space f
a Zocally convex space and naof
E
for every
H(U;Fa)
A continuous fu n ct i o n
HG(U;F)
E
f
then
E
i s bounded).
f
is
i f and only i f
w i t h v a l u e s i n a normed l i n e a r spa c e i s
f
l o c a l l y bounded ( i . e . each p o i n t i n t h e domain o f whose image under
H(U;F)
F
cs(F).
in
a
E,
f
h a s a neighbourhood
The co n v erse i s f a l s e i n g e n e r a l b u t it
i s t r u e f o r G-holomorphic f u n c t i o n s as o u r n e x t r e s u l t shows.
Lemma 2 . 8
U
I f
i s an open subset of a Zocally convex space
a normed Zinear space a d
f
E
HG(U;F)
f
then
E
H(U;F)
E,
F
i f and only if
is f
i s ZocaZZy bounded. Let
Proof
5
E
U
be a r b i t r a r y .
hood of z e r o such t h a t
5+VCU
By p r o p o s i t i o n 2 . 5 , f o r a l l
If
5,+5
a > a
.
as
a + -
m
and and
Choose f(5+V) O < S <1,
t h e n we can ch o o s e
Hence we h av e
T h i s completes t h e p r o o f .
V
ct 0
a convex ba la nc e d neighbouri s a bounded s u b s e t o f
F.
w e have
such t h a t
5-5,
E
6V
for all
59
Holomoiphic mappings between locally convex spaces S i n c e e v e r y l o c a l l y bounded polynomial i s c o n t i n u o u s , we a l s o have shown t h e f o l l o w i n g : C o r o l l a r y 2.9
E
Let
and E
U
be an open subset o f
U
and every p o s i t i v e i n t e g e r
F
be arbitrary ZocaZZy convex spaces. f
and suppose
is ( m ~ F; ) .
Zmf(E)
m,
E
8(mE;e)
E
A
Corollary 2.10
Let
E
be open i n
f
and
E
F
and
E
H(U;F).
Let 5
Then f o r every dmf(E)
and
E
be arbitrary ZocaZly convex spaces,
HG(U;F).
f
Then i f
in
U
i s ZocaZly bounded i t Zies i n
H(U;F). Apply lemmata 2.7 and 2.8
Proof
H
We Zet
LB
(U;F)
denote the vectop space o f a l l G-hoZomorphic ZocalZy
bounded mappings defined on t h e open subset U E
with vaZues i n t h e ZocaZZy convex space
MLB(U;F) C H ( U ; F )
f o r any
and
U,E
o f t h e ZocaZZy conuex space We have j u s t s e e n t h a t
F.
We now look a t t h e r e v e r s e
F.
inclusion.
Lemma 2 . 1 1
is an open subset of a ZocaZZy convex space E and f o r every ZocaZZy convex space F then E i s a normed U
If
H(U;F) = H LB (U;F)
l i n e a r space.
I t suffices t o take
Proof
i d e n t i t y mapping (from
F=E
and to n o t e t h a t t h e r e s t r i c t i o n o f t h e
into itself) t o
E
l o c a l l y bounded i f and o n l y i f
U
i s always c o n t i n u o u s b u t i s
i s a normed l i n e a r s p a c e .
E
There are, however, s e v e r a l n o n - t r i v i a l examples o f p a i r s o f s p a c e s and
F
i n g s from
into
E
F
coincide.
One can o b t a i n some i n f o r m a t i o n a b o u t t h i s
problem by e x t e n d i n g H a r t o g s ' theorem t o l o c a l l y convex s p a c e s .
f;U+G
E
for w h i c h , t h e holomorphic and t h e l o c a l l y bounded holomorphic mapp-
where
U
i s an open s u b s e t o f
ExF,
E,F
convex s p a c e s , i s separateZy hoZomorphic i f f o r each y + f(x,y)
i s holomorphic and f o r each
i s holomorphic.
y
in
F
and
G
x
in
A function
being l o c a l l y E
the function
t h e function x
+
g(x,y)
H a r t o g s ' theorem i n s e v e r a l v a r i a b l e s i m p l i e s t h a t s e p a r -
a t e l y holomorphic f u n c t i o n s a r e G-holomorphic. P r o p o s i t i o n 2.12
Let
E
and
F
be ZocaZZy convex spaces and suppose
60
Chapter 2
every separately holomorphic function defined on an open subset of i s holomorphic.
Proof f(x,$)
Let =
Then HLB(U;F) = H(U;F)
f
E
@(f(x)).
H(U;F).
We define
f o r every open subset
f; UxFb
+
B:
ExF'
U of
B E.
by the formula
f is obviously separately holomorphic and by our hypoth-
esis, it is holomorphic. Since Q: is a locally compact space f is a locally bounded function. Hence, if 5
E
U, we can find a neighbourhood of
5, VE, and a neighbourhood of zero in Fb, Bo,
where
B
is a bounded
Hence sup \@(f (x)) I < m xev for every $ in F' and by Mackey's theorem f(Vg) isc a bounded subset of F. This complets the proof. subset of F,
Example 2.13
Bo such that \ \ f \ \ v E
=
M <
-.
If E is a Frgchet space and
F
is a DF
space (the
strong dual of F is a Frgchet space), then HLB(U;F) = H(U;F) open subset U
for any
of E. This follows from proposition 2.12 since it is
known that separately holomorphic functions defined on open subsets of the product of Frgchet spaces are holomorphic. Example 2.14 If E is a $ 3 2 space and F is an 3 d space then HLB (U;F) = H(U;F) for any open subset U of E. This also follows from proposition 2.12 and the fact that separately holomorphic functions defined on open subsets of a product of $ 3 2 spaces are holomorphic. We do not know if the same result holds for
a3@ spaces although we
do know that separately continuous polynomials defined on a product of PTkyL spaces are continuous. We now look at functions which are holomorphic analogues of the hypocontinuous and the M-continuous polynomials defined in section 1.2 Definition 2.15
A function
ZocalZyconvex space
f defined on an open subset
U of a
E
with values i n a l o c a l l y convex space F i s said it i s G-holomorpkic and continuous on the compact subsets of E. We l e t HHY(U;F) denote the vector space of a l l hypoanalyt i c mappings from U i n t o F. t o be hypoanalytic i f
Some authors give a slightly more general definition of hypoanalytic functions - they consider functions which are G-holomorphic and bounded on the compact subsets of U. The following example shows that this can lead to a strictly larger class of functions.
61
Holomotphic mappings between locally convex spaces Example 2.16 (en)n
E
Let
be an i n f i n i t e dimensional H i l b e r t space and l e t
be a sequence of mutually orthogonal u n i t v e c t o r s i n
I : (E,u(E,E'))
+
/I 11)
(E,
E.
Then
maps compact s e t s onto bounded s e t s b u t i s not
hypoanalytic s i n c e t h e sequence
(en)n
i s weakly b u t not s t r o n g l y
conver-
convergent. If a
E
i s a k-space ( i n p a r t i c u l a r , i f
,33fM,space)
then
HHY(U;F)
f
A function
l o c a l l y convex space
i s metrizable o r i f
E
f o r any open subset
U
of
E
is
E
and
F.
any l o c a l l y convex space Definition 2.17
H(U;F)
c
defined on a
T~
open subset
of a
U
i s said t o be Mackey or S i l v a holomorphic i f
(EJ)
i t i s G-holomorphic and M-continuous. We l e t space of a l l Mackey holomorphic mappings from
%(U;F)
denote t h e vector
U
F.
into
The following r e s u l t gives an a l t e r n a t i v e d e f i n i t i o n of Mackey holomorphic f u n c t i o n s .
We omit t h e p r o o f .
Proposition 2.18
U
Let
be a
open subset of the l o c a l l y convgx
T~
space E and l e t F be a 1ocaZZ.y convex space. following are equivalent:
b)
for each 5 exists E>O
U
f(t+EB)
such t h a t
5
f o r each
E
U
g
and
E
If H(U;F)
(E,T) f o r any
H(V;E),
rdmf(5)
where
In p a r t i c u l a r , space o r a holds f o r
838
C
i s a superinductive space then , T
open subset
%(U;F)
= H(U;F)
space and
F
U
if
of
U
H (U;F) G
then the
B
V
and g(O)= 5, the f u n c t i o n on some neighbourhood of zero i n CC.
of zero i n
E
of E there i s a bounded subset of F,
and each bounded subset
5 E U and each m i n N ,
c ) for each d)
E
If f
E
E
@,("E;?).
i s a neighbourhood fog
T ~ = T
i s hoZaorphic
and hence
%(U;F)
and any l o c a l l y convex space
F.
is an open subset of a Frzchet
is arbitrary.
We do not know i f t h i s r e s u l t
3% spaces.
There are s e v e r a l o t h e r t y p e s of holomorphic f u n c t i o n s t o b e found i n the literature.
=
We s h a l l introduce them i f t h e need a r i s e s .
Our main
Chapter 2
62
i n t e r e s t l i e s i n t h e s t u d y o f holomorphic f u n c t i o n s and a l l o t h e r f u n c t i o n s p a c e s are i n t r o d u c e d s o l e l y t o h e l p o u r s t u d y i n t h i s d i r e c t i o n .
The d i f f -
e r e n t k i n d s of holomorphic f u n c t i o n s we have d e f i n e d s a t i s f y t h e f o l l o w i n g inclusions. E
Let E.
open subset of
and
F
U
be l o c a l l y convex spaces and
an
The following inclusions hold
An i m p o r t a n t q u e s t i o n which w i l l a r i s e i n t h i s book and which i s s t i l l undergoing a c t i v e r e s e a r c h i s t h e f o l l o w i n g :
f o r what
t h i s q u e s t i o n f o r polynomial mappings.
U,E
and
F
are
We have a l r e a d y looked a t
some (or a l l ) o f t h e above i n c l u s i o n s p r o p e r ?
For t h e moment, w e c o n s i d e r o n l y a
few s i m p l e examples. Example 2.19 %(U;F)
= H
If
(U;F)
has property
E
(s)
f o r any open s u b s e t
HY p o l y n o m i a l s i n c h a p t e r 1 can b e e x t e n d e d .
Example 2 . 2 0 of
A G-holomorphic f u n c t i o n
with val ues i n
E
and
F
of
E.
U
f
i s a r b i t r a r y , then The p r o o f g i v e n f o r
d e f i n e d on a n open s u b s e t
U
i s hypoanalytic i f e i t h e r of t h e following condit-
F
ions hold: a)
f i s bounded on compact sets and "m d f ( c ) E 8Hy(mE;F) f o r e v e r y 5 i n and e v e r y p o s i t i v e i n t e g e r
b)
s e p a r a b l e and criterion (i.e.
Since
E
(E,o(E,E'))
T
and
T~
d e f i n e t h e same conver-
E).
i s l o c a l l y convex and h y p o a n a l y t i c i t y i s a l o c a l prop-
e r t y , w e may suppose w i t h o u t l o s s o f g e n e r a l i t y t h a t anced and t h a t a)
Let
F
K C U
Hence there e x i s t s
is
s a t i s f i e s t h e Mackey convergence
E
g e n t sequences i n Proof
m.
i s bounded on compact s e t s ,
f
U
i s convex and b a l -
U
i s a normed l i n e a r s p a c e .
b e compact. X>1
such t h a t
We may suppose t h a t
XKCU.
K
i s balanced.
The Cauchy i n e q u a l i t i e s imply
63
Holornotphic mappings between locally convex spaces
lim sup( 1 mn! 3 f(0) By hypothesis ___ E PH,,(("E;F) and hence f is the uniform limit on K n! of a sequence of continuous polynomials. Thus f is continuous on K and
that
f
E
HHY(U;F). b)
By (a) we may suppose that
Since E
f is an m-homogeneous polynomial.
is weakly separable, the compact subsets of E are metrizable and
hence it suffices to show that f is sequentially continuous. Let be a null sequence in E . such that
(An\
There exists a sequence of scalars,
and
+ +m
m
(x~)~=~
Anxn
+
as n-.
0
Since f i s bounded on com-
pact sets
I An(
is a bounded subset of F. Now n
-t m
+ a
and this implies f(xn)
+
0 as
and completes the proof.
We now look at holomorphic versions of the Factorization Lemma proved for polynomials in the first chapter. The situation is much more complicat-
ed due to the fact the polynomials are always defined on the entire space and continuity at a single point implies continuity at all points. These properties are not necessarily true of arbitrary G-holomorphic functions. Here the topological and geometric properties of the set U and the global continuity properties of the function f have to be taken into consideration. E
Let
Theorem 2.21
be a l o c a l l y convex space and l e t
F be a normed
U i s a connected open subset of E and f E H(U;F) then there e x i s t s an a E cs(E) such t h a t f o r any XEU, YEE for which Zinear space.
If
a(y) = 0 and Ix
ue have
+
f(x+y) Proof
OCX,
xy; =
(*I
f(x). is convex and balanced. Condition (*)
We first suppose that U
is then satisfied if x and x+y
E
U.
Since F
is a normed linear space
there exists an a in cs(E) such that Ba = IxEE;~(x)< 11 C U and = M <=. By the Cauchy inequalities it follows that Ba
64
Chapter 2
"m
Ild)l/
m! Ba ials we have
E
U
for all m
in E
for every x x,x+y
M
6
a(y) = 0
and
"m
m
f(x+y)
=
and all
Cm=o
y
in E
for which a(y)
=
0. Hence if
then
f(o)
m!
and by the factorization lemma for polynom-
(x+y) = 1" m=O
This completes the proof when U
!L@l (x) m!
=
f(x3.
is convex and balanced. For arbitrary U
we choose S E U and V a convex balanced open set such that S + V C U . By the above there exists an a in cs(E) such that if x,y E E , x E V , x+y E V and a(y) = 0 then f(S+x) = f(€,+x+y). Moreover, if x,y E E , XEV, S+x+y E U, a(y) = 0 and (*) function of one complex variable
-
A
f(S+x+Ay)
is satisfied, then we may consider the
- f(S + X I
This function is constant, by the above, on some neighbourhood of zero, and hence it is constant on the connected interval
[0,1]. Hence
f(F+x+y)
=
f(S+x). Let Uo = {xsUlif YEE, a(y) = 0 and (*) is satisfied then f(x) = f(x+y)l. We have just shown that Uo is a nonempty open subset of U.
If xg
E
Uo
+
x
E
U
as
8
+
-, YEE and
{x+AylObh$lj C U , then)
since E i s a topological vector space, x8+Ay E U for all 9 sufficiently large and all A E [0,1]. Hence f(x+y) = lim f(xe+y) = lim f(x8) = f(x)
e-
Thus Uo is a non-empty open and closed subset of U . ected, this implies U=Uo and completes the proof.
e-
Since U
is connru
Our next step motivated by the polynomial case would be to define f on n a ( U ) . There are, however, several difficulties which cannot be surmounted without certain modifications. Without condition
(*)
N
f may not
be well defined.
It is possible to surmount this problem, in the general situation, by considering domains spread over E and using a pullback operator or by restricting oneself to pseudo-convex open sets. To simplify our presentation, we confine ourselves to convex open sets. A second difficulty arises from the fact that ila(lJ)
is not necessarily an open subset of E
65
Holomorphic mappings between locally convex spaces However,
w i l l always b e a
17 (U)
we c a n a s k , assuming
tf
and c o n s e q u e n t l y Ea N i s w e l l d e f i n e d , whether o r n o t f i s a holomor-
N
f
open s u b s e t of
N
The s e t o f p o i n t s o f c o n t i n u i t y o f
phic function.
may n o t b e a l l o f
na(U).
f
w i l l b e nonempty b u t
Example 2 . 2 2 i l l u s t r a t e s t h i s d i f f i c u l t y .
d i f f i c u l t y i s overcome by p l a c i n g e x t r a c o n d i t i o n s e i t h e r on
na
mappings
Example 2.22
We d e n o t e by
H(C)
t h e s p a c e o f e n t i r e f u n c t i o n s of one
Let
complex v a r i a b l e endowed w i t h t h e compact open t o p o l o g y . b e d e f i n e d by f u n c t i o n on
in
a
subset of
H(C)cl
f o r any
cs(H(C))
a
E
such t h a t
suplf(z)l
=
cs(H(E)). B
fn(z) fn(0) = 0,
where
I z1sR
n 2
=
Since If
O t h e r w i s e , t h e r e would
is a closed
z+z
=
R > 2.
let
+...
+ z
(2R) + . . . +
2 fn(2R) = 4R - 2R
n
(2R)
.
n
2
.
(Z
for all n
2
-2R)
n +
and
-.
F ( f + f n ) = ( f + f n ) ( f ( 0 ) + f n ( O ) ) = f (2R)+fn(2R) = 2 F ( f ) = f(2R) = 2R-4R # 0 w e h a v e shown t h a t no such a e x i s t s .
f(z)
U
2R-z
Then
i s convex and b a l a n c e d (or even pseudo-convex)
o f Theorem 2 . 2 1 i s s a t i s f i e d f o r a l l
(*)
C
i s an e n t i r e
{fEH(C);F(f) = 0 )
=
as
0.
+.
Without l o s s o f g e n e r a l i t y we may assume
For e a c h p o s i t i v e i n t e g e r
Let
F
F!H(C)
does n o t f a c t o r as a holomorphic
F
We claim t h a t
H(E),. a(f)
Then
I t is e a s i l y seen t h a t
F(f) = f ( f ( 0 ) ) . H(C).
func ti on through e x i s t an
Ea
and w e g i v e v a r i o u s s u f f i c i e n t c o n d i t i o n s .
This
o r on t h e
x
and
then condition
and we o b t a i n t h e
y
following f a c t o r i z a t i o n r e s u l t . Proposition 2.23 f
E
Proof
be a l o c a l l y convex space and l e t
F
be a
is a convex balanced open subset of E and r) then there e x i s t s an a i n c s ( E ) and f E HG(na(U);F) such
H(U;F)
that
E
Let
normed l i n e a r space.
If
U
J. .
f = fona.
We d e f i n e
ru
f
on
na(U)
by
.V
f(x) = f(2)
if
2E
U
and
66
Chapter 2 rrl
n,(x)
Theorem 2 . 2 .
= x.
f = f e n . Since f .-./ HG(na(U);F). N
shows t h a t
f"
i s w e l l d e f i n e d and by c o n s t r u c t i o n
i s a G-holomorphic f u n c t i o n , it f o l l o w s t h a t
W e now g i v e a s u f f i c i e n t c o n d i t i o n f o r t h e c o n t i n u i t y o f P r o p o s i t i o n 2.24
a directed s e t
If D
f
cs(E)
i s a ZocaZZy convex space and
E
4
of semi-norms which d e f i n e the topology of a in D then
contains
E
nO1
and
i s an open mapping f o r every
li.e.
if
such t h a t
f
E
H(U;F)
then there e x i s t s an
f = faofla)
to lie in
D.
Hence
na(U)
there exists a and
since
Ra
llfll
B
i s a n open s u b s e t o f
HG(na(U);F)
6
l o c a l l y bounded s i n c e
C U
fa
E
U
of
H(TIu(U);F) E
and
We first n o t e t h a t i n t h e p r o o f o f theorem 2 . 2 1 w e may choose
p r o p o s i t i o n 2.23, f a 0
and
F.
any normed l i n e a r space Proof
D
in
01
for any convQx balanced open subset
f
D m.
and
such t h a t
p>O
Hence
w
Ilfa//
BB,s,P
i s open, i t f o l l o w s t h a t
fa
and t h e r e e x i s t s , by
Ea
Now
= f O 1 olla.
i s a normed l i n e a r s p a c e .
F
in <
such t h a t
f
is
Hence f o r each
BB,S,P
d B B , S,P
a
) =
=
5 in IxEE; B(5-X)
IlfiiB
10
and,
B,S,P
i s l o c a l l y bounded and so b e l o n g s
t o H(nO1(U);F). T h i s c o m p l e t e s t h e p r o o f s i n c e t h e o p p o s i t e i n c l u s i o n i s obvious. The above p r o p o s i t i o n c o v e r s t h e case where each s p a c e and y i e l d s t h e f o l l o w i n g examples.
Let
Example 2 . 2 5
nnZl m
Eu
i s a Banach
En where each En i s a Banach s p a c e . n EO1 = II. E . f o r a l l n . I f f E H(E;F) and F i s a normed 1=1 I l i n e a r s p a c e , then t h e r e e x i s t s a p o s i t i v e i n t e g e r n and f n E H(E ; F ) an such t h a t f = f n o Rn where IIn i s t h e c a n o n i c a l p r o j e c t i o n o f E o n t o E
=
Let
Ea
n
.
As a p a r t i c u l a r example, we see t h a t
u ,
H((CN)
=
I
neN H((Cn).
