INTRODUCTION TO HOLOMORPHY
NORTH-HOLIAND MATHEMATICS STUDIES Notas de Matematica (98)
Editor: Leopoldo Nachbin Centr...
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INTRODUCTION TO HOLOMORPHY
NORTH-HOLIAND MATHEMATICS STUDIES Notas de Matematica (98)
Editor: Leopoldo Nachbin Centro Brasiteiro de Pesquisas Fisicas, Rio de Janeiro and University of Rochester
NORTH-HOLLAND -AMSTERDAM
0
NEW YORK
0
OXFORD
106
INTRODUCTION TO HOLOMORPHY
Jorge Albert0 BARROSO UniversidadeFederal do Rio de Janeiro Rio de Janeiro Brasil
1985
NORTH-HOLLAND -AMSTERDAM
NEW YORK
OXFORD
Elsevier Science Publishers B.V., 1985
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
ISBN: 0444 87666 9
Publishers: ELSEVIER SCIENCE PUBLISHERS B.V. P.O. Box 1991 1000 BZ Amsterdam The Netherlands
Sole distributors for the U.S.A.and Canada: ELESEVIER SCIENCE PUBLISHING COMPANY, INC. 52 Van d e r b i It Avenue NewYork, N.Y. 10017 U.S.A.
Library of Congress Cataloging in Publication Data
Barroio, dorge Alberto. Tntroduction to holomorphy. (North-Holland asthmatic studies ; 106) (Notas dc M t m t i C a ; 98) Bibliography: p. Includes index. 1. Named linear epaccs. 2. Damins of holoaorphy. I . Title. I T . Series. 111. Series: Notas dc materdtica (hterdnm, Netherlmdr) ; 98. QAl.N86 no.98 rQA322.21 510 (I r515.7'31 84-22283 ISBN 0-444-87666-9
PRINTED IN THE NETHERLANDS
T o Anna Amalia
with love.
This page intentionally left blank
FOREWORD
This book presents,
011
the one hand,
a set
of
basic
properties of holomorphic mappings between complex normed spaces and between complex locally convex spaces.
These
properties
have already achieved an almost definitive form and should
be
known to all those interested in the study of infinite dimen-
sional Holomorphy and its applications.
On the other hand, for
reasons of personal taste but also (and especially) because of the importance of the matter, some
incursions have been
made
into the study of the topological properties of the spaces
of
holomorphic mappings between spaces of infinite dimension.
An
attempt is then made to show some of the several topologies that can naturally be considered in these spaces. There has been
no concern to establish priorities
relatively few authors are quoted in the text. facts should be pointed out here.
The study of
and
Some historical differential
mapping and holomorphic mapping between spaces of infinite dimension apparently begins with V. Volterra [142],
[143], [ 1441,
[ 1451 , [ 1461 around .L887. Then D. Hilbert, in his work
[ 561 ,
outlines a theory of holomorphic mappings in an infinity of variables, in which the concept of polynomial in such a context already clearly appears.
At the same time (1909), M. Fr6chet
publishes his first work [40] on the abstract theory nomials in an infinity of variables.
vii
of poly-
Later on, the development
viii
F 0R E WO RD
,[ 423)
of the theory of normed spaces led Fre'chet to defirie([41]
real polynomial in a more general situation.
Mention must be
made of R. Ggteauxls works [ 431, [ 441 , in which he proposes definition for complex polynomials.
In the period
mid-6Os, several other names are worthy of note.
a
until the
A historical
vision of the development of the notions of polynomial and holomorphic mapping in this period can be obtained through works of A.E. Taylor, [ 1353, [136].
the
The mid-60s witnessed
a
rekindling of interest and a quickening of the development the study of questions that originate in the notion of
of
holo-
morphic mapping between complex normed spaces and between complex locally convex spaces.
Holo-
Thus, infinite dimensional
morphy appears as a theory rich in fascinating problems
and
rich in applications to other branches of Mathematics andMathematical Physics.
Once again without any desire to establish
priorities, we would like to quote the names of H. Cartan
P. Lelong in France, for their influence and work, as well the French team composed of G. Coeure',J.-F.
Ramis and J.P. V i g d .
as
Colombeau, A. Douady,
M. H e r d , A . Hirschowitz, P. K r g e , P. Mazet, P. Noverraz, Raboin, J.P.
and
P.
Still, we should especial-
ly like to stress the important role played in the development o f this theory by Leopoldo Nachbin and his doctoral students in
Brazil and the United States: Baldino, J.A.
Aragona, R.M. Aron,
Barrroso, P.D. Berner, P.J. Boland, S.B.
S. Dineen, C.P. Pombo, R.L.
A.J.
Gupta, G . I .
Soraggi, J.O.
as T. Abuabara, T.A.W.
Katz, M.C.
R.R.
Chae,
Matos, J. Mujica, D.P.
Stevenson, A.J.M.
Wanderley as well
Dwyer, J.M. Isidro, L.A.
Moraes and D.
Pisanelli, all of whom were directly influenced by him.
ix
FOREWORD
Mention should also be made of the German school, represented by K.-D.
Bierstedt, B. Kramm, R. Meise,
as
&I.
Schottenloher and D. Vogt, the Italian school as represented by E . Vesentini, and the Swedish school, as represented by C .O.
Kiselman. Let us speak a little about the contents of this book.
We begin with a study of algebraic and topological properties of m-linear mappings, m-homogeneous polynomials and power
se-
ries, and then introdlice the concept o f holomorphic mapping between complex normed spaces and complex locally convex spaces. We endorse Weierstrass's point of view, that is,that holomorphic mappings are, in a sense, locally represented Taylor series.
Several
expressions are then
by
their
derived
for
Cauchy's integral formula and for Cauchy' s inequalities; then a study is presented of the convergence of Taylor series of a holomorphic mapping.
The differences with the case of infinite
dimension are stressed, thus leading naturally to a consideration of holomorphic mappings of bounded type.
The relation-
ships are shown between the notions of holomorphic weakly holomorphic mapping and finitely (holomorphic in GGteaux's sense).
mapping,
holomorphic
mapping
We then present the infin-
ite-dimensional versions of the theorem of maximum module and uniqueness
of holomorphic
continuation.
3-bounding sets o f locally convex spaces, Josefson-Nissenzweig theorem:
"If E
In studying we apply
I((pmll = 1
in the weak topology
f o r every o(E',E)".
m E N
the
is a normed space
infinite dimension, there exists a sequence such that
the
and
'
(pm' mE
(pm
-t
0
in
as
m
of
E' -t
This theorem resolves a famous
FOREWORD
X
problem proposed by Banach, and thus enjoys an a p p l i c a t i o n i n an a r e a d i f f e r e n t from i t s i n i t i a l context. i s made of t h e p r o p e r t i e s o f topologies P a r t I and P a r t 11.
T
A d e t a i l e d study
both i n
~ T, ~ ' r,6
The spaces of thebounded type holomorphic
mappings a r e d e a l t with i n d e t a i l and i n t h e spaces we prove t h e Cartan-Thullen base space b e i n g separable,
context of such
theory i n t h e case o f t h e
t h i s bringing Part I t o a close.
P a r t I1 ends with t h e study of bornological p r o p e r t i e s of t h e spaces o f holomorphic mappings. Incomparably more could be s a i d about t h e have been l e f t out.
topics that
P a r t i c u l a r mention should be made o f the
f a c t that no p r o p e r t y of a n u c l e a r n a t u r e i s r e f e r r e d the t e x t , although q u e s t i o n s r e l a t i n g t o n u c l e a r i t y
to
in
are
be-
coming more and more i m p o r t a n t i n t h e study of Holomorphy. P r o f i t a b l e ilse of t h i s book w i l l r e q u i r e some familari t y with t h e b a s i c theorems o f Functional A n a l y s i s ,
in
c o n t e x t o f normed spaces and l o c a l l y convex spaces.
The read-
i n g o f P a r t I1 does not presume knowledge of P a r t I.
the
Theoret-
i c a l l y , i t could be s a i d t h a t t h e study of both p a r t s r e q u i r e s no previous knowledge of s e v e r a l
complex v a r i a b l e s .
however, i n the a u t h o r ' s opinion,
i s a marvellous
This, case
of
w i s h f u l thinking. T h i s work owes much t o t h e experience acquired
during
t h e courses administered a t t h e F e d e r a l U n i v e r s i t y o f R i o de J a n e i r o , a t t h e U n i v e r s i t y of Santiago de Compostela v i t a t i o n o f P r o f e s s o r J.M.
Isidro)
(on i n -
and a t t h e U n i v e r s i t y of
Valencia ( o n i n v i t a t i o n of P r o f e s s o r M .
Valdivia).
But
it
FORE WORD
xi
owes most to what was learned in the classes given by Profess o r Leopoldo Nachbin, to whom we are deeply grateful.
A special word of thanks to Dr. Raymond Ryan
of
the
University o f Galway, Ireland, f o r his English translation of a preliminary version of this book. fessors
s.
Many thanks also to Pro-
Dineen and J.P. Ansemil for their suggestions. O u r
thanks also go to the Mathematics Institute of
the
Federal
University of Rio de Janeiro f o r their financial support, and to Wilson Gdes for his excellent typing services.
Jorge Albert0 Barroso
Federal University of Rio de Janeiro August 1984
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TABLE O F CONTENTS
FOREWORD
............................................... PART I.
THE NORMED CASE
...... CHAPTER 2 . POWER S E R I E S ............................... CHAPTER 3. HOLOMORPHIC MAPPI NGS ....................... CHAPTER 4. T H E CAUCHY I NTEGRAL FORMULAS ............... CONVERGENCE OF T H E TAYLOR S E R I E S ........... CHAPTER 5. CHAPTER 6. WEAK HOLOMORPHY ............................. FINITE HOLOMORPHY AND GATEAUX HOLOMORPHY ... CHAPTER 7. T O P O L O G I E S ON S P A C E S O F HOLOMORPHIC CHAPTER 8. M A PPI NGS ................................... CHAPTER 9. U N I QUENESS OF ANALYTIC CONTINUATION ........ CHAPTER 10. T H E MAXIMUM P R I N C I P L E ...................... CHAPTER 11. HOLOMORPHIC MAPPI NGS O F BOUNDED T Y P E ....... ................... CHAPTER 1 2 . DOMAINS O F #,,-HOLOMORPHY CHAPTER 1.
CHAPTER
NOTATION AND TERNINOLOGY.
13. THE CARTAN-THULLEN O F #b-HOLOMORPHY
P A R T 11.
THEOREM
POLYNOMIALS
CHAPTER
17.
CHAPTER 18. CHAPTER 19. CHAPTER 20.
CHAPTER 2 1 . CHAPTER 22.
1
17 25 31 45 57 69 81 111
115 119 127
FOR DOMAINS
...........................
139
THE LOCALLY CONVEX CASE
1 4 . NOTATION AND M U L T I L I N E A R MAPPINGS........... CHAPTER 15. POLYNOMIALS CHAPTER 1 6 . T O P O L O G J E S ON S P A C E S O F M U L T I L I N E A R
CHAPTER
vii
................................ M A PPI NGS AND HOMOGENEOUS POLYNOMIALS ....... FORMAL POWER S E R I E S ........................ HOLOMORPHIC MAPPI NGS ....................... S E P A R A T I O N AND PASSAGE T O T H E QUOTIENT ..... #-HOLOMORPHY H-HOLOMORPHY .............. E N T I R E MAPPI NGS ........................... . SOME ELEMENTARY P R O P E R T I E S O F HOLOMORPHIC M A PPI NGS ................................... AND
xiii
155 159 167
173 177 183 185
187 191
TABLE OF CONTENTS
xiv
CHAPTER 2 3
. HOLOMORPHY.
. 25 .
................................ SETS .............................. THE ............................... ....................... LOCAL CONVERGENCE OF THE .............................. C O N T I N U I T Y AND AMPLE
BOUNDEDNESS
CHAPTER 24
BOUNDING
CHAPTER
THE CAUCHY INTEGRAL INEQUALITIES
CHAPTER 26
. THE TAYLOR
CHAPTER
. 28 . 29 . 30 . 31. 32 . 33 . 34 . 35 .
CHAPTER
36
CHAPTER CHAPTER
CHAPTER
CHAPTER CHAPTER
CHAPTER CHAPTER CHAPTER
27
.
AND
CAUCHY
REMAINDER
COMPACT AND TAYLOR S E R I E S
........................ ............... .............
THE M U L T I P L E CAUCHY I N T E G R A L AND THE CAUCHY I N E Q U A L I T I E S DIFFERENTIALLY LIMITS
STABLE
OF HOLOMORPHIC
UNIQUENESS
SPACES
MAPPINGS
OF HOLOMORPHIC
CONTINUATION
.....
........... ............... LIMITS ................................. ....................... OF ..................
HOLOMORBHY AND FINITE
HOLOMORPHY
THE MAXIMUM SEMINORE1 T H E O m M P R O J E C T I V E AND I N D U C T I V E HOLOMORPHY
195 197 209
215 221
229
233 237 241
245 249
AND
253
T O P O L O G I E S ON # ( U ; F )
263
BOUNDED S U B S E T S
273
#(U;F)
279
................................ ...........................................
AN I N D E X O F D E F I N I T I O N S
297
AUTHOR I N D E X
301
PART I THE NORMED CASE
This page intentionally left blank
CHAPTER 1
NOTATION AND TERMINOLOGY.
We denote by
IN, R
and
POLYNOMIALS
the systems of non-negative
0:
integers, real numbers and complex numbers respectively. Throughout this book, all vector spaces considered will
have
CC
as their field of scalars unless explicitly stated otherwise.
E
and
F
will denote complex normed spaces, and
empty open subset of
If 5
U
a non-
E.
is a point in a normed space and
p
a positive
real number, the open ball (respectively, the closed ball) in this space with centre B~
(respectively
(5 )
DEFINITION 1.1
Let
f
and radius
will be denoted by
p
Cp ( 5 1 ) .
...,Em
(m E N,
E1,E2,
m > 0)
be a finite
sequence of normed spaces.
...,E,;F) denotes the vector space of m-linear mappings fi Ei = El x . . . ~ Em into F, where addition and multii=1
La(E1, ~~
of
plication by scalars are defined pointwise.
s. (El,...,Em;F)
denotes the subspace of
Xa(E1,
?
...,Em;F) m
of
Ei into F. ( Ei being i=l i d In the case m = 0 we endowed with the product topology). c ontinuous m-linear mappings of
identify
S(E1,
S,(E~,
...,E,;F)
...,E,;F) with
F
with
F
as vector spaces, and
as normed spaces.
1
CHAPTER 1
2
inf{M z 0: 1IA(x1
,...,xm )I/
L
MIlx11I ...I( x , )
for all xl€E1
,...,xmEEm],
and 2)
denoting by
IIAII
the common value of the expressions
which appear in l), the mapping
is a norm. REMARK 1.1:
a)
We commit an abuse of notation, using the
same symbol to represent the norms on spaces which m a y be distinct b)
IIA(xl, c)
.
...,E,;F),
If A E S(E1,
-
,xm)ll
5
IIAll
If A E Ca(E1
it is easy to see that
.I1 Xmll
lIxlll
,...,Em;F)
for every
then
xlEE1,.
A E C(E1
..
, xf
Em
,...,Em;F)
if
and only if
...,
s UP
xl+o, d)
xdo
In the case in which
the space
C(E1,
...,E,;F),
In the case in which
IIA(X1’
YXm
Ill
11 ~111. 11 Xmll F
< =.
is a Banach space, s o too is
with the norm above. El = E2 =...=
Em = E
the space
of m-linear mappings (respectively, of continuous m-linear m a p
NOTATION AND TERMINOLOGY.
3
POLYNOMIALS
m
7
Em = G...xE
pings) of
(respectively,
order
Sa(%;F)
(m > 0 )
Sas(%;F)
the subspace of
In other words, if
m,
spaces and
Sm
is the symmetric gro-Jp of
then
In the case
m = 0,
'as (%;F)
= Cm(E;F) = F
S,(%;F)
We denote by
Ss(%;F)
=
s~("'E;F) =
F
as vector
as normed spaces.
the subspace of
S(%;F)
sisting of the continuous symmetric m-linear mappings of into
Sas(%;F)
for
Em
Ss(%;F)
is a closed subspace
S(%;F). With each
of
con-
F. Thus,
It is easy to see that
of
Em
consisting of the symmetric m-linear mappings of
F.
into
will be denoted by
x (%;F)).
We denote by
Sa(%;F)
F
into
A,
xl,
A E Sa(%;F)
which we denote by
is associated an element of As,
and call the symmetrization
defined by:
...,xm E
E.
The mapping
A
E Sa(%;F)i--.As
is linear and surjective, and is a projection of the subspace
gas (%;F);
thus
= As
(As) S
E Las(%;F) Xa(%;F)
onto
for every AEX~(~E;@
4
CHAPTER 1
By restriction we obtain a projection of
C(%;F)
onto
Cs(%;F)
s IIAll
for every
IIA,II
which is continuous since
A 5 C(%;F).
If A(x,.T.,x)’ x
m > 0, and
m E N,
m Ax ;
by
if
A E Sa(%;F),
m = 0
we denote
Axo = A
we write
for every
E E.
DEFINITION 1.2
m E N.
Let
an m-homogeneous polynomial from
A E Ca(%;F)
such that
P: E + F
A mapping
P(x) = A x
E m
F
into
is called
if there exists
for every
x E E.
When
A
P
and
A
are related in this way we write
REMARK 1.2:
If A E Ca(%;F)
restriction of
A
and
to the diagonal of
-.L = A 6 x). If Am: E + Em
,...,
Am(x)
r-7
=
lent to
P =
i,
P = A. then
Em, since
P
is the
P(x) =
is the diagonal mapping, x E
for every
E,
then
P =
is equiva-
P = AoAm.
We denote by nomials from
E
ba(%;F)
into
F.
the set of m-homogeneous polyThis set forms a vector space,
addition and multiplication by scalars being defined pointwise. DEFINITION 1 . 3
Let
m E N.
A mapping
P: E + F
continuous m-homogeneous polynomial from exists
A E S(%;F) We denote by
such that
P(%;F)
eous polynomials from
E
P(x) = Ax
E rn
into
is called a
F
for every
if there
x
E E.
the set of continuous m-homogen-
into
F.
This set forms a vector
space, addition and multiplication by scalars being defined pointwise, and is a subspace of
Pa(%;F).
NOTATION AND TERMINOLOGY.
P E r(~;F)
It is easy to see that if IIp(x)l
sup xEE,x~O
=
II xIIm
II p(x)1I
inf{N ;:: 0
5
then
sup II p(x)1I xEE,1I xll!':l
!': Nllxll
P E P(~;F)
and that the mapping
POLYNOMIALS
m
for every
~ Ilpli
x E E} <
ex> ,
is a norm, where IIpli
denotes the common value of the expressions above. REMARK 1.3:
a)
In the case in which
r(~;F)
so too is the space b)
If
F
is a Banach space,
with the norm above.
P E P(~;F),
then
IIp(x)ll!': Ilpllllxll
m
for every
x E E.
c)
m = 0,
In the case
Pa(OE;F)
r(~;F)
as a vector space, and
is identified with
is identified with
F
F as a
normed space. EXAMPLE 1.1
In the case in which
geneous polynomial fro~ = aA
m
E
A E~,
for every
generally, taking
E
=
=F
E
into
F
=~,
every m-homo-
is of the form
and
~
A E~,
for every
F
where
In fact, every mapping A(A l,··· ,Am) = Al ••• Amb is some element of
REMARK 1.4:
If
F,
~.
where a is some element of
b
for all
and
P(A) =
is some element of
A E £a(m~;F)
and thus
A E £a(~;F)
More
an arbitrary normed space,
then every m-homogeneous polynomial is of the form
= bAm
P(A) =
is of the form
AI' ••• ,Am E V, P =
As
.. A
F.
where
b
takes the form
is its symmetrization,
6
CHAPTER 1 A
then
-
A = AS
and taking
.
x
1
T o see this, let
= x2 =...=
PROPOSITION 1.1 Let for
xl,
X
E.
Then
yields
(The Polarization Formula):
m E IN,
...,xm E
xm = x
,.., m E
X ~ ,
m
2
1, A E eas(%;F)
and
P =
i.
Then
E,
We omit the proof of this formula which is purely algebraic in nature. a)
PROPOSITION 1.2:
The mapping
A E .Ca(%;F)
k-i E
is linear and surjective for every
Pa(%;F) m E IN.
b) The mapping A
E SaS(%;F) 1-2 E Pa(%;F)
is an isomorphism of vector spaces for every
PROOF:
a)
m E IN.
Surjectivity of this mapping is an immediate con-
sequence of the definition of an m-homogeneous polynomial;
NOTATION AND TERMINOLOGY.
7
POLYNOMIALS
the proof of linearity is trivial. b)
This mapping is certainly linear, being the restriction
of the mappinggiven in a) to the subspace
X,,(%;F)
of
.Ca(%F) The mapping is surjective since, given by part a) there exists by Remark 1.4,
is = i
Xa(%;F)
A
= P,
and
P E
P =
such that
As E
Pa(%;F),
fi,
and
Xas(%;F).
T o see that this mapping is injective, let
A E .Cas(%;F).
By the polarization formula, 1
A(x~,...,x~) =
for all A(xl,
xl,
..., m ) x
...,xm = 0
c m 2 m! ci=fl
E E.
€
1.
.
C
m
fi ( C lX1+.
.
+& , X , )
A
Thus if
for all
xl,
A = 0, it follows that
...,xm E
E,
and hence
A = 0.