67
Holomotphic mappings between locally convex spaces
Let X be a completely regular Hausdorff topological space
Example 2.26 and let E
=
& (X).
Then we can choose our directed set D
such that
Ea Y &,(K), K compact in X, for each a in D . Hence each E, is a Banach space and we obtain a factorization result for normed linear space valued holomorphic functions defined on convex balanced open subsets of
.
&(XI
Ea, is given in the following proposition. This condition also arises in various other A further sufficient condition, this time on the spaces
parts of infinite dimensional holomorphy, for instance analytic continuation and we show (theorem 2.28) that it is satisfied by any Banach space.
In
fact, we prove a more general result which we shall use later. This, then, I n more general
gives an alternative proof for examples 2.25 and 2.26.
factorization theorems it is applied to prove results unobtainable by using proposition 2.24. Let
E be a l o c a l l y convex space and l e t F be a normed linear space. I f there e x i s t s a directed family D of semi-norms which define t h e topology of E and each a i n D s a t i s f i e s the following condition: Proposition 2.27
f
i f
E
where
H(U,;F)
i s an open subset of
U,
then t h e s e t of points o f c o n t i n u i t y o f
Ea
f i s open and
ctosed. Then
H(U;F)
=
u
H(n,(U);F)
C~ED
f o r any convex balanced open subset
Proof
u
~ E C (SE l
U of
H(lla(U);F)
E.
This follows immediately from theorem 2.21.
Theorem 2.28 let
=
Let
U be a connected open subset o f t h e Banach space
F be a normed l i n e a r space and l e t f E HG(U;F). SEU and every p o s i t i v e integer m then f
f o r some
Proof
pf(E) H(U;F).
If E
Without l o s s of generality, we may assume that U
balanced set and that 5 = 0.
Let
E
E,
6(mE;/\F)
is a convex
Chapter 2
68
Am
Since each
i s continuous,
d f(0)
6
Vn = U s i n c e n=l say V some Vn,
i s a closed subset of
Vn
f
i s G-holomorphic.
,
has nonempty i n t e r i o r .
"0
V C V
no
.
n
1-m=O zm
L
f
qeU "0
and
i s a convex
V
then lemma 1.13 implies
2 f (0)
SUP
L
-V 2
Thus
If
n+VcV
Hence
Ilf Ill
and
By t h e Baire category theorem
balanced neighbourhood of zero such t h a t that
U
2no.
=
i s l o c a l l y bounded and continuous a t t h e o r i g i n .
By using t h e
Taylor s e r i e s expansion of G-holomorphic functions, we s e e t h a t
f o r any
0
in
f o r any
n
and any
that
i s continuous a t
f
have shown t h a t
and any
U
f
0
E
in
x
in
U.
Since
8.
H(U;F).
By lemma 1.19,
E.
Pf(0)
E
(P(nE;$)
By t h e f i r s t p a r t of o u r p r o o f , i t follows (3
was a r b i t r a r i l y chosen i n
U
we
This completes t h e proof.
6332
Since Frzchet spaces and
spaces a r e superinductive l i m i t s of
Banach spaces, one can e a s i l y prove a r e s u l t s i m i l a r t o theorem 2.28 f o r such spaces. So f a r we have been d e s c r i b i n g f a c t o r i z a t i o n r e s u l t s which use contin-
uous semi-norms on t h e domain space. solving t h e Levi problem on s o r t of f a c t o r i z a t i o n .
C e r t a i n s i t u a t i o n s ( f o r example, i n
&3mspaces
with a b a s i s ) r e q u i r e a d i f f e r e n t
We g i v e some r e s u l t s i n t h i s d i r e c t i o n .
A topological space
X
i s a L i n d e Z Z f space i f every open cover o f
contains a countable subcover.
X
i s s a i d t o be h e r e d i t a r y LindeZb'f i f
X
69
Holomotphic mappings between locally convex spaces every open subset of
i s Lindelbf.
X
533%
Separable FrGchet spaces and
spaces a r e examples of h e r e d i t a r y Lindelb'f l o c a l l y convex spaces. Proposition 2.29
F f
be a hereditary Lindelb'f l o c a l l y convex space,
H(U;F)
IJ
Proof
a metrizable l o c a l l y convex space
then there e x i s t f
U),
and
H(nf(U);F)
E
IIf(U)
in
5
t h e r e e x i s t s an
U
where
<
f
The semi-norms l o c a l l y convex topology on TIf
E.
Let
a5
U
cs(E)
If
(which E
onto
and
and
f
f = fonf.
such t h a t
5
5
it contains a countable sub-
g e n e r a t e a semi-metrizable
denote t h e a s s o c i a t e d m e t r i z a b l e
denote t h e q u o t i e n t mapping from
c o n s t r u c t i o n nf(U)
Ef
E. Ef
Ba (1)= { x E E ; ~(x) < 1 ) .
(a5 , )mn = l Ef
E
E
from
i s an open subset o f
I1 115+B a g ( l ) + Ba5(1) cu and Since { 5 + B a _ (1)IEEu i s an open cover of
c.
space and l e t
nf
a continuous s u r j e c t i o n
such t h a t
For every
a convex balanced open subset of
U
a normed l i n e a r space and E
depends on f
E
Let
i s an open s u b s e t of
Ef.
E
onto
Ef.
By
on
We now d e f i n e
i n t h e usual manner and s i n c e it i s l o c a l l y bounded, it l i e s i n
nf(U)
H(nf(U);F).
This completes t h e proof. Corollary 2.30 E
F
and
If U
space
i s a Banach space, then H(U;F)
Proof
is a convex balanced open subset of a
A bjq
=
u
~ E C (E) S
H(na(U);F).
space i s a h e r e d i t a r y Lindelb'f space and a l s o a DF-space.
The r e s u l t now follows e a s i l y by using t h e c o n s t r u c t i o n of p r o p o s i t i o n 2.29 and t h e following property of
DF
spaces:
if
m
i s any sequence o f
continuous semi-norms on a DF-space, then t h e r e e x i s t s a continuous semi-
norm
a
an 6 c na
on
E
and a sequence of p o s i t i v e r e a l numbers
for all
m
( c ~ ) ~such = ~t h a t
n.
Corollary 2.30 may be strengthened i n t h e c a s e of e n t i r e functions (see e x e r c i s e 2.105). Our f i n a l example f i r s t a r o s e i n f i n d i n g a counterexample t o t h e Levi problem.
The proof i s q u i t e d i f f e r e n t from t h o s e j u s t given and v a r i a t i o n s
of t h e technique used w i l l appear i n chapter 5 . Example 2.31
Let
r
denote an uncountable d i s c r e t e s e t .
If
Chapter 2
70 x = (x )
CL CiEr
c0(r)
E
( x E c o ( r ) ; s ( x ) cI',) /If11
and If
J
m
where
...,a n1
{al,
=
rl
f o r any s u b s e t
M <
=
s(x) = {acr;xa # O } .
we let
of
co(rl)
Let
Now suppose
I'.
i s t h e open u n i t b a l l of
B
r
i s any f i n i t e s u b s e t o f
=
f
E
(co(r),
H(2B;C)
11 1 1 ) .
t h e n , by u s i n g a monomial
e x p a n s i o n i n s e v e r a l v a r i a b l e s , w e see t h a t f(z) If
e,
has i t s
zero,lthen
where
IkENjw(k)zk
=
M2
z
for a l l
in
ith c o o r d i n a t e e q u a l t o 2 sup{)f(z)I2; z =
1
i=l
1
2Bn c o ( J )
and a l l o t h e r c o o r d i n a t e s
Ziei,\Zij=
1
il
all
I 1
i s n o r m a l i s e d Haar measure on { z i e i , z . =1} f o r i = l , .. . , n . iBm zm = e , m = l , ..., n, it f o l l o w s t h a t
dzi
By u s i n g the change of v a r i a b l e
Since
Jcr
where
N(')
Hence
{k;w(k)#O}
was a r b i t r a r y , i t f o l l o w s t h a t
= { $ : r + N, $ ( a ) = O
r l ={azr\ 3 s u b s e t o f r. I t
(x,A),
x
E
2B
and
EN'^),
k
E
2B,
co(r)
c o n t i n u o u s , we have shown t h a t E
B.
and
w(k)#O
is e a s i l y seen t h a t
x+Ae
i s a dense subspace o f
(xa)aer
a
in
r}.
is countable.
Let
able
f o r a l l e x c e p t a f i n i t e number o f
if
a
E
laer 1
i s a count-
ufor
f(x+Xea) = f(x)
r\rl.
Since
( i n t h e norm t o p o l o g y ) ( ("a
rl
k(a)#O}.
JCT , J f i n i t e C o ( J ) and f i s
= f((xaIaErl)
for a l l
By u s i n g t h e p r i n c i p l e o f a n a l y t i c c o n t i n u a t i o n ( i n s e v e r a l
71
Holomotphic mappings between locally convex spaces complex v a r i a b l e s ) one c a n e a s i l y e x t e n d t h e above p r o o f t o show t h e f o l l owing : if
then t h e r e e x i s t s a countable subset &
such t h a t onto
co (r I+ and
i s a convex b a l a n c e d open s u b s e t o f
U
f = f on
co(rl).
rl
where
'r
rl
of
r
and
f
E
f
E
H(U;K)
H(U n Co(i-l);E)
i s t h e c a n o n i c a l s u r j e c t i o n of c o ( r )
Many o f t h e above f a c t o r i z a t i o n theorems can be extended t o pseudoconvex domains (and t h i s e s s e n t i a l l y means t o a l l open s e t s ) by v i r t u e of t h e following r e s u l t : if and
U
U
i s a pseudo-convex open s u b s e t o f t h e l o c a l l y convex s p a c e
contains an a - b a l l , a
f i n i t e l y open s u b s e t o f
,E
E
and
cs(E),
then
n
(U)
E
i s a pseudo-convex
U = Uil(II,(U)T.
T h i s r e s u l t i s u s e d i n studying pseudo-convex domains, h o l o m o r p h i c a l l y convex domains and domains of holomorphy i n l o c a l l y convex s p a c e s . F a c t o r i z a t i o n r e s u l t s f o r Mackey holomorphic f u n c t i o n s a r e r e q u i r e d i n Chapter 6 .
The c o n c e p t s and methods needed t o p r o v e t h e s e r e s u l t s w i l l b e
given later. 52.3
LOCALLY CONVEX TOPOLOGIES ON SPACES OF HOLOMORPHIC FUNCTIONS Two t o p o l o g i e s a r e u s u a l l y c o n s i d e r e d on t h e s p a c e o f Mackey holomor-
phic functions;
t h e t o p o l o g y o f u n i f o r m convergence on t h e f i n i t e dimen-
s i o n a l compact s u b s e t s of t h e domain and t h e t o p o l o g y o f uniform convergence on t h e s t r i c t l y compact s u b s e t s o f t h e domain.
Since w e s h a l l not use
a n y r e s u l t s d e r i v e d by u s i n g t h e s e t o p o l o g i e s , we c o n f i n e our i n t e r e s t i n t h e s e t o p o l o g i e s t o some e x e r c i s e s a t t h e end of t h e c h a p t e r .
On t h e
s p a c e o f h y p o a n a l y t i c f u n c t i o n s , t h e n a t u r a l t o p o l o g y i s t h e compact open topology. Definition 2.32
Let
U
be an open subset of the l o c a l l y convex space
E and l e t F be a l o c a l l y convex space. The compact open topology ( o r t h e topology of uniform convergence on the compact subsets of U l on HHY(U;F) i s t h e l o c a l l y convex topology generated by t h e semi-norms
12
Chapter 2
ranges over the compact subsets of
K
where
F.
continuous semi-norms on Naturally
U
and
ranges over the
$
We denote t h i s topoZogy by
T
induces a l o c a l l y convex topology on
T~
again c a l l t h e compact open topology and denote by
T
~
~
.
which we
H(U;F) .
This i s t h e most
n a t u r a l topology t o consider on spaces o f holomorphic f u n c t i o n s .
We f i n d ,
however, it does n o t always possess very u s e f u l p r o p e r t i e s and f o r t h i s reason w e introduce t h e
topology.
T~
This topology has good topological
p r o p e r t i e s b u t can be d i f f i c u l t t o d e s c r i b e i n a concrete fashion and i t s r e l a t i o n s h i p with t h e
topology may not always be c l e a r .
T~
duce a f u r t h e r topology,
T
w’
We a l s o i n t r e
whose d e f i n i t i o n was motivated by c e r t a i n
p r o p e r t i e s o f a n a l y t i c f u n c t i o n a l s i n several complex v a r i a b l e s . topology i s intermediate between t h e
T
0
and t h e
f u l l y it w i l l s h a r e t h e good p r o p e r t i e s o f both
topologies.
T~
and
T~
This
r o l e appears t o be a s a t o o l i n proving r e s u l t s about
T~
T
6 and
Hope-
but i t s main (see
T~
f o r i n s t a n c e chapter 5 ) . D e f i n i t i o n 2.33 F
and l e t
Let
U
be an open subset of a ZocaZZy convex space
be a normed Linear space.
to be ported by t h e compact subset V,KCVcU,
The
T~
C(V)>O
there e x i s t s
topoZogy on
H(U;F)
K
A semi-norm
of
U
p
on H ( U ; F )
E
i s said
i f , f o r every open s e t
such t h a t
i s t h e LocaZZy convex topology generated by
a22 seminorms ported by compact subsets of U . D e f i n i t i o n 2.34
Zet
F
D e f i n i t i o n 2 .3 5
and l e t
Let
u
be an open subset of a ZocaZZy convex space and
be a ZocaZZy convex space.
F
Let
U
We d e f i n e
T~
on
H(U;F)
by
be an open subset of a ZocaZly convex space
be a normed l i n e a r space.
A semi-norm
p
on
H(U;F)
E
i s said
13
Holomorphic mappings between locally convex spaces
t o be (Vn);=l’
T&
P(f)
The
6
C
I1 f l l V
n
topology on H ( U ; F )
T&
the
U,
continuous i f for each increasing countable open cover o f there e x i s t s a p o s i t i v e i n t e g e r no and c , O such t h a t f o r every
f
in
H(U;F).
0
i s t h e l o c a l l y convex topology generated by
continuous seminoms.
Definition 2.36 F
and l e t
be an open subset of a l o c a l l y convex space
U
Let
be a l o c a l l y convex space.
(H(U;F)
We d e f i n e
T&
on H ( U ; F )
E
by
=
,T&)
The g e n e r a l r e l a t i o n s h i p between t h e t h r e e t o p o l o g i e s j u s t d e f i n e d i s g i v e n i n t h e f o l l o w i n g lemma.
Lemma 2 . 3 7
E.
F
and
be l o c a l l y convex spaces and l e t
On H ( U ; F )
we have
T
~
~
T
~
~
We may suppose, w i t h o u t l o s s o f g e n e r a l i t y , t h a t
Proof
l i n e a r space.
Since
follows t h a t on
E
Let
open subset o f
H(U;F)
T~ 2 T
0
.
IlfllK
<
llfllv
Now suppose
p
is a
exists
no
such t h a t
such t h a t
c>O
shows t h a t
T
p
is
T&
containing of
K
Since
U.
i s a neighbourhood o f
Vn
p(f)
cll€lIvno
F
be an
~
i s a normed K
it
c o n t i n u o u s semi-norm
w
which i s p o r t e d by t h e compact s u b s e t
d e n o t e an i n c r e a s i n g c o u n t a b l e open cover o f can choose
v
f o r every
U
T
f o r every
f
U. K
Let
(Vn)i=l
i s compact w e
K.
Hence t h e r e
in
H(U;F).
This
c o n t i n u o u s and completes t h e p r o o f .
Our n e x t r e s u l t shows t h a t
T&
has good t o p o l o g i c a l p r o p e r t i e s when
t h e r a n g e s p a c e i s a Banach s p a c e ( a s l i g h t l y less g e n e r a l r e s u l t h o l d s when t h e r a n g e s p a c e i s a normed l i n e a r s p a c e ) . a l t e r n a t i v e description of t h e Proposition 2 . 3 8
Let
U
T&
T h i s r e s u l t a l s o g i v e s an
topology.
be an open subset o f t h e l o c a l l y convex space
E and l e t F be a Banach space. Then ( H ( U ; F ) , T & ) is an inductive l i m i t o f Frgchet spaces and hence i t is barrelled, bornological and ultrabornol-
ogical .
Chapter 2
74 Proof
For each i n c r e a s i n g c o u n t a b l e open c o v e r o f
l e t HU(U;F) = I f EH(U;F); llfll < m Vn t o p o l o g y g e n e r a t e d by t h e semi-norms
m e t r i z a b l e l o c a l l y convex s p a c e .
n).
all
L') = (Vn);=l,
U,
We endow
.
p n ( f ) = Ilflb
HV(U;F) w i t h t h e
H,(U;F)
is then a
I t i s i n f a c t a 8 r & h e t space s i n c e
F
i s a Banach s p a c e and l o c a l l y bounded G-holomorphic f u n c t i o n s a r e holomor-
phic.
We c l a i m
H(U;F) = U$Ho(U;F)
c o u n t a b l e open c o v e r s o f i s open and 0
Wn
Since
f
H(U;F)
x m Hb(U;F).
( H ( U ; F ) , T ~ )=
follows t h a t
H(U;F) w e l e t
E
Wn
=
CXEU; [lf(x)(\ < n3
We now l e t
w e h a v e proved our claim.
i n d u c t i v e l i m i t t o p o l o g y on i.e.
f
ranges over a l l increasing
i s an i n c r e a s i n g c o u n t a b l e open c o v e r o f
= (Wn)mnZ1
Hw(U;F)
E
If
U.
where
T~
d e f i n e d by a l l t h e s p a c e s
Since
Hv(U;F)
U.
denote the H,(U;F),
i s a F r g c h e t s p a c e it
( H ( U ; F ) , T ~ ) i s a n u l t r a b o r n o l o g i c a l s p a c e and hence it i s
b a r r e l l e d and b o r n o l o g i c a l . We complete t h e p r o o f by showing t h a t
mapping from
Hw(U;F)
into
a b l e open c o v e r CJ o f Let not
p
of
denote a
(Vn):=l
U,
/lfnlIVn$ Let
Wn
=
~
.
Since t h e i d e n t i t y
i s c o n t i n u o u s f o r any i n c r e a s i n g count-ii
and a sequence
p(fn) > n
for all
(1
n
CXEU; ( ( f m ( x )
denotes t h e i n t e r i o r o f
W,"
T
>
7
6' H(U;F).
+,
(U;F) (fn)iz1
i s continuous.
(fn)EZ1
for all Since
Wn.
ml.
in
Let
]]fmIIVQ
H(U;F)
is
for a l l
n.
such t h a t
= (W")"
where n n=l 1 f o r a l l min, it
s i p llfmlb ,<
BY c o n s t r u c t i o n
i s a bounded s u b s e t o f
p(fn) > n
fact that
p
n.
f o l l o w s t h a t D i s an i n c r e a s i n g c o u n t a b l e opt% c o v e r o f 'IH Hence
Suppose
Then t h e r e e x i s t s an i n c r e a s i n g c o u n t a b l e open c o v e r
,
=
1 and
=
c o n t i n u o u s semi-norm on
T i
continuous.
T&
H(U;F)
we have
U
T.
Ho(U;F).
Hence
7.
=
7&
U.
n
Hence f o r every n .
This Fontradicts t h e and we have completed
t h e proof. Since 'o,b with
H(U;F)
and T~
T~
and
w,b and T
T
w
a r e n o t i n g e n e r a l b o r n o l o g i c a l t o p o l o g i e s , we l e t
d e n o t e t h e b o r n o l o g i c a l t o p o l o g i e s on
T
w
respectively.
H(U;F)
associated
(Note t h a t t h e t o p o l o g y induced on
by t h e b o r n o l o g i c a l t o p o l o g y a s s o c i a t e d w i t h t h e compact open
topology o f
HHY(U;F) need n o t b e
o g i e s t h a t can b e p l a c e d on
H(U;F)
T ~ , ~ T ) h. e r e a r e a l s o f u r t h e r t o p o l -
-
such a s t h e t o p o l o g y o f uniform
convergence of t h e f u n c t i o n and i t s f i r s t
n
d e r i v a t e s on t h e compact
75
Holomotphic mappings between locally convex spaces
...,
U, n = 1 , 2 ,
s u b s e t s of
but s i n c e we s h a l l not u s e t h e s e topologies
we w i l l not go i n t o any f u r t h e r d e t a i l s . A g r e a t p o r t i o n of t h i s book i s concerned with f i n d i n g conditions on
U,E
and
F
which imply e i t h e r
=
T~
T
w
, T~
= T~
or
= T
T~
(together
6
The remainder of t h i s s e c t i o n
with t h e i m p l i c a t i o n s of t h e s e c o n d i t i o n s ) .
is devoted t o a number of b a s i c f a c t s , concerning t h e s e t o p o l o g i e s , which we s h a l l f r e q u e n t l y u s e and t o a few examples and counterexamples which w i l l prove u s e f u l i n l a t e r c h a p t e r s .
lVe f i r s t n o t e t h a t t h e compact open topology i s a sheaf topology, i . e .
it i s l o c a l l y defined. T&
We do not know i f t h i s i s t r u e f o r t h e
T
and t h e
w
topologies and any r e s u l t s i n t h i s d i r e c t i o n would c e r t a i n l y h e l p t h e
general development of t h e t h e o r y .