Q.E.D. PROPOSITION 1.3:
a)
The mapping
A E X(%;F)+
fi
E P(%;F)
is linear, surjective and continuous. b)
The mapping A E Xs(”E;F)W
E P(%;F)
is an isomorphism of vector spaces and a homeomorphism. Furthermore,
8
CHAPTER 1
A E Cs(%;F)
for every a)
PROOF:
and
m
E
IN.
This mapping takes its values in P(%;F)
and is
surjective by the definition of a continuous m-homogeneous polynomial; subspace
it is linear since it is the restriction to the
C(%;F)
Xa(%;F).
on
I f i l l ?:
ga(?E;F)
of
of a linear mapping defined
Continuity is a consequence of the inequality
l/All, whose verification is immediate. This mapping is certainly linear, being the restriction
b)
of the mapping given in a) to the subspace
Ss(%;F)
and is continuous for the same reason.
L(%;F),
The mapping is surjective since, given by part a) there exists know that if
A E
of
is = i
X(%;F),
A E P(%;F)
P E P(%;F),
such that
P =
i. We
(Remark 1.4), and it is easy to see that
then
As E Xs(%;F).
Injectivity is a con-
sequence of part b) o f Proposition 1.2, since this mapping is the restriction of the mapping considered there to the subspace
x,(%;F)
of
x,~(~E;F).
Finally, we prove the inequality
\(ill i
m m IIAll s m! I IiII
,
from which it follows that the given mapping is a homeomorm
phism.
We show that
IlAll 5
5I l i l l - as we have already in-
dicated, the other inequality is immediate. and
xl,.. .,xm
whence
E.
Let
By the polarization formula,
A € Xs(”E;F)
NOTATION AND TERMINOLOGY.
9
POLYNOMIALS
1s i s m
1sism
1s is m
Thus, if
/Ixl\l=...=
1lxml( = 1, then
1sism
and hence
REMARK
1.5:
The mapping
in general an isometry. which satisfies:
m,
P(%;F)
is not
In fact the smallest constant
IIAIl s C l I i l I
depending only on
iE
E XS(%;F)++
A
is
m m
-.m !
independently of
E
and
C
F,
This is shown by the follow-
ing example. EXAMPLE 1.2
Let
E
x = (x1,x2,...,xm,...) m
Let
m
be
4,
,
1
the vector space of all sequences
of complex numbers for which m
be a positive integer,
m
2
1,
and let
,...,xm1,...) , x2 = ( x2 p 22 ,...,xm,...),...,xm 2 m m m (x1,x2 ,...,xm,...) be elements of E. We define
x1 = (x1,x2 1 1 =
Am:Em
=
+
d:
CHAPTER 1
10
by :
It is easy to see that from
Em
into
6.
Am
is a symmetric m-linear mapping
We have
1
m!
IIx 1IIIIx 2I1
*..IIXrnll
Thus
Furthermore, taking
,...,xm =
..., (4
(0,
.. .. ) ,
x1 = (l,O,.
0,1,0,...),
,O,.
we have
1 2 x ,x
(1) and (2) together imply that
Now let
im(x)
x E E,
which implies that
IIA,ll
,...,x
m
=
..., ,...),
E E
0
and
1
mT.
.,.,
x = (x1,x2,
?--=--= xlx2...xm. = Am(~,...,~)
x2 = (O,l,O,
xm,...). Then
Thus f o r
x f E
we have
NOTATION AND TERMINOLOGY.
POLYNOMIALS
11
m A
'1 1
ilxl! = 1
1'
x = (m,m,...,m,O,...,O,...)
Taking and
E
Z(E;F)
F
and
112m !I
= - l G .Therefore m
Iim(x)I
Thus we have shown that
If
E E,
1 = - - -m- .
m
,.
mm
we have
IjAmll = xIIAm!l.
are vector spaces we shall denote by
the set of all mappings from
E
F.
into
This set
f o r m s a vector space, addition and multiplication by scalars
being defined pointwise.
REMARK 1.6:
Let
B
be a vector space and
a se-
C Bm the set o f mE IN the finite sums which can be formed with elements chosen from
B.
quence of subspaces of
the subspaces exist integers
We denote by
. In other words, x E m€C ml,m2, ...,mk and elements
Bm
Brn N x
,...
such that ,X E Bm m E Bmk 2 k 2 C Bm is a subspace o f B Then m€ N the algebraic sum of the subspaces xm
E Bm , 1 1 x = xm + x xm l m2 k it is referred to as m
+...+ .
-
Bm.
An algebraic sum o f subspaces
Bm
a direct algebraic sum if, whenever the
m E N,
9
m
j
implies
N,
are pairwise
xm
@ Bm , and we say that the subspaces rnE [N are linearly independent.
notation m
is called
= x =...= x = 1 m2 mk In the case of a direct algebraic sum, we employ the x = 0
distinct, the condition
= 0.
if there
Now let
E
F
and
sider the sequences their algebraic sums
C
of
B,
be normed spaces, and let us con-
pa(%;F) m€N
Bm
and
Pa(%;F)
fJ(%;F), and
C
mEIN
rn € I N ,
P(%;F)
and within
12
CHAPTER 1
the vector space
1.4.
PROPOSITION
3(E;F). The subspaccs
m E N,
Pa(%;F),
of
5(E;F)
are linearly independent. PROOF:
It suffices to prove the following statement f o r
every
m E
P = P
+ P1 +...+
for
j
m = 0
P j E pa('E;F),
if
(N:
the condition
Pm
= 0,1,...,m.
P = 0
m
m-1,
2
1,
and prove it for
P
j
= 0
For
m.
According-
and suppose that
X E a!
m
c
implies
We assume the truth of the
l y , let P . € P(jE;F), j = 0,1,..., m, J m C P j = 0. Then for all x E E and j=O
(a)
and
We s h a l l prove this by induction.
there is nothing to prove.
statement for
= 0,1,..., m ,
j
we have
m
o
pj(x) =
and
(b)
j=O
Pj(Xx) = 0 .
C j=O
Multiplying equation (a) by
XIn
and subtracting the
result from equation (b) we obtain: m-1 C (hm-hj)Pj(x) = 0
(c)
f o r all
X E
d:
and
x E E.
j=O
We now choose a value o f tion of any of the
m
equations
X E CC Xm-XJ
which
i s
not a s o l u -
= 0, j = 0,1,...,m - 1 .
Then the induction hypothesis applied to the relation m-1
C
(hm-hj)Pj
= 0
j=O j
= O,l,...,m-1,
Pm = 0.
given in (c) shows that P = 0 for j rn and then, since C P . = 0 , we also have j=O
J
Q.E.D.
COROLLARY 1.1
The subspaces
P(%;F),
m E IN,
of
3(E;F)
are linearly independent. PROOF:
This is a straightforward application of the proposi-
NOTATION AND TERMINOLOGY.
tion, using the fact that
P(%;F)
POLYNOMIALS
C
13
for every m EN.
Pa(%;F)
Q.E.D.
1.7
REMARK
In the language of algebraic s u m s and direct al-
gebraic s u m s Proposition 1.4 and its Corollary state that
and =
C 63(%;F) m€ Ui
Pa(E;F)
We denote by
Pa(%;F)
63
subspace
CB P(%;F). mEN
(respectively, P(E;F))
(respectively,
m€ N
DEFINITION 1.4 E
from
into
P(%;F)).
CB
m€N
An element o f
F,
pa(E;F)
is called a polynomial
P(E;F)
and an element of
is called a con-
tinuous polynomial from
E
into
Thus to say that
P
is a polynomial from
means that either
P = 0, or if
F.
#
P
written in a unique way in the form where
Pj E Pa(JE;F),
j
P.
m
+
P = Po and
#
into
F
can be
+...+ 0.
Pm, I n the
is called the degree o f the
P = 0 is
By convention the polynomial
assigned the degree
P1
Pm
E
P
0, that
= 0,1,..., m,
latter case the natural number polynomial
the
-1: m
PROPOSITION 1.5 pj
pa( j E ; F ) ,
If P E Pa(E;F) then
P E P(E;F)
and
P =
C j=O
Pj,
if and only if
where
P . E P(jE;F) J
for every
j
= 0,1,...,m.
The p r o o f of this proposition is similar to that of Proposition
1.4 and will be left to the reader to carry out,
as will the proof of the following:
14
CHAPTER 1
PROPOSITION 1.6
E, F
Let
(a)
If
P € Pa(E;F)
(b)
If
P E p(E;F)
and
G
be normed spaces.
Q E P,(E;G),
and
then
Q E P(F;G),
and
then
E Pa(E;G).
QoP
QoP t P(E;G).
We state without p r o o f the following:
If
llPROPOSITIONA :
m E
A € Cs(%;F),
N,
the following are
equivalent : a)
A
is continuous.
b)
A
is continuous at one point of
c)
There exists a neighborhood
such that
A
is bounded in
V
Em. Em
of the origin in
V."
This is used to prove PROPOSITION 1.7
If
P E pa(E;F)
the following are equivalent:
a)
P
is continuous.
b)
P
is continuous at one point of
c)
There exists a non-empty open subset
that
is bounded in
P
PROOF:
E. U
of
E
such
U.
The implications a)
3
b) a c) are easily verified.
We prove the implication c) a a). By c), there exists a non-empty open subset
M
and a constant 1)
I\P(X)~ Since
t: E
-t
E
2
s M
U
0
of
such that
for every
x E U.
is non-empty there exists
be the translation:
1) is equivalent to
U
t(x)
= x-x
0 )
x
0
E U.
x E E.
Let Then
E
NOTATION AND TERMINOLOGY.
llPot-'(y)/I
2)
5
M
f o r every
neighborhood of the origin in
E
polynomial from sition 1.6, that
ll~(~)ll
s M
Q =
t
is a
is a continuous
y E V.
P = Qot,
and
Q.
continuity of
P
We shall prove that
is Q
is
We begin with the following assertion:
j = O,l,...,m,
and
bounded in a subset if
Since
V
and
is a polynomial, and, by 2),
equivalent to continuity of continuous.
t(U)
( o f degree 1) we have, by P r o p o -
-1
P o t
for every -1 Q = Pot
AS
E
into
y f V =
E.
15
POLYNOMIALS
Qj
Qm
#
V
of
is bounded in
V
0,
then
Q
is
if and only
E
for every
j,
j = 0,1,..., m."
If the Q
Q j , j = O,l,...,m,
is certainly bounded in
by induction.
The case
V
are bounded in
also.
m = 0
V
then
We prove the converse
is trivial.
Assuming the
truth of the statement for all natural numbers less than
m
2
1, we consider the case
m.
m,
Then:
m
3)
Q(x)
=
c
and
Qj(X)
j=O
for all
x E E
and
1 E
C.
From 3 ) and
4) we obtain
m-1
5)
im Q(~>-Q(A~) = c
(~"-x~)Q~(X)
j=O
for all
x E E
and
1 E
C.
We choose a value for
which is not a solution of any of the equations
X E
Xm-Xj =
(c
0,
16
CHAPTER 1
j = O,l,...,m-1.
The polynomial m-1
C
(h"-XJ)Qj(x)
j=O
has degree at most
Q
hypothesis,
is bounded in
V,
is bounded in
V
V
and is bounded in
m-1,
and hence
for each fixed value of
since, by
h"Q(x)
-
With
h
1.
Q(Xx)
chosen
as indicated, it now follows from the induction hypothesis is bounded in
that j'
V
for
j = O,l,...,m-1,
and then
siricc m-1
Q,
V.
is also bounded in
This proves assertion ( * ) . a
A j E Sas(jE;F)
Let
with
= A J.
j '
for
j = O,l,...,m.
Then
7)
Q(x)
=
m C
m Qj(x)
=
j=O
Now let
W
C A.xj. J j=O
be a balanced neighborhood o f the origin in
E
such that
,--. w
for every
...,m.
j = 1,
j
+...+w c
Then if
v
...,xj) E
(xl,
j
WJ
w e have
.
C cixi E V if e i = k l , f o r every j = l,.. ,m. Using i=l the polarization formula, we conclude that A is bounded in j the neighborhood Wj of the origin in Ej, for j = 1,. ,m.
..
A.
is constant, and hence is continuous. It now follows from Proposition A above that
continuous for every
7),
Q
is continuous.
j = O,l,...,m.
Q.E.D.
A
j
is
Therefore, by relation
CHAPTER 2
POWER S E R I E S
D E F I N I T I O N 2.1
A power series from
point
5 E E
where
Am E X s ( % ; F )
E
into
m = O,l,...;
about the
x E E
is a series in the variable
for
F
of
the form
if we prefer to use
polynomials the series can be written: m
72
Pm(X-%)
m=O A
where
Pm = Am
for
sponding polynomials
m = O,l,...
.
The
Am
,
o r the corre-
P m , are often referred to as the coef-
ficients of the power series in question. D E F I N I T I O N 2.2
The radius of convergence, or the radius of
uniform convergence, of a power series about the point is the supremum of the set of numbers
0 s r .s
r,
that the series is uniformly convergent in p,
0
gp(S)
5 E
E
such for every
s p < r. A power series is said to be convergent, or uniformly
convergent, when its radius of convergence is greater than z e r o , that is, when there exists
converges uniformly in
Ep (5 )
.
p > 0
such that the series
CFIAPTER 2
18
C
F o r power series from
into
(c,
the radius of con-
vergence can be calculated by means of the Cauchy-Hadamard formula.
E
When
is a normed space, and
F
a Banach space,
we have the following: PROPOSITION 2.1 (Cauchy-Hadamard).
The radius of uniform con-
W
c ~ ~ ( x - 5 from ) m= 0 is given by
vergence o f a power series
5 E E
about the point
In this expression,
REMARK 2.1:
r
into
E
F
is taken to be 0 or
if
is infinite or zero respectively.
lim SUP lIP,l/ m-tm
PROOF:
a)
Given
LE
for which each
lim sup l\Pm]ll’m = m . m-w M(L) the set of all m E N
We considcr first the case El,
L > 0 we denote by
llPml\l/m > L .
m E M(L)
\lPm(tm)/l > Lm.
Then tm E E
choose Let
m E M(L).
Then
for every
m E M(L).
p
M(L)
is an infinite set. )Itml/ = 1,
with
= 1/L, and let
xm =
such that
5 + ptm
for each
This shows that the power series in ques-
tion does not converge uniformly in
Bp(5),
since its general
term does not converge uniformly to zero in this ball.
L
For
Since
is an arbitrary positive real number, we conclude that the
series does not converge uniformly in any ball p > 0.
b)
gp(s),
p E R,
Hence the radius of convergence o f the series is zero. Suppose now that
lim sup llPm\\l’m = 0 . m-w
POWER SERIES
Given
p E R,
number of values of
t E E,
E = -
> 0, let
p
/(tll s 1,
m,
2P
m E IN,
m,
f o r all but finitely many
For all but a finite
*
we have
x = 5
and let
19
+
pt.
l\Pmlll’m s
E
.
Let
Then
rn E IN,
and hence the given
B p ( ~ ) . Since
power series converges uniformly i n
p
is an
arbitrary positive real number, we conclude that the radius of convergence is infinite. c)
If
Finally, we consider the case
p E R
and
8 E R,
exists
0
< p < l/h,
then
0 < 8 < 1,
A < l/p,
jlPm// s (f3/p)m.
Thus if
A < e/p.
such that
all but a finite number of values of t E E,
and so there
m,
Hence, for
m E N,
we have
and
x = 5 + pt,
)/tl/s 1,
then
for all but finitely many
m,
m E N,
power series converges uniformly i n radius o f uniform convergence,
r,
and hence the given
Ep(5).
satisfies
If, on the other hand, we take and so there exists a n infinite subset
/IPmlI> ( l / ~ ) for ~ I(tm(l = 1,
tm E E ,
xm = 5
+
ptm
,
m E M.
For every
such that
we then have
Therefore the
llPm(tm)l/
r ;r l / A .
p > l/A,
M
of
m E M
then [N
> l/p,
such that
there exists
> ( l / ~ ) ~ .Taking
CHAPTER 2
20
for every ly in
m E M,
g,(g).
and so the series does not converge uniform-
r s l/A.
Thus
r = 1/A.
Therefore
Q.E.D. m
REMARK 2.2:
C Pm(x-5) from E into F m= 0 the radius of Convergence is unchanged
F o r a power series
5 E E,
about a point if the norm on
F
is replaced by an equivalent norm;
E
this radius can change if the norm on
however,
is replaced by an
equivalent one. One can not, in general, replace the polynomials
Pm
A
by the corresponding
Am,
Pm = A m ,
in the formula for the
radius of convergence given in the preceeding proposition. This is illustrated by the following example: EXAMPLE 2.1
E
Let
be the space
C1
of absolutely summable m
IIxII =
sequences of complex numbers, with the norm
x = (x1,x2 ,...) E L 1 ,
for
we define
where
Am: Em
j x j = (xl
3
F
F = a.
and let
,...,xm’j ...) E
E,
Am
j = 1
A.
=
m
,...,
m.
is m-linear and symmetric,
and, as in Example 1.2, we find that
am^^
Am
is continuous, and
m
inT9
= 0, and s o
m E
,...
by
It is easy to see that
that
For
C lxml m= 1 m = 1,2
(N,
IIAo/l = 0 .
m 2 1. For
m = 0, we take
POWER SERIES
21 A
The continuous rn-homogeneous polynomial sociated with
Am
x
= (xl,
as-
is given by
P,(x) for
Prn= Am
...,xrn,...) E
m
= rn xl...x
E
rn
rn = 1,2,...
and
.
If
rn = 0,
Pm =O. Again following Example 1.2 we find that \lPm/l= 1 f o r
.. .
rn=1,2,.
Thus
but
m
m
COROLLARY 2.1 series about
Let
E E.
=
C P,(x-g) rn=O
C
rn
Arn(x-5)
be a power
rn= 0
Then the following are equivalent:
a)
The series is uniformly convergent.
b)
The sequence
{ \lPmll'/"'I
m E N,
is bounded.
c)
The sequence
[~~Aml~l/rn}, rn E N,
is bounded.
PROOF:
,
Statement a), that the radius of convergence is
lim sup \lpml/ l/m E R, rn-w and this in turn is equivalent to the assertion that the se-
greater than zero, is equivalent to
{~lP,,,~l~m], m E N,
quence
is bounded.
Hence a) and b) are
equivalent. The equivalence of b) and c) i s a consequence of the relation
and the fact that the sequence
[(rnm/m!)l'm],
m E IN, which
CHAPTER 2
22
converges to
PROPOSITION
e,
Q.E.D.
is bounded.
2.2
Let
E P,(%;F),
pm
5 c
m E IN,
E,
p
7
0,
a
C Prn(x-5) = 0 m= 0 for every m E N.
and suppose that Pm = 0
Then
PROOF:
x E Bp(5)-
for every
We begin by proving the following:
c 'lm3 mELN
"If
is a sequence o f elements o f m
6 > 0, and
rn
X 'um = 0
C
F,
X E G,
for every
m=O
1x1
s 6,
then
urn = 0
u 0 = 0.
= 0, w c obtain
Taking uo = u1 = * " =
k-1
--
0
for
m E IN."
for every
k
We shall prove that
1.
2
Suppose that
uk = 0, and our claim follows by induction: m
Since
m
rn
6 um = 0, lirn 5 urn = 0, and hence
C In= 0
In+-
1)
m
L = sup I/Um116 <
+m.
mE N By the induction hypothesis, 0
<
1x1
5
6,
m
C m=k+l
m-ku m
for
,,
uk = 0.
t E E,
let
t
#
0, h E C ,
a
hypothesis,
-
and s o by l),
it follows that
NOW
uk =
0
=
C m= 0
and
x =
5 + At- BY
m
Prn(x-S) =
C hmPm(t) m= 0
when
\lx-gI/=
POWER SERIES
= [XI/Itl] < p ,
t h a t i s , when
result proved above w i t h f o r every Pm = 0
m E N.
for e v e r y
23
< Plltl1-l.
urn = P , ( t )
we f i n d t h a t
Since t h i s holds f o r a l l
m E IN.
Applying t h e
Q.E.D.
t E E,
P,(t)
= 0
we have
This page intentionally left blank
CHAPTER 3
HOLOMORPHIC MAPPINGS
DEFINITION 3 . 1 in
5 E
if f o r every
U
f: U
A mapping
Am E X s ( % ; F ) ,
m €
.-t
F
is said to be holomorphic
there exists a sequence
U
and a number
(N,
p E R,
p
Bp
(5 )
U
C
Am(x-5 )
C
and the power series
mEN
9
> 0, such
m
that
C
m
from
E
m=O into
converges uniformly to
F
f(x)
Bp(5).
in
It follows as a consequence of Proposition 2.2 that if f: U + F
is holomorphic in
the sequence
U,
{A } nc(N
Am
E es(%;F),
associated to each
5
E U
m A
Pm = A m
If
C
then
=
A,(X-S)~
m=O
called the Taylor series of
is unique. a
f
C P,(x-S) m=O
about the point
is
g E U,
and
we write m
f(x)
-2
m
C Am(x-5)m m= 0
The set
H(U;F)
are holomorphic in
U
or
f(x)
c ~~(x-5).
I
m=0
of mappings from
U
into
F
which
forms a complex vector space, the
operations of addition and multiplication by scalars being defined pointwise. REMARK 3 . 1 :
We can also formulate a definition of holomorphy
at a point:
f: U
-t
F
is said to be holomorphic at
g E U
CHAPTER 3
26
if there exists a sequence p E IR,
and
>
p
Bp(5) c U
such that
0,
and the power
m
c
series
~ ~ ( x - 5converges ) ~ uniformly to
f(x)
in
1.
1 3 (~5
m=O
It is shown in Nachbin [ 9 0 ] that when f: U + F ,
space and
the set of points in
is a Banach
F
U
at which
f
is holomorphic is open.