The l o c a l c h a r a c t e r o f t h e compact open
topology i s contained i n t h e following lemma. Lemma 2 . 3 9 (Ui)iEI
Let
U
U
an open cover o f (H(U;F),T~) i n t o
from
and
F
a locally convex space.
IIisI(H(Ui;F),~o)
i s t h e r e s t r i c t i o n of f (where f ' U i onto a subspace of IIiEIH(Ui;F)
H(U;F) Proof
I t suffices t o note t h a t
K
only i f t h e r e e x i s t s a f i n i t e subset of a compact subset of
E,
be an open subset of a l o c a l l y convex space
f o r each
UL.
1
j,
which maps
to
.
Uil
f
to
The mapping (f(Ui)isI
i s an isomorphism o f
i s a compact subset of I,
Ll,
..., Ln
such t h a t
K
and
U
i f and
(Kj)Y=l,
=U3,1Kj.
K.
1
I t i s obvious t h a t a s i m i l a r r e s u l t holds f o r hypoanalytic f u n c t i o n s . We now show t h a t (H(U;F),ro)
( p ( m E ; F ) , ~ o ) i s a closed complemented subspace of f o r any open subset U of t h e l o c a l l y convex space E , any
complete loca?.ly convex space (H(U;F),ro) spaces then
F
m.
Hence i f
( 6 ( m E ; F ) , ~ o ) must a l s o have t h e same property. U
i s an open subset of a ZocalZy convex space E i s a complete locaZZy convex space then ( 6 (mE;F) ,TO) i s a cZosed If
compZemented subspace o f Proof
and any p o s i t i v e i n t e g e r
has any property i n h e r i t e d e i t h e r by subspaces o r by q u o t i e n t
Proposition 2 . 4 0
and
F,
Since
(H(U;F),T~) for any p o s i t i v e integer m.
(H(U;F),T~) = (H(U-S,F),T~) f o r any
5
E
E
we may suppose
76
Chapter 2 As u n i f o r m convergence on t h e compact s u b s e t s o f
O E U.
is equivalent t o
E
uniform convergence on t h e compact s u b s e t s o f some neighbourhood o f z e r o f o r
elements o f
(mE;F)
it f o l l o w s t h a t
t h e compact open t o p o l o g y . A
dm -
H(U;F)
:
m!
f
( H ( U ; F ) , T ~ ) i n d u c e s on @(mE;F)
Now c o n s i d e r t h e mapping
-
H(U;F) ;;mf ( 0 )
_ _ _ _ _ f -
m!
A
T h i s i s a l i n e a r mapping and s i n c e it i s a p r o j e c t i o n from
p(mE;F)
dm - (P)(O) = P f o r every P i n m! H(U;F) o n t o @ (mE;F): To complete t h e
p r o o f w e must show t h a t it i s a c o n t i n u o u s p r o j e c t i o n . convex b a l a n c e d open s u b s e t o f balanced s u b s e t o f
f o r every
in
E
such t h a t
V C U .
Let If
denote a
V
i s a compact
K
t h e n , by t h e Cauchy i n e q u a l i t i e s
V
and hence t h e p r o j e c t i o n i s c o n t i n u o u s .
cs(F)
This
completes t h e p r o o f . For a r b i t r a r y
w e do n o t have any u s e f u l r e p r e s e n t a t i o n o f t h e
U
@
t o p o l o g i c a l complement o f
(%;F)
in
H(U;F)
b u t we s h a l l see, i n t h e
n e x t c h a p t e r , t h a t t h e T a y l o r s e r i e s r e p r e s e n t a t i o n o f holomorphic f u n c t i o n s g i v e s u s a means o f i d e n t i f y i n g a u s e f u l t o p o l o g i c a l complement when
We now p r o v e t h e a n a l o g u e o f p r o p o s i t i o n 2.40 f o r t h e
balanced.
xS
topologies.
T
is
U
w
and
Our p r o o f i s f o r Banach s p a c e v a l u e d mappings b u t t h e same
r e s u l t f o r a n a r b i t r a r y complete l o c a l l y convex r a n g e s p a c e c a n be proved i n a similar f a s h i o n . Proposition 2.41
and
F
If
U
((3 ( m E ; F ) , ~ w ) is a cLosed complemented
i s a Banach space then
subspace of (H(U;F),-cw ) and of ( H ( U ; F ) , T ~ ) . I n particular induce the same topoZogy on 8 (mE;F). Proof
We f i r s t show t h a t
denote a
T~
T~
i n c r e a s i n g c o u n t a b l e open c o v e r o f N
and
c o n t i n u o u s semi-norm on
b a l a n c e d neighbourhood o f z e r o i n integer
and
C>O
E
is an open s u b s e t of a l o c a l l y convex space
such t h a t
E. U
T~
c o i n c i d e on
H(U;F)
and l e t
The sequence
l$'(mE;F). V
T
w
and
T6
Let
p
d e n o t e a convex m
(UnnV)n=l
i s an
and h e n c e t h e r e e x i s t a p o s i t i v e
71
Holomorphic mappings between locally convex spaces
for a l l
f
E
H(U;F).
'i;
Hence
Ptopology I PPE;F)on
is a
continuous semi-norm on
T , ~
4
=
8(
i)/p(mE;F) m ~ ; ~f o) r a l l
it follows t h a t
T~
H(U;F) and
and s i n c e induce t h e same
T~
m.
The above a l s o shows t h a t t h e mapping given i n p r o p o s i t i o n 2.40 i s a continuous p r o j e c t i o n f o r both
T~
and
T ~ .
We now look a t t h e l o c a l l y bounded o r equicontinuous s u b s e t s of H(U;F). D e f i n i t i o n 2.42 E
F
and l e t
be an open subset of the locally convex space
U
Let
be a ZocaZly convex space.
locally bounded i f f o r every c,Vg,
Lemma 2.43
A subset
2
of H(U;F)
is
there e x i s t s a neighbourhood of
E
where
Proof
F.
A locally bounded subset of
(H(U;F),T6)
and
F
U
an open subset of
are l o c a l l y convex spaces, i s a bounded subset of
We may assume, without l o s s of g e n e r a l i t y , t h a t Let r) be a l o c a l l y bounded subset of
continuous semi-norm.
wn and l e t
H(U;F),
*
l i n e a r space. T~
in U
such t h a t
i s a bounded subset of
E
5
=
IXEU;
For each p o s i t i v e i n t e g e r
(IfCx)((
Vn = I n t e r i o r (Wn).
c n f o r every f Since
3
i s a normed
F
H(U;F), n
and
p
a
let
i n 31
i s l o c a l l y bounded
(Vn)n
i s an
78
Chapter 2
i n c r e a s i n g c o u n t a b l e open c o v e r of
Hence t h e r e e x i s t s
U.
and
C>O
N,
a p o s i t i v e i n t e g e r such t h a t
cllfll
p (f) s
f E
f
in
H(U;F)
and t h i s completes t h e p r o o f .
sup p ( f ) s C.N
Hence
f o r every
vN
3
If U i s an open subset of a ZocaZZy convex space E, F is a l o c a l l y convex space and every bounded subset of (H(U;F),ro) is l o c a l l y bounded then T ~ , T and ~ T & have the same bounded subsets in C o r o l l a r y 2.44
H(U;F). In particular
etc. i n place o f
C o r o l l a r y 2.45
then
and
(H(U),-ro)
H(U;F) = H(U;Fu)
H(U;C), YIY(U;E), e t c . F
be ZocaZly convex spaces.
Let
Fa = ( F , o ( F , F ' ) ) .
where
.3
= (@of)@EB
lies i n
H(U).
Thus
:y
Consequently
in
Vg
f(Vg)
containing
U
5
f @
i s a l o c a l l y bounded f u n c t i o n .
H(U;Fu).
E
in
(H(U),ro)
Hence f o r each such t h a t
sup
then
U
F'.
The
Hence
5 E U there V5 < a.
exists
and once more
i s a bounded s u b s e t of
f(Vg) Hence
and s o by
I($of((
i s a weakly bounded s u b s e t o f @ E B F
by Mackey's theorem it f o l l o w s t h a t f
f o r every
C
i s a bounded s u b s e t o f
o u r h y p o t h e s i s , it i s l o c a l l y bounded. a n open s e t
and l e t
and by Mackey's theorem i t i s
F
i s a weakly bounded s u b s e t o f
s t r o n g l y bounded.
i s a normed
F
i s a compact s u b s e t o f
K
If
@cf(K) = @ ( f ( K ) ) i s a bounded s u b s e t o f f(K)
F' B
be t h e u n i t b a l l o f
B
I f the
are ZocaZZy bounded f o r every open subset U
We may assume w i t h o u t l o s s o f g e n e r a l i t y t h a t
l i n e a r space. set
E
Let
bounded subsets of
Proof
.
T
I f t h e r a n g e s p a c e i s t h e f i e l d o f complex numbers w e w r i t e
Notation H(U), HHy(U)
of E
is t h e bornoZogicaZ topoZogy associated with
T~
F, i . e .
i s a holomorphic f u n c t i o n and
f
t h i s completes t h e p r o o f s i n c e t h e c o m p o s i t i o n o f holomorphic f u n c t i o n s i s holomorphic and so we always h a v e Example 2.46
Let
s p a c e and l e t subset,
3,
F of
U
H(U;F)
C
b e a n open s u b s e t o f a m e t r i z a b l e l o c a l l y convex
We claim t h a t any
b e a normed l i n e a r s p a c e . H(U;F)
H(U;F,).
i s l o c a l l y bounded.
SEU such t h a t f o r e v e r y open s e t
V,
,€ E
T~
bounded
I f not, then t h e r e e x i s t s
VCU,
w e have
sup I l f l ) fE
3
=
a.
79
Holomotphic mappings between locally convex spaces Hence we can choose IIfn(Sn)II > n
S,,
n.
for a l l
5,
U,
E
+
Since
5
n
as
and
+ m,
(fn)n
c3such
that
i s a compact subset of
{5n}nUIC)
U
t h i s i s impossible and we have proved our claim. The remaining examples given i n t h i s s e c t i o n d e a l with holomorphic and on l o c a l l y 8 3 % spaces, Banach spaces, aN x C")
f u n c t i o n s on
convex spaces which do not admit a continuous norm.
These examples a r e
elementary i n s o f a r a s t h e proofs a r e r a t h e r d i r e c t .
However, they a r e of
i n t e r e s t s i n c e they show t h e divergence between l i n e a r and holomorphic f u n c t i o n a l a n a l y s i s and a l s o because many of t h e examples and methods encountered h e r e have e x p l i c i t l y and i m p l i c i t l y motivated t h e development These examples a l s o provide a good
of t h e theory a s o u t l i n e d i n t h i s book.
i n t u i t i v e guide t o t h e t y p e of behaviour we may look f o r i n d e l i c a t e situations. Example 2.47 F
Let
U
be an open subset of a
b e a normed l i n e a r space.
We show t h a t t h e
space
and l e t
E
bounded s u b s e t s of
a r e l o c a l l y bounded (we have already proved t h i s r e s u l t f o r homo-
H(U;F)
geneous polynomials i n chapter 1 ) . ity, that
We may assume, without l o s s of genera
i s a convex balanced open subset of
U
bounded subset
T~
83 T~
3
of
H(U;F)
E
and we show t h a t t h e
i s l o c a l l y bounded a t t h e o r i g i n .
Let
be a fundamental system of convex balanced compact s u b s e t s of
(BJn
L-
E.
As we have previously noted, i t s u f f i c e s t o f i n d a sequence of p o s i t i v e
r e a l numbers,
(in),., such t h a t
B1
f E 3
B = kl 1
If
11 IIA
1.1=1A.B. 1 1
=
6>0
choose and
we l e t
f
M < m .
L(6) = L + 6Bk+l.
such t h a t
in
inequalities.
Now suppose
H(U;F) Hence
L(S1)CU
B1 B2L(61)
where
c(B1)
11
<
m.
'i;=l~n%
il> 0 such
...,
that
Hence
AIBICU.
i2, ik have been chosen
is. a compact s u b s e t of
such t h a t
61>0
B 2 '1,
for a l l
f~3-
i s compact, we can choose
Since
L
sup Ilf
U
Let
E
and >O
sup fE3
((fl(
S M +
be a r b i t r a r y .
and next choose
B1
so t h a t
I,,=,
--
-MI.
2"
We f i r s t
and
i s a l s o a compact subset of
B2, U.
i s derived by using t h e Cauchy
B1>
1
Hence
Chapter 2
80
continuous semi-norm on
___ dnf(o)
Let
n! If
f
E
H(U;F)
and s i n c e
H(U;F)
we can f i n d a p o s i t i v e i n t e g e r
and 6 > 0
s2
T
-bounded and
61,
B2>1
fin d f(O)/n!.
then
n. <
0,
is
be t h e symmetric n - l i n e a r form a s s o c i a t e d with
f o r any non-negative i n t e g e r
62
3
such t h a t
N
m
f o r each
n
and
6
we can choose
so t h a t
Hence, by induction, we can choose a sequence of p o s i t i v e real numbers,
3 and
i s a l o c a l l y bounded family o f f u n c t i o n s . T
6
d e f i n e t h e same bounded s u b s e t s o f
This implies t h a t H(U;F).
T ~ ,T~
Since B 8 3 b s p a c e s
a r e h e r e d i t a r y Lindelzf spaces and contain a fundamental sequence of compact s e t s it follows t h a t every open subset of a 3 3 M s p a c e contains
81
Holornorphic mappings between locally convex spaces a fundamental system of compact s e t s . Since
T6
that
T~
643%
spaces a r e
=
This, i n t u r n , implies t h a t
i s a metrizable, and hence a bornological l o c a l l y convex space. i s a l s o a bornological topology on H(U;F) we have i n f a c t shown
(H(U;F),ro)
T
-
w
-
on
6'
H(U;F).
F i n a l l y we remark t h a t open s u b s e t s of
k-spaces and s o
i s a Frgchet space i f
(H(U;F),ro)
F
is
a Banach space. Example 2.48
Let
Banach space m
E
is a
T
0
c0
be an open subset of an i n f i n i t e dimensional
U
Let
E.
5
continuous semi-norm on
be t h e u n i t b a l l of
If
E.
H(U).
H(U) Hence
which i s not continuous f o r t h e (H(U) ,
T ~ )#
(H(U) ,.cw)
f o r any open
o f any i n f i n i t e dimensional Banach space.
U
Example 2.48
Let
CN
E = CN
((? (2E) , T ~ ) .
( @ (2E) , T ~ )#
s e t of
B
( t h e space of n u l l sequences of complex numbers) then
compact open topology on subset
and l e t
U
E
x
C(N).
x
a").
Hence
We have a l r e a d y seen t h a t (H(U),ro) # (H(U),ru)
Example 1.39 shows, a l s o , t h a t
T
f o r any open sub~
#
,T
~~
on ,
~H(U).
For our next example we need a concept which f r e q u e n t l y a r i s e s i n i n f i n i t e dimensional holomorphy
-
t h e concept of very s t r o n g s e q u e n t i a l '
convergence - b u t which does not a r i s e i n l i n e a r f u n c t i o n a l a n a l y s i s .
Since
t h e dual concept - very weak s e q u e n t i a l convergence - w i l l a l s o be needed l a t e r , we t a k e t h e opportunity of giving i t s d e f i n i t i o n h e r e .
Further
information on t h e s e concepts i s o u t l i n e d i n t h e e x e r c i s e s .
D e f i n i t i o n 2.50
A sequence
( x ~ )i n~ a locally convex space
said t o be very strongZy convergent if sequence o f scalars x
Xnxn+ 0
in
E
as
n-
E
is
f o r every
The sequence is said t o be nontrivial i f
n # 0 f o r each n. A sequence i s obviously very s t r o n g l y convergent i f and only i f f o r
each
p
p(xn) = 0
in
cs(E)
for a l l
there exists a positive integer,
n 2 n(p).
n(p),
such t h a t
A metrizable l o c a l l y convex space
a n o n t r i v i a l very s t r o n g l y convergent sequence i f and only i f
E
E
admits
does n o t
-
82
Chapter 2
admit a c o n t i n u o u s norm.
un
(0, ...,1 , O ...)
=
Definition 2.51
CN
For example, i n
t h e sequence
i s a n o n t r i v i a l v e r y s t r o n g l y convergent sequence.
nth position
For example, i n
E
( x ~ )i n~ a l o c a l l y convex space
A sequence
t o be very weakly convergent if A n xn + 0 i n of non-zero scalars ( A ~ ) ~ .
E
as
i s said
f o r some sequence
M
u = (0 ,..., 1,0 ,...) i s n o t a n L- n t h p o s i t i o n In f a c t , i f E i s any Fre‘chet s p a c e which
t h e sequence
v e r y weakly convergent sequence. i s n o t a Banach s p a c e t h e n
c o n t a i n s a sequence which does n o t converge
E’
v e r y weakly.
Let
Example 2 . 5 2
b e a l o c a l l y c o n v e x s p a c e which c o n t a i n s a non-
E
( x ~ ) ; = ~ . Let
t r i v i a l v e r y s t r o n g l y convergent sequence
E.
For each
If
f
f
f(x+w) = f ( x )
Since
00
such t h a t
n > n
for all
a
f + If(ny+x )
uous semi-norm on
n H(E)
a b a r r e l l e d t o p o l o g y on semi-norm on
f o r every
x
and
w
and
-
for all
a(xn) = 0 p(f)
E
C,
all
defines a
f(ny)(
?
H(E)
n,.
it f o l l o w s t h a t
Hence f
in
and hence a
-co
B
be a
H(E), T~
i.e.
n
2
no
p
is a
= 0.
p
f(ny+xn) = H(E).
-c6
nu and s i n c e is a
-i6
The
continT~
is
continuous
and a l l
f
fn(xny + xn) # f n ( x n y )
i s bounded on t h e
T~
c o n t i n u o u s semi-norm on o,b H(E). We b e g i n by showing t h a t
n
such t h a t
in
B.
for all
p
‘I
bounded s u b s e t o f
f(iy+xn) = f(hy)
for all
I f t h i s were n o t t r u e , t h e n by
u s i n g subsequences i f n e c e s s a r y , w e can choose that
n
f o r every p o s i t i v e i n t e g e r
there exists a positive integer h
cs(E)
.(w)
H(E).
bounded s u b s e t s of Let
CL E
such t h a t
E
i s f i n i t e f o r every
W e now improve t h i s r e s u l t by showing t h a t
H(E).
in
i s a v e r y s t r o n g l y convergent sequence t h e r e e x i s t s a p o s i t -
(Xn)n=l i v e i n t e g e r n, function
w e c o n s i d e r t h e sum
H(E)
such t h a t
f(ny)
belong t o
t h e n , by t h e F a c t o r i z a t i o n Lemma, t h e r e e x i s t s an
H(E)
E
in
y#O
n.
An E C and f E B such n For each p o s i t i v e i n t e g e r n l e t
83
Holomorphic mappings between locally convex spaces
By t h e i d e n t i t y theorem f o r f u n c t i o n s o f one complex v a r i a b l e we may
s e l e c t a sequence o f complex numbers, gn(Xh) # 0.
Now
hn
For each i n t e g e r
H(E)
E
f o r each
n,
n.
Hence
and each
and s i n c e m
sequence o f complex numbers, for all
n
IXAl
(wn)n=l,
Ifn(X;y+wnxn)I
>
n
let
C
w E
6
% , n
hn(0) # h n ( l ) , for all
we c a n choose a
I hn(wn) I
such t h a t n.
such t h a t
Since
I
I
n+ fn(X,!,y)
>
( x ~ )i s~ a
v e r y s t r o n g l y c o n v e r g e n t sequence, (w x ) i s a n u l l sequence. Since n n-n IhAl s - f o r a l l n it f o l l o w s t h a t XAy+wnxn -t 0 as v and hence n2 03 K = I01 u IA;Y+wnxnln=l i s a compact s u b s e t o f E . As IlfnllK > n f o r
all
n
t h i s contradicts the fact that
B
is a
H(E).
Hence t h e r e e x i s t s a p o s i t i v e i n t e g e r
f(Xy)
for all
n
We now show t h a t
n
5
p
all 1
0'
C
is not a
otherwise, i . e . t h a t
p
T~
n
and a l l
f
bounded s u b s e t o f
such t h a t in
T h i s shows t h a t
B.
c o n t i n u o u s semi-norm on
T
i s p o r t e d by t h e compact s u b s e t
H(E).
$,(y)
# 0,
+ o ( ~ n )= 0
for all
n,
$,(y)
of
K
u s i n g subsequences i f n e c e s s a r y we c a n choose a s e q u e n c e i n such t h a t
f(Xy+xn) =
= 0
Suppose By
E.
E', for all
n>O
$ n ( ~) # 0 i f and o n l y i f n=m. Let m $,,, = $o$m n f o r any p o s i t i v e i n t e g e r s n and m . $n,m E P("+'E) and n n P($n,m) = Im $ o ( y ) $ m ( x m ) ~ f o r a l l n and lil. I f V i s any neighbourhood and f o r
of
H(E).
ll~olIvm < m'
O
m($,(Y)
I
follows t h a t
# 0.
$,(y)
Let
V
=
ll$mllvm <
II$,1IVm.
llgoIIvm.
6
c(V) > 0
such that.
Choose a n a r b i t r a r y neighbourhood and
II$ I l n
c(V,,,)
positive integers
then there e x i s t s
K
in
f
m,n
p
Hence
Taking Since
ml+o(y)I 6 Hence
-.
nth
Vm
II$oIlx.
is not
{f € H ( E ) ; p ( f ) ,< l}.
p ( f ) 6 c(V) llfllv Vm
of
K
w
V
such t h a t
mnl~o(Y)lnl$m(xm)l 6
r o o t s and l e t t i n g
we get
n-
was a n a r b i t r a r y neighbourhood o f T h i s cannot hold f o r a l l
T
f o r every
m
K
it
since
continuous.
i s a convex b a l a n c e d
T
(and h e n c e
T
w
.)
Chapter 2
84
bounded subset
of
Since
E.
neighbourhood of zero (because that neither spaces.
(H(E) ,
p
nor
T ~ )
i s not a
V
i s not
T T
w
(and hence not a
T ~ )
continuous) we have shown
(H(E) , T ~ ) a r e i n f r a b a r r e l l e d l o c a l l y convex
The above can e a s i l y be modified t o show t h a t t h e same r e s u l t
holds f o r
H(U:F\,
an a r b i t r a r y open subset of
U
E,
and
F
any l o c a l l y
convex space.
12.4
GERMS OF
HOLOMORPHIC
FUNCTIONS
We now introduce t h e space of holomorphic germs on a compact subset of Apart from i t s c l o s e r e l a t i o n s h i p with spaces of
a l o c a l l y convex space.
holomorphic f u n c t i o n s defined on open sets, t h e space
of germs i s a l s o an
important t o o l i n developing a s a t i s f a c t o r y d u a l i t y theory.
The problems
t h a t a r i s e i n studying t h e topological vector space s t r u c t u r e of t h e space of germs a r e of a d i f f e r e n t kind from those which a r i s e i n function space theory and t h i s d i f f e r e n c e a r i s e s p r i m a r i l y from t h e d i f f e r e n c e between p r o j e c t i v e and inductive l i m i t s . Let
K
be a l o c a l l y convex space.
relation which
f
4
u
be a compact subset o f a l o c a l l y convex space
f
where g
and
N
g
On
H(V;F) V 3K V open
We denote by
H(K;F)
and l e t
F
we define the equivalence
i f there e x i s t s a neighbourhood
are both defined and
E
W
K
of
on
flw = g l w .
the r e s u l t i n g vector space o f equivalence classes
If f is and the elements of H(K;F) are called holomorphic germs on K. an F-valued holomorphic f u n c t i o n defined on an open subset of E which contains
K
then we a l s o denote by
determined by
f.
f
t h e equivalence c l a s s i n
The natural topology on
H(K;F)
H(K;F)
i s given by t h e
l i m (H(V;F),rcw) ( t h e inductive l i m i t being taken i n t h e --f V 3 K Vopen category of l o c a l l y convex s p a c e s ) .
inductive l i m i t
for
I f F i s a normed linear space we l e t Hm(V;F) = {f E H(V;F);I) f l I V < m j V open in E and on t h i s space we define a topology by means
of the norm F
11 11 v.
i s a Banach space.
see that
H"(V;F)
i s a normed l i n e a r space which i s complete i f
Using t h e same equivalence r e l a t i o n s h i p we e a s i l y
85
Holomorphic mappings between locally convex spaces
=
H(K;F)
V3 K V open
If K i s a compact subset o f a l o c a l l y convex space
Lemma 2 . 5 3
E and
F i s a normed l i n e a r space then
V3K V open
V3K V open
Proof If V is an open subset of E which contains K then the natural injection from Hm(V;F) into (H(V;F) ,T,,,) - is continuous and hence the identity mapping from lirn (Hm(V;F), 11 ) into lirn (H(V;F) , T ~ )
IIv
-+
VDK V open is also continuous. Conversely, if p lim
(Hm(V;F),
IIv)
11
then for each V
3
V> K V open is a continuous semi-norm on open, V 2 K
there exists
c(V)>
0
-----f
V 2K V open such that p(f)
c(V) IIfllV for every f in Hm(V;F). If then llfllv = m and the same inequality holds. Hm(V;F) ,<
f E H(V;F)\ Hence the restriction of p to H(V;F)
is a
ported by the compact subset K
Thus p
norm on
lim
+
(H(V;F),T~)
of V.
T
continuous semi-norm is also a continuous semi-
and this shows that the two topologies coincide
V> K V open
on H(K;F)
and completes the proof.
It follows that H(K;F) is a bornological space if F is a normed linear space and an ultrabornological (and hence a barrelled) space if F is a Banach space.
If E
is a metrizable space and F
is a Banach space
then H(K;F)
will be a countable inductive limit of Banach spaces and hence a bornological DF-space. Thus we see that the space of germs will always have some good topological properties since it is an inductive limit and indeed the main topological problems connected with H(K;F) are those generally associated with inductive limits (as opposed to those connected with projective limits) such as completion, description of the continuous semi-norms (sometimes we only need a description of sufficiently many
86
Chapter 2
continuous semi-norms) and a c h a r a c t e r i z a t i o n of t h e bounded s e t s . encounter a l l of t h e s e problems i n l a t e r c h a p t e r s . s e l v e s t o c h a r a c t e r i z i n g bounded s e t s when E = limE
Let
E
We s h a l l
Here we confine our-
i s metrizable.
be an i n d u c t i v e l i m i t of l o c a l l y convex spaces.
The
4
a
inductive l i m i t i s said t o be regular i f each bounded subset of E contained and bounded i n some
is
Ea.
Lemma 2 . 5 4 W
let B
=
Let K be a compact subset o f a l o c a l l y convex space be a convex balanced open subset of E . Then
{f EH(K+W); ( ( f ( \ K + ,W < 1) i s a cZosed subset o f
Proof
Let
{falaEA be a convergent n e t i n
H(K)
which l i e s i n
show t h a t
{fa)aEA i s a Cauchy n e t i n K+W
L CK+pW.
By using t h e Cauchy i n e q u a l i t i e s we s e e t h a t
(H(K+W) , T ~ ) .
'm, K , L
p,
If
L
O
H(K)
and a l s o on
We
i s a compact
such t h a t
(H(K+W),
YEL
f o r every nonnegative i n t e g e r
m.
{ f a l a E A C B t h e Cauchy i n e q u a l i t i e s imply t h a t
Hence, f o r some
B.
XEK YEL
d e f i n e s a continuous semi-norm on
Since
and
H(K).
subset of
then t h e r e e x i s t s a r e a l number
E
00
and a l l
a,B E A ,
T ~ ) .
Hence
87
Holomorphic mappings between locally convex spaces this
t e n d s t o z e r o as in
a+=
H(K).
ci,B
since
+ m
By t h e above, t h e r e e x i s t s a
u n i f o r m l y on t h e compact s u b s e t s of every non-negative i n t e g e r m h f o l l o w s t h a t s m f ( F ) = d g(C) g
Proof
Since
in
Since
K+W.
m
for all
+
such t h a t
B
as
f fa
+
g
Pm,K,L(f) = Pm,K,L(g)
and a l l H(K)
fa
5
of
L
K.
in
K+W
it
Hence
f
and
and t h i s completes t h e p r o o f .
i s a compact subset of a rnetrizable locally i s a regular inductive Z i m i t .
H(K)
i s m e t r i z a b l e it c o n t a i n s a c o u n t a b l e fundamental
E
for
K
If
convex space then
g
and e v e r y compact s u b s e t
d e f i n e t h e same e q u i v a l e n c e class i n
Proposition 2.55
Now suppose
O
neighbourhood system a t z e r o ,
(Vn)n,
and hence
\In)
H(K) = l i m (Hm(K+iTn),ll --f
n
i s a bornological
DF
space.
Hence each bounded s u b s e t o f
c o n t a i n e d i n t h e c l o s u r e o f a bounded s u b s e t o f some
is
Hm(K+Vn). S i n c e t h e
Hm(K+Vn) i s a l s o a c l o s e d s u b s e t o f
closed u n i t b a l l of
H(K)
H(K), by lemma
2 . 5 4 , t h i s completes t h e p r o o f A semi-norm c h a r a c t e r i z a t i o n o f t h e bounded s u b s e t s o f
H(K)
is given
i n t h e following p roposition Proposition 2.56
convex space
E.
K
Let A subset
be a compact subset of a rnetrizabze ZocalZy B
of
H(K)
i s bounded i f and onZy i f it
s a t i s f i e s each of the foZlowing conditions: ( a ) f o r each continuous semi-norm
(b) i f
03
( x ~ ) ~and = ~ (x;):=~ m
p
on
H(0)
are two convergent sequences i n
m
(ynInz1
and (yA)n=l are nu22 sequences i n E, (kn);=l a s t r i c t l y increasing sequence of p o s i t i v e integers and xn+yn
Proof
=
xA+y'
n
for a22
n
K, is
then
W e f i r s t show t h a t t h e semi-norms g i v e n i n
(*)
and
(**)
are
Chapter 2
88
continuous with r e s p e c t t o t h e i n d u c t i v e l i m i t topology
T
on
H(K).
Using t h e above n o t a t i o n w e l e t
f o r every
f
Since and
n
in (H(K)
i s a b a r r e l l e d l o c a l l y convex s p a c e and b o t h
T)
induce t h e
H(0)
T~
t o p o l o g y on
8
(nE)
H(K)
f o r each p o s i t i v e i n t e g e r
it s u f f i c e s t o show each o f t h e above semi-norms i s f i n i t e .
Let such t h a t
f
E
f
H(K
be a r b i t r a r y .
c a n be i d e n t i f i e d w i t h an element o f
M = sup lf(x+4y) XEK,YEV
Since
p
C(V) > 0
i s a c o n t i n u o u s semi-norm on
such t h a t
p(g)
n>N
o f zero
Hm(K+4V). L e t
(H(O),T)
t h e r e e x i s t s a constant
c C(V) / / g / / v f o r e v e r y g i n H(0).
Hence
is continuous.
(*)
Now choose a p o s i t i v e i n t e g e r For a l l
V
I .
and each semi-norm o f t h e form
n>N.
There e x i s t s a neighbourhood
N
such t h a t
yn
and
yn '
E
V
for a l l
89
Holomorphic mappings between locally convex spaces
and hence
Thus each semi-norm o f t h e form
set of
H(K)
i s c o n t i n u o u s and any bounded sub-
(**)
s a t i s f i e s c o n d i t i o n s ( a ) and ( b ) . i s a s u b s e t o f H(K) which s a t i s f i e s (fa)aar Using semi-norms o f t h e form (*) we see t h a t
Conversely, suppose ( a ) and ( b ) .
$"fa (x) (-
n!
1
a~y,x~Kn , arbitrary
i s a bounded s u b s e t o f
Since
H(0).
t h e r e e x i s t s a neighbourhood o f
f o r every If
a XEK,
in
r,
x
and
YEW
in aE.T
0
H(0)
in
and a l l
K
E,W,
i s r e g u l a r ( p r o p o s i t i o n 2.55) and
such t h a t
&O
n.
let
To complete t h e p r o o f , it s u f f i c e s t o show t h a t t h e r e e x i s t s a neighbourbood x,x'
V E
o f zero i n
K, y,yf E V
E,
with
VCW,
such t h a t
x+y = x ' + y '
f,(x)(y)
and e v e r y
a in
= fa(x')(y')
r.
for all
90
Chapter 2
If not, there exist sequences in E ,
xn+yn
such that
two sequences in m
(yn)n=l
and
(Y;);=l,
= x’+y’ f o r all
n
K,
n
n
m
03
( x ~ ) ~ =and ~
( x A ) ~ = ~ ,two null
and a sequence
(an):=1
in
r
and
Now choose inductivelz a strictly increasing sequence of positive integers m
(kn)n= 1
such that
2 nSn > n+2M
for all n.
Let
Since
(faIaEr
satisfies condition ( b ) ,
sup q(fa)
<
m.
c1
On the other hand,
This is a contradiction, and completes the proof Condition (a) says that bounded subsets of H(K)
satisfy Cauchy
K while condition (b) says that the Taylor series expansions
estimates on
satisfy certain coherence properties. If K
satisfies certain connected-
ness conditions, then condition (a) is sufficient. That some conditions on K are necessary for boundedness is shown by the following example. Example 2.57
Vn Let
=
Let
IzEE; z
=
E
= C.
x+iy, x <
F o r each positive integer
-1
E
H(V
n
u Wn)
let
and let tun = I z E C ; z=x+iy, x
n + -2
fn
n
be identically one on Vn
1 1 n+z
> __
and identically zero on Wn
91
Holomorphic mappings between locally convex spaces 1 K = { --} n ~ ul { O }
If
n.
for all
then
The sequence
i s a compact s u b s e t o f
K
{fn};=l
and
C
fn
E
H(K)
s a t i s f i e s c o n d i t i o n (a) o f p r o p o s i t i o n
2.56 b u t i s n o t bounded by p r o p o s i t i o n 2.55. We s h a l l u s e p r o p o s i t i o n 2 . 5 6 and i t s method o f p r o o f a g a i n i n a l a t e r
chapter.
The f o l l o w i n g r e s u l t f o l l o w s e a s i l y from t h e c o r r e s p o n d i n g r e s u l t
f o r holomorphic f u n c t i o n s on open s e t s .
a compact subset of a l o c a l l y convex space i s a cornpZete ZocaZly convex space, then ( 8 ( m E ; F ) , ~ , ) is a
P roposi t i on 2.58 E
and
F
K
If
is
closed complemented subspace o f H(K;F)
f o r every p o s i t i v e i n t e g e r
I n d i s c u s s i n g holomorphic f u n c t i o n s on a compact s u b s e t l o c a l l y convex s p a c e complete.
A
If not, l e t
E
b a l a n c e d open s u b s e t o f into
6
d e n o t e t h e completion o f
E.
If
of a
K
w e may always suppose t h a t t h e s p a c e
E
is
E
i s a convex
V
t h e n t h e n a t u r a l r e s t r i c t i o n from
m.
Hm(K+V)
H m ( K + V ~ E ) i s a b i j e c t i v e i s o m e t r y and hence t h e s p a c e s
H(KE)
H(KA) are isomorphic as l o c a l l y convex s p a c e s , ( K E ( r e s p e c t i v e l y E t h e s e t K c o n s i d e r e d as a s u b s e t o f E ( r e s p e c t i v e l y g ) ) .
Kg)
and is
We s h a l l a l s o need h y p o a n a l y t i c germs i n l a t e r c h a p t e r s and t h e d e f i n i t i o n s are e x a c t l y as one would e x p e c t .
If
K
u
i s a compact subset of a l o c a l l y convex space
ence r e l a t i o n s h i p f
N
and
v
by g
and
F
is a
HHY(V;F) 7')e define t h e equivalKCV, V open fNg i f there e x i s t s a neigkbourhood W of K
l o c a l l y convex space then on
on which both
E
are defined and coincide.
open
We l e t
/
and t h e elements of t h i s vector space are caZled F-valued hypoanaZytic germs
on K.
On
Yly(K;F)
we d e f i n e t h e l o c a l l y convex i n d u c t i v e l i m i t t o p o l o g y
W e have used an i n d u c t i v e l i m i t o f t o p o l o g y on
H(K;F).
T
topologies t o define t h e natural
I t i s n o t known i f t h e
as a p r o j e c t i v e l i m i t o f t h e s p a c e s
H(K;F).
T
topology can be recovered
T h i s problem i s r a t h e r impor-
t a n t and a p p e a r s t o b e i n t i m a t e l y r e l a t e d t o t h e l o c a l i z a t i o n problem
92
Chapter 2
is
mentioned p r e v i o u s l y :
T
a l o c a l topology?
w
i n q u i r y , w e c a n d e f i n e a "new'' t o p o l o g y on
a l o c a l l y convex s p a c e and T
.
D e f i n i t i o n 2.59 F
and
U
If
H(U;F),
a n open s u b s e t o f
U
a l o c a l l y convex s p a c e , which w e d e n o t e by
F
I t i s conjectured t h a t
Following t h i s l i n e o f
and
T~
always c o i n c i d e .
T
E
i s an open subset o f a locaZly convex space
i s a Zocally convex space then the
defined as the projective l i m i t o f
topoZogy on H(U;F)
T~
is
H(K;F)
where
on
H(U;F).
I n c h a p t e r s i x , we s h a l l
and
coincide.
K
ranges over the
compact subsets of U. One e a s i l y sees t h a t
T ~ '
~
TS
g i v e examples o f s i t u a t i o n s i n which
52.5
EXERCISES
2.60
Show t h a t
HG(U;F)
dimensional space E .z; (E"). 2.61*
T
w
f o r any open s u b s e t
and a n y l o c a l l y convex s p a c e
E
Show t h a t
H(U;F)
=
T
(E,tf)
U
o f an i n f i n i t e
i f and o n l y i f
F
i s a t o p o l o g i c a l v e c t o r s p a c e i f and o n l y i f
E
has a countable algebraic b a s i s . 2.62 E
onto
If
and
E
TI : ( E , t f )
2.63
If
f o r any 2.64* E.
f o r every 2.65*
@
i s a l i n e a r mapping from
fl
Let
E
i s a n open mapping.
a r e v e c t o r s p a c e s and
E
E
and
A
E
Show t h a t
f
E
it(mE;F)
show t h a t
n6m.
b e Banach s p a c e s and l e t
F
HG(U;Fi). in
(F,tf)
y1 ,..., yn
and
E E
---t
F
E,
in
Let f
and
E
x
Let
a r e v e c t o r s p a c e s and
F
show t h a t
F
H(U;F;)
U
b e a n open s u b s e t o f
i f and o n l y i f
@ o f EH(U)
F. and
F
b e Banach s p a c e s and
U
a connected open s u b s e t
93
Holomorphic mappings between locally convex spaces of
E.
2.66 -
E.
If
by
f(U)
f : U
Let
f(V) C F '
Let f
2.67* -
F*
-f
and suppose
@
0
f EH(U)
f o r some nonempty open s u b s e t
H(U;F)
E
F
and
E
f o r every
of
V
b e Banach s p a c e s and l e t
Let
show t h a t t h e c l o s e d v e c t o r s u b s p a c e o f
and
E
F
b e Banach s p a c e s .
such t h a t
f(Vx)
F
A mapping
generated
F
g e n e r a t e d by
is said
f : E-tF
i f t h e r e e x i s t s a neighbourhood
x
and
F
f EH(U,; Fb).
b e a n open s u b s e t o f
U
i s equal t o t h e c l o s e d v e c t o r subspace o f
t o b e compact a t t h e p o i n t
in
@
Show t h a t
U.
i s a r e l a t i v e l y compact s u b s e t o f
F.
If
of
Vx
f
x
H(E;F)
E
show t h a t t h e f o l l o w i n g are e q u i v a l e n t : f
i s compact a t e v e r y p o i n t o f
(b)
f
i s compact a t some p o i n t o f
(c)
t h e r e e x i s t s a compact s u b s e t
(a)
f(E)
E, E,
K
of
F
such t h a t
i s c o n t a i n e d i n t h e v e c t o r s p a c e spanned by
K,
(d)
t h e t r a n s p o s e mapping
2.68* Suppose
Let fn
f * ( @ ) = $of)
dnf(0)
i s compact f o r a l l
A
(e)
E
=
H(En)
E
f*; (F',To)
(where
m
where each
En
f o r each
n
I:=,
(H(E),ro)
n.
U
i s a l o c a l l y convex s p a c e .