In the prcceeding definitions of a holomorphic mapping, whether in an open subset or at a point, one can dispense with the condition that f
that
Am,
m E IN,
is continuous on
In fact, in this case there
U.
BD({),
exists an open ball
be continuous, if we assume
for
5 E
m
bounded in Pm
f.
Therefore, by Proposition 1 . 7 we have that
Bp(S).
(and hence also
is continuous for every
Am)
We note too that the set the norms o n DEFINITION 3 . 2 m
C A,(x-Z) m=O
where
and
E
is bound-
m
C Am(x-5) = C Pm(x-5) converges m= 0 m=0 This implies that each P m , m f N, is
ed and the power series uniformly to
f
where
U,
m
F
remains unchanged if
are replaced by equivalent n o r m s .
f E #(U;F),
Let
g(U;F)
m E N.
5 E U, and let
m
m
=
C
Pm(x-s)
be the Taylor series of
f
at
5 ,
m=O *
Pm = A m , m E N.
Then
a)
dmf(S)
= m!A,
b)
Z"f(5)
= m!im = m!Pm E P ( % ; F )
E 2 ("E;F)
and
are called the differential of order
m
of
f
at
g.
In a)
the differential is viewed as a continuous symmetric m-linear
ISOLOMORPHIC M A P P I N G S
27
mapping, and i n b ) a s a c o n t i n u o u s m-homogeneous polynomial.
;mf(5 )
Note t h a t
6 ' 1
i s an a b b r e v i a t e d n o t a t i o n f o r
The T a y l o r s e r i e s o f
at
f
s
d
f(5).
can n o w be w r r i t e n a s
or m
f(x)
1
c
2
m= 0
m T
m,f,5
C
m
k=0 of f
degree
If f
1
(x) =
f
o r order
zkf(S)(x-4;)
5
at
E
m
we can c o n s i d e r t h e d i f f e r e n t i a l s of
N,
d m f : 4;
a s mappings d e f i n e d o n
E
5 E
;t"f:
i s t h e T a y l o r polynomial o f
a
E 3J(U;F), m,
m! 2 " f ( S ) ( X - S ) .
UH
d m f ( 4 ; )E Ss("E;F)
U H
;zmf(g)
U:
E P(%;F).
W e can go a s t e p f u r t h e r and d e f i n e t h e d i f f e r e n t i a l
m
o p e r a t o r s of o r d e r
on
S((U;F):
dm: f E 3 J ( U ; F ) w dmf E S ( U ; S s ( % ; F ) )
;m:
f
E
m > 0,
DEFINITION 3.3
Let
ed s p a c e s , and
A E Sa(E1
and
x1
1)
E
E1,...,xh
If
h = m,
E
If
h
< m,
Eh,
A(xl,
value o f t h e mapping 2)
zmf E
#(U;F)I+
A
A(xl,
let
,...,
Z(U;P(%;F)).
El,
Em;F).
A(xl,
...,Em For
...,
xh)
...,
xh) = A(xl,
a t t h e m-tuple
...,
xh)
and h
E
N,
F
be norm0 :E h i m ,
i s defined a s follows:
...,
xm) E F
(X
l,...,~m).
i s t h e mapping f r o m
i s the
CHAPTER 3
28
X
...x
Em
F
into
given by
It is easy to see that a)
x...x
...,xh)
A(x~, Em
b)
last
F,
into
If El =...=
m-h
Em = E
A(xl,. h = 0
we set
A E Sa(%;F)
Axo
for every
= A.
h E N,
0
,..., r E hl hr Ax1 ... x r h
DEFINITION f
N,
3.4
1
L
is symmetric in the
A
Ii
x1 =.
..=
E N,
1
xh = x E E L
h
5
m,
and for m E IN,
Axh E La(m-%;F)
then
we write
is defined
h s m.
?;
1
hl
+...+
r
4
h
r
L
m,
= h
5
x1 m,
,...,xr E
E
and
we define
to be the mapping
A mapping
is holomorphic in
E.
f: E + F
#(E;F)
space of entire mappings of operations.
or, more gen-
With this convention, if
If A E ga(%;F), hl
and
x E E,
and
and
X
A(x l,...,~h)E gas (m-hE;F).
Eh = E
..,xh) = Axh
for every
Em = E
Eh+l
is continuous.
A
A E eas(%;F)
and
variables, then
If El =...=
mapping of
which is continuous if
=...=
Eh+l
erally, if
is a n (m-Ii)-linear
E
is said t o be entire if
denotes the complex vector
into
F,
with the usual
29
HOLOMORPHIC MAPPINGS
PROPOSITION
3.1
cisely, if
m E IN,
if
k E
[N,
0
if
k E
[N,
k
P(E;F) A
E .@,(%;F),
k < m,
: 8
P =
i
and
More pre-
E E,
then
and
> m. P(%;F)
PROOF:
It suffices to show that
rn E
[N.
This is obvious in the cases
rn
2,
let
2
#(E;F).
is a subset of
A E Es(%;F)
and
P =
c #(E;F)
m = 0
i.
for every m = 1.
and
Then for every
For
3 E
E
we have m C
P(x) =
m
( k ) Agmmk(x-5)
k
k=O
x E E.
for every
5
f
of
E, P
P
This shows that
is holomorphic at every
and that the radius of convergence of the Taylor series at every
5 E E
is infinite, since this series is fiP(x)
nite and hence converges uniformly to every of
P
in
g,({)
for
> 0. The expressions for the differentials
p 5 R,
p
at any
g E E
follow directly from the relation above.
Q.E.D. REMARK 3 . 2 :
In the case in which
f E #(E;F),
the radius of convergence of the Taylor series of
5
has finite dimension and
5 E E.
If the dimension of
f
at
E
is infinite, the radius of convergence of the Taylor series
of
is infinite for every
E
f E #(E;F)
is either finite at every point of
finite at every point of
E.
E
or in-
The second possibility charac-
terises the entire functions of bounded type, which we shall
CHAPTER 3
30 consider later. COROLLARY 3.1
Let
m f IN,
P E p(%;F)
and
E
k
N,
0 S lc s m.
Then
P R O P O S I T I O N 3.2
Let
E, F
non-empty open s u b s e t of t o f E S((U;G)
Then
5 E
U
and
PROOF:
If
and
E,
and f
be normed spaces,
G
E
= to(dmf(g))
dm(taf)(g)
5 E U,
Bp(Z) C U
uniformly i n
there exists
p
E IR,
uniformly in
Z ) ,
L(F;G).
for every
dmf(5)
Bp(5).
dm(tOf)(s)
m € N
t: F + G
is linear and continu-
F,
1) implies that
It is easily seen, using Proposition 1.6,
E Ss(%;G)
tof E #(U;G),
f o r every
and
B p ( % ) . Since
to(dmf(g))
we have
E
> 0 and a con-
p
o u s , and hence is uniformly continuous in
that
t
a
m E N.
tinuous symmetric m-linear mapping such that
and
S((U;F)
U
for every
m E IN.
Therefore, by
and by the uniqueness of the Taylor series
= to(dmf(5))
for every
5 E U
and
rn E N.
QeEoD.
CHAPTER
4
THE CAUCHY INTEGRAL FORMULAS
REMARK
4.1:
In this chapter we make use of some properties of
integrals o f functions defined on a subset of
IR
CC
or
with
values in a real or complex Banach space (see Dieudonne' [ 2 8 ] and Hille [ 5 7 ] )
.
PROPOSITION 4.1 (The Cauchy Integral Formula). Let
f
E
(1-X)g
such that
REMARK 4.2:
5 E U, x E U
#(U;F),
+ Ax E
Although
U
F
for every
and
p E R,
h E C,
1x1
p
s p.
> 1 Then
is not necessarily complete, the
existence of the integral in ( * ) is guaranteed, since we may consider
f
identifying
as taking its values in the completion
F
with its image in
metric inclusion Iof,
and as
of
F,
F
under the natural iso-
I. The relation ( * ) will be proved for
(Iof)(x)
= f(x)
tegral in (*) is an element of PROOF:
i
A
F.
By the preceeding remark,
sider the case in which
F
it follows that the in-
F,
it is sufficient to con-
is a Banach space.
use the following well-known result:
We shall make
32
CHAPTER
4
a) Let V be a non-empty open subset of 6, z E V , 6 E IR, 6 > O , such that 6 ( z ) c V , E 6 ( z ) being the usual closed ball of radius 6 p and center z in C, and Set g
5 , x,
With V = (h E
EP(O)
(I:
V
Z
and
+
h x E U]
E D6(z).
is an open subset of
1 E Dp(0).
If
+ Ax]
1 E V,
g E #(V;F).
T
Then
as in the statement of the proposition,
: (1-1)s
g ( h ) = f[(l-l)z
that
p
E s(V;F) and
for
g: V
Applying a) to
-t
F
C,
is defined by
then it is easy to see g,
T = 1,
with
we obtain
PROPOSITION 4.2 (The Cauchy Integral Formulas). Let
f
E
3S(U;F),
such that
5 + l x E
for every
m E IN.
PROOF:
U
g E U,
x E E
for every
p
and
h E 6,
1x1
E iR, s p.
p > 0
Then
A s in the preceeding proposition, it is sufficient to
consider the case in which
F
is a Banach space.
We shall make use of the following well-known result: a)
Let
r,R E R, c V.
V
be a non-empty open subset of
0 < r < R,
Then
a E V,
such that
C,
g E #(V;F),
E C: r s
/),-a\ s R]
THE CAUCHY INTEGRAL FORMULAS
NOW
subset of
v C,
by
64x1
If
0< 0 < p
=
[A E V
and
= f (hm+l s+xx)
#
C: 2
,
P
(0)
h E V,
-
(01.
is a non-empty open
If
g: V
-I
V
is defined
it is easy to see that g E #(V;F).
{A E C :
then
E U]
0 , s+Xx
33
0
1x1
s
I;
p} c
V
and s o , by a),
we have
f o r every
m E N.
The Taylor series o f
at
f
5 ,
C
P&(Z-5;),
con-
&=O
f(z)
verges uniformly t o With
x f 0, choose
E
in a ball
E IR,
0
<
0
Bu(5;),
< p,
u E iR,
u
7
0.
sufficiently small
m
so that the series
C
P,(Z-T)
converges uniformly to
f(z)
&=O
in the closed ball with centre
5
and radius
~llxll. Then if
Therefore, from (*), we have
Since the series converges uniformly in this yields
{hEC;
IXI=c],
4
CHAPTER
34
x = 0, the proposition is trivial.
For
REMARK lc.3:
In the case
corollary: p 7 0
if
m = 0, we obtain the following
f E #(U,F),
x E E,
E U,
g+Xx E U
such that
Q.E.D.
for
x
1x1
E 6,
and
p E R,
s p,
then
f
The following proposition is a generalization of P r o postion 4.2:
4.3
PROPOSITION
k E IN,
and
1
Let
4,
+
(xl
S
pj,
such that
E
N,
m z 1,
a k-tuple o f elements of
k
c
Xjxj E U
f o r every
j=1
lhjl
m
E,
a k-tuple o f strictly positive real numbers
(pl, . . . , p k )
such that
,...,xk)
k s m,
5
E H(u;F), 5 E U,
f
j = l,,..,k.
Then if k In( = C n . = m, j=1 J
-
nl!
..
n =
n dmf(6)x1 1
1
.nk
...,h k ) (nl,...,
...
(hl,
E Ck with
nk) E Nk
n
x
k
-
-
is
35
THE 1:AUCEIY INTEGRAL FORMULAS
REMARK
4.4:
In the case
k = 1
the expression which appear
in Proposition 4.3 reduces to that o f Proposition 4.2, the expression for the coefficient o f order
m
of the Taylor
series taking the form o f a homogeneous polynomial, and the
In the other ex-
integral being a simple contour integral. k = m,
treme case,
we obtain m
d mf ( 5 ) (xl,.
..
r(5+ c 1
,X,)
XjXj)
j=1
= - - -( ~ n i ) ~
x, J
2
... xm
dX l...dhm
.
lf;jsm
The following proposition is a consequence of the Cauchy integral formulas: PROPOSITION 4.4 (The Cauchy Inequalities). { E U
and
every
m E IN,
PROOF:
Let
p
E El,
t E E,
p
>
0
such that
Let
gp(s) c
IItlI = 1, and let
U.
f E #(U,F), Then for
x = 5 + Pt E
Bp(5:) cu.
Then, by Proposition 4.2, f
and hence
Since this holds for every
t E E
with
IItl( = 1
we have
4
CHAPTER
36
Q.E.D. REMARK 4.5:
Taking into account the inequalities mm
1/41s /IAll valid f o r every
m! ll All 6
9
we can write the Cauchy ine-
A E dis(%;F),
qualities in the f o r m
where
f 6 #(U;F)
and
The constants
rn E IN.
1 --
and
Pm
m m . -which appear respect-
1 -pm
m!
ively in the Cauchy inequalities and in ( * ) are the least possible universal constants in these inequalities.
PROPOSITION 4.5 p
Let
> 1, such that
Then f o r every
g E
f E W(U;F),
(1-1)s + Xx E U
U,
x E U
f o r every
X 6
m E IN,
where
PROOF:
For
f o r every
1 E C,
m E IN.
#
0
and
7,
#
1, we have
and C,
p E R,
1x1
4
p.
37
THE CAUCEIY INTEGRAL FORMULAS Multiplying ( * ) by
2~1
f[ (1-X)g +Ax]
1x1
the resulting equation over the circle
fc
1
(**)
2n i
( ' 1
)5+xx1
x -1
and integrating = p,
we obtain
dX =
I 1 I'P m
'
k=0
f
f
fC-_
1 Zni j *
(1-X -
)g+xx1
xk+l
dl
fr c?I?L)g 5x1d h .
1
+
2ni
I x I=P
P+l(bl)
IxI=p
Applying Proposition 4.1 to the left hand side of (**) and Proposition 4.2, with
(x-5)
in place of
x,
to the
first term of the right hand side (noting that (1-1)s
5 + X(X-~)),
we obtain the desired relation.
COROLLARY 4.1
+
Ax = Q.E.D.
Under the hypotheses o f Proposition
4.5 we have
the following estimate f o r the norm of the Taylor remainder of order
for
m
PROOF:
f
at
5:
This follows immediately from Proposition
4.5, using
the usual inequality for the norm of an integral and the fact that
inf
Ix I = P
11-11 = p - 1 .
COROLLARY 4.2 r > 0,
Let
such that
f E #(U;F),
Q.E.D.
m E N,
5 E
B r ( 5 ) c U, and let x E
U
and
Br(5).
r E R,
Then
38
CHAPTER 4
x = 5 ,
If
PROOF:
=ma r
let
1x1
the inequality is trivial.
?;
p.
Then
p
(1-X)5 +
> 1 and
Thus, by Proposition
For
Ax E U
x
# 5,
if
X E
C,
4.5, r-
Applying the usual inequality for the n o r m of an integral, and the fact that
which implies
we obtain:
The corollary now follows when
-
II
r ~
p.
is substituted for
X-lli
Q.E.D.
REMARK
4.6:
mappings o f
C o n s i d e r the vector space U
into
F.
C(U;F)
of continuous
For each compact subset
K
of
Uy
the mapping
is a seminorm on
c(u;F).
The separated locally convex topology defined on C ( U ; F )
39
THF: CAUCHY INTEGRAL .FORMULAS
(pK: K
by the family of seminorms
compact,
K c U)
is known
as the compact-open topology. PROPOSITION 4.6
F
#(U;F)
is complete, and
then
#(U;F)
is a vector subspace of
C(U;F)
If
C(U;F).
carries the compact-open topology,
is a closed vector subspace of
C(U;F),
and
hence is complete in the induced topology.
If
PROOF:
f E #(U;F),
that the Taylor series of ly in that
Bp(5).
if
f(x)
Ilx-5ll < p
ous at
1 1 5 2"f (5 )I1
Po(x-%) = P o =
formly to
5
,
5
at
f
and
converges to
in
and
s C C
and
< 1.
Pm(x-5)
It follows that
F
is a Banach space.
complete by the following result: locally compact space, and
Y
"If X
Topologie General, Chapter 10).
m
2
1,
g E U.
we define
m
[N,
such 2
1.
converges uni-
f
is continu-
C(U;F)
is
is a metrizable or C(X;Y)
(see Bourbaki [ 1 9 ] ,
To prove that
#(U;F)
is
for the compact-open topology, we take
E #(U;F) and show that Let
uniform-
C > 0
a Banach space, then
is complete in the compact-open topology"
f
f
such
#(U;F) c C(U;F).
and therefore
C(U;F)
m E
> 0
m= 0 we have
B,(g),
p OC
for every
p
a
f(s),
Suppose now that
closed in
p E R,
By the Corollary 2.1, there exists
llPm( l/m =
Since
5 E U,
let
f E a(U;F).
We set Am: Em
-b
f(s),
A.
=
F
in the following manner:
and f o r each
m
E
[N,
40 if
CHAPTER
,...,xm)
(xl
E E
m
,
4 (pl
there exists an m-tuple
,...,
)p,
of strictly positive real numbers, which we shall call an m-tuple associated to
(xl,
m
c
5 +
ljxj E
u
...,xm),
xj
if
j=1
such that
Jijl
E C,
j = ~~...,m.
s pj,
W e then define
m a)
...
A ~ ( x ~ , ,xm) =
__
1 -
m
m ! (2ni)
[
f(%+ I
’Ixjl=Pj
_
c
x .x.)
j=1 J J
- =xm)
-
(11.
dX1e..dXm
IS j s m We note first that the integral in a) is defined since f
is continuous and
F
is complete.
We show next that the
value of this integral is independent o f the choice of the ( p l , ...,p,)
m-tuple clear for
associated to
f E #(U;F)
as was noted in Remark
fixing
(xl,
...,
...’xrn).
(xl,
This is
since, in this case,
4.4.
xm) E Em,
Consider the following mapping: and an associated m-tuple
1s j r m This mapping i s easily seen to be continuous, and by Remark4.4,
THE CAUCHY INTEGRAL FORMULAS
,..., (xl, ...,
if to
and
p,)
(pl
,...,
are two m-tuples associated
p&)
then
xm),
Since
( p i
41
TPl,
...,Pm;xl’...7x
T
and
m
,p;ixl,
PLY
,xm
are continuous, it follows that
Am
Therefore Am
is symmetric and m-linear. P1’
prove only that
(xl,
elements
(xl,
E E
J
(pl, ...,p
...,x
j
g E W(U;F).
and
(xl,
J
...,
xm)
of
Em,
..., .,....,xm ) X‘
J
(g)
,Pm;xl’
9
xj *
xm ( g )
Tpl
+
m
Let
,...,
..., ,...yxm
pm:x1y
..,pm;xl,. ..,x .+XI.,...,x T ...,x .,...yx T ply ..., ...,x;, ...,x T p l,.
J
p1,”’,Pm;Xl’
PmiX1,
and
=
X I
j
Therefore b)
we shall
we have
p l , ...,~m;xl,...,xj+x~,...,x
9
; YX m
Choosing an
T
T PI’
.
which is associated to each of the three
m)
...,x.,...,~~),
+ x>,
,PmiX1’
is additive in each variable.
Am
...,x.,x>, ...,xm
m-tuple
This is proved using
..
T
the continuity of the mapping
x1,x2,
m E N.
is well-defined f o r every
-
J
+
J
W(U;F)
0
(8).
42
CHAPTER
4
From b ) , and t h e c o n t i n u i t y o f t h e t h r e e mappings which appear, w e have TP
I , . . . ,p r n ; x l
,...,x
,...,xm / w ; F j -
+x;
----
j
...,
x
Am(xl,
j
,
f E #(U;F)
Therefore, as
+X’.,...,X
we have
rn ) = T
pl,.
..,
pm;xl,...,x
Analogous c o n s i d e r a t i o n s show t h a t v a r i a b l e , and symmetric.
Thus
W e prove next t h a t P
r e a l number and 3 ) sup
II t-5lIhP
>
0
such t h a t
1)
0
5:
M,
and l e t
pm)
s t r i c t l y p o s i t i v e real numbers with
g,(5) c
if
U,
(xl,
...,
x ) E Em m
m then
5 +
ljxj E U
C
< p < 1,
...,
(pl,
p1
with
(Al,
for every
j=l
lXjl
s Pj,
associated j = l,...,m.
we have
1 c j
5
m.
t o every Thus for
...,xm) ...,
(X1,
E Em
Xm)
2)
(5)
I=
U,
b an m-tuple o f
+...+ (Ixl(l
= p.
p,
E Cm (p,
f o r which
Since
= 1,
IIXmll
=.me=
...,A m )
E Em,
P
Fix a
such t h a t
Therefore the m-tuple (xl,
for every mElN.
i s continuous.
M > 0
t h e r e e x i s t s a real number Ilf(t)li
eas(%;F)
m s 1,
Am,
(f) =
i s homogeneous i n each
Am
Am E
m
j+X>,...,X
with
,...,
p,)
is
IIxjl/ = 1,
/Ixll(= a * . =
II
Xmll
=
1
9
THE CAUCHY I N T E G R A L FORMULAS
1
1 -
2np1...2np
~.
(2ny
--
rn
rn
1 2
P1...Prn
43
2
Ix J+ P j
1s j s m continuous. Now let x
E B <
Pa
-l(g),
be a real ninnber,
0
(1-X)g
+ Ax E B ( 5 )
P
f
U
> 1; then if for every
X E
C,
We claim
U.
~-
for
E
3
E H(U;F),
-l(s)
x E B
and
rn E IN.
PO
We have already proved this for tion
4.5).
If
/Ix-511 < Po-',
f E #(U;F)
the following mappings are con-
tinuous : i)
Vx: h E C ( U ; F )
Therefore c ) holds for F r o m c ) we have
I--
Vx(h)
(Proposi-
= h(x)
f E #(U;F).