En
and l e t
If t h e r e e x i s t s a neighbourhood
-f
i s continuous,
gn
E
o f zero i n
8 E
Ek)
such t h a t
f o r each
n.
Ilgnllu
+
for all
n
2.69
Show t h a t , t h e c o m p o s i t i o n o f holomorphic ( r e s p e c t i v e l y
show t h a t
F =
gnfnE
c
m
H(E).
h y p o a n a l y t i c ) f u n c t i o n s i s holomorphic ( r e s p e c t i v e l y
M-holomorphic,
M-holomorphic, h y p o a n a l y t i c ) .
2.70
If
i s a l o c a l l y bounded holomorphic f u n c t i o n ,
f
c o n t i n u o u s holomorphic f u n c t i o n and
gof
i s a Mackey
g
i s d e f i n e d , show t h a t
gof
is a
( c o n t i n u o u s ) holomorphic f u n c t i o n . 2.71 -
Let
U
be a n open s u b s e t o f a l o c a l l y convex s p a c e
b e a l o c a l l y convex s p a c e .
If
f
HG(U;F)
and e i t h e r
E
or
and l e t
E F
is
F
Chapter 2
94
separable, show t h a t
f
HHY(U;F)
E
if
gof
E
H(U)
f
HG(E) show t h a t
If
E
i s a Frgchet space and
and only i f
f
i s a Bore1 measurable f u n c t i o n .
2.72*
E
f o r every
g f
in
H(F).
H(E)
E
if
2.73* L e t U be an open subset o f a Banach space E and l e t F be a Banach space with a b a s i s , I f f i s a mapping from U i n t o F
then
f(x)
The mapping of
m
=
f
fn(x)en
where
fn : U
+
C.
i s s a i d t o be normal i f , f o r each compact subset
U
and each p o s i t i v e
fn
E
6,
there exists a positive integer
n(K)
K
such
that
If
H(U)
n
for a l l
show t h a t
f
H(U;F)
E
i f and only i f
f
is a
normal mapping. Let
__ 2.74*
in
be a l o c a l l y convex space and l e t
E
(a)
If
E
i s a Frgchet space o r a83fLvl space show t h a t
l z = l ( $ n y ~ H ( E ) i f and only i f f o r every (b)
If of
x
in
($n)n
show t h a t
bounded s e t s
8
such t h a t
i s a connected
being
as
E
H(E)
n-tm
i f and
contains a fundamental system of
E
( t h e vector span o f
EB B
in 6
HG(U;F)
in
f
f
E
%(U;F).
normed by t h e gauge E
is
TM
complete.
complete l o c a l l y convex show t h a t
f
E
%(U;F)
and every p o s i t i v e i n t e g e r
complete l o c a l l y convex spaces, and
U
is a
U
If
continuous show t h a t
where
T!,
f c HG(U;F)
T~
E
B
then we say t h a t
open subset of t h e
T
space E , F i s a Banach space and if 'dhf ( x ) E 6 M ( n E ; F ) f o r some x 2.76*
0
i s a n u l l sequence and t h e vector span ($n)n i s f i n i t e dimensional.
of B ) i s complete f o r every
F
+
1"n=l( @n )"
I f t h e l o c a l l y convex space
2.75* -
U
@,(x)
E.
E = (E,o(E,E'))
only i f
If
($n)n be a sequence
E'.
T~
open subset of f
ExF, E
n. and
i s s e p a r a t e l y Mackey
95
Holomoiphic mappings between locally convex spaces __ 2.77
If
i s an open s u b s e t of a l o c a l l y convex s p a c e
U
l o c a l l y convex s p a c e , show t h a t f o r each
f
in
HG(U;F)
E
and
is a
F
t h e following are
equivalent: (a)
f E H(U;F)
(b)
f o r each
(respectively in
5
HLB(U;F)),
U
is a n e q u i c o n t i n u o u s ( r e s p e c t i v e l y l o c a l l y bounded) f a m i l y of
mappings. 2.78 -
Let
b e a Banach s p a c e and l e t
E
sequences i n
2.79 -
If
(WE) ,
T~)
f
H( II Ea)
E
where each
If
J
fon
and &(Y), that
If
m
(0,)
n
.
of
A1
i s a l o c a l l y convex s p a c e , show A
and
II
%(afAEa) __ 2.80*
L(($n)n) =
f
i s t h e n a t u r a l p r o j e c t i o n from A1 i s m e t r i z a b l e f o r each a i n A show t h a t
where
A1
Ea
X
E
H( JI Ea) a d1
TI Ea a EA
such t h a t
onto
aZAIEa.
=
and
Y
a r e c o m p l e t e l y r e g u l a r Hausdorff s p a c e s
and
& , (X)
each w i t h t h e compact open t o p o l o g y , are i n f r a b a r r e l l e d , show HLB(,&(X); QCY);)
2.81 __
Ea
~ E A
t h a t there e x i s t s a f i n i t e subset f =
b e d e f i n e d by
i s w e l l d e f i n e d and holomorphic.
L
Show t h a t
-
be t h e s p a c e o f a l l n u l l
endowed w i t h t h e sup norm t o p o l o g y .
E'
L : co(E')
Let
co(E')
Show t h a t
and any open s u b s e t
=
H(U;F)
=
of
& ,
U
H(A(x);
,&(Y)~l-
H (U;F) f o r any l o c a l l y convex s p a c e F HY (X) where X i s a paracompact t o p o l o g i c a l
space. __ 2.82
Show t h a t
any open s u b s e t
U
H(U;F) = %(U;F) of
@,
(X)
where
f o r any l o c a l l y convex s p a c e X
i s a Lindel'df s p a c e .
F
and
96
Chapter 2
If
2.83*
r
is an uncountable discrete set and F
is a separable
Banach space, show that
where the topology on co(r)
is given by the sup norm and
r 'cr T ' countable 2.84* -
F
Let U be a
T~~
open subset of a locally convex space and let
be a locally convex space. Let
T~
denote the topology on $(U;F)
of uniform convergence on the finite dimensional compact subsets of U.
is metrizable if and only if E
(%(U;F),rF)
contains
a countable fundamental system of finite dimensional compact sets and F is metrizable. Show that $(U;F)
T~
induces on each
T~
bounded subset of
the topology of pointwise convergence.
Show that
(%(U;F),T~) is a semi-Monte1 space if and only if F is a semi-Monte1 space. Show that ($(U;F); T ~ ) is quasi-complete if and only if F is quasi-complete. is an arbitrary set, show that the bounded subsets of locally bounded. 2.86 -
If E
is a complete locally convex space and the
subsets of H(E 2.87*
the T~ U of E.
Let E and T
are locally bounded, show that E =
T~
bounded
is barrelled.
En where each En is a Banach space. Show that bounded subsets of H(U) coincide for every open subset
We say that a countable increasing cover 3 = (Fn)n=l of a topological space U is k-dominating if each F, is closed and each compact subset of U is contained in some 'n. Let U be an open subset of 2.88*
50
97
Holomorphic mappings between locally convex spaces a l o c a l l y convex s p a c e all
n}.
H
"
(U)
t h e semi-norms
E
and l e t
Hg(U)
I f cHG(U);
=
llfllF<
pn(f) =
Ildg .
Let
n
(H(U),T
6K
)
l i m H3(U)
=
2+
r a n g e s o v e r a l l c o u n t a b l e i n c r e a s i n g k-dominating c o v e r s o f T
6K
is t h e bornological topology a s s o c i a t e d w ith
(H(U),T
i s a n u l t r a - b o r n o l o g i c a l space i f
)
6K
-
for
i s endowed w i t h t h e l o c a l l y convex t o p o l o g y z e n e r a t e d by
U
T
0
on
U.
where
.3
Show t h a t
H(U).
Show t h a t
i s a k-space.
2.89 On H(U), U a n open s u b s e t o f a l o c a l l y convex s p a c e , l e t T P d e n o t e t h e t o p o l o g y o f p o i n t w i s e convergence. I f = (Fn):=l is a
3
countable i n c r e a s i n g cover o f
c o n s i s t i n g o f c l o s e d sets and
U
H3(U)
is
t h e a s s o c i a t e d m e t r i z a b l e s p a c e o f holomorphic f u n c t i o n s , show t h a t
is t h e bornological space associated with
l i m H 3(U)
(H(U) , T ~ ) . (
3
-3+ r a n g e s o v e r a l l p o s s i b l e c o v e r s o f t h e above t y p e ) .
If
2.90 and
i s a n open s u b s e t o f a m e t r i z a b l e l o c a l l y convex s p a c e
U
i s a Banach s p a c e , show t h a t
F
E
(H(U;F),T&) i s a r e g u l a r i n d u c t i v e
l i m i t of Frgchet spaces. If t h e l o c a l l y convex s p a c e
2.91*
c o n v e r g e n t sequence, show t h a t
Ei
E
contains a nontrivial very strongly
c o n t a i n s a sequence which i s n o t v e r y
weakly c o n v e r g e n t . __ 2.92
If
i s a s e p a r a b l e l o c a l l y convex s p a c e i n which e v e r y sequence
E
i s v e r y weakly c o n v e r g e n t , show t h a t 2.93
Let
E
be a F r g c h e t s p a c e .
EL
a d m i t s a c o n t i n u o u s norm.
Show t h a t t h e f o l l o w i n g are e q u i v a l -
ent : (a)
E
c o n t a i n s a n o n t r i v i a l v e r y s t r o n g l y c o n v e r g e n t sequence,
(b)
E
c o n t a i n s a s u b s p a c e isomorphic t o
(c) (d)
i s a quotient of
CN,
E'
a'
e v e r y c o n t i n u o u s semi-norm on
E
has an i n f i n i t e dimensional
kernel. Show t h a t ( a ) and (d) a r e n o t e q u i v a l e n t f o r a r b i t r a r y l o c a l l y convex spaces.
Chapter 2
98
Let
2.94*
~
Let
E,
E A
{xgE;
=
3 (x~):=~ E
.
c o n v e r g e n t sequence}. f
E
A
b e a m e t r i z a b l e l o c a l l y convex s p a c e w i t h completion
E
such t h a t
i s a v e c t o r subspace o f
E,
Show t h a t
E.
( x ~ - x ) i~s a v e r y s t r o n g l y
-
A
E.
If
show, by u s i n g t h e f a c t o r i z a t i o n lemma o r o t h e r w i s e , t h a t t h e r e
H(E),
d
e x i s t s a unique
f
in
H(E,)
such t h a t
flE = f .
Generalise t h i s r e s u l t
t o a r b i t r a r y l o c a l l y convex s p a c e s .
~
Let
2.95 (H(U),-ro)
b e a n open s u b s e t of a l o c a l l y convex s p a c e .
U
i s a l o c a l l y m-convex a l g e b r a .
m u l t i p l i c a t i v e semi-norm on
~
Let
2.96*
f
Let
E
H(U;F).
F-valued holomorphic germs a t rJ
f(x)
show t h a t
N
f
If
2.97"
T~
Show t h a t e v e r y
=
{germ o f
If in
0 f
?: E)
at
T~
continuous
continuous.
be a n open s u b s e t of a Banach s p a c e
U
a Banach s p a c e .
is
H(U)
Show t h a t
U
-f
E
H(0;F)
F
and l e t
be
( t h e s p a c e of
i s d e f i n e d by
x
translated t o the origin)
HG(U;H(O;F)). i s a compact s u b s e t o f a l o c a l l y convex s p a c e
K
i s a l o c a l l y convex s p a c e , show t h a t
H(K;F)
and
E
F
i s a Hausdorff l o c a l l y convex
space. Show t h a t
__ 2.98*
H(K)
i s n e v e r a s t r i c t i n d u c t i v e l i m i t when
is a
I<
compact s u b s e t of a l o c a l l y convex s p a c e . If
__ 2.99*
space
E
neighbourhood
f o r every 2.100*
i s a convex b a l a n c e d s u b s e t of a m e t r i z a b l e l o c a l l y convex
K
show t h a t
5 If
V
in U
B CH(K)
i s bounded i f and o n l y i f t h e r e e x i s t
of z e r o and two p o s i t i v e numbers
K
and e v e r y n o n - n e g a t i v e i n t e g e r
c
and
C
2.101"
F' B
'2
If
such t h a t
m.
i s a n open s u b s e t of a Banach s p a c e , show t h a t
a d u a l Banach s p a c e ( i . e . show t h a t t h e r e e x i s t s a Banach s p a c e that
a
Hm(U)
F
is
such
Hm(U)).
K
i s a compact s u b s e t o f a m e t r i z a b l e l o c a l l y convex s p a c e ,
99
Holomorphic mappings between locally convex spaces show t h a t
m -convex a l g e b r a .
is a l o c a l l y
H(K)
Show t h a t a convex b a l a n c e d s u b s e t of
2.102
o f a l o c a l l y convex s p a c e
is a
E,
T&
i f it a b s o r b s e v e r y e q u i c o n t i n u o u s s u b s e t o f 2.103*
Let
integer
n
rn
let
an open s u b s e t
U
H(U).
b e a l o c a l l y convex space.
E
H(U),
neighbourhood o f z e r o i f and o n l y
d e n o t e t h e t o p o l o g y on
For each n o n - n e g a t i v e g e n e r a t e d by t h e semi-
H(E)
norms
where
K
r a n g e s o v e r t h e compact s u b s e t s o f
bounded s u b s e t s o f
B
ranges over t h e
E.
rn =
Show t h a t
and
E
T
~
+f o r~
some (and hence f o r a l l )
n
i f and o n l y i f
each bounded s u b s e t o f
E
i s c o n t a i n e d i n t h e c l o s e d convex h u l l o f a
compact s e t . A subset
2.104"
K
of a l o c a l l y convex s p a c e
i s s a i d t o be s t r i c t l y
E
compact i f t h e r e e x i s t s a convex b a l a n c e d bounded s u b s e t that
K
i s c o n t a i n e d and compact i n
EB.
On
%(E)
B
let
of rS
E
t o p o l o g y o f uniform convergence on t h e s t r i c t l y compact s u b s e t s of Show t h a t
(%(E),rS)
2.105"
If
f
t h e r e e x i s t s a neighbourhood
52.6
E.
i s a semi-Monte1 s p a c e .
i s a n e n t i r e f u n c t i o n on t h e
every p o s i t i v e i n t e g e r
such
denote t h e
n.
V
a3w
o f z e r o such t h a t
space
/If
lLv
E,
show t h a t
is f i n i t e for
NOTES AND RtPIARKS
The h i s t o r i c a l n o t e s o f A . E . T a y l o r [680] a r e p r o b a b l y t h e most comprehensive g u i d e t o t h e development o f i n f i n i t e dimensional holomorphy up t h e mid-nineteen f o r t i e s a v a i l a b l e and w e have made e x t e n s i v e u s e o f t h e s e n o t e s i n 11.6 and s h a l l do s o a g a i n i n t h i s s e c t i o n .
In another
p a p e r , [681], A . E . T a y l o r documents t h e r o l e of a n a l y t i c i t y i n o p e r a t o r and s p e c t r a l t h e o r y and h i s h i s t o r y of t h e d i f f e r e n t i a l i n t h e n i n e t e e n t h
Chapter 2
100
and t w e n t i e t h c e n t u r i e s [682] c o n t a i n s much o f i n t e r e s t c o n c e r n i n g t h e growth o f i n f i n i t e d i m e n s i o n a l a n a l y s i s .
E . H i l l e and R.S. P h i l l i p s ( [ 3 3 4 ] ,
s e c t i o n 3.16 and c h a p t e r 26) a l s o p r o v i d e a summary o f t h e fundamentals known a t t h a t t i m e
-
1958
-
t o g e t h e r w i t h a s h o r t b i b l i o g r a p h y on t h e sub-
j e c t ( c h a p t e r 26 was w r i t t e n i n c o l l a b o r a t i o n w i t h M.A.
Zorn).
D. Pisanelli
[578] g i v e s a s u r v e y o f most o f t h e d i f f e r e n t c o n c e p t s o f holomorphic f u n c t i o n t h a t are c u r r e n t l y i n u s e and documents t h e i r o r i g i n . V.I. Averbukh and O . G . Smolyanov ( [ 4 0 ] , § 2 ) g i v e a comprehensive a c c o u n t of t h e development o f t h e concept of t h e d i f f e r e n t i a l i n t o p o l o g i c a l v e c t o r spaces.
They a r e m a i n l y concerned with the r e a l t h e o r y b u t n a t u r -
a l l y t h i s h a s many consequences f o r t h e complex c a s e .
An i n t e r e s t i n g p o i n t
t h a t emerges from [40] i s t h a t t h e s e a r c h f o r a d e f i n i t i o n o f d i f f e r e n t i a b l e f u n c t i o n between r e a l l i n e a r t o p o l o g i c a l v e c t o r s p a c e s h a s been i n p r o g r e s s more o r l e s s c o n t i n u o u s l y f o r t h e l a s t s e v e n t y y e a r s w i t h many d i f f e r e n t d e f i n i t i o n s b e i n g proposed - t h e a u t h o r s l i s t t w e n t y - f i v e i n [40] - and s i n c e a number o f d e f i n i t i o n s were n o t s e e n t o b e e q u i v a l e n t and i n d e e d , v a r i o u s a u t h o r s were n o t aware o f one a n o t h e r ' s work, t h i s l e d t o a c e r t a i n
a mount o f chaos - o r a t l e a s t a p p a r e n t chaos - which h a s o n l y been r e c t i f i e d i n [40].
F o r t u n a t e l y f o r d i f f e r e n t i a t i o n o v e r complex s p a c e s we
have power series e x p a n s i o n s and t h i s h a s l e d t o a f a i r l y unanimous a c c e p t a n c e o f Frgchet d i f f e r e n t i a b i l i t y as t h e s t a n d a r d d e f i n i t i o n w i t h minor r o l e s b e i n g p l a y e d by GGteaux holomorphic, Mackey o r S i l v a holomorp h i c and h y p o a n a l y t i c f u n c t i o n s . The names c h i e f l y a s s o c i a t e d w i t h t h e development o f r e a l i n f i n i t e d i m e n s i o n a l d i f f e r e n t i a l c a l c u l u s a r e ( s e e [40] f o r more d e t a i l e d i n f o r mation) V . V o l t e r r a , J . Hadamard, M . F r g c h e t , R . Ggteaux, P . L&y,
A.D.
Michal, L . M . Graves, E.W. Paxson, J . G i l d e Lamadrid, J . Sebasti'go e S i l v a , H.H.
Keller and G . Marinescu.
I n p a r t i c u l a r , we would l i k e t o mention
J . Hadamard, M. F r g c h e t and A . D . Michal.
Hadamard was one o f t h e f i r s t t o
r e c o g n i s e t h e importance o f V o l t e r r a ' s work, h e e x e r c i s e d a d e e p i n f l u e n c e
on h i s s t u d e n t s FrGchet and Gsteaux and was always v e r y i n s i s t a n t on l i n e a r i t y b e i n g i n c o r p o r a t e d i n t o any d e f i n i t i o n o f t h e d e r i v a t i v e .
M . F r g c h e t and A . D . Michal b o t h s p e n t a c o n s i d e r a b l e p o r t i o n of t h e i r l i v e s developing t h e t h e o r y
-
r o u g h l y f o r t y y e a r s by F r g c h e t - and opened many
avenues s t i l l being explored. b e found i n P . L&y
([442],
Readable a c c o u n t s o f t h e r e a l t h e o r y a r e t o
b o t h t h e 1922 and 1950 e d i t i o n s ) , A . D . Michal
101
Holomoiphic mappings between locally convex spaces [491], V.I. Averbukh and O . G .
Smolyanov [39], J . P . Penot [567], A.
Frb'hlicher and W . Bucher [247] and S. Yamamuro [718]. Surveys of a more s p e c i a l i s e d n a t u r e but which a r e more r e l e v a n t t o t h e t o p i c s discussed i n t h i s book a r e L . Nachbin [518], M . Schottenloher B i e r s t e d t and R . Meise [70], P . Krce [405] and S . Dineen [198].
[642], K.D.
We now p r e s e n t our b r i e f h i s t o r y . We have a l r e a d y noted i n 51.6 t h a t t h e d e f i n i t i v e s t e p s i n t h e c r e a t i o n of r e a l and complex i n f i n i t e dimensional d i f f e r e n t i a l c a l c u l u s were taken by V . V o l t e r r a (1887) and D . H i l b e r t (1909) r e s p e c t i v e l y . P r i o r t o t h e s e , t h e works of J . Bernoulli on t h e curve of quickest descent, L . Euler on t h e c a l c u l u s of v a r i a t i o n s (see J . Hadamard [298]) and H. Von
Koch [393], on i n f i n i t e systems of d i f f e r e n t i a l equations may a l s o be mentioned a s of p a r t i c u l a r importance f o r l a t e r developments.