E F,
44
CHAPTER
f o r every
1x1
=
m E N.
(I
\lx-~llL pa",
Since
we have for
X E 6,
DY
and s o , by 3 ) ,
sup
I x l=u
//f[(l-h)s+hx)I/
s M.
Hence, by
(*),
m
Since to
f(x)
U
> 1,
uniformly i n
it f o l l o w s that
-'(s),
B PO
C Am(x-g)" rn= 0
and t h e r e f o r e
converges f E #(U;F).
CHAPTER 5
CONVERGENCE OF THF: TAYLOR SERIES
In this chapter we consider the following problem: Given
f E #(U;F)
sets of
U
in which the Taylor series of f;
uniformly to
at
5
E = Cn,
5
at
f
represents
converges uniformly to
DEFINITION 5.1
x
f
If E
A
converges
In the case in
and contained in
1x1
(1-X)S;
+
A t E
?,x E A
and
E r,
If A
is 5-balanced and non-empty, then
5;
A subset
A
E
of
= {x-5 : x E A]
5
C
E,
and
U.
5 E E,
f o r every
5 E A.
The
are the simplest examples
E.
is 5-balanced if and only if the i s 0-balanced.
are referred to simply as balanced sets.
If 5 E A
f
1.
of g-balanced sets in a normed space
A-5
5
in every compact subset of
5
open and closed balls with centre
set
f.
is a normed space,
is said to be 5-balanced if
E
at
it is well known that the Taylor series of
the largest open ball centred at
A
f
that is, we wish to know to what extent the
Taylor series of which
5 E U, we seek to determine the sub-
and
the set
45
The 0-balanced sets
45
5
CHAPPER
is the largest {-balanced set contained in the 5-balanced kernel of an open set then
9
If
is never empty.
A.
is
4, E A c E
If
it is called
A;
A
is
Open*
where
B c A,
is {-balanced, and
A
then r.
= r(l-x)g+Xx
B
5
B"
Thus
s 11 c
A.
is the smallest %-balanced set containing
5
PROPOSITION 5.1
E
1x1
E c,
is called the %-balanced hull of
5
f
x
: x E B,
Let
an open set
V,
K c V c U,
K C U,
K,
and a number
v, x
: x E
{(l-x)g+Xx
B.
be an open 5-balanced set and let
U
F o r every compact set
Sf(U;F).
B.
1x1
E @,
p
there exists
> 1, such that
P I c
u
and
PROOF.
Denoting by
and radius
P,
5
the closed ball in
P
is continuous, and s o Since
set of
F,
p
3
+ hx E E
is a compact set contained in
T(ElxK)
f E S((U;F) c C ( U ; F ) ,
f[T(E1xK)]
and hence is bounded.
bounded neighbourhood number
with centre
the mapping
T: (X,X) E C X E H ( 1 - X ) C
U.
(I:
A
of
that 5,XV
C
Therefore there exist a
f[T(61XK)]
> 1, and an open subset
T-'[f-l(A)]
is a compact sub-
V
in of
F,
a real
E, K c V c U , such
CONVERGENCE O F THE TAYLOR SERIES
47
which i m p l i e s t h a t
T(5
P
x V ) c f-l(A)
f-l(A) c U ,
Since
f[T(EpxV)]
and
C A.
we have
(1-1)s + Xx E U
x E V
if
1x1
1 E G,
and
s p,
and sup(/IfC(l-X)S+hx]I) : x E
v, X
1x1
E c,
<
P I
Q.E.D. REMARK
5.1:
i n g way: and
5 . 1 can a l s o be s t a t e d i n t h e f o l l o w -
Proposition
5
"Let
f E #(U;F).
€ U,
a compact s - b a l a n c e d s u b s e t o f
K
Then f o r e v e r y open s e t
W,
t h e r e e x i s t s an open 9 - b a l a n c e d s e t r e a l number
p
Let
f
open, 5 - b a l a n c e d s e t .
E
#(U;F)
Taylor s e r i e s of
f
at
V
5
Given a compact s e t
e x i s t s a r e a l number and
W
C
C
V,
and
5 E
V c U, and a
P
> 1
of
K,
V c U,
K,
K c U,
such t h a t t h e
converges u n i f o r m l y t o
K c U,
i s an
U
where
U,
Then f o r e v e r y compact s e t
t h e r e i s a neighbourhood
K c V c U
C
K
> 1 such t h a t
P R O P O S I T I O N 5.2
PROOF.
K
V,
U,
f
in
V.
by P r o p o s i t i o n 5.1 t h e r e
and an open s e t
V
such t h a t
CHAPTER 5
48
By Corollary
4,
for every
4.1 we have
and every
0
;r
x E V.
position follows immediately. COROLLARY 5.1
open 5-balanced set. converges to COROLLARY
5.2
series o f
f
f
Q.E.D.
f E W(U;F)
Let
and
at
5 E U, where
Then the Taylor series of
at every point of
Let
> 1, the pro-
p
Since
f E W(U;F)
5
and
U
f
is an
at
5
U.
5 E U.
converges uniformly to
bourhood contained in
U
Then the Taylor in a neigh-
f
of every compact set contained in
U
5'
REMARK 5 . 2 :
Corollary 5.2
shows that for convergence of the
5
Taylor series, the largest open hall with centre
in
U
of the finite dimensional case is replaced i n the in-
finite dimensional case by the s-balanced set DEFINITION 5.2 bounded at
V c U,
5 E
A mapping U
such that
locally bounded in of
contained
f: U + F
.
U
5
is said to be locally
if there exists a neighbourhood
f
is bounded in
V;
f
V
of
5,
is said to be
if it is locally bounded at every point
U
u.
DEFINITION 5 . 3
Let
f: U + F
The radius of boundedness of the set of real numbers is bounded in every ball
be locally boiinded at f
r > 0
Bp(f)
at
5
is the supremum of
such that with
5 E U.
Br(Z)
c U
0 < p < r.
and
f
49
CONVERGENCE OF THE TAYLOR SERIES
PROPOSITION 5.3
g E U.
Let
F
f E #(U;F)
be a Banach space,
The radius of boundedness
rb
of
R
to the minimum of the radius of convergence series of boundary o f
5
at
f
and the distance
rb
0 < p < r b , then
is a real number, finite.
is equal
of the Taylor
5
from
to the
U.
From Definition 5.3 we have
PROOF.
d
5
at
f
and
?;
L =
d,
and also, if sup
l\f(x)ll
I1 x-5 II =P
p
is
Applying the Cauchy inequalities, we have
m E
for every
'Im
Therefore
(N.
l z l\Pml]
s
1 , P
and so by the
Cauchy-Hadamard formula
Hence
rb
g
R,
and therefore
(")
rb
5
min(d,R).
To prove the reverse of inequality (*),
< min(d,R).
p
Since
the Taylor series o f and since
:,(5)
converges to
Since
f
at
1
f
at
5
f
zp(5).
5.1 this series
Therefore the
converges uniformly to m E N,
is bounded in
rb
2
8,(5),
converges uniformly in
at every point of
Pm E P ( % ; F ) ,
It follows that
(""1
f
0< p <
by the Cauchy-Hadamard formula
is 5-balanced, by Corollary
Taylor series of
gp(g).
< R,
let
Pm
Bp(S),
min(d,R)
.
f
in
is bounded in
and therefore
Bp(0).
50
5
CHAPTER
REMARK
5 . 3 : In the classical case in which
given
5 E U,
and
p f OR,
if
E = C
< p < dist(5 , a U )
0
n
,
n
1,
2
G,(s)
then
c U
g p ( 5 ) is compact. Therefore <
llf(x)ll
sup
//X-5/I~P
f o r every
f E W(U;F) c C ( U ; F ) .
From the definition of the
radius of boundedness and Proposition 5 . 3 it follows that rb = d
5
this explains why, in the finite dimensional case,
R;
the concept of radius o f boundedness is not of great significance.
5.1
EXAMPLE
E
Let
be
Thus for each
x,
ing on
the space of all sequences
of complex numbers which are eventually
( X l,...,~m,...)
zero.
Coo,
is
rb = R < d.
infinite, it is possible to have
x =
E
However, in the case in which the dimension of
x E Coo
such that
= x
xm
there is an index
~
-
... =+
~
m
0,
depend-
0. ~ The norm of
0
this space is given by
..., ...
x= (xl,
x ,
) w IIXII
rn
This normed space is not a Banach space.
Pm: E
define
-t
F
,...,x ,...)
x = (xl
rn
Prn(x) = xl. ..x m
by E C
It is easy to see that nomial from
E
into
and
00
Pm
F,
bml.
= SUP
rn E N,
Taking
F
= 6,
we
if m
5.
1, and
Po(x) = O .
is a continuous m-homogeneous poly-
and that
l\Pml/ = 1, rn E
N,
m
2
1.
m
Consider the power series
C
P,(x).
We claim:
m= 0
a)
This series converges for every
b)
If
for every
0< p < 1
t E E;
x E E;
the series converges uniformly in
-
Bp(t)
51
CONVERGENCE OF TIIE TAYLOR SERIES m
c)
x E E,
f: E -+ F
The mapping
E,
is holomorphic in
F o r every
x E E,
f(x) =
defined by that is,
P,(x)
f
C P,(x), m= 0
is entire.
= 0
after a certain index;
E E
and
this proves a). Let 0
< p < 1.
t = (tl Then
x = t+y,
tm = 0
such that
,...,tm,...)
for
IIyl/ s p ,
x E ep(t),
and if
m
m > mo,
we have, with
ll..'IYm
+kl
where
is an index k = O,l,2,...:
k
cop
s
9
0
where
c=(
tll+P)
I
I +PI
.( tm 0
m
Hence
C
IPm(x)
I
S
C
t
C*p
+ Cop 2 +
m=m O
the series proves b)
.
...,
which shows that
m
C
P,(x)
converges uniformly in
Rp(t).
This
m=0
F r o m this it follows, using a classical compactness
argument, that C(E;F)
f E C(E;F);
since
#(E;F)
is closed in
for the compact-open topology, we then have f E #(E;F),
and c) is proved.
In this example, with the Cauchy-Hadamard formula,
5 =
0, we have
R = 1.
d =
50,
and by
Therefore
rb = min(m ,I) = 1. REMARK
5.4:
In Example 5.1 we have a phenomenon which cannot
occur in the finite dimensional case
-
a holomorphic, and con-
sequently continuous, function which is not bounded on every
CHAPTER 5
52
bounded subset of its domain.
is not bounded in El(0): for each m E N, m ,-. 'IL consider the point x = (1,. f , 0 , 0 , ...) E a B , ( O ) .
directly that
m
f
..,
1,
2
We note that it can be proved
We have
f(xm) = m
and s o
,...,Pm(x m )
= 0, P,(X")
p0(x")
EXAMPLE 5 . 2
x = (xl,
Let
= 1
m E N,
for
co
m
2
=
= 1, pm+,(x")
1.
*
0,=**
Therefore
be the vector space of all sequences
...,xm,...)
complex numbers such that
of
lim xm = 0,
m-tm
with the norm x E
IIxI/ = sup
C0l+
xml
m c0
is a Banach space.
Taking
F = CC
= (x,)"
m E IN,
m
and
P,(x)
x = (xl,
...,xm,...)
if E E.
geneous polynomial from
Then
E
2
Pm
into
F
we define
p o w = 0,
and
1
is a continuous m-homo-
for every
m E N.
Con-
0)
sider the power series
C
Pm(x).
We have:
m= 0 a)
The series converges for every
b)
If
for every
0< p < 1
x
E E;
the series converges uniformly in
P
(t)
t E E; m
c)
The mapping
f:
E + F, defined by
f(x) =
C
Pm(x),
m= 0
x E E,
is holomorphic in Taking
E.
5 = 0 we again have
rb = R = 1 < d.
Thus we
have another example of an entire function which is not bounded on every bounded subset of its domain.
Note that
co
is
53
CONVERGENCE OF THE TAYLOR SERIES c'0 =
separable but not reflexive, since the space of bounded sequences.
E
occur even when
EXAMPLE 5 . 3
( 4 ' ) '
and
is
However, this phenomenon can
is separable and reflexive.
E = LP,
Let
L1
E
p
F = 6.
p > 1, and
IN,
Using
the same polynomials and the same function as in Example 5 . 2 ,
rb = R = 1 < d
we again find that
5.5:
REMARK
5 = 0.
at
We call the attention of the reader to Chapters
11, 12 and 13. PROPOSITION dmf
5.4
If f f # ( U ; F )
E #(U;Ss(%;F))
and
zmfE
and
m E LN
then
#(U;P(%;F)).
If
f(x)
CD
C
Pm(xs)
is the Taylor series of
f
5 6 U,
at
then
m= 0
the Taylor series of
dmf
imf at
and w
dmf(x)
are
m d Pk+m(x-5 )
C
=
5
k=0 m . .
;mf(x)
PROOF.
Let
5
X
= f(x)
6 E n x-5 E E ,
and 3.1 we have
Qi
and
gx = f
-
U,
m+X C Qi, i=0
Since
-
.
be defined by
m+ h
c Pi(X'5),
x
E u;
i=O
Qi = PiOtg. BY Propositions
E P(E;F) c 8(E;F)
Using the same notation, polynomial to
*m C d Pk+m(x-S ) k=0
gx : U -+ F, 1 E IN, g,(x)
t :
a
Qi,
for every
E
for every
i
E #(U;F)
for every
h E IN.
zm: #(U;F)
3
i 6 IN.
for the restriction of this
Q i E #(U;F) gX
1.6
S(U;P(%;F))
N,
and since
is linear we have
CHAPTER 5
54
A
i=O
and tlierefore
f o r every
m+X Am C d Pi(x-s) i=m
= Pf(x)
-
C ZmPk+m(x-5) k=O
rb
U = p'
-
every
x E X,
m! . -
om
x E
cp (s )
X = [x E U : g,(x)
-
su llf(t) Ilt-xf=u
at
5 ,
c U].
Then, by the
5
m+ X
c
and
dist(5 , a U ) ,
I/ t-x/l =
u
~~(t-s)lI for every
x
E X.
i=O Bo(x)
which implies that
(**) we have
f
and hence
< rb
x E gp(5),
X E N.
be real numbers such that- 0 < p < p f < r b ,
and
p,
x
be the radius of boundedness o f
pf
and
pf
N.
-
Let
Since
X E
Zmgx(x) = 2mf(x)
x E U,
for every
p
U,
( * ) and Proposition 3.1 we have
From
let
E
x
then
C
E p (~5 ) c
E,(s) c X.
I] t-51)
I;
U
for every
Furthermore, if
p' ;
therefore, by
55
CONVERGENCE OF THE TAYLOR SERIES
for every
x C-
zp (g ).
Since
this inequality tends to 50
that
C
Am d Pk+m(x-5)
p'
as
0
< r b , the second term of ?,-
tends to
converges uniformly to
which shows
m ,
zmf(x) in
gp(5).
k=O This proves the statement in the proposition concerning
zmf, since
imPk+m E P ( % ;
(Corollary 3.1). dmf,
P(%;F))
for every
k E
(N
T o prove the corresponding statements for
it suffices to note that the equation
can be written as
and, by Proposition 1-39
COROLLARY
PROOF.
5.3
If
f E #(U;F)
and
5 E
U,
then for
In the notation of the proposition we have
k,m E N ,
56
COROLLARY
CHAPTER
5.4
If
f E SI(U;F)
and
5
5 E
U,
then f o r k , m E
[N,
CHAPTER 6
WEAK HOLOMORPHY
In this chpater we shall prove the following propositions : PROPOSITION 6.2
If F
f: U
is a Banach space, and
4
F
a
mapping, the following are equivalent: 1)
f E #(U;F).
2)
$ o f
E #(U;(c) for every
G
complete,
f: U
U
-t
Let
U
E, F
and
G
F’
is the to-
be normed spaces,
a non-empty open subset of
a mapping.
L(F;G)
E F‘, where
F.
pological dual of PROPOSITION 6.3
(I
E,
F
and
and
Then the following are equivalent:
#(u;c(F;G)).
a)
f E
b)
The mapping
f o r every
x E U n tu[ f (x)(y)] E C
y E F
PROPOSITION 6.4
and
If F
is holomorphic in
tu E G‘.
is a Banach space and
f: U
-t
F’
a
mapping, the following are equivalent:
H(u;F’)
a)
f E
b)
The mapping
for every
x E U H f (x)( y ) E C
y E F.
We begin with:
57
is holomorphic in
U
CHAPTER 6
58 LEMMA 6.1
M
Let
be a metric space, f: M
or complex) and
F
-t
a normed space (real
F
a mapping.
The following are equi-
valent: 1)
f
is locally bounded (Definition 5.2).
2)
f
is bounded on every compact subset of
M.
PROOF : This is proved by a simple compactness argument.
1) 3 2 ) .
We note that this implication is valid if
M
is an arbitrary
topological space. 2)
5
Let
1).
3
E M,
any neighbourhood of
m E BJ,
each
By a),
u EX,; K,
5
xm
-I
2
11
m
1,
2
1 < m
4 5 ,x,>
a)
on
m
as
and suppose that
5.
and
xm E M
b)
m,
Ilf(xm)ll
M.
mapping.
F
Suppose that
ed if and only if
$(Y)
> m. K = {C] U
But by b),
which is a contradiction. Let
such that
and hence the set
is compact in
PROPOSITION 6 . 1
is not bounded on
Then, by recursion, we obtain for
a point
m -+
f
f
Q.E.D.
be a Banach space and
Y c F'
is unbounded
is such that
is bounded for every
f: U
Y
C
F
(I € Y .
3
F
a
is boundThen
the following are equivalent: 1)
f
E #(U;F)
2)
~
4
Ef H ( U , C )
for every
JI E Y.
PROOF :
1) Y,
3
2).
This implication is true without any restrictions on
by virtue of Proposition 3.2.
59
WEAK HOLOMORPHY
1)
2)
Y
a) y E F,
separates the points of y
#
0, there exists
y f 0, the set
if
J,
I
J,(Cy) = CJ,(y) is unbounded;
ber
IJi(y)I
IJi(y)I s dllJiII
Ji E Y
and if
J,
F,
Ji E Y
#
such that
d
there exists a real num-
0
21
f o r every
0,
y E F
E Y}.
Y
such that
Y = [y E F :
I n fact, let
sup I $ ( y ) I
we have
0. For
$(y)
Ji E Y.
for all
#
$(y)
is unbounded in
there exists
IIyIl < cd
such that
dl/J,l/for a l l
5;
such that
hence
F o r every real number
c 2 0
E Y
Cy = [ x y : 1 E C]
and so by the definition of
b)
that is, f o r every
F,
0 E Y,
is non-empty, since
d]lJi/I,which implies, by
L
YEY
Y,
the definition o f
that
Y
it follows that
c = d-l sup I/yI/ if
d
#
YEY y E Y.
=
0,
every c)
f
If
d
0,
is bounded in
IIyII
K
C U:
Ji
f o r every
for every
E Y,
d)
f
f
$ o f
for
Thus
$ o f
is
Y,
$of
is
$[f(K)]
is bounded
Y,
f(K)
is
F. This follows from c) and Lemma
is a metric space.
is continuous in
there exists
4.2 to
Y.
is locally bounded:
6.1, since u e)
J, E
and so, by the definition of
a bounded subset of
cd
h
Since J, E
holomorphic, and hence continuous, f o r every K
Taking
the assertion is trivial.
is bounded on every compact
bounded on
F.
r
>
0
E W(U;C)
U:
such that with
Let
f E U.
G r ( 5 ) c U.
m = 0,
Ji E Y ,
Since
u
is open,
Applying Corollary we have
CHAPTER 6
60
x E Br(5)
for every
By d)
f
Q E
and
Y.
i s l o c a l l y bounded, and s o , t a k i n g
f i c i e n t l y small, s o t h a t
)
f o r every
x E B,(f
f o r every
x E Br(5),
Q E Y.
and
x
Ilf(t)ll
11 t
# 5,
x E Br(S),
f o r every
It follows t h a t f)
f
f
5+Ax E
x E E U
# 5,
JI E Y .
c
2
Pm: E
-t
U: F
Let
E
6,
It follows by b )
such t h a t
0
1x1
5. g E U.
For each
i n t h e f o l l o w i n g way:
we choose a r e a l number
f o r every
we have
and hence
i s continuous a t
i s holomorphic i n
we define a mapping each
x
+m,
suf-
Therefore
and
t h a t t h e r e e x i s t s a r e a l number
= d <
r
a: p ,
p
>
0
such t h a t
and s e t
m E IN For
61
WEAK HOLOMORPHY
This integral exists since by e)
F
and by hypothesis
is complete.
5
max[p,p'],
is continuous,
We show first that the
value of the integral is independent of positive real numbers such that
f
P.
5+Xx E U
p,
p'
be
X E 6,
1x1
Let for
5
and let r
Pm(x> = -2rri
B y hypothesis,
by Proposition
Qof
,
E #(U;C)
$ E
Y
and
m E IN.
f o r every
lJ E Y
and
m
for every
Q E Y
and
m E N.
F,
Q E 1,
and hence,
4.2 (the Cauchy integral formula),
for every
of
for every
and hence
E
IN.
Similarly,
Therefore By a)
Pm(x) = Pk(x)
Y
separates the points
for every
We show next that the mappings
= Q[Pk(x)]
$[Pm(x)]
m
Pm, m E
E
IN. N,
are con-
62
CHAPTER 6
tinuous.
If
xo E E ,
is continuous at the point such that and
g+Xx E U
IIx-xoII
for every
5
p.
(O,xo),
for all
x E Bp(xo).
E
and so there exists p > 0 and
x E E
with
N o w , using the fact that
is continuous at
We now show that
Pm E P ( % ; F )
1x1
L p
is con-
f
xo
for every
m E
for every
m E IN.
For
[N.
this is immediate, since
and applying Proposition 4.2 the functions
for every
$of
x E E
the points of
E
W(U;C),
and
$ E
F, P o ( x ) =
(the Cauchy integral formula) to $ E
Y.
f(5)
Y,
Since, by a),
Y
for every
x E E.
separates Therefore
E P('E;F). For
by
C
5 +Ax E
-B
and the usual inequalities for integrals,
Pm
it follows that
p0
1 E
CXE
Then
tinuous (part e)),
m = 0
(1 , x ) E
the mapping
m
N,
m 2 1, we define a mapping
k : Em
-B
F
63
WEAK HOLOMORPHY
L: ei=±l
&le 2 ••• e m Pm(elxl+ ••• +emXm)
1" i"m where
m E tN,
(Xl, ••• ,x ) m m :2: 1,
E
Em.
are symmetric and m-linear, and that
It is easy to see that to prove that
~
E 'l',
Am
Am
A- m
Pm .
is symmetric, and i t then suffices
is linear in the first variable.
Am(Xl+X~'X2""'Xm)
only that
We shall prove that the mappings Am'
= Am(xl
We show
, x 2, ••• ,xm) +
we have
HAm(Xl+X~,X2"",xm)J
=
1
L: el ••• em(*oPm)(el(Xl+x~)+e2X2+ 2 mm.I &i=±l
•• .+emxm) =
l"i:s:m
=
L:
&l"'&m
~!
am(*of)(O(&1(Xl+X~)+&2X2+··.+emXm)
e i=±l 1" i:s;m
= 2
1 mm!
L:
1
&1 • • • e m
Ii1T am(*of) (~)
e 1 ••• & m
Ii1T am(W o f) (~)
&i=±l
(e lX l+"
'+&mXm) +
(e lX~+"
'+&mxm)
1" i:s;m 1 2 mm.,
L:
e i=±l
1
l"i:s;m 1 =-L: &1'''& (wop )(e x m m m l 1+ . . . +& mx m ) + 2 m.I &i=±l 1" i"m
=
64
CHAPTER 6
-m 1 C 2 m! E =il
el...s m
($0
Pm) (E
r+C
lX;+
mxm)
i 1s i s m
16 i s m
14 i s ; m
Therefore
= $[Am(x19x2,.. Since
Y
$[Am(x1+x;,x2,.
.,xm )
..,xm)]
=
+ A m (x‘1 ,x2,. . . , x m )] for every
separates the points o f
F,
Jr E Y .
it f o l l o w s that
A
Next, we show t h a t that
Pm E P ( ? E ; F ) .
$Cim(X)l
Am = P m , from which i t follows
We have
,L = $[Am(%
,X)l
1si s m
for every
E Y,
x E E
and
m E N,
m
2
1,
and hence
WEAK HOLOMORPHY
im(x) = Pm(x)
f o r every
x E E
rn E N,
and
m z 1.
To complete the proof of the proposition we show that m
C Prn(x-5) m=O
of
converges uniformly to
in a neighbourhood
f
5. Since
M > 0
numbers sup
f
llf(t)ll
is locally bounded (part d)) and
L
II t-s I1 =o such that
r < ap
> 0 such that
0
M.
.
to the functions
f o r every
$[ Pk( x)]
=
1
;Ik(
$ 0
r > 0
and
C
Prn(x-5)
m= 0
5;
uof,
x E B,(g)
p > 1
We claim that the series f
p,
~ ~ ( 9 ) .
in
x E Br(Z)
1x1
E C,
for all
and
U
m
converges uniformly to
For every
gu(g) t
Choose real numbers
-1
there exist real
we have
(1-x)g
+ xx E
fi ( 5 )
C
U
and s o , applying Proposition
u
4.5
$ E Y,
and
rn E
f)( 5 ) (x)
,
Using the fact that
[N.
this yields r
for every
$ E Y,
the points of
for every
F
x E Br(s)
and
m E
[N.
it follows that
x E Br(C)
and
rn E N.
Hence
Since
Y
separates
66
CliAPTER
6
m f o r every
x E B,(g)
m E
and
It follows that
!N.
C
P,(x-~\
k=0 converges uniformly to
in
f
Q.E.D.
Br(g).
A s a corollary to Proposition
If F
PROPOSITION 6.2
6.1 we have
is a Banach space and
f: U
-t
F
a
mapping, the following are equivalent:
t
W(U;F)
1)
f
2)
$of E a ( U ; C )
topological dual of PROOF:
for every
$ E F'
F
is bounded, and so
Y = F'
PROPOSITION 6.3
U
complete, .S(F;G)
E, F
Let
satisfies the con-
and
Q.E.D. G
be normed spaces,
a non-empty open subset of
U
a mapping.
E,
F
and
and
Then the following are equivalent:
~(u;c(F;G))
a)
f E
b)
the mapping
for every
PROOF:
is the
Thus Proposition 6.2 is a par-
ticular case of Proposition 6.1.
-t
F'
F.
ditions of Proposition 6.1.
f: U
where
By the Banach-Steinhaus Theorem every weakly bounded
subset o f
G
,
For
y
E
F
y E F
x
E U
w[ f( x ) (y)]
E d:
is holomorphic in
w E G'.
and and
-t
w E G'
J I ~ , ~u: E C ( F ; G ) is linear and continuous.
I-
Let
the mapping
L U U C ~E( (C~ ) I
WEAK HOLOMORPHY
If x E U ,
y E F
w E G’,
and
then
and hence condition b) is equivalent to E H(U;C)
of
L
Y
for every
w E G‘,
y E F;
this can be written: ct)
Jiof
~ ( u ; c ) for
E
By hypothesis, complete.
G
every
$ E Y.
is complete, and so
x(F;G)
is also
Thus by Proposition 6.1 the equivalence of a) and
c1) is established once we have proved the following: B c C(F;G) for every
is bounded if and only if
is bounded
Q E Y.
To say that that
$(B)
$(B)
I
sup Iw[u(y)]
<
is bounded for every
w E G’
f o r every
m
and
6 E Y y E F;
means by
u€ B
the Banach-Steinhaus Theorem this is equivalent to sup IIu(y)// <
y f F,
for every
m
and applying the Banach-
u EB
Steinhaus Theorem once again (since
F
is complete), this
in turn is equivalent t o the condition that di ( F ; G )
.
B
is bounded in
Q.E.D.
PROPOSITION 6.4
If
F
is complete and
f: U
-t
F’
is a map-
ping, the following are equivalent:
u
E
a)
f
b)
The mapping
#(U;F‘)
for every
PROOF:
Take
x E U
I+
f(x)(y)
E
C
is holomorphic in
y E F. G = CC
in Proposition 6.3.
Q.E.D.
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CHAPTER 7
FINITE HOLOMORPHY AND GATEAUX HOLOMORPHY
A mapping
DEFINITION '7.1
f: U
-I
F
is said to be finitely
holomorphic if for every finite dimensional subspace
S
for which
n
U
into
F
of
E
f/SnU E g(UnS;F).
we have
@
gfh(U;F)
The set
U
#
S
of finitely holomorphic mappings of
is a complex vector space, the operations of addi-
tion and scalar multiplication being defined pointwise.
We
have :
#(U;F) c gfh(U;F)
a) from
S
E
into b)
into
F
since the restrictions of polynomids
S
to subspaces
of
E
are polynomials from
F.
f E gfh(U;F)
f E #(U;F). f E g(U;F)
does not necessarily imply that
A s we shall see presently, the condition
f E gfh(U;F)
is equivalent to
together with one
other condition. We shall denote by
fS
the restriction of
f
to
s n u. REMARK
7.1 Let
f E Zfh(U;F),
mensional subspace of
E
n S and m imfs(s) E b(mS;F).
we have
5 E U
subspaces of
E
E
[N
If S1 with
and let
such that
and
S1 c S 2
69
S
n
S
U
be a finite di-
#
d.
Then for
dmfs(5) E .Cs(mS;F) and S2
and
are finite dimensional U
n
S1
#
@,
and
CHAPTER 7
70
5 E
U
n
S1 c U
n
the following compatibility relations
S2 '
hold:
m E N
This guarantees the existence for every
b"f(5)
E aas(%;F)
S
finite dimensional subspace =
f ( 5 ) E CaS(OE;F).
bmf(5)
If
associated to
every
5 E
bmf(5)
or
U,
tmf(5)
f E #(U;F)
then
5 E
every
m E N.
U
E.
of
a^"f(l)
For
for every
m = 0, bof(5)
=
denotes the polynomial
bmf(c),
then
a^"f(Z)
E Pa(?E;F)
for
In general we have no guarantee that
will be continuous. d"f(5)
and
= dmfs(5)
bmf(p)Is
such that
of a mapping
= 6"f(s)
m E N.
However, if zmf(<) =
and
i"f(5)
for
This follows from the uniqueness of
the Taylor expansion about the point
5,
and the fact that
the restriction of a continuous symmetric m-linear mapping of
Em
into
F
to a subspace
Sm
is itself a continuous sym-
metric m-linear mapping of
Sm
into
dmf,(5 of
) = Smf(g ) E,
1
and hence
EXAMPLE 7.1
F;
thus we have
for every finite dimensional subspace dmf(5) =
bmf(5)
for every
Every polynomial from
E
into
S
m E M.
F,
continuous or
not, is finitely holomorphic.
This follows from the fact if
A
Em
is any m-linear mapping of
nite dimensional subspace o f is m-linear and continuous.
E,
into
F, and
S
the restriction of
is a fiA
to Sm
71
FINITE HOLOMORPHY AND GATEAUX HOLOMORPHY
is an infinite dimensional normed space, and E*,
If E
E'
are its algebraic and topological duals respectively, then
f
E*\E' cp
B
Let
E, F
and
a non-empty open subset of
and
a E F.
into
F
PROOF:
pa
be complex normed spaces,
G
f E W(U;G),
F,
pa(x)
+
= v(x)
Em + Fm
and
[N
E.
Then
fova
we have
A E Ss(%;F)
4.. A o p = AoM E f f ( % ; G )
then
f o r every
m;
Aoym E
thus
the second equation follows from the first. m
5
If g = ka(C),
E V,
we have
uniformly in a neighbourhood in 1 Am = m! dmf(g),
m E
[N.
m
m
C m= 0
U
Thus
m
=
A,OP
E
is given by
It is easy to see that if
E es(%;G)
E
and suppose that
a,
5 E V, m E
and for every
vm:
p E S(E;F)
be the continuous affine mapping of
is a non-empty open subset of
#(V;G),
where
Let
defined by
V = p,l(U)
E
but
SI(E;C).
PROPOSITION 7.1
U
cp E #,,(E;C)
cp E E*\E',
F o r each
ip.
m
(z-c)"
,
of
f(t) =
C
m=O 5 = Ua(C),
Am(t-5)" where
72
CHAPTER
uniformly in a neighbourhood in fopa
E
7 of
V
5.
Therefore
and
#(V;G)
PROPOSITION 7 . 2
f
Let
be a mapping from
U
into
F.
U.
The
following conditions are equivalent: 1)
f
is holomorphic in
2)
f
is finitely holomorphic and continuous in
3)
f
is finitely holomorphic and locally bounded in
The implications 1)
PROOF:
We shall prove 3 ) Let
h
E
(tx)5,
=
Vx = { k E CC : g+kx E U)
c U
n
Sx;
and
E
tx E X(CC;E),
tx: 6: + E
5 E
by
U.
=
t,(k)
Tx
the
defined, as in the preceeding Propo-
+
1 E 6.
The set
is open, and
0 E Vx.
Xx,
generated by
5
and
x
If
then
Sx
is
c
Tx(Vx)
thus we may form the composition f
TX
Sx
are trivial.
We denote by
vx~~-unsX-Since
3)
2)
be locally bounded, and let
we define a mapping
(tX){(k)
the subspace of
2)
U.
1).
Clearly
C.
affine mapping sition by
3
f E afh(U;F)
x E E
For each = hx,
U.
is finite dimensional and
-
F.
f E gfh(U;F),
E fSX
E #(UnSx;F),
and so by Proposition
Using the fact that
7.1,
0 E Vx,
7.1,
o T x E #(Vx;F). fSX we have, again by Proposition
FINITE HOLOMORPHY AND GATEAUX HOLOMORPHY
73
m
c
=
uniformly in some
Er
(5)~"
dmfS
!m
m= 0
X
(0). By the definition of the mappings
X
8 mf(5 )
and
a^"f(5 ) ,
uniformly i n some
this yields
Er
(0). X
Now since
f
is locally bounded in
strictly positive real numbers
SUP
and
1x1
II t-5 114 p 5
p,
f o r every foTx
Ilf(t)ll
then
4
E SI(Vx;F)
Thus
M
IIxIl
gp(g) C
such that 4
X
1, and
iip(0) = { X E
Q: :
1x1
h
we have
x E E,
IIxII
( fOTx)(O+X 1)
=
5;
1
dX
Xm+ 1
and
m E IN.
Thus f
1
m!
for every
x E E,
I X I'P IIxIl
4
1
and
m E IN.
Xm+1
U
E C,
Applying Proposition 4.2 to
1 ~"(fOT,)(O)(l) m!
for every
If x E E ,
M.
t+Xx E U.
x E gl(0).
and
p
there exist
U
dh
This shows that
p)
cvx
74
7
CHAPTER
f o r every
a^"f(5)
r > 0
m E IN.
E P(%;F),
Let
U
and
and
ur
h
r
1 E Vx
for
be real numbers such that
we know that
1x1
E C,
(foT
=
Gr(5) c Ep(5)
4.5 to the points X
Y-5
)(1)
-
s
0,
m
C
k=0
U).
Let
y E
foTx E #(Vx;F).
0
and
1
of
1 -k k-! d (foT
Y-5
)(0)(1)
m E IN,
Thus
for every
y E
Er(5),
rn E N.
It follows that
m
M
k=0
the series
Urn(U-l) rn E IN.
r n l
C
m! Cmf($)(y-5)
m= 0
in
Gr(5).
Q.E
s ince
we have
gr(5 ) ,
isr($),
fjr(5).
Vx,
y E
y E
1,
Applying
for every
for every
CJ
We shall prove that the series
p.
x = y-5,
Proposition
is continuous, that is, m E IN.
f o r every
(Note that Taking
i"f(5)
Thus
O D .
Since
u > 1 we conclude that
converges uniformly to
f(y)
75
FINITE HOLOMORPHY AND GATEAUX HOLOMORPHY
DEFINITION 7 . 2
A mapping
f: U
phic (Gateaux-holomorphic) in b E E
-b
F
is said to be G-holomor-
if for every
U
a E U
and
the mapping
1 E (X E C : a+Xb E U} c C
f(a+Xb)
E F
( X E C : a+Xb E U]
is holomorphic in the open subset REMARK 7 . 2
t--
of
This definition is equivalent to the following
statement: a)
If x
and
y
are points in
A = { A E C : x+Xy E U}
E
such that the set
is not empty, then the mapping
X E A c CH
f(x+hy) E F
is holomorphic in the open set
A.
It is clear that a) implies the condition given in Definition 7.2. points in
E
Conversely, suppose that
x
and
y
are
such that the mapping
Xo
is not holomorphic at a point
of its domain.
Then
a = x+Xoy E U , and the mapping
x
E
(X
E
(I:
: a+Xy E U}
is not holomorphic at the point existed
bm E F,
m E N
and
c CH
f(a+Xy)
1 = 0. F o r , suppose there
p > 0
such that a
C.
76
CHAPTER 7
= Po.
B (0)
uniformly in
P
(*)
We may write
E C
as: co
= f(x+>" o y) +
f(x+>.. o Y+AY)
L: m=l
co
whence
f(x+>" 0 y) +
B (0).
formly in
p
L: b mAm converges to m=l Thus the function x+>..y E U} c
AE{>..E
>"0'
f(x+(A o +A)Y)
uni-
'-,> f(x+>"y) E F
contradicting our hypothesis.
It
follows that Definition 7.2 implies condition a).
Let p ~ 1,
F
U
be a non-empty open subset of
a complex normed space, and
CPo
nonical basis for
We denote by
Zk = (Zl""'Zk_l,o,zk+l'."'Zp) E CPo k = 1,2, ••• ,p
J
z,k
pEN,
{el, ••• ,e p} Jk
1 :s;
CP,
the ca-
the mapping
k :s;
p,
Then for
>.. E C
and
we have
denotes the continuous affine linear mapping
xE
DEFINITION
7.3
Let
A mapping
f: U
~
J
U
F
= Zk + >.. e k E C P •
z, k (>.. )
be a non-empty open subset of
is separately holomorphic in
satisfies the following condition: Z = (zl""'zp) E CP,
Uz,k = J-lk(U) ~ ¢ z,
and
we have
U
For every
k = 1,2, ••• ,p foJ z" k E
~(uz
such that kIF).
CPo if it
77
FINITE HOLOMORPHY AND GATEAUX HOLOMORPHY
We remark that this condition is equivalent to the
..,
z = ( zl,.
requirement that for every k = 1,2,...,p,
the restriction of
determined by
zl,. . . , Z
zp) E Cp
f
to the sectaon of
z
is holomorphic.
* P
k-1,Zk+l’
PROPOSITION 7.3 (Theorem of Hartogs): open subset of
U
ping of
F
(Cp,
F.
into
U
be a non-empty
a complex normed space, and
f
a map-
The following are equivalent:
U.
f
is holomorphic in
b)
f
is separately holomorphic in
7.3
U
Let
a)
REMARK
and
U.
The proof of the theorem of Hartogs may be found
in [ 6 O ] .
7.4 Let U be a non-empty open subset
PROPOSITION f
a mapping of
F.
into
f
is G-holomorphic in
2)
f
is finitely holomorphic in
1)
=
such that
for
S.
Let
2).
V = S
n
S
U
E and
The following are equivalent:
1)
PROOF:
E
U
of
U. U.
be a finite dimensional subspace of
#
and let
@
[al,
...,ap]
be a basis
Since the mapping cp:
(al,
...,zp) E
(C
P
PH
c
zkak E
s
k=1
is a homeomorphism and an isomorphism between the vector spaces (Cp (Cp.
and
S
the set
W = cp -‘(V)
is a non-empty open subset of
Consider the mapping
,..,.,zP)
g: (zl
We have
f = gorp S
-1
E
w
P
t--
f(
c
zkak) E F.
k=1
and s o , by Proposition
7.1, to show that
78
CHAPTER
fS E g(UnS;F)
7
it suffices to prove that
position 7 . 3 this is equivalent to
g
t H(W;F).
By P r o -
being separately h o l o -
g
morphic. For
...,
z = (zl,
zp) E Cp
k E N,
and
1 s k s p,
such that
we must show that the mapping
is holomorphic in
Wz,k
k
We have
k
= ( f o c p ) ( z +Xek) = f[p(z
g ( z +Xek)
k )+lak],
and
from the definition of a G-holomorphic mapping and Remark 7 . 2 , we k n o w that the mapping
is holomorphic in 2)
1).
3
Let
E [A E
and
b b
E
If
E.
then
a
E
is the subspace of
S
U
n
S
E
and by 2),
Applying Proposition 7 . 1 to the composition
fS E #(UnS;F).
X
a E U,
a
generated by
Wz,k
(I:
: a+Xb E U } H
this mapping is holomorphic.
a+Xb E U Thus
n
f
SH
f(a+Xb) E F,
is G-holomorphic. Q.E.D.
COROLLARY 7.1
Let
f
lowing are equivalent:
be a mapping of
U
into
F.
The fol-
79
F I N I T E HOLOMORPHY A N D GATEAUX HOLOMORPHY
a)
f
is holomorphic in
b)
f
is G-holomorphic and continuous in
c)
f
is G-holomorphic and locally bounded i n
PROOF:
U.
U. U.
T h i s corollary is an immediate consequence of P r o p o -
sitions 7.2 and
7.4.
Q.E.D.
This page intentionally left blank
CHAPTER 8
TOPOLOGIES O N SPACES OF HOLOMORPHIC MAPPINGS
DEFINITION 8.1
A non-empty subset
to the boundary of
px: f E
into
F.
is a U-bounded subset of
E,
Wb(U;F).
U
sup{Ilf(x)IJ
t h e mapping
: x E X] E
The family of seminorms
ranges over all the U-bounded subsets of rated locally convex topology on ~
U.
will denote the vector space o f holomorphic
w~(u;F)I+ pX(f) =
norm on
X
is said t o 5 e of b o z n d e d type if
mappings of bounded type from
If X
is said to be
is positive.
U
is bounded on every U-bounded subset of
UL,(U;F)
T
E
of
is bounded and the distance from
f E J$(U;F)
DEFINITIOW 8.2 f
X
X t U,
U-bounded if
X
Hb(U;F),
where
px
E,
IR is a semiX
defines a sepawhich we denote by
This topology is known as the natural to2ology o n Wb(U;F).
.
PROPOSITION 8.1 topology PROOF:
F
is complete then
Sb(U;F),
with the
is a Fre'chet space.
Tb
n = 1,2,...,
For
xn
If
= Ex E
There exists
u
no E N
let
. )/xIIs *
n
and
such that for
81
dist(x;aU) n
5 no ,
5.
Xn
1
1.
is non-empty
CHAPTER 8
82
For
and hence U-bounded. m
m
u
u =
n=1
where
1
2
Vm = { x E U : dist(x;hU)
and
there exist positive integers
Vmo
unO
which
m= 1
Un = Ex E U : IIxI/ s n]
3
n
n'
#
0.
#
X
u vm
u =
and
m,,
for
n
such that
I t follows that there is an index
a 1
n
for
0. E
Moreover, every U-bounded subset of Xn
n0
and
5
sufficiently l a r g e .
n = 1,2,...
is contained in
c pxn 3
Thus the sequence
determines the topology
r
.