D. Hilbert
[332] discussed a n a l y t i c continuation and t h e composition of holomorphic f u n c t i o n s but h i s work i s not easy t o read, and a t times r a t h e r vague. Next came R. G2teaux [251,252,253] who looked a t both r e a l and complex analysis
-
a s opposed t o M. Frgchet who only s t u d i e d t h e r e a l c a s e
s e t out t o c l a r i f y and extend t h e works of Frgchet and H i l b e r t . he succeeded b r i l l i a n t l y .
-
and
In t h i s
I n [252], he defined complex polynomials and
n o t i c e d t h e i r r e l a t i o n s h i p with b i l i n e a r and q u a d r a t i c forms, defined t h e "G$teaux" d e r i v a t i v e , extended t h e Cauchy i n t e g r a l formula and t h e Cauchy i n e q u a l i t i e s , proved an i d e n t i t y theorem and v a r i o u s convergence theorems f o r holomorphic functions, e s t a b l i s h e d t h e now standard correspondence between d e r i v a t i v e s of a holomorphic function and t h e homogeneous polynomials i n i t s Taylor series expansion and gave various r e s u l t s about a n a l y t i c continuation and power s e r i e s expansions - a l l i n an i n f i n i t e dimensional s e t t i n g .
H i s o t h e r paper [253] i s a l s o q u i t e i n t e r e s t i n g and
contains b r i e f o u t l i n e s of a number of t o p i c s which h i s untimely death prevented him from developing.
CN,
&[a,b]
and
G2teauxIs work was confined t o t h e spaces
.Q2.
In 1920, S . Banach gave, i n h i s t h e s i s , t h e axioms f o r a complete normed l i n e a r space (a Banach space) and i n 1923, N.Wiener
[715], who
i n c i d e n t a l l y gave independently t h e same axioms t h r e e months a f t e r Banach,
102
Chapter 2
n o t i c e d t h a t t h e Cauchy i n t e g r a l formula e x t e n d s t o Banach v a l u e d holomorp h i c f u n c t i o n s o f one complex v a r i a b l e and i n t h i s s e t t i n g , many o f t h e c l a s s i c a l r e s u l t s such as M o r e r a ' s theorem, A b e l ' s theorem and t h e r e s i d u e theorem are a l s o v a l i d .
L . FantappiGIs long a r t i c l e [234] appeared i n 1930
I n i t h e proposed t h a t a c o n t i n u o u s f u n c t i o n aBanach s p a c e be c a l l e d holomorphic i f
f
d e f i n e d on a domain
f o $ i s holomorphic ( a s a f u n c t i o n
o f one complex v a r i a b l e ) f o r a n y holomorphic f u n c t i o n @ t h e complex p l a n e i n t o
D.
in
D
from a domain i n
F a n t a p p i g ' s work s e r v e d as m o t i v a t i o n f o r A . E .
T a y l o r 116751 and was developed and p u t i n a modern c o n t e x t u s i n g t o p o l o g i c a l v e c t o r s p a c e s by J . S e b a s t i r o e S i l v a [649,650,651,652,653]
(see a l s o
D. P i s a n e l l i [578]).
I n t h e n i n e t e e n t h i r t i e s , an i n t e n s i v e s t u d y o f t h e whole f i e l d o f a n a l y s i s a n d geometry i n a b s t r a c t s p a c e s was c a r r i e d o u t by A . D . Michal and h i s s t u d e n t s R . S . M a r t i n , A . H . C l i f f o r d , I . E . Highberg and A . E . T a y l o r .
In
1932, R . S . M a r t i n [449] developed t h e t h e o r y o f holomorphic mappings f o r Banach s p a c e s u s i n g a power series approach ( s e e a l s o A . D . Michal and A.H. C l i f f o r d [492]).
The f i n a l s t e p towards t h e c u r r e n t d e f i n i t i o n of holomor-
p h i c mapping between Banach s p a c e s was t a k e n i n d e p e n d e n t l y by L.M. Graves i n 1935 [275] and A . E . b r i e f , p.649-653,
T a y l o r i n 1937 [ 6 7 5 ] .
Graves' t r e a t m e n t i s r a t h e r
and i n it h e n o t e s t h a t Ggteaux d i f f e r e n t i a b i l i t y p l u s
c o n t i n u i t y are e q u i v a l e n t t o F r g c h e t d i f f e r e n t i a b i l i t y ( d e f i n e d f o r normed l i n e a r s p a c e s by F r g c h e t i n 1925, "2451) when d e a l i n g w i t h f u n c t i o n s between complex normed l i n e a r s p a c e s and a l s o t h a t t h i s l e a d s t o a s a t i s f a c t o r y t h e o r y o f holomorphic f u n c t i o n s between normed l i n e a r s p a c e s . T a y l o r ' s work i s much more e x p l i c i t .
H e d e f i n e s a holomorphic f u n c t i o n as
a c o n t i n u o u s f u n c t i o n whose one d i m e n s i o n a l s e c t i o n s are holomorphic.
This
d e f i n i t i o n i s a n a t u r a l outgrowth o f t h e work o f G2teaux and b r i n g s t o g e t h e r t h e i d e a s o f many o f h i s p r e d e c e s s o r s .
H e proved t h e f o l l o w i n g
result: If
f
i s a mapping between normed l i n e a r s p a c e s , t h e n t h e f o l l o w i n g
are equivalent a)
f
i s c o n t i n u o u s and h a s a Ggteaux d e r i v a t i v e a t each
point; b)
f
h a s a F r g c h e t d e r i v a t i v e a t each p o i n t ;
c)
f
h a s a power s e r i e s expansion which converges
103
Holomorphir mappings between locally convex spaces u n i f o r m l y i n a neighbourhood o f each p o i n t . T a y l o r goes on t o g e n e r a l i s e Riemann's theorem on removable s i n g u l a r i t i e s , M i t t a g - L e f f l e r ' s theorem, L i o u v i l l e ' s theorem and t h e CauchyRiemann e q u a t i o n s .
He a l s o shows [675,p.292] an awareness o f t h e f a c t t h a t
t h e r a d i u s o f uniform convergence i s n o t e q u i v a l e n t t o t h e r a d i u s o f a n a l y t i c i t y and t h i s i s t h e f i r s t s t e p i n a n e x t e n s i v e s t u d y o f t h e r a d i u s o f boundedness which w e u n d e r t a k e i n c h a p t e r 4 .
I n a f u r t h e r p a p e r [678],
A . E . T a y l o r d i s c u s s e s a number o f i n t e r e s t i n g examples o f holomorphic
f u n c t i o n s on
s p a c e s and g i v e s a g e n e r a l i z a t i o n of H a r t o g s ' theorem on
kP
separate analyticity.
The P o l i s h s c h o o l o f f u n c t i o n a l a n a l y s i s (S. Banach,
S. Mazur, W . O r l i c z , e t c . ) a l s o c o n t r i b u t e d d u r i n g t h i s p e r i o d ( s e e c h a p t e r 1) and a l t h o u g h t h i s t r a d i t i o n a l
interest
i n non-linear functional
a n a l y s i s was m a i n t a i n e d u n t i l r e l a t i v e l y r e c e n t l y ( s e e A . Alexiewicz and W . O r l i c z [ 7 ] , W .
[563,564,565]),
f o r instance
Bogdanowicz [76] and A . P e l c z y n s k i
it was l a r g e l y overshadowed by t h e r a p i d development of
t h e l i n e a r theory.
T h i s predominance o f l i n e a r f u n c t i o n a l a n a l y s i s may b e
t r a c e d t o t h e i n f l u e n c e o f S. Banach's book [44], p u b l i s h e d i n 1932, a l t h o u g h Banach h i m s e l f hoped t o d e v e l o p t h e n o n - l i n e a r t h e o r y i f we a r e t o j u d g e from h i s comment [ 4 4 , p . 2 3 1 ] on complex v e c t o r s p a c e s
"Ces espaces c o n s t i t u e n t Ze point de de'part de l a the'orie des op&ations
Zingaires cornpZexes e t d ' m e cZasse, encore pZus
vaste, des op6rations anazytiques, qui prgsentent une g'enerali s a t i o n des fonctions anaZytiques ordinaires icf. p . ex. L . Fantappi;,
I . funz?:onaZi a n a l i t i c i , G i t t a d i Caste220 1 9 3 0 ) .
Nous m u s proposons d 'en exposer Za thebrie dans un autre vo ZTme
".
See a l s o t h e p r e f a c e t o [ 4 4 ] . M . A . Zorn [723,724,725] made a number o f i m p o r t a n t c o n t r i b u t i o n s i n
t h e m i d - f o r t i e s and w e refer t o h i s r e s u l t s l a t e r i n t h e t e x t .
The n i n e -
t e e n f i f t i e s saw t h e a p p e a r a n c e of A . G r o t h e n d i e c k ' s memoir [287] on t o p o l o g i c a l t e n s o r p r o d u c t s and n u c l e a r s p a c e s and t h i s h a s had and w i l l c o n t i n u e t o h a v e a c o n s i d e r a b l e i n f l u e n c e on t h e development o f i n f i n i t e d i m e n s i o n a l holomorphy.
During t h i s p e r i o d , J . S e b a s t i z e S i l v a a l s o
developed h i s t h e o r y o f i n f i n i t e d i m e n s i o n a l holomorphy and H . J .
Bremermann
[103,104,105] proved v a r i o u s r e s u l t s on pseudo-convex domains and t u b u l a r
104
Chapter 2
domains o f holomorphy i n Banach s p a c e s . T h i s b r i n g s u s t o t h e modern p e r i o d , t h e l a s t s i x t e e n y e a r s , d u r i n g which time most o f t h e work we d i s c u s s h a s been d i s c o v e r e d .
The con-
t r i b u t i o n s o f t h e various authors during t h i s period are d e t a i l e d i n t h e f i n a l s e c t i o n o f each c h a p t e r . The concept o f Mackey holomorphic f u n c t i o n i s u s u a l l y a t t r i b u t e d t o J . S e b a s t i a o e S i l v a [652], a l t h o u g h an e a r l i e r e q u i v a l e n t d e f i n i t i o n (con-
d i t i o n 2 . 1 8 ( d ) ) i s due t o L . F a n t a p p i g [234].
The t h e o r y o f Mackey holo-
morphic f u n c t i o n s f i n d s i t s most n a t u r a l e x p r e s s i o n w i t h i n t h e language of b o r n o l o g i e s - s e e f o r i n s t a n c e D . P i s a n e l l i [578], D . Lazet [423] and J . F . Colombeau [141]. E . H i l l e [333].
L o c a l l y bounded holomorphic f u n c t i o n s were i n t r o d u c e d by I t i s d i f f i c u l t t o locate t h e o r i g i n of hypoanaliticity
( d e f i n i t i o n 2.15) b u t t h e name u s u a l l y a s s o c i a t e d w i t h t h e r e a l analogueh y p o c o n t i n u i t y i s J . L . K e l l e y who a l s o i n t r o d u c e d t h e concept of k-space ( s e e , f o r i n s t a n c e , S. W i l l a r d [716] o r Z . Semadeni [654]).
Examples 2.16,
2.19 and 2 . 2 0 t o g e t h e r w i t h f u r t h e r examples o f a similar k i n d are g i v e n i n S. Dineen [190] ( s e e a l s o c h a p t e r 5 ) .
The c o n n e c t i o n between H a r t o g s '
theorem and l o c a l l y bounded holomorphic f u n c t i o n s ( p r o p o s i t i o n 2 . 1 2 , examples 2 . 1 3 and 2.14) a p p e a r s i n [190] and i n f i n i t e dimensional g e n e r a l i z a t i o n s o f H a r t o g s ' theorem a r e d i s c u s s e d i n t h e comments on e x e r c i s e 2.76. F a c t o r i z a t i o n r e s u l t s (most o f which depend on L i o u v i l l e ' s theorem c o n c e r n i n g bounded e n t i r e f u n c t i o n s ) are i m p l i c i t i n A. Hirschowitz [335],
C . E . R i c k a r t [605] and L . Nachbin [514]. S. Dineen [188,189,190,191]
and
E . Ligocka [443], developed i n d e p e n d e n t l y a g e n e r a l t h e o r y o f f a c t o r i z a t i o n
which we d i s c u s s i n more d e t a i l i n 16.2 and a p p l i e d i t t o such t o p i c s as t h e Levi problem, RungeS theorem, H a r t o g s ' theorem, holomorphic c o m p l e t i o n , Zorn's theorem and t o p o l o g i e s on s p a c e s o f holomorphic f u n c t i o n s .
Factor-
i z a t i o n methods h a v e a l s o been a p p l i e d t o t h e Levi problem by Ph. Noverraz [540,544,546],
S. Dineen [183,186],
B. J o s e f s o n [358], M . S c h o t t e n l o h e r
[640], J . F . Colombeau and J . Mujica [154], J . Mujica [506],to t h e c o n s t r u c t i o n o f t h e envelope o f holomorphy by M . S c h o t t e n l o h e r [640] and P. Berner [59,60], i n s t u d y i n g meromorphic f u n c t i o n s and t h e Cousin I problem by V . Aurich
[35] and S. Dineen [192], t o t h e t h e o r y of c o n v o l u t i o n o p e r a t o r s
by P. Berner [ 6 2 ] and M.C. Matos [463,467] and i n s t u d y i n g l o c a l l y convex t o p o l o g i e s on s p a c e s of holomorphic f u n c t i o n s by P . Berner [61], S. Dineen
105
Holomotphic mappings between locally convex spaces
[194], and Ph. Noverraz [552]. Theorem 2 . 2 1 and proposition 2.33 are given in [190;. Example2.22and proposition 2.24 aredue to L. Nachbin [514] and a generalization can be found in S. Dineen [190]. subset U
in (EN and
A. Hirschowitz [335] gives an example of an open
f &H(U)
which admits a local but not a global
factorization. This example shows that condition (*) of theorem 2 . 2 1 is not sufficient to obtain a global factorization result. The difficulties posed by Hirschowitz's example were overcome by P. Berner [59] and M. Schottenloher [640]. Examples 2.22, 2.25 and 2.26 are due to L. Nachbin [514], and the proof of example 2 . 2 2 appears in [193].
Proposition 2.27 is
due to S. Dineen [190]. Theorem 2.28 is due to M.A. Zorn [724].
Generalizations to other
locally convex spaces are due to Ph. Noverraz [536,537], D. Pisanelli [578], A. Hirschowitz [341], D. Lazet [423], J.F. Colombeau [141], E. Ligocka [443]
and S. Dineen [190].
A locally convex space E
for which the con-
clusion of proposition 2.29 is valid is called an w-space (S. Dineen
[184],
[190]). E . Grusell [292,293] and M. Schottenloher [645] give examples of locally convex spaces which are not w-spaces. Corollary 2.30 is due to J.F. Colombeau and J. Mujica [154], who use it together with the solution to the Levi problem for Hilbert spaces (L. Gruman [290]) to solve the Levi problem in strong duals of Frgchet nuclear spaces (&3?:spaces). Example 2.31, which arose in constructing a counterexample to the Levi problem, is due to B. Josefson [358] andthe proof given here is taken from
S. Dineen [193].
The inequality of example 2.31 is used in A. Renaud
([604, proposition 4) to obtain a vector-valued Schwarz lemma.
Other
factorization results may be proved by considering locally bounded holomorphic functions and in such cases one gets results for non-normed range spaces [190]. The topological vector space structure of H(U),
U an open subset of
Cn, has been studied by a number of authors including A. Grothendieck [285], G. Kothe [396] and A. Martineau [451]. The compact open topology was first investigated on spaces of holomorphic functions in infinitely many variables by H. Alexander [S] and L . Nachbin [509]. They found, however, that it did not possess very good topological properties and s o , motivated by properties of analytic functionals in several complex variables
Chapter 2
106
due t o A. Martineau [450], L . Nachbin I5091 i n t r o d u c e d t h e ( d e f i n i t i o n s 2 . 3 3 and 2 . 3 4 ) .
The
topology
T~
t o p o l o g y ( d e f i n i t i o n s 2.33 and 2.34)
T&
was f i r s t i n t r o d u c e d f o r holomorphic f u n c t i o n s on s e p a r a b l e Banach s p a c e s by G . Coeur;
[128,129] i n c o n n e c t i o n w i t h problems o f a n a l y t i c c o n t i n u a t i o n
and t h e g e n e r a l d e f i n i t i o n i s due t o L. Nachbin [510]. and 2.41 are g i v e n i n S. Dineen [185]. known and a p r o o f i s g i v e n i n K - D . c o r r e s p o n d i n g problems f o r t h e
T~
P r o p o s i t i o n s 2.38
Lemma 2.39 a p p e a r s t o b e w e l l
B i e r s t e d t and R . Meise [ 6 9 ] . and
T~
The
topologies, i . e . are t h e s e
l o c a l o r s h e a f t o p o l o g i e s ? are p e r h a p s t h e most i n t e r e s t i n g c u r r e n t open problems i n t h e g e n e r a l t h e o r y o f l o c a l l y convex s p a c e s of holomorphic f u n c t i o n s i n i n f i n i t e l y many v a r i a b l e s .
A r e l a t e d open problem i s t o
c h a r a c t e r i z e t h e open sets i n which
and
l6
7
continuous m u l t i p l i c a t i v e
l i n e a r f u n c t i o n a l s are p o i n t e v a l u a t i o n s (by e x e r c i s e 2 . 9 5 t h e the
T
continuous m u l t i p l i c a t i v e linear f unc t i o n a l s c o i n c i d e ).
T~
and This
problem arises i n s o l v i n g t h e Levi problem and i n c o n s t r u c t i n g t h e envelope R e s u l t s d e a l i n g w i t h t h i s problem are t o be found i n
of holomorphy.
H. Alexander [S], G . Coeur:
641,6431, J . M .
[129,131], M. S c h o t t e n l o h e r [632,633,634,635,
I s i d r o [352] and J . Mujica [502,503,504,505].
Lemma 2.43,
c o r o l l a r y 2.44 and example 2.46 are proved i n S. Dineen [185]. 2 . 4 5 and example 2.47 a r e t o be found i n S . Dineen [194].
For
Corollary
832
( s t r o n g d u a l s o f F r g c h e t Schwartz) s p a c e s , example 2.47 i s due t o J . A . Barroso, M . C .
Matos and L . Nachbin [ 5 0 ] .
Reference [194] i s devoted t o t h e
t h e o r y o f holomorphic f u n c t i o n s on &$hl ( s t r o n g d u a l s o f Fr‘echet Monte1 s p a c e s ) and i t i s i n t e r e s t i n g t o n o t e t h a t a l t h o u g h many r e s u l t s on 8 3 8 s p a c e s ( f o r i n s t a n c e , example 2.47) can be extended t o holomorphic f u n c t i o n s on83K;spaces, t h e methods o f proof a r e q u i t e d i f f e r e n t .
OnB&f
s p a c e s , one u s e s f r e q u e n t l y t h e p r o p e r t y t h a t such s p a c e s are i n d u c t i v e l i m i t s o f Banach s p a c e s w i t h compact l i n k i n g mappings i n t h e c a t e g o r y o f
t o p o l o g i c a l s p a c e s ( t h i s i s n o t t r u e o f &J-h’L
s p a c e s ) w h i l e on
8 3%
s p a c e s , one u s e s t h e f a c t t h a t such s p a c e s a r e k-spaces and h e r e d i t a r y Lindelb’f s p a c e s .
Example 2.48 i s due t o L . Nachbin [SO91 and example 2.49
can b e found i n S . Dineen [185]. Very s t r o n g l y convergent s e q u e n c e s ( d e f i n i t i o n 2.50) were i n t r o d u c e d by S . Dineen [184] i n s t u d y i n g holomorphic completions ( s e e 5 4 . 4 ) .
Subsequent-
l y t h e y were used by S . Dineen [185] i n counterexample 2.52 and i n [190] t h e y were a p p l i e d t o g e t h e r w i t h v e r y s t r o n g l y convergent n e t s and v e r y weakly c o n v e r g e n t sequences ( d e f i n i t i o n 2 . 5 1 ) i n o t h e r areas o f i n f i n i t e
107
Holomorphic mappings between locally convex spaces dimensional holomorphy.
F u r t h e r a p p l i c a t i o n s a r e t o b e found i n P . Berner
[58,59] and S. Dineen and Ph. Noverraz [ 2 0 5 ] . T
0
= T
0
A on H(C )
example 2 . 5 2 ,
i f and o n l y i f
A
J.A.
Barroso [46] shows t h a t
i s c o u n t a b l e and t h i s , t o g e t h e r w i t h
completes t h e p i c t u r e o f t h e r e l a t i o n s h i p between t h e d i f f e r -
e n t t o p o l o g i e s on
A
H(C*),
a r b i t r a r y ( s e e a l s o c h a p t e r 5, J . A .