Therefore
'rb
is
a rnetrizable topology.
be a ~,,-Cauchy sequence i n
Let
is U-bounded,
Bb(U;F). Erm3mcN
Since every compact subset of
U
a ~ - C a u c h ysequence, where
is the topology induced on
T
Ub(U;F) by the compact-open topology of tion
4.6,
#(U;F)
is closed i n
topology, and hence for this topology. every
x E U.
then since m
0
C(U;F)
C(U;F).
B y Proposi-
for the compact-open
{fm]mE(N converges to some
f E B(U;F)
f(x) = lim fm(x) m-tm is a U-bounded set, and
for
I t follows that
N o w if
X c U
is
E
> 0,
ifm]mt'N is a ' ~ ~ - C a u c hsequence, y there exists
E IN* = [ 1 , 2 , . ..]
such that
llfm(x)-fn(x)l/ < E Therefore
which implies that
for every
x E X,
m,n
2
m0
.
TOPOLOGIES ON SPACES O F HOLOMORPHIC MAPPINGS
sup /lf(x)ll xE x
f
f o r the topology
DEFINITION 8.3
rb.
(x)ll
U
+ c
<
fa-
0
converges to
f fm’ mEN
9.E.D. denotes the vector space o f holornor-
WB(U;F)
phic mappings o f
I fm
and ( * ) s h o w s that
f E Wb(U;F);
Thus
SUP
x€x
83
into
F
which are bounded on
U.
The
mapping f E Slg(U;F)H
IIflI
=
SUP
x€u
i\f(x)lI E IR
s~~(u;F).
is a n o r m on
If
PROPOSITION 8.2
F
is complete then
aB(U;F)
with the
norm above is complete. PROOF:
Then, given m,n
2
‘
Let
fm’mcN
be a Cauchy sequence in
c > 0, there exists
n0 E IN
(SIB(U;F),ll
such that f o r
n0 ’
A s in the proof of Proposition 8.1, it follows that
f E a(U;F)
converges to s o m e
compact-open topology on f E
11).
w~(u;F).
Since
‘
for the topology induced by the
C(U;F).
W e m u s t show that
lim fm(x) = f(x)
for every
x E U,
m-tm
follows from (*) that there exists
for every
n
2
n
0
.
Hence
fm’ mEN
no E IN
such that
it
84
CHAPTER 8
Therefore in
f
E
WB(U;F);
( q m ,I1
11).
and we conclude from ( * ) that fm
-t
f
Q.E.D.
We now consider the varioas ways of topologizing SI(U;F). DEFINITION
8.4
will denote the topology on
T~
duced by the compact-open topology of separated locally convex topology o n
C(U;F). 3I(U;F),
#(U;F) T o
in-
is then a
defined by the
seminorms
PK: as
K
E
f
#(U;F)M
ranges over the compact subsets of
DEFINITION
8.5
If
m E N,
cally convex topology on
as
K
#(U;F)
logy on
U.
will denote the separated lodefined by the seminorms
U,
and
n
over
.. .
[0,1,2,.
DEFINITIOlV
K
Tm
ranges over the compact subsets of
the set
as
PK(~) = SUP Ilf(x)li xE K
8.6
#(U;F)
,m}
r m denotes the separated locally convex topo-
defined by the seminorms
ranges over the compact subsets of
REMARK 8.1:
a)
and
n
over
IN.
It follows from Proposition 1.3 that the to-
pologies defined in
is replaced by
U,
d.
8.5 and 8.6 are unchanged if the symbol
;
TOPOLOGIES ON SPACES OF HOLOMORPHIC MAPPINGS
b)
It is obvious that
T~
Tm
S
S
s rm
Tm+l
85
for every
m E N.
We introduce next the topology DEFINITION 8.7
A seminorm
L3
T
on
W
is said to be port-
#(U;F)
3n
U(U;F).
if f o r every neighbourhood
ed by a compact subset
K
of
V
U
there is a real number
of
K
contained in
U
C(V) > 0
such that p(f)
c(v)
5
sup Ilf(x)ll
f o r every
E #(u;F).
f
xE v T
W
denotes the separated locally convex topology on
defined by the seminorms subset
K
(K
U
of
p
[9l],
which are ported by some compact
may vary with
REMARK 8.2: The topology
T
#(U;F)
p).
was introduced by Nachbin in
UI
having as a motivation Martineauts concept of a linear
analytic functional ported by a compact subset.
See Martineau
1761 REMARK 8.3: If the seminorm
p
compact
is another compact set with
KIC U
K1 c K2 c U,
and if
then
p
K2
on
is ported by a
#(U;F)
is also ported by
K2.
On the other hand, if the seminorm each of two compact subsets of in general, true that
p
U,
K1
is ported by
and
K1
is ported by
p
n
K2,
it is not,
K2.
In other words, a seminorrn which is ported by a compact set does not, in general, possess a minimal compact porting set. EXAMPLE 8.1
P(f) =
If(%)
Let
,
U = E =
f E w(C,C).
F = C,
and for
5 E
It is clear that
C, p
let is a semi
86
CHAPTER 8
norm o n
la-51 < r,
and
for every
f
Therefore
p
r'
tlien
= l r ( 5 ) 1 s s u p ( If(x)l
p(f)
E )$(C,C),
where
c
= (x E
~,(a)
p
then
n
K2
since
n
K~
K2. K~
sr(a)}
= Sr(a).
/%-a1 < r' < r ,
is also ported by K1
not be ported by
: x E
: [ x - a \ = r].
K1
is ported by the compact set
is a real number such that
= Srl(a),
a E C
By the maximum modulus theorem, if
)$(C,C).
If K2 =
and
However,
= 6 , and
can-
p
p
is not
the zero seminorm. We note that the unique seminorin ported by the a m p t y set is the seminorm which is identically z e r o . REMARK
8.4:
say that
3
Let
3
for every real number
x E
f E SI(U;F)
converges to
> 0
g
g E M
such that if
)$(U;F)
be a filter i n
thcn
uniformly on
if,
M = M(c) 6 F
therc exists
s E
(jf(x)-g(x)(l
X
We
for every
x.
P R O P O S I T I O N 8.3
p
Let
a)
p
b)
If
V
K
of
PROOF:
a)
x
a b).
2
03 Let
W(U;F)
V,
U
U.
be a filter i n
The family
b'
p(3)
3
conin
0.
#(U;F)
V
which conof
K
con-
o f sets of the form
W
= [O&],
verges uniformly to zero on a neighhourhood tained in
such that
the filter base
converges to
3
for which there exists
contained i n
verges to zero u n i f o r m l y o n IR+ = {x E [R :
K
and let
The following are eqnivaleat:
is a filter i n
a neighhourhood
W(U;F)
K.
is ported by
3
be a seminorm on
U.
be a compact subset of
E
X c U.
and
E
ranging over the set of positive real numbers, is a base
87
TOPOLOGIES ON SPACES OF HOLONORPIIIC MAPPINGS
for the filter C(V)
of neighbourhoods of
b
3
M E 3
such that if
x E V.
converges to zero unif'ormly on
E M,
g
g E M.
Therefore a).
p(3)
converges to
Suppose that
K
of
V
V
which
is finer than
b.
0.
3
while
#(U;F)
in
U
contained in
zero uniformly in
=
s C(v)*c/C(V)
satisfies b) but is not ported by K.
p
We shall construct a filter hood
p(5)
G
for every
= [ p ( g - ) : g E M} c W E ,
p(M)
Hence
< €/C(V)
8.7, p(g)
Therefore, using Definition
for
a
Let
there exists
V
lig(x)ll
then
shows that the filter generated by
b)
R+.
> 0 be the number given by Definition 8.7, and let
Since
= c
in
0
and a neighbour-
such that
3
converges to
does not converge to z e r o ,
p(3)
contradicting b).
If
V
K
of
p
fk
E #(U;F)
gk = fk/p(fk)
k 2 1.
U
contained in
there is an
Then
3
Let
for every
.n *
2
k
N-{O},
k]
k E IN-CO]
,
E
k
2
for every k
E
k
bJ,
1,
for which
and
k E
E
= 1
p(g,)
#(U;F)
be the filter in
Fr6che-t filter in
there is a neighboarhood
such that for every
t SI(U;F)
image under the mapping
Nk = [n E N
K,
is not ported by
N-[O}
H
gk
E
N,
generated by the #(U;F)
of the
which has as base the sets [N,
k
and every
converges to zero uniformly on
2
1.
x
E V.
V.
From ( * ) we have
(**) shows that
However,
p(3)
3
does not
CHAPTER 8
88
0
converge to k
; I
1.
in
R+,
since
This contradicts b),
= 1
p(g,)
and so
p
for every
is p o r t e d by
k E N,
K. Q.E.D.
EXAMPLE 8 . 2
K
Let
be a compact subset o f
U,
a n d let
p
be defined by p: f E H ( U ; F ) H
p(f)
= sup l l f ( x ) l l . xE K
I t is easy to see that the seminorin EXAMPLE 8.3
a =
Let
b m l m C Nbe
khat
K
p
is ported by
U
be a compact subset of
K.
and let
a sequence of non-negative real numbers such
lim
=
Then the mapping
0.
m-t-
is a seminorm on let
V
H(U;F)
be a neighbourhood of
is a real number
r,
K.
which is ported by
K
contained i n
r > 0, such that
Vr
=
T o see this,
U.
Then there
'J Er(x) c V. xE K
Applying the Cauchy inequalities, Proposition
and s o
Then
4.4,
we have
89
TOPOLOGIES ON SPACES OF HOLOMORPHIC MAPPINGS
I t follows that if
K.
is ported by f E #(U;F), mediate.
is a seminorm on
We show that
K' ,a
(f)<
#(U;F),
+m
then it
f o r every
the remaining properties of a seminorm being imf E #(U;F)
Since
neighbourhood
< w.
K' ,a
Vf
of
K
is locally bounded, there is a
in
such that sup{I/f(t)l]
U
:
t E Vf]
Therefore
REMARK 8.5:
We may replace
by
d
d
in this example without
is a sequence of Cam' mEW = 0, non-negative real numbers for which lim a and m m-w B m = am(mm/m!), m E PJ, then 8, 2 0 for every m E N, and altering the conclusion, since, if
'Im
lim m+m PROPOSITION 8.4 PROOF: norm
Let
= (lim (am)l/m)e
= 0.
m-w
To, < 7
K c U on
'K,n
(Pm)'/" w
on
#(U;F).
be compact and
#(U;F),
If we define the sequence
n E N.
Consider the semi-
defined by
{am}
by a m = O if m f n and m€N an = 1, then Example 8.3 shows that 'K,n is ported by K. Since
T~
is defined by the family of seminorms
follows that REMARK 8.6:
7-
i 7
W
.
'K,n
'
it
Q.E.D.
Example 8.3 gives us a method for constructing
an infinite number of seminorms on by compact sets.
Denoting by
r
#(U;F)
which are ported
the set of seminorms which
arise in this way, a natural question to ask is whether
is
90
CHAPTER 8
a fundamental system of seminorms f o r the topology
other words, if
is a T -continuous seminorm on
p
does there exist
W
r
E
K' ,U
p s pK
such that
?
,a
In
TuJ
S((U;F),
A positive
answer to this question would give us an explicit, manageable expression for the seminorms on compact subsets of
which are ported by
#(U;F)
A negative answer would imply the
U.
existence of another locally convex topology
T'
defined by the family have
s Tr s
T,
PROPOSITION
T
8.5
~
of seminorms.
T
r
on
#(U;F),
In this case we would
.
Let
p
be a seminorm on
#(U;F).
The fol-
lowing are equivalent: a)
p
b)
F o r every real number
number
is ported by a compact subset
c(e),
c(c)
U.
e > 0, there exists a real
c,
> 0, such that
F o r every real number
neighbourhood number
of
f E #(u;F).
for every
c)
K
V
K
of
c(c,V) > 0
E
>
and for every open
0,
contained in
U
there exists a real
such that
f E #(U;F).
for every
The proof of this proposition requires the following two lemmas. LEMMA 8.1
such that
Let
B
2P
f E 3((U;F)
(g) c U
then
and
5 E
U.
If
p E R,
p
>
0
is
91
TOPOLOGIES ON SPACES OF HOLOMORPHIC MAPPINGS
f o r every
x E
Bp(5),
k E IN.
,...
By Proposition
PROOF OF LEMMA 8.1: f o r every
k = 0,1,2
Since
5.4,
’-Akf k!
E W(U;P(kE;F))
g2p(g) is 5-balanced, the Corollary
5.1 shows that
for
,...,
k = 0,1,2
x E B2p(5),
wise with respect to the n o r m o f
the convergence being point-
P(kE;F),
where
1
For
defined by
m € N,
m
2 k,
consider the mapping
Pk(y) = Pm(y-5).
Pk E W ( U ; F )
Pk: U
-P
F
and so, applying
the Cauchy inequalities,
for
x E
i p (k{= ) 0,1,... ,
and
x E
g,(g)
and
m
5
k.
But if
then
Therefore
for
k
= 0,1,...,
rn E IN,
m
2 k
and
x E ip(5).
/Ix-yll = p
CHAPTER 8
92
From 1) and 2) we conclude that
for
= 0,1,...
k
LEMMA 8.2
Let
is such that
and f
x E gp(5).
E W(U;F)
fi,p(X)
c U,
and
Q.E.D.
X c U.
If
p
E
R,
p > 0
then for every real number
E
> 0,
we have
Let
PROOF O F LEMMA 8.2:
such that
y E
B'
(x),
P
for
k
= 0,1,...
.
for
k
= 0,1,...
.
Let
and
y E
gp(X);
then there exists x E X
and s o b y Lemma 8 . 1 ,
Therefore
93
T O P O L O G I E S ON S P A C E S O F HOLOMORPHIC M A P P I N G S
Then
c zm pm-k
'
,
Mk
Mk myk k E N we have
and
m
C
k 6
and hence for every
m
oa
C ( C e k=O m=k
M k s
k=O
k
2
m
p
m-k
MA)
which is the statement of the Lemma 8.2. PROOF O F P R O P O S I T I O N
a) E
c).
3
U.
in
p
* >
Let
b). 0,
p
E
< e/4.
c BZp(K) c U,
E R,
Then and
<
0
Bp(K)
set
K,
to
gZp(K)
and to
m
(e/4Im
E
E E R,
Let K
contained
> 0 such that
< dist(K,aU),
p < 2P
Applying Lemma 8.2
m= 0
C(V)
c ) follows f r o m 1) with
K.
C
K.
But since
ing
2)
is ported by
be an open neighbourhood of
f E #(U;F).
f E #(U;F),
for
p
There is a real number
for every
c)
V
Q.E.D.
8.5:
Suppose that
> 0, and let
E R, e > 0
6
sup
x€Bp(K)
c(e,V)
= c(V).
and let
< e / 2 < dist(K,aU).
is an open subset of to
e/4 1 llz
f € #(U;F),
we have ^k
d f(x)ll
p E R, Bp(K) c
U
contain-
to the compact
CHAPTER 8
94
c/4 >
N o w , by h y p o t h e s i s , g i v e n t h e r e a l number
V = B (K) P
t h e open s u b s e t
a r e a l number (since by
depends on
p
f E #(U;F).
f o r every
f
*
let
a).
p
E
containing
t h e r e exists
E
and which we may a c c o r d i n g l y denote
F r o m 2 ) and 3) we o b t a i n :
E H(u;F).
Let
be an open s u b s e t o f
V
with
R,
0
< p < dist(K,aV).
U
containing
f o r every
E g(U;F),
f
m E
[N.
t h e r e i s a r e a l number
f o r every
f E #(U;F).
f o r every
f E #(U;F).
depends on
V,
p o r t e d by
K.
Choose .(€)
Thus, f r o m
As
we may t a k e
Q.E.D.
E
>
gp(K)
Then
s o , by t h e Cauchy i n e q u a l i t i e s ( P r o p o s i t i o n
thesis,
K,
which i n f a c t depends o n l y on
0,
c)
U
such t h a t
c(a),
f o r every
b)
>
c(c,V)
of
and
0
4.4),
c = p/2. 0
K,
and and
c V
we have
By hypo-
such t h a t
4),
depends on
C(V) = 2c(g).
p,
which i n t u r n
Therefore
p
is
95
TOPOLOGIES ON SPACES OF HOLOMORPHIC MAPPINGS
PROPOSITION 8.6 lanced subset o f be p o r t e d by V
number
>
c(V)
f o r every PROOF:
f
U,
and l e t
K,
be a compact 2-ba-
I n o r d e r t h a t a seminorm
U.
of 0
which c o n t a i n s
U
p
on
U(U;F)
K,
there i s a r e a l
such t h a t
E U(U;F).
The c o n d i t i o n i s n e c e s s a r y :
ed by
K
i t i s n e c e s s a r y and s u f f i c i e n t t h a t f o r every
K
open s u b s e t
5 E
Let
and t h a t
Suppose t h a t
i s an open s u b s e t o f
V
p
i s port-
c o n t a i n i n g K.
U
Let
w
=
K C W.
v
: (1-h)S
x
+ xx E v ,
E c,
1x1
.s
i s t h e l a r g e s t open 1 - b a l a n c e d s u b s e t o f
W
Then
[ X E
13. V,
and
B y C o r o l l a r y 5.1 we have m
f(x) =
c
Pm(x-l)
m= 0
for e v e r y m E IN. C(W)
>
x E W
Since 0
p
and
f E #(U;F),
i s p o r t e d by
K,
where
pm =
1
m! :"f(5)
,
t h e r e i s a r e a l number
such t h a t
f or e v e r y
f E #(U;F).
Using
f o r every
f E #(U;F).
Since
(*), we have
W
depends o n l y on
V,
we may
96
CHAPTER 8
for every
f
E x(u;F).
The c o n d i t i o n i s s u f f i c i e n t : of set
containing
U W
of
K.
By Remark
containing
V
(1-1)s + Ax E V
that
K,
Let
5.1
V
be an open s u b s e t
t h e r e e x i s t s an open subp
and a r e a l number
for e v e r y
x
E
W
> 1, such
E C,
and
Applying t h e Cauchy i n t e g r a l f o r m u l a , P r o p o s i t i o n 4.2,
for
f
1x1
s p.
we have,
E ~(u;F),
f o r every
x E W,
a r e a l number
C(W)
m = 0,1,...
>
0
.
By h y p o t h e s i s , t h e r e e x i s t s
such t h a t for e v e r y
f
E #(U;F),
and s o
Taking
C(V)
for every
f
-c(w)p 7 we have '
E
#(U;F).
Hence
p
i s p o r t e d by
K.
Q.E.D.
TOPOLOGIES ON SPACES O F HOLOMORPHIC MAPPINGS
PROPOSITION 8.7
5
PROOF:
is 5-balanced, the Taylor series about
f E S(U;F)
every
of
If U
converges to
f E g(U;F),
Let
and let
W = { g E kl(U;F)
for the topology subset of
Vf
C(Vf)
U
of
5
about
f
T
: p(g)
UJ
.
<
in the topology r
f
.
K c U.
Given
> 0
E
the
is a neighbourhood of zero
E}
By Proposition 5.2
K
containing
w
be a seminorm on S f ( U ; F )
p
which is ported by the compact set set
97
there is an open
such that the Taylor series
converges uniformly to
f
in
Vf
.
Let
> 0 be such that
Now there exists
for every
m
2
m o E IN
mo
and
E W
if
such that
Applying (*) to
x E Vf.
-
f
T
m,f&
we have
thus m
-t
m
f-T
m,f&
in the topology
m
2
TUJ
mo
.
Therefore f E
We now introduce another topology on DEFINITION 8.8
Let
empty open subsets.
#(u;F).
We denote by U
f
as
HI(U;F)
into
F
Q.E.D.
S((U;F).
I be a countable cover of
of holomorphic mappings of each open set of
-t
m,f,S
for every
'
T
U
by non-
the vector space
which are bounded on
I.
The natural topology of
gI(U;F)
is the separated local-
ly convex topology defined by the seminorms
CHAPTER 8
98
where
V
ranges over
If
PROPOSITION 8.8
I. is a Banach space then
F
with
aI(U;F)
the natural topology is a Fre‘chet space. PROOF:
Since
I
is countable, and the natural topology is
clearly separated,
F
whether or not
WI(U;F)
with this topology is metrizable,
is complete.
be a Cauchy sequence in
e > 0 and
given
m,n E N,
m
I
mo
V E I and
n
there exists
Then
WI(U;F).
mo E N
such that if
m o t then
B
(*> If K
is a compact subset of
argument shows that f o r every
U,
a classical compactness
e > 0 there exists
mo E
if
mo.
[N
such that
(**) Thus
sup l\fm(x)-fn(x)II x€K
‘
E
2
mo
and
n
2
is a Cauchy sequence for the topology gI(U;F)
by the compact-open topology on
the sequence to a function and that
‘
‘
fm’ mEN f
r
in-
C(U;F).
is complete, it follows from Proposition 4.6
F
that
converges in the compact-open topology
E W(U;F).
We must show that
converges to fn’mEN It follows from (**) that
x E V,
m
fm’ mEN
duced on Since
<
and s o , by ( * ) , if
f
in the natural topology.
= lim fm(x) for each m-tand E > 0 , there exists
f(x)
V E I
f E WI(U;F),
TOPOLOGIES ON SPACES OF HOLOMORPHIC MAPPINGS
m
E
such that
N
Therefore
llf(x)ll
sup
XE v
sup
5
a)
I1 and
Let
by non-empty open subsets. if f o r every c
v 1.
If
then given
+
(X)II
G
<
Q.E.D.
I2 be countable covers of
I2 is said to be finer than Vl E I1
I1 and if
I 2 is finer than
V 2 E 12,
So
converges
f m 3 mEN
SI(U;F).
E I2 there exists
V2
and
+-t
0
in the natural topology of
f
REMARK 8.6:
v2
\Ifrn
XEV
Finally, (***) shows that
f E SI(U;F).
to
99
there exists
V1
E I1
U
I1
such that f
E S
(U;F),
I1 such that V2 cV1,
and it follows that
in other words,
f
E sII
(U;F)
and
There-
2
fore
#I (U;F) c sI
(U;F), and the inclusion mapping is con1 12 tinuous for the respective natural topologies. b)
Let
I1 = [Vm : m E N}
countable covers o f
I~ =
{vm n wn
: m
E
countable cover o f
U
I 2 = {Wn : n
and
by non-empty open subsets.
vm n wn # 01
LN,
n E N
U
by non-empty open subsets.
and
the set of all countable covers of for
Then c)
11,12E R
E
define
I1 5 I2
if
U
IN]
Then
is also a Let
#(u;F) =
u
H~(u;F);
f o r if
I2
is finer than
f E H(U;F)?
let
I€R vk =
{X
E
u
: Ilf(X)II
< k]
for
k
R
be
by open subsets, and
is a filtered, or directed, partial order on
L
be two
E IN, k
2
1.
Then
I1 8,:
-
CHAPTER 8
100
cvk3k=l,2,. U
..
is an increasing sequence o f open subsets o f
whose union is
that
#
Vk
I E rn
for
@
and
and there exists
k
ko. Let
2
k o 2 1 such
;
then
I = EVklkEN,krko
HI(U;F) c W(U;F)
for which the inclusion mappings
U
countable covers of
WI(U;F)
REMARK 8.8
Gi,
is the ficest locally convex topology on
T~
I
are all continuous, where
each
ko E
E u~(u;F).
f
DEFINITION 8.9 S(U;F)
U,
ranges over the set
?6
by non-empty open subsets o f
of U
and
carries its natural topology.
The topology
T
6
was introduced independently
and at the same time by Coeure [22] in the separable case and by Nachbin [93] in the general case. REMARK 8.9
a)
T~
The following are equivalent descriptions o f
'
is the final locally convex topology of the natural
WI(U;F)
topologies of
a I ( u ; ~ )c H ( u ; F ) , b)
' 6
I1
Given
with respect to the inclusions
I ranges over R .
as
I2 belonging to
and
with
6?
I1 s 1 2 ,
the inclusion 8 (U;F) c (U;F). Then j12,11 I1 I2 with the topology is the inductive limit of the
denote by #(U;F)
inductive system
(U1(U;F), jI,J 1I,JEB
PROPOSITION 8.9 PROOF:
Let
p
T~
K
continuous f o r every W
S
T
6 .
Fix
7
b'
be a seminorm on
a compact subset
T
h
I E R.
of
U.
I E R,
#(U;F)
which is ported by
We prove that
is
from which it follows that
Then there exist elements
V1,. ..,Vk
of
TOPOLOGIES ON SPACES OF HOLOMORPHIC MAPPINGS
I by
K,
Vk = V ,
K c V1 U...u
such that
C(V) > 0
there is a real number p(f) s C(V)
sup Ilf(x)ll XE
Now, for every mapping
and in particular, if SUP
of
U
is ported
f E
#(u;F).
we have
uI(U;F),
f
=
llf(x)I/l
{SUP
xEVi
14irk
F
into
p
such that
f o r every
v f
and since
101
SUP EPV ( f > 3 1s isk i
Therefore (*) can be written: p ( f ) s c(V)
which shows that REMARK 8.10:
p/gI(U;F)
T~ s T,,~
5
f o r every
C gI(U;F),
f
Q.E.D.
is continuous.
We have the following relation between the
various topologies o f T~
(f)}
sup { P 1s isk ‘i
PROPOSITION 8.10
s
Let
w(u;F): T,
S:
xc
T
W
s
f o r every
T*
g(U;F).
m E IN.
The following are equi-
valent: 1)
X
is bounded f o r the topology
2)
X
is bounded for the topology
PROOF: 1) 9 2)
= 1).
2)
Let
pK: S((U;F) + IR
is trivial, since
K
‘b T
T o 5 T
~
6
be a compact subset of
be the seminorm
pK(f)
. U,
E #(U;F).
and let
= s u p IIf(x)ll, xEK
f
.
CHAPTER 8
102
Then
= {f E # ( U ; F )
p;'([O,l])
neighbourhood of
in
0
> 0
p
bounded, there exists
x
and so, since
#(U;F)
is a
s 13
: pK(f)
T 0
is
-
0-
px c pKl([O,l]),
such that
that is,
X
Hence
E
: x
sup{jlf(x)ll
I<,
x}
f E
s p-I
< +=.
is bounded on every compact subset of
and it
U,
follows by methods analogous to those used in the proof of
x
Lemma 6.1 that
k z 1,
k E IN,
For
X)
fE
every
is locally bounded in
and let
..
Then cvk'k=1,2,.
sets.
Fk = { x E U : l\f(x)ll s k
let
f?,
be
Vk
x E U.
Since
there exists an open neighbourhood
Wx
sup{\lf(y)l/ : f E
is contained in
since
wX
and then U
Fk,
ko E IN,
ko
k
,
x
in other words, k E N,
llf(x>ll
8
2
1, and then
x E Vk. Vk
#
@
k
2
ko.
on
#(U;F)
Thus for
k 2ko,
is a countable cover of f E
x
and every Vk
and
SUP xE Fk
llf(X>II
and
(f) s k for every "k is bounded in g I ( U ; F )
Hence
X
Since the inclusion
is continuous for the natural topology on T
contained in U I t follows that
k
such that
F o r every
c gI(U;F)
the natural topology.
topology
x
we have, by the definitions of SUP
x,
of
k E IN,
IN, k 2 ko}
XE Vk
'
Fk
is locally bounded,
VlC, and so
ko 2 1,
I = {Vk : k E
Wx
x
for
by open sub-
U
y E Wx] < +=.
for some
by non-empty open subsets.
k E IN,
f E
x,
w X c gk =
is open,
there exists
the interior of
is a countable cover of
T o see this, let
such that
U.
we have that
x
Fk
'
k;
for
XI(U;F) c # ( U ; F ) HI(U;F)
and the
is T -bounded in # ( U ; F ) . 8 Q.E .D
103
TOPOLOGIES ON SPACES OF HOLOMORPHIC MAPPINGS
COROLLARY 8.1 (m 5 IN)
To,
T
~ T, ~
+ T~m ,,
Tw
and
T
6
all have the same bounded sets.
REMARK 8.11:
E
The topologies
A locally convex topology
o n a vector space
T
is bornological if the following condition is satisfied: F o r every locally convex topology
the same bounded sets as
T
,
s
T’
on
T’
E
which has
T.
It can be shown that for every locally convex topology T
E
on a vector space
pology ed sets. T
.
if
=
T
T
Thus
is
PROOF:
,
T 5 T,,
and
have the same bound-
Tb
and
is bornological if and only
T
b’
PROPOSITION 8.11 T O
such that
is known as the bornological topology associated
Tb
with
E
on
T~
there is a unique bornological to-
T
~
The bornological topology associated with
;that
= (To),.
is,
I E R,
F o r every
the natural topology of
HI(U;F)
is locally convex and metrizable, and hence is bornological. Theref ore
‘6
being the inductive limit of bornological to-
’
pologies, is bornological. Proposition 8.10 that
T~
O n the other hand, we know f r o m and
I t follows by Remark 8.9 that associated with COROLLARY 8.2 with
T~
,
F =
where
lo],
m E LN.
~
6
T~
have the same bounded sets is the bornological topology
Q.E.D.
.
is the bornological topology associated
T6
PROPOSITION 8.12 if
T
T
U {w].
i E
[N
a)
If the dimension of
then
T~
U
=
{a]
T m = T’m+l = T m
=
E T
UI
is finite, or
=
T6
for every
104
CHAPTER 8
'0
<
'm+l
'1"
is obvious.
#
F
<
then
E
Since
I
countable cover
then
{O}
m E IN.
= fO]
sf(U;F)
and the assertion
E
Suppose n o w that the dimension of
{O].
#
F
is infinite and
for every
< 'w
'm
If F = { O ]
a)
PROOF:
and
E
If the dimension of
b)
is finite
is finite dimensional there exists a
U
of
'by non-empty open subsets such
that
v
The closure
2)
F o r every compact subs t
such that
K
c
of each
f E #(U F)
SUP
XE v
I, # ( U ; F )
-
V
= gI(U;F).
topology coincides with
#
{ O}
V E I,
and every
is compact.
Hence, for every such
It follows that
coincides 8 and since this natural
HI(U;F),
with the natural topology of
b)
these exists V E I
U
of
c U.
l l f ( 4 l l =
since by construction cover
K
is compact and
V.
Then for every
F
V E I
1)
for such an
T~
I, we have
Suppose that the dimension of
E
.
m-1 < T,
We shall prove first that
T
T
8
--
To*
is infinite and for every
mrl.
m E N ,
Suppose that
g E U,
T
~
- -T~
~ f o r some
m E iN,
m 5: 1; if
the linear mapping
amf(g)
f E #(U;F)++
is continuous when
#(U;F)
hence is continuous for a real number
c > 0
T
E P(%;F)
carries the topology ~
- -T
T,,
and
~ .Therefore, there exists
and a compact subset
K
of
U,
such
105
TOPOLOGIES ON SPACES O F HOLOMORPHIC MAPPINGS
that
f E
for every
Choose dual o f
E,
H(u;F). '0 E F ,
b
#
and
0,
Ji E E'
,
(I%:
and consider the mapping
the topological
E -+ F
defined by
m
($%)(x)
= [Ji(x)]
-b, x E E.
Then
E
m-homogeneous polynomial f r o m triction to the diagonal of
Em
JImb is a continuous
into
F,
being the res-
o f the symmetric continuous
m-linear mapping
Let
f = Ji%/U.
L. i!
for
i = 0,1,..., m,
Applying (*) to
Since
f
Then
2if(x)
1
x E U.
In particular,
we obtain
/Ibll # 0, m-1 i=O
which implies that
= (m) [~)(x)]~-~*Ji~b 1 ^m -m! d f(x)
m = (I b.
CHAPTER 8
106
Therefore
which can be expressed as:
where
The set
M = hK
is compact;
B = { x E E : IIxII 5 11,
for every
x E B.
SUP
JI E
that
(I
MO
E Bo;
then
Therefore
xt B Then
if we set
= ( 4 E E' thus
l4+4l * : I+(x)
Mo c Bo,
I
SUP
l$(X>l*
XE M g
1
for every
x E MI
implies
from which we deduce.
B o o c Moo.
(**)
In the sequel we shall need the following results from the theory of topological vector spaces: 1)
If E
is a locally convex space and
is the closed convex balanced hull of
2) space
3)
Rieszt Theorem:
E If
c E,
then A o o
A.
If the closed unit ball o f a normed
is precompact, then
M
A
E
is finite dimensional.
is a compact subset of a locally convex space,
then the closed convex balanced hull of
M
is precompact.
107
TOPOLOGIES ON SPACES OF HOLOMORPHIC MAPPINGS
Applying 1) to (**) we find that B
B = Boo c Moo,
By 3 )
is closed, convex and balanced.
i s precompact,
Moo
and since a subset o f a precompact set is precompact, But then by 2) the dimension of
precompact.
since
E
B
is
must be fi-
nite, contradicting our initial hypothesis. We deduce f r o m this that if
=
T
€ o r some
T~
m
m0
since
‘m
0
0
E N,
m E N, f o r
for every
=
T~~
+1 ,
T~ 0
+I‘ T m *
”m 0
a = (a,]
which
T-
we would have
Finally, we prove that let
c
T~
C
rw
.
Assuming the contrary,
be a sequence of positive real numbers for m€ IN
lim a l l r n = 0, m
let
5 t U,
and let
p
be the semi-
m-tm
norm
Example 8.3 shows that
=
p
Therefore, if
T
a real number
c > 0, and
W
T
~
f € #(U;F).
for every
is ported by the compact set
, there
exists a compact set
m E IN,
rn
5
1,
(g}.
K C U,
such that
F r o m the definition of
p
we have
Combining ( * ) and (**),
f o r every
f
E SI(U;F). f = gmb,
Proceeding as before, with tradiction.
Therefore
T-
<
T
~
.
we obtain a conQ.E.D.
CHAPTER 8
108
In the case i n which
REMARK 8.12:
and TW
<
#
F
{O}
E
is infinite dimensional
it can happen either that
7
= r6
W
or that
T 6 *
To illustrate this, we first recall some definitions: A sequence
in a Banach space
E
Schauder basis for
x E E
if every
E
is a
can be written uni-
m
x =
quely in the form
an E
C anxn, n=1
6:
for each
n E N.
..
A Schauder basis
is unconditional when fXnIn=1,2,. m every convergent series o f the form C anxn is unconditionn=1 ally convergent, that is, for every permutation TT of m
m
The following result is due to Dineen [32]: Let
E
and
F
be complex Banach spaces,
empty open subset of and if
E
pologies
and
4;
E
U.
If
U
a non-
is 5-balanced
has an unconditional Schauder basis, then the toT~
8.13:
REMARK
E,
U
and
7
b
coincide on
#(U;F).
Since every Banach space with an unconditional
basis is separable, it is natural to ask the following question: Let
E
and
open subset of
E.
more generally, if and
7
6
F
If E
E
coincide on
REMARK 8.14:
be Banach spaces, and
U
a non-empty
has an unconditional basis, o r ,
is separable, do the topologies #(U;F),
The spaces
that is, is
co
and
unconditional Schauder basis; conditional Schauder basis.
.CP
C[O,l]
The space
(p E
r
W
T~
bornological?
(R,
p z 1)
have
does not have an unLp[O,l]
( p E 1R, p r 1)
109
TOPOLOGIES ON SPACES O F HOLOMORPHIC MAPPINGS
has a n unconditional Schauder basis but, i n general, does not have a n unconditional Schauder basis.
Lp(0,8,p) EXAMPLE 8.4
logies
T
W
If
and
E =
.ern,
r6
do not coincide, that is,
then on
#(E) = #(E;C),
the topor
W
< '6.
See Dineen [ 3 1 ] . PROPOSITION 8.13
for some
c301
1.
5 E U,
#(E;C)
then
is r #(U;C)
W
-complete; if is
T
W
U
-complete.
is 5-balanced (See Dineen
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CHAPTER 9
U N I Q U E N E S S OF A N A L Y T I C C O N T I N U A T I O N
PROPOSITION
9.1
Let
f E #(U;F).
and
be a non-empty
U
open connected s e t
The f o l l o w i n g a r e e q u i v a l e n t :
a)
f
b)
There e x i s t s a non-empty
i s i d e r t i c a l l y zero i n
U.
open s e t
such t h a t
V c U
flv = 0. c)
There e x i s t s a p o i n t
f o r every d)
k
PROOF: a
4;
Suppose t h a t
E
S
U
such t h a t
b)
3
=)
c)
& akf(4;) =
dkf(4;) = 0
1 *k - d f(4;) = 0 k!
so
k E W.
f o r every
0
1 *k k!
S,
d) are t r i v i a l .
3
= {x E U : - d f(x) = 0
f o r every
k
i s open i n
U:
x
if
E
S,
y E BR ( x )
and
f
Rx
S
k
E N.
IN}.
at
i s t h e r a d i u s of
x
L k!
and s o by P r o p o s i t i o n 2 . 2 ,
Thus
i s closed i n
i s a t l e a s t one i n d e x
y
U: ko
E
:kf(y)=O
S.
if
E
f(y) = 0
then
X
f o r every
E
Let
i s n o t empty.
S
convergence o f t h e T a y l o r s e r i e s o f f o r every
1
k E IN.
S
Then
E
4;
The i m p l i c a t i o n s a )
a).
such t h a t
lN.
There e x i s t s a p o i n t
f o r every
d)
E
4; E U
W
x
d o e s n o t lie i n
such t h a t
111
-k1 -oT
S,
$0
f(x)
there
#
0.
CHAPTER 9
112
Consider the mapping -__
%,
ko
d
k !
1 ko!
A k 0
f: y E U n ___ d
0
f(y) E 63(
k0
E;F).
5.4 this mapping is holomorphic.
By Proposition
In particular,
vx
it is continuous, and so there exists a neighbourhood
x
in
U
on which it does not vanish.
Thus
k
1
5 - 7 d Of(y)
of
#
0
0
for every
y E Vx,
closed in
U.
Since
U
is connected,
U.
ly zero in
f E #(U;F).
Let
U
and s o
f
be a non-empty open connected set
F: U
f(U),
the image o f
b)
f(V),
the image o f a non-empty open subset
f.
A
where
5
d)
CZf1
[1za"f(S)(x)
by
f.
...,
xm) : m E IN,
V
(xl,...,xm)
of
U
E Em)
U. : rn E
[N,
x E E]
where
6
is any
U. Denoting by
generated by since
dmf(6)(x1,
is any point o f
B =
point o f PROOF:
=
is
is identicdl-
a)
c)
S
Then the following sets all generate the
same closed subspace o f
by
S = U,
which shows that
Q.E.D.
PROPOSITION 9.2 and
Vx c U\S,
and s o
V c U.
f(U) Let
canonical mappings.
FU
and
and
f(V)
FV
respectively, we have
n u : F + F/FU Since
rrv
and i s
F
the closed subspaces o f
nv: F + F/FV
FV c F U ,
be the
linear and continuous, by
and zm(nvof)(g) = Proposition 3.2 n v o f E SI(U;F/FV), *m = nVod f ( 9 ) for every 5 E U , m E IN. Since f ( V ) c FV
,
113
UNIQUENESS OF ANALYTIC CONTINUATION
nvof
vanishes on
identically on
V
and so by Proposition 9.1
This implies that
U.
f(U)
c Fv
nV o
,
f
vanishes
and hence
FU C F v . Now let nerated by
FA
and
and
B
A
FB
denote the closed subspaces ge-
respectively:
it is easy to see that
F c F A . On the other hand, the polarization formula shows B
that every vector in in
B.
Thus
A
is a linear combination o f vectors
FB = FA. FU =
Finally, we show that
FB
.
By Proposition 3.2
= nuo~mf(~).
nUof E g(U;F/FU)
and
f(U) c , F u , nuof
is identically zero in
= 0
im(.ilUof)({)
f o r every
every
f o r every
m E N,
rn E N,
;"(n,of)(s)
x
m E N.
E
Hence
E.
FB
x E E,
nB(&
1 i"(.il,of)(s)(x) m!
= 0
position 9.1, therefore FUCMARK
9.1
;"f(s)(x))
nBof =
F c FB. U
for every 0
in
U,
1 nU(x imf(5))=0
1 m ! Gmf(s)(x)
E FU
for
F U'
C
1 On the other hand, since m! z"f(g)(x)
m E N,
so that
U,
Therefore
which shows that
Since
=
0,
E FB
f o r every
and so
m E N,
x E E.
Now by Pro-
which implies that f ( U ) c F B;
Q.E.D.
It is easy to show that Propositions 9.1 and 9.2
are equivalent.
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CHAPTER 10
THE MAXIMUM PRINCIPLE
PROPOSITION 10 1 (The maximum modulus theorem) f E W(U;(c).
open connected set and
If(S)l
IfW
2
PROOF:
for every
E =
(C.
x
U
be an
is such that
is constant on U.
E
Let
be any norm-
a positive real number such that
r
x E B,(5),
E U
If
E U, then f
We assume known the case
ed space, and For
x
Let
Br(S)
c U.
# 5,
We define a function
then
g E W(D;C)
1 E D.
every 1 E D,
Therefore
we have
Since f
and by hypothesis
x
f
REMARK 10.1:
that is,
Ig(X)l
2
is constant on
= g(l),
g(0)
D, =
f(x)
was an arbitrary element of
is constant on
connected,
g
Ig(0)l
is constant in
and since
f(5).
B,($),
~ ~ ( 9 )BY. Proposition 9 . 1 ,
for
since
x
# 5,
u
is
Q.E.D.
U.
The following simple example shows that Propo-
sition 10.1 is false in general for vector valued functions. EXAMPLE 10.1 mum norm.
Let
E =
Thus, f o r
(c
and let
F
be
C3
with the maxi-
Y = ( Y ~ ~ Y €~ F~yY ~ II) yII =
115
116
CHAPTER 10
= max{/yll,l~21,1~31}. Let f: U
define
connected,
-t
f
F
by
,/XI
2
llf(x)l/
c
E 2
2
x E U.
x E U,
Then
U
is
and
U,
) = 1 for every
for every
< 11, and
: 1x1
f(x) = (l,x,x ) ,
is holomorphic in
i;f(x)l/ = max{l,lxl llf(O)]l
Ex
u =
x E U.
In particular,
so that hypothesis ana-
logous to those of Proposition 10.1 are satisfied, while
U.
is not constant in
REMARK 10.2
f
F o r further information on this question we refer
the reader to the works of Thorp and Whitley
[137] and of
Vasentini [ i 3 9 ] . PROPOSITION 10.2 (The maximum norm theorem)
E,
bounded open subset of E #(U;F).
SUP
X€B
PROOF:
and
f E C(6;F)
Let
be a
such that
Then
I l f ( 4 I I = SUP xE u
llf(x>ll =
SUP Ilf(x)ll XE
au
We note first that the equation sue Ilf(x)li xc u
= SUP llf(x)ll
XE u
follows f r o m the continuity of
f
in
-
U.
And since
we have
SUP
XE au
Ilf(x)ll
g
SUP llf(x)ll XE 0
Thus we must prove that
We shall divide the proof into three parts. a)
U
E = F = C:
we assume this case as known.
a U c
c,
THE MAXIMUM PRINCIPLE
E = 6, F
b)
117
#
F
an arbitrary normed space,
{O]
.