Barroso
and L . Nachbin [53] and V . Aurich [ 3 3 ] ) . Holomorphic germs on compact s u b s e t s o f Banach s p a c e s were f i r s t i n v e s t i g a t e d by S.B. Chae [119,120] and A . Hirschowitz [339,343].
Subse-
q u e n t l y , J . Mujica [503] developed t h e t h e o r y on m e t r i z a b l e l o c a l l y convex spaces ( s e e a l s o R.R. 3. Mujica [41], K-D.
Baldino [ 4 3 ] , A . J . M .
Wanderley [714], P . A v i l e s and
B i e r s t e d t and R . Meise [69,70] and E . Nelimarkka [525])
Lemma 2.54 and p r o p o s i t i o n 2.55 are due t o J . Mujica [503].
Earlier,
S.B. Chae [120] and A . H i r s c h o w i t z [339] had g i v e n incomplete p r o o f s f o r t h e Banach s p a c e case.
In p r o v i d i n g a complete p r o o f , A. Hirschowitz
i n t r o d u c e d semi-norms s i m i l a r t o t h o s e which a p p e a r i n p r o p o s i t i o n 2.56 ( s e e a l s o S. Dineen [ZOO]).
The q u e s t i o n o f whether o r n o t c o n d i t i o n (b)
o f p r o p o s i t i o n 2.56 i s redundant i s answered n e g a t i v e l y by example 2.57 (due t o R.M. W.R.
Aron) and h a s l e d t o some i n t e r e s t i n g i n v e s t i g a t i o n s by
Z a m e [721], K. Rusak [617] and J . T . Rogers, J r . and W.R.
f i n i t e dimensions and by R . L .
Z a m e [609] i n
S o r a g g i [664,665,666] i n i n f i n i t e dimensions.
Rogers and Zame show t h a t c o n d i t i o n (b) i s n o t n e c e s s a r y f o r compact subs e t s of
which have o n l y a f i n i t e number o f connected components and
(c
t h i s r e s u l t does n o t extend t o h i g h e r dimensional s p a c e s .
For a r b i t r a r y
l o c a l l y convex s p a c e s , s u f f i c i e n t c o n d i t i o n s f o r t h e removal o f (b) a r e o b t a i n e d by p l a c i n g r a t h e r t e c h n i c a l l o c a l c o n n e c t e d n e s s c o n d i t i o n s on t h e compact s e t . The
T~
t o p o l o g y was i n t r o d u c e d by S.B. Chae [120] and it h a s a l s o
been s t u d i e d by J . Mujica [SO31 and K . D . chapter 6).
B i e r s t e d t and R . Meise [70], ( s e e
This Page Intentionally Left Blank
Chapter 3
HOLOMORPHIC FUNCTIONS ON BALANCED SETS
I n many b r a n c h e s o f f u n c t i o n t h e o r y t h e r e a r e c o l l e c t i o n s o f s e t s which a r i s e n a t u r a l l y and p o s s e s s p r o p e r t i e s which e n s u r e t h a t t h e y p l a y an i m p o r t a n t r o l e - f o r e x a m p l e c o n v e x s e t s i n f u n c t i o n a l a n a l y s i s and e f f i l g s e t s i n p o t e n t i a l theory.
I n t h e t h e o r y o f s e v e r a l c o m p l e x v a r i a b l e s many
d i f f e r e n t k i n d s o f s p e c i a l s e t s a r i s e - pseudoconvex s e t s , p o l a r s e t s , p o l y n o m i a l l y convex s e t s , p o l y d i s c s and S t e i n manifolds. B a l a n c e d open s e t s a r i s e a s t h e n a t u r a l domain o f convergence o f t h e Taylor s e r i e s expansion a t t h e o r i g i n f o r holomorphic f u n c t i o n s .
I n t h i s c h a p t e r we c o n s i d e r a l l t h e
h o l o m o r p h i c f u n c t i o n s on a b a l a n c e d o p e n s e t U a n d show t h a t t h e T a y l o r s e r i e s e x p a n s i o n s l e a d t o a t o p o l o g i c a l decomposition of H(U) chapter 2.
f o r t h e d i f f e r e n t t o p o l o g i e s we d i s c u s s e d i n
We t h e n u s e t h i s d e c o m p o s i t i o n t o d e d u c e t o p -
-0logica1 properties of H(U)
and t o e x t e n d r e s u l t s a b o u t
s p a c e s o f homogeneous p o l y n o m i a l s t o s p a c e s o f h o l o m o r p h i c functions.
Our main t o o l s a r e a s s o c i a t e d t o p o l o g i e s and
Schauder d e c o m p o s i t i o n s and s i n c e t h e s e t o p i c s a r e n o t v e r y w e l l known we d e v o t e t h e f i r s t s e c t i o n t o t h e i r e x p o s i t i o n . The t h e o r y o f h o l o m o r p h i c f u n c t i o n s on b a l a n c e d o p e n s e t s may b e r e g a r d e d a s a l o c a l t h e o r y a n d i f we c o u l d show t h a t T~
and T& were l o c a l
(or sheaf) topologies then the
r e s u l t s o f t h i s c h a p t e r would e x t e n d t o a r b i t r a r y open s e t s .
109
110
Chapter 3
1 3 . 1 . ASSOCIATED
TOPOLOGIES
DECOMPOSITIONS I N
AND
GENERALISED CONVEX
LOCALLY
SPACES
I n l i n e a r f u n c t i o n a l a n a l y s i s we e n c o u n t e r t w o m u t u a l l y exclusive but not exhaustive types of properties
-
those
p r e s e r v e d under l o c a l l y convex i n d u c t i v e l i m i t s and t h o s e preserved under p r o j e c t i v e limits.
For example t h e l o c a l l y
convex i n d u c t i v e l i m i t o f b a r r e l l e d l o c a l l y convex spaces i s b a r r e l l e d
(respectively bornological) (respectively bornological)
and t h e p r o j e c t i v e l i m i t o f complete l o c a l l y convex s p a c e s i s complete
(respectively nuclear)
(respectively n'uclear).
blany o t h e r p r o p e r t i e s a r i s e a s a c o m b i n a t i o n o f p r o p e r t -
i e s from t h e s e two t y p e s , as semi-reflexivity infrabarrelledness
f o r example r e f l e x i v i t y i s d e f i n e d
(preserved under p r o j e c t i v e l i m i t s ) plus (preserved under l o c a l l y convex inductive
l i m i t s ) and t h e combined p r o p e r t y i s n o t p r e s e r v e d under A g e n e r a l method used i n showing t h a t
e i t h e r kind of l i m i t . a l o c a l l y convex s p a c e , associate with
(EJ)
has a given property is t o
(E,T),
a n o t h e r l o c a l l y convex s p a c e ( F , T ' ) with
t h e g i v e n p r o p e r t y a n d t h e n t o show t h a t
(E,T)
= (F,T').
For
e x a m p l e we c a n a s s o c i a t e w i t h a g i v e n l o c a l l y c o n v e x s p a c e i t s completion and a l s o i t s a s s o c i a t e d b o r n o l o g i c a l s p a c e ,
A
s y s t e m a t i c t h e o r y h a s been developed f o r p r o p e r t i e s which are p r e s e r v e d u n d e r l o c a l l y c o n v e x i n d u c t i v e l i m i t s a n d we now d i s c u s s t h i s t h e o r y and a p p l y i t i n l a t e r s e c t i o n s t o t h e theory o f holomorphic functions. Definition 3.1
i s a f a m i l y ?of (a)
A Q f a m i l y of
ZocaZZy c o n v e x t o p o l o g i e s
l o c a l l y convex topologies such t h a t
3 i s s t a b l e under l o c a l l y convex i n d u c t i v e Z i m i t s ,
( b ) on any v e c t o r s p a c e t h e f i n e s t ZocaZly c o n v e x t o p o l o g y
belongs t o 3 . L e t ( E , T ) b e a Z o c a Z l y c o n v e x s p a c e and l e t Q f a m i l y of
ZocalZy c o n v e x t o p o l o g i e s .
3 be a
The ZocaZly c o n v e x
i n d u c t i v e Z i m i t o f a l l t o p o l o g i e s on E w h i c h l i e i n s a n d a r e f i n e r than
T
also l i e s i n
3.
We d e n o t e t h i s t o p o l o g y by
and c a l Z i t t h e 3 t o p o l o g y a s s o c i a t e d w i t h
T.
T~
111
Holomorphic functions o n balanced sets The t o p o l o g y
c a n a l s o b e c h a r a c t e r i z e d as t h e s o l u t i o n 3 t o a universal problem. S p e c i f i c a l l y i f (E,T) i s a l o c a l l y T
is
convex space and 3 i s a Q-family t h e n
t h e unique
;C
t o p o l o g y on E s u c h t h a t a n y c o n t i n u o u s l i n e a r m a p p i n g f r o m (F,T!) into E,
~ ~ € f a3c t o,r s
For c e r t a i n Q-families
through
(EJ.rg).
t h e r e a r e a l s o o t h e r known
c h a r a c t e r i z a t i o n s which can be u s e f u l . Let b denote t h e family o f a l l bornological
Example 3 . 2 topologies.
Then b i s a Q - f a m i l y and f o r a n y l o c a l l y convex
topology,
we l e t
T ,
topology.
If
T~
denote the associated bornological
( E J ~ )i s a l o c a l l y c o n v e x s p a c e a n d B i s a
c l o s e d c o n v e x b a l a n c e d b o u n d e d s u b s e t o f E t h e n we l e t E B d e n o t e t h e v e c t o r s u b s p a c e o f E g e n e r a t e d by B and normed s o that B i s its closed unit b a l l . ( E , T ~ )=
I t i s w e l l known t h a t
l i m
7E B
where B r a n g e s o v e r a l l c l o s e d convex b a l a n c e d bounded s u b s e t s of E. Example 3 . 3
Let ub d e n o t e t h e f a m i l y o f a l l u l t r a -
b o r n o l o g i c a l l o c a l l y convex t o p o l o g i e s .
Then ub i s a Q - f a m i l y .
F o r a n y l o c a l l y c o n v e x s p a c e ( E , T ) we l e t
T
a s s o c i a t e d u l t r a b o r n o l o g i c a l t o p o l o g y on E .
-
example 3 . 2 ,
ub
denote t h e
We h a v e , a s i n
( E , T ~ ~= ) l i m EB B where B r a n g e s o v e r a l l c o m p l e t e b a l a n c e d convex bounded subsets of E. Example 3 . 4
A l o c a l l y convex space,
(E,T) is c a l l e d a
K e l l e y s p a c e i f a n y f i n e r l o c a l l y convex t o p o l o g y on E h a s l e s s compact s e t s . topology.
An u l t r a - b o r n o l o g i c a l
topology is a Kelley
The c o l l e c t i o n o f a l l l o c a l l y convex K e l l e y
t o p o l o g i e s i s a Q - f a m i l y a n d we l e t
T~
denote the Kelley
topology a s s o c i a t e d w i t h t h e l o c a l l y convex topology
T .
Chapter 3
112
The b a r r e l l e d a n d i n f r a b a r r e l l e d t o p o l o g i e s a l s o f o r m Q - f a m i l i e s a n d f o r a g i v e n l o c a l l y c o n v e x s p a c e ( E , T ) we l e t and T d e n o t e t h e a s s o c i a t e d b a r r e l l e d ( t o n n e l e ) and t i infrabarrelled topologies respectively. An a l t e r n a t i v e
T
description of these topologies, induction,
by means o f t r a n s f i n i t e
i s given i n proposition 3.5.
The c o l l e c t i o n o f a l l
b a r r e l l e d and b o r n o l o g i c a l l o c a l l y convex t o p o l o g i e s ,
bt,
a l s o a Q - f a m i l y a n d we d e n o t e b y
and
T
b a r r e l l e d topology associated with
the bornological
bt T.
For any Hausdorff l o c a l l y convex s p a c e ( E , T )
-
is
we h a v e t h e
following i n c l u s i o n s between t h e various a s s o c i a t e d topologies, where a
b means t h a t a i s f i n e r t h a n b ,
We c o m p l e t e o u r d i s c u s s i o n o f a s s o c i a t e d t o p o l o g i e s b y
showing t h a t t h e b a r r e l l e d topology a s s o c i a t e d w i t h a complete topology i s a l s o complete. Proposition 3.5
I f
( E , T ) is a c o m p l e t e l o c a l l y c o n v e x s p a c e
t h e n (E,-rt) i s a l s o a complete l o c a l l y convex space. Proof
We f i r s t c o n s t r u c t a n o r d e r e d f a m i l y o f l o c a l l y c o n v e x
Let
t o p o l o g i e s on E . number and (E,T,)
=
l i m B Z
T
Now s u p p o s e a i s a n o r d i n a l
~ T= .
h a s b e e n d e f i n e d f o r a l l o r d i n a l s B
T~
(E,T ) B
d e f i n e d by a l l
-
i.e.
.
Let
Ta i s 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
T ~ , B < ~ . We d e f i n e - r a a s t h e t o p o l o g y w h i c h h a s
a neighbourhood b a s e a t z e r o c o n s i s t i n g o f a l l T a c l o s e d convex balanced absorbing subsets of E.
By t r a n s f i n i t e i n d u c t i o n
i s defined f o r every ordinal a.
Since t h e c a r d i n a l i t y of t h e
T
s e t of a l l convex balanced absorbing subsets o f E i s less than o r equal t o
*IE1
it f o l l o w s t h a t t h e r e e x i s t s an o r d i n a l
number 8 s u c h t h a t
T~
=
'eel
for all
el 2
0.
Since a locally
convex s p a c e i s b a r r e l l e d i f and o n l y i f each c l o s e d convex balanced absorbing s e t i s a neighbourhood o f zero it follows that
( E , T ~ ) i s a b a r r e l l e d l o c a l l y convex s p a c e .
S i n c e T t ->
T
113
Holomorphic functions on balanced sets
t h e above c o n s t r u c t i o n shows t h a t
T
> T ~ . t -
w e a k e s t b a r r e l l e d t o p o l o g y on E f i n e r t h a n
T
is the t it follows t h a t As T
We now s h o w b y t r a n s f i n i t e i n d u c t i o n t h a t
= Tee (E,T") i s a c o m p l e t e l o c a l l y c o n v e x s p a c e f o r e a c h o r d i n a l n u m b e r a. Tt
By h y p o t h e s i s ( E , T ~ ) = ( E , T ) i s c o m p l e t e . is complete for a l l
c1<
al.
Now s u p p o s e
(E,T )
Since the projective l i m i t of
complete l o c a l l y convex spaces i s complete t h e space (E, Ta ) =
lim "<"l
c -
1
(E,T")
is complete.
Let (x,) B E
.
B
b e a Cauchy
) Hence (xg) i s a l s o a Cauchy n e t i n t h e "1 B E B c o m p l e t e s p a c e (E,T ) and s o c o n v e r g e s t o an e l e m e n t x i n "1 Now l e t V b e a TC1 - n e i g h b o u r h o o d o f z e r o i n E . With(E,Tal). 1 closed o u t l o s s o f g e n e r a l i t y we may s u p p o s e t h a t V i s a T "1 convex balanced s u b s e t o f E . Since ( x ) is T - Cauchy B E B "1 t h e r e e x i s t s P o i n B s u c h t h a t x B 1 - x B 2 E V f o r a l l B ~ , B ~B ~ . Since x x i n ( E , T ) a s B2and V i s T closed 82 "1 "1 it follows t h a t xB x E V for all B 13,. H e n c e x -+ x i n B T h i s shows t h a t ( E , T " ~ ) i s c o m p l e t e . ( E , T a l ) a s -8 By
n e t i n (E.T
-
-.
-
-
t r a n s f i n i t e i n d u c t i o n (E,.ra) i s complete f o r each o r d i n a l number a and h e n c e ( E , T ~ )= (E,rt) i s a c o m p l e t e l o c a l l y c o n v e x space.
This completes t h e proof.
A m o d i f i c a t i o n o f t h e above proof
yields the following
r e s u l t and a l s o g i v e s an a l t e r n a t i v e d e s c r i p t i o n o f t h e associated infrabarrelled topology. Proposition 3.6
L e t ( E , T ) be a l o c a l l y c o n v e x s p a c e and
b e , r e s p e c t i v e l y , t h e b a r r e l z e d and i n f r a l e t T~ and T~ b a r r e l l e d t o p o l o g i e s a s s o c i a t e d w i t h T. I f ( E , T ) i s compZete ,
then (E,ri) ( E , T ~ )and
is a l s o c o m p l e t e . (E,.ri)
I f
( E , T ) i s quasi-complete
are both quasi-complete.
sequentiaZZy complete then (E,.rt)
I f
then
(E,T) i s
and ( E , T ~ )a r e b o t h s e q u e n t -
i a l l y complete. We now l o o k a t v a r i o u s k i n d s o f S c h a u d e r d e c o m p o s i t i o n s .
These are g e n e r a l i z a t i o n s o f t h e c o n c e p t o f a b a s i s and i n d e e d a l o c a l l y convex s p a c e h a s a Schauder b a s i s i f and o n l y i f it
Chapter 3
114
has a Schauder decomposition c o n s i s t i n g of one dimensional subspaces. Definition 3.7
conoex space
(E,
[Enjn
of a ZocaZZy i s a S c h a u d e r d e c o m p o s i t i o n of ( E , T ) i f
A s e q u e n c e of s u b s p a c e s T)
e a c h x i n E can be w r i t t e n i n a u n i q u e way as x =
-
C m x ) w h e r e xn n = l 11
l i m m--.,
urn : E
u m (:zl
E,
E
E,
xn) =
If t h e s e q u e n c e of m a p p i n g s ,
;zl
xn ( i . e .
a 2 2 n and if t h e p r o j e c t i o n s
zm
xn a r e aZZ c o n t i n u o u s .
n =1
(um),",,,
f o r m an e q u i c o n t i n uous f a m i Z y t h e n we s a y t h a t t h e d e c o m p o s i t i o n i s e q u i -
If E c o n t a i n s a f u n d a m e n t a 2 f a m i Z y of s e m i - n o r m s ,
Schauder.
'1,
such t h a t p(E" xn) = Yl p ( x n ) f o r e v e r y p i n n=l
and e v e r y
C" Xn i n E t h e n we s a y t h a t t h e d e c o m p o s i t i o n i s a b s o b u t e . The n =1 decomposition i s s a i d t o be s h r i n k i n g i f ( ( E ) ' ) " - 1 i s a nB n Schauder d e c o m p o s i t i o n f o r E h .
I t i s e a s i l y s e e n t h a t e v e r y a b s o l u t e S c h a u d e r decompos-
i t i o n i s an equi-Schauder decomposition.
s e t of aZZ s e q u e n c e s of c o m p t e x n u x b e r s
We l e t (an)
A
denote t h e
such t h a t n
l i m s i p lanll'n 5 1. If i s a Schauder d e c o m p o s i t i o n nfor ( E , T ) t h e n we s a y { E n I n i s an A-Schauder d e c o m p o s i t i o n i f W
Xn E E and imply anxn E . The d e c o m p o s i t i o n n=l n=l i s an A - a b s o Z u t e d e c o m p o s i t i o n i f f o r e v e r y p E c s ( E ) and (an)E=1
A
E
t h e semi-norm ;(c"
continuous. Proposition 3.8
n=1
xn) = xm n=l
a fundamentaZ s y s t e m of semi-norms = SU nP P,(Un(x))
f o r e v e r y a i n A and x
Proof
is
A S c h a u d e r d e c o m p o s i t i o n {En}n of a ZocaZZy
convex space (E,T) i s equi-Schauder P,(x)
bnI P ( X n )
i f and onZy i f t h e r e e x i s t s A for
(E,-r)such t h a t
(*I i n E.
We f i r s t s u p p o s e t h a t { E n ) ,
i s an equi-Schauder
-
d e c o m p o s i t i o n o f E a n d p i s a c o n t i n u o u s s e m i - n o r m on ( E , T ) .
-
Let q ( x ) as n
= sup p(un(x)) f o r every x i n E. Since un(x) x n and p i s c o n t i n u o u s i t f o l l o w s t h a t q ( x ) i s f i n i t e
115
Holomorphic functions on balanced sets f o r every x i n E. w h e r e m 5 n we h a v e q(x) = Let
SIIP
B = {x E E;
Then B = { x
E
E;
S i n c e u (u x)=u ( x ) f o r e v e r y x i n E m m n
q(un(x)) f o r every x i n E. q ( x ) 5 11 and
8
= {x; p(x) 5 1).
p(un(x)) 5 1 f o r a l l n}
=
n n Ix e E ;
=
Qun-'
p ( u n ( x ) ) 5 1)
($1.
-
-
i s an e q u i c o n t i n u o u s f a m i l y o f mappings q u n - l i s a neighbourhood o f zero i n E and hence q i s a continuous
S i n c e (un),
semi-norm on E .
Since un(x)
i n E t h i s shows t h a t q 2 p .
x as n
As p r a n g e s
m
(5)
f o r every x
over cs(E) q ranges
o v e r a fundamental f a m i l y of semi-norms which s a t i s f y ( * ) . This completes t h e proof i n one d i r e c t i o n . C o n v e r s e l y i f E c o n t a i n s a fundamental f a m i l y o f semi-
*,
n o r r n s , ( ~ ~ ) w ~ h~i c h s a t i s f y ( * ) t h e n f o r e v e r y a i n A . I x E E ; p a ( x ) 5 11 = a n d h e n c e (un),"=,
Ba
=G
un-'
(B,)
i s an equicontinuous f a m i l y o f mappings.
This completes t h e proof. Proposition 3.9
A S c h a u d e r d e c o m p o s i t i o n ~ f 'a b a r r e Z Z e d
ZocaZZy c o n v e x s p a c e is an e q u i - S c h a u d e r Proof
Let {En),
decovposition.
be a Schauder decomposition f o r t h e
b a r r e l l e d l o c a l l y c o n v e x s p a c e (E,T).
If p i s a continucus
semi-norm on E l e t q ( x ) = s x p p ( u ( x ) ) f o r e v e r y x i n E .
n
As
i n t h e p r e v i o u s p r o p o s i t i o n q i s a semi-norm on E ,
B = {x EE;
u
q ( x ) 5 11 = O { x E E ; p ( u n ( x ) ) 5 1 ) .
i s a c o n t i n u o u s l i n e a r mapping
Let
Since each
I x E E ; p ( u n ( x ) ) 5 11 i s a
n c l o s e d s u b s e t o f E and hence B i s a c l o s e d convex b a l a n c e d absorbing subset of E.
S i n c e E is b a r r e l l e d B i s a n e i g h b o u r -
hood o f z e r o and h e n c e q i s a c o n t i n u o u s semi-norm on E .
As
i n t h e p r o o f o f t h e p r e c e d i n g p r o p o s i t i o n i t now f o l l o w s t h a t q L P a n d q ( x ) = S,UP q ( u n ( x ) ) . An a p p l i c a t i o n o f p r o p o s i t i o n 3 . 8 now c o m p l e t e s t h e p r o o f . The a b o v e method o f p r o o f i s a l s o u s e d i n t h e f o l l o w i n g proposition.
Chapter 3
116 Proposition 3.10
i s a n /J1-SchazAer d e c o m p o s i t i o n for t h e b a r r e Z Z e d ZocaZly c o n v e x s p a c e ( E , r ) t h e n {En}, i s a n A-absoZute decomposition. Proof
Let p
E
If
cs(E) and l e t
E
4.
The s e q u e n c e
(n 2 a n r 2 a l s o l i e s i n a a n d s o i f 1" x E E then n a x n n n=l n n=l 2 also lies i n E. H e n c e En a x la' i s a bounded s u b s e t c f E . n I? n = l - 1 2 T h e r e f o r e s x p p ( n a x ) = M
ifl
%
%
shown t h a t p i s a s e m i - n o r m on E . S i n c e B = { X EE ; p ( x ) 5 1 ) n n m = l x m E E ; ;z1 l a m l p i x , ) 5 11 i s a c l o s e d c o n v e x
=n{c-
b a l a n c e d a b s o r b i n g s u b s e t o f E and E i s a b a r r e l l e d s p a c e i t 2,
f o l l o w s t h a t B i s a neighbourhocd o f z e r o and hence p i s a c o n t i n u o u s s e m i - n o r m on E .
This completes the proof.
We now l o o k a t a s s o c i a t e d t o p o l o g i e s f o r s p a c e s w i t h a n A-absolute decomposition. I f i s an j - c b s o Z u t e d eco mp o s itio n Proposition 3.11 for ( E , T ) t h e n { E n , ~ b } nis a Z s o an $ - a b s o l u t e d e c o m p o s i t i o n
for
(E>Tb).
Proof of
For each A i n
(E,T).
p is a
T
p
xn) =
'L
[4zl
A
Let B = (x ) A s r
d e n o t e an a r b i t r a r y bounded s u b s e t
r
l e t xA =
;Z1
x:
c o n t i n u o u s s e m i - n o r m o n E a n d (a , )
norm on E .
:zl
kzl
n21an/ p(xn)
T
E
(E,T).
4
If
then
n continuous semi-
is a a n d {n2 a x A } n n n,X I f q i s a rb c o n t i n u o u s semi-norm b o u n d e d s u b s e t o f (E,T) o n E t h e n t h e a b o v e s h o w s t h a t s u [ n 2 l a n l q ( x n1 ) ] = M 1 < n x a 1 and h e n c e s p xn) 5- M1 $ = 1 < m . Hence t h e
Hence syp
2,
n
2
is also a
in
,K
ifl
x
+
-
-
lanl
q(Xn)
o as A m.
w
-
-*
i s b o u n d e d on t h e
so i s rb continuous.
bounded s u b s e t s o f ( E , T ~ )a s A
M <
.
semi-norm q ( C m n = 1X n ) t h a t x:
=
l a n l p[x:)
in(E,Tb)
T
T h i s a l s o shows
i f xA
o in
T o c o m p l e t e t h e p r o o f we m u s t s h o w t h a t
PI
xm
If p is a
T
Zx i n ( E , T ~ )a s n -m. m=l m c o n t i n u o u s semi-norm on E and x =
&zl
x m €E then
117
Holomorphic functions on balanced sets
i s a bounded s u b s e t o f
(E,Tb). q(x
-
as n
i!yl
If q is a
-
n2 q ( n 2
Xm) = q ( m c"= n Xm) = 1
-.
(E,'I)
and o f
c o n t i n u o u s semi-norm on E t h e n
Tb
zrn
m=n xm) __+
O
This completes t h e proof.
The s i t u a t i o n f o r t h e a s s o c i a t e d b a r r e l l e d t o p o l o g y i s s l i g h t l y morc c o m p l i c a t e d .
T h i s i s d u e t o t h e f a c t t h a t we
d o n o t , i n g e n e r a l , know i f a S c h a u d e r d e c o m p o s i t i o n ,
i s a l s o a S c h a u d e r decompos-
(E,T),
f o r a l o c a l l y convex s p a c e ,
i t i o n f o r the associated barrelled space (F,Tt). a p o s i t i v e answer t o t h i s problem i f t o p o l o g i e s on E i . e . and only i f o(E',E).
(E,'I)'
(E,T)'
if
=
'I
and
'I
We o b t a i n
are compatible
t ( E , T ~ ) 'a n d t h i s o c c u r s i f
i s q u a s i - c o m p l e t e f o r t h e weak t o p o l o g y
On t h e o t h e r h a n d we d o h a v e t h e f o l l o w i n g u n i q u e i s a Schauder decovposition
n e s s r e s u l t ; i f {En), and i f each E
for (E,r)
i s a b a r r e l l e d s p a c e t h e n t h e r e e x i s t s a t most
n one b a r r e l l e d t o p o l o g y on E which c o i n c i d e s w i t h and h h i c h h a s {En}
n
T
on e a c h En
a s a Schauder decomposition.
A l t h o u g h we d o n o t know i f s u c h a t o p o l o g y e x i s t s t h e a b o v e s a y s t h a t when i t d o e s i t m u s t b e t h e a s s o c i a t e d barrelled topology.
he p r o v e a l e s s g e n e r a l r e s u l t which i s
adequate f o r our applications. Proposition 3.12
~ e {t E ~ be ) ~a r ~ R - . d e e o m p o s i t i o n for (E,T)
and suppose each En i s a b a r r e l l e d l o c a l l y convex space.
I f
t h e r e e x i s t s G b a r r e l l e d t o p o l o g g T~ o n E w h i c h c o i n c i d e s w i t h 'I on each. Erl and i.f { E n } n i s c S c h a u d e r d e c o m p o s i t i o n fcr ( E , T ~ )t h e n
Tl
=
Tt
a x d a f u n d c m e n t a l s g s t e n : of sen;<-norms
f o r ( E 3 r t ) i s g i v e n by G Z Z s e m i - n o r m s , following conditions; (a)
PIE,
is ~ - e o n t i n ~ o u s
( b ) p ( , ? l l x n ) = ;:l Cm x i n E. n=l n Proof
P , w h i c h satisfy t h e
p(xn) f o r every
By h y p o t h e s i s
'I
5
T~
5
' I ~ and
hence
Chapter 3
118
=
"En
T
=
.
T
S i n c e {En}n i s a S c h a u d e r decomposit follows t h a t it is also a
i t i o n f o r ( E , T ) and ( E , r l )
S i n c e I E n I n i s an
Schauder d e c o m p o s i t i o n f o r (E..rt). A-decomposition
f o r (E,T) it i s a l s o an 2-decomposition f o r
a n d ( E , T ~ )a n d h e n c e b y p r o p o s i t i o n 3 . 1 1 i t i s a n
(E,rt)
d - a b s o l u t e and i n p a r t i c u l a r an a b s o l u t e d e c o m p o s i t i o n f o r both of these topologies.
Hence b o t h o f t h e s e t o p o l o g i e s a r e
g e n e r a t e d by semi-norms which s a t i s f y ( a ) and ( b ) .
On t h e
o t h e r hand i f p i s a semi-norm on E which s a t i s f i e s ( a ) and p(x) 5 1 ) is a
(b) then V = {x E E ;
closed c o n v e x b a l a n c e d
a b s o r b i n g s u b s e t o f E a n d h e n c e p i s a -rt c o n t i n u o u s s e m i norm.
Hence
' I ~ =
r t and
T~
i s t h e l o c a l l y c o n v e x t o p o l o g y on
E g e n e r a t e d by a l l t h e semi-norms on E which s a t i s f y ( a ) and
(b). We now l o o k a t d e c o m p o s i t i o n s o f t h e s t r o n g d u a l . Proposition 3.13
E
a
a n d BI' = { n 2 C m a m=n 'm)n,X
n
= TIE
n
i s bounded
E
El
i s an
Since
decomposition it follows t h a t
I f T E E ' we l e t T
I f T8
are bounded s u b s e t s
f o r every positive integer n .
f o r e a c h 8 i n s o me d i r e c t e d s e t R a n d B
projection o f B i n E then A n 1 IiTenIIB = s u p I T e ( x n ) 5 -2 I l ~ ~ l l Bf t o r a l l n * n a an Hence, s i n c e En i s a complemented s u b s p a c e o f E , T i
-
i f Te
B
& ( w e may a n d s h a l l a s s u m e w i t h o u t l o s s o f
B ' = ( n 2 an ' n ) n , a of E. B"
T) I .
a
g e n e r a l i t y t h a t an 2 1 f o r a l l n ) . A-absolute
decomposition f o r
(x )aET.b e a bounded s u b s e t o f E and l e t
Let B
a = (a,),
{ E n I n is a n & a b s o l u t e
I n is an ~ - a b s o Z u t e d e c o m p o s i t i o n for ( E ,
{ (En);
(E,T) t h e n Proof
I f
o i n (E o in
I
) I
nB
as 0
-
m
n
Since
is the
f o r each n
E l .
B
T h i s shows t h a t i s a Schauder decomposition f o r To c o m p l e t e t h e p r o o f i t s u f f i c e s t o n o t i c e t h a t
E l .
B
Holornorphic functions on balanced sets
119
C"
n=1
< C"
n=
f o r every T i n E'.
A n d - a b s o Z u t e d e c o m p o s i t i o n of a ZocaZZy
Corollary 3.14
convex space i s a shrinking d e c o m p o s i t i o n . F u r t h e r p r o p e r t i e s o f a s s o c i a t e d t o p o l o g i e s and S c h a u d e r decompositions a r e given without proof a s t h e need a r i s e s . For f u r t h e r d e t a i l s we r e f e r t o t h e N o t e s a n d Remarks a t t h e e n d of t h i s chapter,
Apart from t h e immediate a p p l i c a t i o n ' s o f t h e above r e s u l t s w h i c h we g i v e i n t h i s c h a p t e r we w i l l a l s o f i n d t h a t t h e techniques developed t o o b t a i n t h e s e r e s u l t s a r e useful i n s t u d y i n g h o l o m o r p h i c f u n c t i o n s on c e r t a i n n u c l e a r s e q u e n c e spaces.
13.2
EQUI-SCHAUDER DECOMPOSITIONS O F ( H ( U ; F )
,T).
In t h i s s e c t i o n U w i l l d e n o t e a b a l a n c e d open s u b s e t o f a l o c a l l y c o n v e x s p a c e E and F w i l l d e n o t e a Banach s p a c e (Most o f t h e r e s u l t s ,
quasi-complete
however, can be e x t e n d e d t o a r b i t r a r y
l o c a l l y convex s p a c e s ) .
P r o p o s i t i o n 3.15
If U is a b a l a n c e d o p e n s u b s e t o f a Z o c a l Z y c o n v e x s p a c e E , F i s a Banach s p a c e , f E H ( U ; F ) and a =
(an)*
A then g = 1" a n=o n
E
Let K C
Proof
s
E
bT e Uc o m p a c t .
H(U;F) We may s u p p o s e w i t h o u t l o s s
o f g e n e r a l i t y t h a t K i s b a l a n c e d , t h e n t h e r e e x i s t A > 1 and V ,
a c o n v e x b a l a n c e d n e i g h b o u r h o o d o f z e r o , s u c h t h a t A(K+V) C U and l
l
IlfIIA(K+v) = ~
\
<
"'
By t h e C a u c h y i n e q u a l i t i e s
l (K+V) A < M for a l l n.
S i n c e (an),
E
3
t h e r e e x i s t s C > o such t h a t
120
<
Chapter 3 n
1+A :zo(x)
C.M
<m.
Thus g i s a l o c a l l y b o u n d e d f u n c t i o n on U a n d s i n c e i t i s G-holomorphic i t
in H(U;F).
is
Proposition 3.16
This completes t h e proof.
I f U i s a baZanced o p e n s u b s e t of a ZocaZZy
c o n v e x s p a c e E , F is a Banach s p a c e and f a H ( U ; F )
then the
TayZor s e r i e s e x p a n s i o n o f f a t t h e o r i g i n c o n v e r g e s t o f i n (H(U;F)
*
For each positive integer m l e t
Proof =
"m
,T6)
{x E U ;
n
2 d f o )
the interior of V i t i o n shows t h a t
m
.
(W
Since
5 m f o r a l l n l a n d l e t Wm d e n o t e 2 "
(n ) n = l
Ed t h e
preceding propos-
i s an i n c r e a s i n g c o u n t a b l e open c o v e r
)" m m=l
of U.
Now l e t p b e a
exist
a positive integer m
c o n t i n u o u s semi-norm on H ( U ; F ) .
T~
for every f i n H(U F).
\I
(x)
a n d C>o s u c h t h a t p ( f ) < C
Ilfll
There
wm 0
IIence f o r a l l n w e h a v e
^.
J = O
j!
d j f (0) j!
Ic
Fl'n+l
"Wmo
m0
j2
c
*
Therefore p(f
- j=o
djf(0) )j!
0
as n
-
and t h i s completes t h e p r o o f . Theorem 3 . 1 7
L e t U b e a baZanced o p e n s u b s e t of a ZocaZZy
c o n v e x s p a c e E and Z e t F be a Banach s p a c e . {&"E;
m
F ) ,~~l~~~
d e c o m p o s i t i o n for Proof
Then
i s a n , # - d e c o m p o s i t i o n and an A - a b s o Z u t e (H(U;F)
,T
By p r o p o s i t i o n 2 . 4 1
complemented s u b s p a c e o f
6) (P("E;F),T,)
(H(U;F)
, T )~
is a closed
and s o p r o p o s i t i o n
121
Holomorphic functions on balanced sets m
3.16 implies that {!?(nE;F),~w}n=O
for (H(U;F),T&).
is a Schauder decomposition
Proposition 3.15 implies that it is an
,$-decomposition
and since
T~
is a barrelled topology
proposition 3.10 implies that it is an A - a b s o l u t e decomposition.
This completes the proof.
We now obtain the same result for the
-c0
and
T~
topologies and
their associated bornological topologies. L e t U b e a baZanced o p e n s u b s e t of a
Proposition 3.18
l o c a l l y c o n v e x s p a c e E , l e t F be a Banach s p a c e , l e t K b e a Then t h e semi-norm compact s u b s e t of U - a n d l e t ( a n ) n E 8 . n f (0 p(f) = I I W b K
Efola,l
is
T
Proof
continuous on H ( U ) .
We may assume without loss o f .generality that K
is a balanced subset o f U . compact subset o f U and n lanl 5 ( I +T A)
for all n 1. n
For any f in H(U;F)
.
we have by the Cauchy inequalities
for every f in H ( U ; F ) Theorem 3.19
0
Choose X > l such that X K is a a positive integer such that
and this completes the proof.
L e t U b e a b a l a n c e d o p e n s u b s e t of a ZocaZZy
c o n v e x s p a c e E and l e t F b e a Banach s p a c e . Then {' 6 (nE ;F) ,T ~ } : = ~ i s a n 8 - d e c o m p o s i t i o n and an d - a b s o Z u t e
d e c o m p o s i t i o n f o r (H(U;F) , T ) . By proposition 3.16, since T 6 > T the Taylor Proof 0' series expansion at the origin o f a holomorphic function
converges t o the function in the compact open topology. Since (@("E;F) , T ~ )is a closed complemented subspace o f n (H(U;F) , T ~ ) (proposition 2.40) this s h o w s that I(?( E;F) ,
OD
T ~ } ~ = ~
is a Schauder decomposition for (H(U;F) , T ~ ) . Proposition 3.15 implies that it is a n A -decomposition and proposition 3 . 1 8 shows that it is a n d - a b s o l u t e decomposition. This
Chapter 3
122 completes t h e proof. Corollary
3.20
L e t U b e a b a l a n c e d o p e n s u b s e t o f a ZocaZZy
c o n v e x s p a c e E and l e t F be a Banach s p a c e . i s and-decomposition
and an 4 - a b s o l u t e
( H ( U ; F ) , T ~ , ~i ) f e a c h p ( " E ; F )
Then { P ( n E ; F ) l z = o
d e c o m p o s i t i o n for
i s given t h e bornoZogical
t o p o l o g y a s s o c i a t e d w i t h t h e compact o p e n t o p o l o g y . Proof
Apply p r o p o s i t i o n
3.11 and theorem 3 . 1 8 .
L e t U be a b a l a n c e d o p e n s u b s e t of a
Proposition 3.21
l o c a l l y c o n v e x s p a c e E , Z e t F b e a Banach s p a c e , be a
T
w
Zet p
c o n t i n u o u s semi-norm on H ( U ; F ) and l e t ( a )
n n
Then t h e semi-norm
E
A.
Proof
Suppose p i s p o r t e d by t h e compact b a l a n c e d s u b s e t K
o f U.
We s h o w t h a t p i s a l s o p o r t e d b y t h e same c o m p a c t s e t .
%
Let V b e a n e i g h b o u r h o o d o f K which l i e s i n U .
ChooseX> 1
and a b a l a n c e d neighbourhood o f z e r o W such t h a t K A(K+W)CV i+x n s u c h t h a t la 5 (-1 for all
I
Choose a p o s i t i v e i n t e g e r n n > no.
2
T h e r e e x i s t s a p o s i t i v e n u m b e r C(W) s u c h t h a t Hence, f o r e v e r y f o r e v e r y f i n H(U;F) p ( f ) 5 C(W) llf
IIK+W
.
f i n H ( U ; F ) , we h a v e
%
and p i s a
T~
c o n t i n u o u s semi-norm on H(U;F).
Theorem 3 . 2 2
L e t U b e a baZanced o p e n s u b s e t o f a ZocaZZy
c o n v e x s p a c e E and Z e t F be a Banach s p a c e . n
Cp(
E; F), T
m
~
}
i s =a n ~ A-decomposition ~
decomposition f o r (H(U;F)
Then
and a n / j - a b s o Z u t e
,T~).
Proof
By p r o p o s i t i o n 3 . 1 6 , s i n c e T 6 > T the Taylor w' series expansion a t t h e o r i g i n o f a holomorphic function
converges t o t h e function i n t h e m
T
w
t o p o l o g y . BY P r o P o s i t i o n
2.41 {@(nE;F),~w)n=O i s a Schauder decomposition f o r I t i s an,J'-decomposition
by p r o p o s i t i o n
3 . 2 1 shows t h a t i t i s a n d - a b s o l u t e
(H(U;F)