We
can reduce this to case a) by using the Hahn-Banach theorem. x E U,
Fixing
I/.
Since
x
$ E F'
this theorem yields a
such that
Now, by Proposition 3 . 2 ,
U
was an arbitrary element o f
this completes the
proof in case b). c)
E
F
and
are normed spaces,
F
#
[O]
.
We leave it
to the reader to show that this can be reduced to case b). Finally, the case REMARK 10.3
F =
E
f: 6
f E C(f?;C),
Let 6
-I
is essential, even when the di-
U
is finite.
EXAMPLE 10.2 let
Q.E.D.
is trivial.
The following simple example shows that the hypo-
thesis o f boundedness o f mension of
[O]
be
E = F = 6, U = f(z)
= eZ
.
and
f E #(U;C).
sup
If(z)l = 1
{ z E
Then
C : Re z > 01,
f E g(C;G),
However, since
we have
zeau
< sup I f ( z ) l = +". zE u
and
and s o
a U = [ix:x€!R},
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CHAPTER 11
HOLOMORPHIC MAPPINGS OF BOUNDED TYPE
DEFINITION 11.1 type if gb(E;F)
f
A mapping
f: E
-t
F
is said to be of bounded
is bounded on every bounded subset of
#(E;F)
denotes the subspace of
E.
consisting of the
entire mappings of bounded type. When
E
is finite dimensional, or
entire mapping of
E
into
F
F =
(03,
is of bounded type.
every Examples
5.1, 5.2 and 5 . 3 s h o w that there exist entire mappings which are not of bounded type. PROPOSITION 11.1
F
Let
be complete.
For
f
E
#(E;F),
the
following are equivalent: a)
f
b)
There exists
is of bounded type.
5 E E
such that
lim
I\&
zmf(g)II l/m
= 0.
m-rm
ll&
c)
lim m-tm
d)
There exists
e)
lim
Zmf(x)\l
I/-&
l/m
=
5 E E
dmf(x)l/
l/m
o
for every
such that
= 0
x E E.
lim m-tm
for every
x
1 1 1 dmf(5))I ~
l/m
= 0.
E E.
m-tm
f)
F o r every
Taylor series o f PROOF:
of
f
For
at
x E E
the radius of convergence of the
f
x
x E E, x,
at let
and let
is infinite.
rb(x)
R(x)
be the radius of boundedness
be the radius of convergence of
CHAPTER 11
120
the Taylor series of rb(x) =
states that
at
Then condition a), which
X.
x E E,
for every
+m
is equivalent
5.3, to the statement R(x) =
by Proposition
x E E.
f
Therefore a)
o
f o r every
f m
f).
The Cauchy-Hadamard formula (Proposition 2.1) shows that f) is equivalent to c), and so a) e f)
Q
c).
By Proposition 1.3,
x E E
for every
and
m
Since
N.
mm l / m lim (-)m! = e,
we
m-w
have
e) and b)
c ) e
e d).
Clearly, c )
a
d), and so the proof
is complete if we show that b) * a).
The Cauchy-Hadarnard formula (Proposition 2.1) applied to b) shows that
R(g) =
every bounded subset therefore
f(X)
B M A R K 11.1:
and
g E 6,
X
9 ,
of
and thus
E
r,(S)
=
+=.
Now
is contained in a ball Bp(S);
is bounded for every bounded
X c E.
Q.E.D.
is a sequence of complex numbers,
If
m
C
the power series
a,(z-g)
m
is the Taylor
rn= 0 series at
5
o f a function
lim lamll/m = 0 .
f E #(C;C)
if and only if
We shall see that in the general case the
m+
situation is not quite the same. PROPOSITION 11.2 quence with
Let
F
be a Banach space,
Am E C s ( % ; F ) ,
and
4; E E.
EAm’mEN
a se-
Then the power
m
series
c
A , ( X - ~ ) ~
about
g
is the Taylor series of a
m=O
holomorphic mapping of bounded type f r o m
E
into
F
if and
HOLOMORPHIC MAPPINGS OF BOUNDED TYPE m
PROOF:
C
Suppose that
Am(x-g)
m
is the Taylor series at
m= 0 of an entire mapping of bounded type.
tion a)
121
Then the implica-
e) of Proposition 11.1 and the uniqueness of the
Taylor series shows that
= 0.
lim llAml\ m+w
Conversely, suppose that
Given p > 0,
lim llAml\l’m = 0. m-rm
mo
there exists
m > mO.
mo
N,
5.
x E Bp ( 5 )
Then if
1,
such that
c
llAm(x-5)mll
m
c
h
m= 0
IIA,I//IX-511
+
m0
I/AmI/pm+
C
F
C
2-m
m>m
m=0 Therefore, since
for
l l ~ m l l l l ~ - ~ l ml
c
m>m 0
m=O
=
r: 1/2p
,
m0
W
llAm\ll’m
<
0
is complete, the series
converges uniformly in every ball
In particular,
Bp(g).
this series converges uniformly in every bounded subset of E. Define 0)
=
f(x)
C
Am(x-g)m,
x E E.
m=O V
If gv(x) =
C
m
A,(x-%)
,
v E IN,
then since compact sets
m=O
are bounded, the sequence
converges uniformly on
[g,] VEIN
every compact subset o f
v E IN,
f E ?(E;F),
E.
complete,
#(E;F)
Therefore
f E #(E;F)
d)
gv E #(E;F)
the closure being taken in
the compact-open topology.
series,
Since
for every C(E;F)
By Proposition 4.6, since
with
F
is
is complete in the compact-open topology.
1 Am = - dmf(5) rn!
and, by the uniqueness of the Taylor for every
m E N.
The implication
a) of Proposition 11.1 shows that f is of bounded type.
Q.E.D.
CHAPTER 11
122
We state the following results without proof: Let
PROPOSITION A : E’
on
U(E’,E)
be a normed space.
E
The weak topology
and the topology of uniform convergence on
compact subsets of
E
induce the same topology on each bound-
E’.
ed subset of
PROPOSITION B (The Josefson-Nissenzweig Theorem):
Let
a real o r complex normed space of infinite dimension. there exists a sequence 1)
ll$ml/
2)
lim qm(x) = m-w
= 1
REMARK 11.2:
{Ji }
m mEN
for every
o
c E’
be
E
Then
such that
m E IN.
f o r every
x E E.
Proposition B provided the solution to a long-
standing conjecture.
The p r o o f can be found in Josefson [63]
and Nissenzweig [112]. We shall use Propositions A and B to prove PROPOSITION 1 1 . 3 #b(E;F) 2)
B rn
E
1).
is finite,
If E
m
every exists
is infinite and
m 0 E IN
#
{O}.
o r if the dimen-
B ~ ( E ; F )= #(E;F).
{$,I,
in the weak topology m E IN.
F = {O}
F
has infinite dimension, then by Proposition
there is a sequence
+
E
1) = 2) is clear, since if
sion of 3
#(E;F)
The dimension of
PROOF:
2)
#
The following statements are equivalent:
For each
c E’ U(E‘ ,E),
and
If+,(l
6, +
0
= 1
for
as
lim $,(x) = 0, there m-w Iqm(x) < 1/2 for m z mo. There-
x E E,
such that
such that
since
I
123
HOLOMORPHIC MAPPINGS OF BOUNDED TYPE m
[$,(x)lm
C
fore the series
is absolutely convergent for
m= 0
x E E,
every
f: E
and we may define a mapping
C
-t
by
cn
c
f(x) =
x E E.
r$m(x)lmY
m= 0 m
C
We claim first that the series
[$m(x)]m
is uni-
m= 0
formly convergent on the compact subsets of we note that the set
E'
,
ll$mll
since
= 1
the weak topology
o(E',E)
and since
qm
-t
0
as
m -+
lows that
qm
-t
0
as
m
BY Proposition A ,
and the topology of uniform con-
-
-b
E
must then agree on
B,
in the weak topology, it fol-
m
in the topology of uniform con-
vergence on compact subsets o f
E,
To see this,
is bounded in
N] U [ O ]
m E N.
for every
vergence on compact subsets o f
subset o f
E
B = [$m : m
E.
E.
Thus if
ml E N
there exists
K
such that
is a compact sup I$,(x)I
<
XE K < 1/2
if
E
m
m z ml.
N,
Then
l$,,,(x)lm
< 1/2m
for every
0)
x E K,
m 2 ml,
and s o the series m
formly on
K.
($!,)"
C
Therefore
C ($m)m converges unim= 0 converges uniformly to f
m= 0 on the compact subsets of Since
E
E.
is metrizable,
C(E,C)
is complete for the
topology of uniform convergence on compact subsets o f By Proposition
4.6, #(E;C)
is a closed subspace of
for this topology, and hence is complete. sums
gm =
m C
(Jlk)k
E. C(E,C)
But the partial
are holomorphic mappings of
E
into
C,
k=0 and since
gm + f
as
m +
in the topology of uniform con-
m
vergence on compact subsets o f f
E,
it follows that
E #(E;C). We now claim that
f
is not o f bounded type.
By virtue
124
CHAPTER 11 l/m zmf(0)l/
of Proposition 11.1 it suffices to prove that does not converge to
m E tN
For every
as
0
rn
tends to infinity.
the mapping
is a continuous m-homogeneous polynomial from
E
into
C.
a
It is easy to see that the series ly in
x €
for if
Gl/2(0),
B'
I/ 2
C $: converges uniformm= 0 (0), then
I t follows from the definition of
f
and the uniqueness of
m
q:
C
the Taylor series that
is the Taylor series of
f
m= 0
at
0; thus
Therefore
f
Now if mapping
@ ab(E;C). F
fob: x
#
(O},
let
b E F,
E
#
E E H f*b(x) = f(x)b E F.
but is not of bounded type, since if of
b
f o r which
is not bounded on
f(X)
X.
X
0, and consider the
Then
f
is entire,
is a bounded subset
is not bounded, it is clear that This concludes the proof of 2)
fob
I).
3
Q.E.D. REMARK
11.3:
In Definition 8.2 we have defined
to be of bounded type if
f
f E #(U;F)
is bounded o n every subset
X
125
HOLOMORPHIC MAPPINGS OF BOUNDED TYPE
of
U
X
such that
dist(X,aU) > 0.
is bounded and
#(U;F)
denotes the subspace of
which are of bounded type.
ab(U;F)
f E #(U;F)
of mappings
We note that when
U = E
this
definition agrees with Definition 11.1, since every bounded
E
subset of
E.
is a positive distance from the boundary of
PROPOSITION 11.4 The following statements are equivalent:
#
#b('iF)
b)
The dimension of
F = {O]
finite or if 3
a).
f
which
5 E U,
,
X
Let
is not bounded.
E R,
0
0
= up
-1
x
+
s
is bounded.
= Cap
-1 x
Moreover
Let
M = sup[]/xll : x E
M/p
< 1.
+ 5
:
X'
E
is
f E #(E;F)
be a bounded subset of
> 0, such that B,(5)
and we may assume that
x E X
{O].
zb(U;F) = #(U;F).
then
bounded there exists a real number
x'
F f
is infinite and
By Proposition 11.2 there exists
fa' ab(E;F).
that
E
a) * b) is clear, since if the dimension of
PROOF:
b)
#(U;F)*
a)
p
c U.
> 0
such
E
on
X]
Since
and let
X
such that X
is
c B (f), P
Then the set
x f X]
is a subset of
is U-bounded.
U
which
In fact, for every
we have
Therefore dist(X',hU) Let fohot,
X'
c Br(f)
and since
g0(g) c U
it follows that
> 0. g
be the restriction to
U
of the composition
where t: x E E M t(x) =
X-5
E E
126
CHAPTER 11
and
h: x Thus
E
E W h(x) = U
g(x) = f[u-lp(x-5)],
g
E
W(U;F),
g
q
W&J;F).
FIEMARK 1 1 .4 :
and s i n c e
h[t(X’)]
P X E E.
I t i s easy t o s e e t h a t
= X,
g(X’)
= f(X).
Hence
I n many problems c o n c e r n i n g i n f i n i t e dimensional
normed s p a c e s t h e space t h a t of
x E U.
-1
W(U;F)
51 ( U ; F )
b
plays a r o l e s i m i l a r t o
i n t h e case o f f i n i t e dimensional spaces.
One example o f t h i s i s t h e s t u d y o f domains o f holomorphy and t h e Cartan-Thullen theorem.
CHAPTER 12
DOMAINS O F ab-HOLOMORPHY
We wish now to consider the following problem: Given a non-empty open subset exist an open subset every U
f
and
E
with
E,
V a U,
E,
V $ U,
E, with
does there such that
f E U(V;F)
are connected, Proposition 9.1
V
of
U
of
of
extends to a mapping
#(U;F)
?
(If
shows that this ex-
Equivalently, given a non-empty open sub-
tension is unique). set
V
U
does there exist an open subset
V
of
E,
such that the linear restriction mapping
is surjective? It is easy to see that, without any further conditions, this question has a negative answer? even when the dimension of
E
is
= C-{g},
1.
For, if we take
E = F = C,
g E E
and
U =
the function f: x E U H f(x)
is holomorphic in
U
1 = €(I: x-5
and has no holomorphic extension to
6.
In the general case, this problem is vary difficult, and there are few satisfactory results. the problem by substituting
ab(U;F)
We shall simplify
for
W(U;F).
More pre-
CHAPTER 12
128
cisely, we shall characterise a large class of non-empty open subsets
U
E
of
V $ U,
V,
for which there is n o open set
such that the linear mapping E Hb(V;F)
f
€ H(U;F)
+ f/U
is surjective. We shall restrict o u r attention to the case
F = 6.
Thus most of the definitions and results which follow will refer to the space
REMARK 12.1:
which we abbreviate by
#,(U;C),
#,(U).
F o r the problem ofhalomoqhic extension which we E
have posed, the case in which the dimension of
is 2 o r
more is totally different from the one-dimensional case. E
Let
P R O P O S I T I O N 12.1
be a complex normed space of dimen-
sion greater than o r equal to 2, and let
If V
Banach space. and
,
U = V-{g]
extension
be a complex
E, 5 E V
is a non-empty open subset of
then every
f E #(U;F)
has a unique hdLcmorphic
#(v;F).
g E
REMARK 12.2:
F
This proposition states that when
E
is a
complex normed space o f dimension greater than or equal to 2 and
F
is a Banach space, it is not possible for a h o l o m o r p h i c
mapping to have an isolated non-removable singularity. PROOF OF P R O P O S I T I O N 1 2 . 1 :
Choose
the Hahn-Banach theorem there exists
I I J r l l = 1 and hyperplane in ly in the f o r m
$(e)
E,
3 :
1.
The kernel
and s o every
x = hxe
+
e E E
J, E E’
S
x E E
s where x)
with
of
1) ell
= 1.
By
such that J,
is a closed
can be written unique-
hx € G
and
sx E S
DOMAINS OF gb-HOLOMORPHY
(xx
129
= JI(x))p > 0
Let
x E BE ( 5 )
For
be such that
B p ( 5 ) C V,
and let
E
= p/3.
let
f( Xe+sx) ’ Ix-1)(5
1 I=e
-Gjq-q- dl
T o see that the first integral is defined, let
x E Be(5:).
and
and s o then
Xe
+
Since
Finally,
= e *
Therefore
X-f(x) = 0
,
5
is impossible, since
IIJ,(IIIX-~(I = g(x)
every
c U.
X E C
The p r o o f that
as the p r o o f that
with
IX-J,(E;))
with
g
+
sx
= 5
;
sx
s e,
he+sx E B p ( 5 )
#
.
And since
for every x E B (g). 0
g E “Be ss
s U.
1 = J , ( x ) implies
IIx-5(1< c .
E F
We assert, without p r o o f , that
x E Be ( 5 )
Xe
Xe+sx E B p ( S ) - { 5 } c V-{S}
is complete, it follows that
Now let
E ,
which is impossible since
e = Il-1)(9)1 = IQ(x)-l)(g)l s F
s
Suppose that
sx E B p ( 5 ) c V.
h = $(Xe+sx) = J , ( 5 ) ,
IX-$(S)I
+
1 = $(5)e
=
lh-$(X)l
( 5 ) ;F).
We claim fhat for
we have
Xe+sx E B p ( 5 ) - [ 5 ) c
is done in the same way
is well defined.
Suppose then that
CHAPTER 12
130
Xe+sx = f
.
sx = s
direct sum, )i
E C,
5 = $ ( f )e + s 5
Then, since
g
,
and
Ge
which is a contradiction.
IX-$(f)I
and
,
+
is a
S
Hence if
then. Xe+sx E B p ( S ) - { S ] .
s E,
E
Now the hypothesis that
has dimension greater than
o r equal to 2 implies that the set
D = { x E Be({) F o r if
is not empty. x E Be(g), x
D =
@
: sx#
s5)
s = s x s
we have
and so
= $(x)e
+
sx = $(x)e
= 5 x E Be(5).
f o r every
+
+
E
f = q(x)e
$(x-g)e
+
E f
W =
@.Then
D
is dense in
s = s Y g
= [@(y)e+sy]
that
y-5
fore
W c 5
@(f)e
Ce
Thus the linear variety
a non-empty open subset of
n
-
f
5 + Ce =
+
Ce
has
E = Ce,
has dimension 1, contrary to o u r hypothesis.
We claim that
D
+ 5
s
non-empty interior, which implies that that is,
f o r every
+
Ce,
-
Be(5)
-
Bc(f)
[$(g)e+s5]
y
Let
W
E W,
which implies
= $(y-f)e E Ce.
which is impossible, since
E
There-
has dimen-
sion greater than or equal to 2 . Now given
x E D,
It is easy to see that F o r every
x E D
consider the mapping
h E #(Be($(g));F)
we have
be
and suppose that
{f},
for every
- 153.
n C(Ge($(g));F).
131
DOMAINS OF Hb-HOLOMORPKY
and hence
@(x)
E
Therefore, applying the Cauchy
Be($(g)).
integral formula to
h, f
for every
x E D.
Hence f
for every f(x)
x E D.
= g(x)
It follows from the definition of x E D.
for every
Now since
lomorphic, and hence continuous, in coincide in the dense subset throughout
Be
-
(5)
[g}
is defined by setting
g’Iu
(5),
with
= f.
0
-
(53,
(5)
and
f
and
Hence, if the mapping
g‘
= g
Bg(5)
in
< el < e l
then
g’
g
that
are ho-
and they
must coincide
g
.
and
g’ : V -+ F
g‘
E #(V;F)
= f
in
and
Q.E.D.
REMARK 12.3:
If the mapping
to
then
ab(U;F),
DEFINITION 12.1 subsets of
D,
Be
f
g
E
Let
E Bb(V;F).
g’ U
such that
morphic extension of
of Proposition 12.1 belongs
f
U
and
V
be non-empty connected open
U c V.
V
if every
(necessarily unique) extension
g
is said to be a ab-holof
E
U
and
V
+
U
possesses a V
w,(V).
a proper #b-holomorphic extension of morphic extension of
E #,(U)
if
V
is said to be is a zb-holo-
U.
We remark that the definition of a #b-holomorphic extension can be rephrased as follows:
the mapping
CHAPTER 1 2
132
is a surjective isomorphism between the algebras
#,(V)
and
w,(u) If E has dimension greater than o r equal to 2 ,
EXAMPLE 12.1
V is a connected open subset of E, 5 E V, and U = V-lg}, then by Proposition 12.1 and Remark 12.3, V is a proper Wb-holomorphic extension of U. REMARK 12.4:
In the case where the dimension of
E
is 1,
the definition of a proper #b-holomorphic extension is of no interest, since the situation described cannot arise.
U
and
V
U $ V,
are non-empty connected open subsets of
For if
E
with
5 E V n aU, then the function
and if
2*
f: x 6 U H f(x) = - E belongs to Thus
ub(U),
and has no holomorphic extension to
is not a proper Ib-holomorphic extension of
V
V.
U.
Next, we introduce some terminology which will help to simplify the definition of a domain of holomorphy.
DEFINTION 12.2
Let
open subsets of
E
U, V
and
such that
W
W c U
a Ib-holomorphic prolongation of f E #,(U)
V by and
there exists
be non-empty connected
g E Wb(V)
U
n
V.
by
V W
is said to be if f o r every
such that
flW = q w is said to be a proper gb-holomorphic prolongation of
W
if
V
is a kl -holomorphic prolongation of b
U
by
U
W,
V $ U. We say that a non-empty connected open subset U
has a oroper #,-holomorohic
of
E
wrolonaation if there exist non-
133
DOMAINS OF Sb-HOLOMORPHY
empty connected open subsets,
V
by
W,
g
b)
If V
W,
f E ab(U)
such that
U
W.
by
f
which contains
g E ab(V)
there exists
in the connected component
W.
We point out that Definition 12.1 is a special case of
Definition 12.2, in the following sense:
(proper) gb-holomorphic extension of
U,
(proper) gb-holomorphic prolongation of
then
U
by
DEFINITION 12.3 (Domains of Hb-holomorphy). U
nected open subset
of
E
U
if
V
and
E, U c V
non-empty connected open subsets of
U
E,
of
is a #b-holomorphic prolongation of
coincides with
n V
U
of
a)
then f o r every
such that
Wo
and
is a proper ab-holomorphic prolongation of
REMARK 12.5:
U
V
V
and
V
are is a
is a
U.
A non-empty con-
is a domain of gb-holomorphy if
has no proper ab-holomorphic prolongation.
Explicitly,
this means that there does not exist a pair of connected open sets
V
and
W
such that
1)
o#wcunv.
2)
F o r every
that
f E Wb(U)
g E gb(V)
such
gIw = flW.
EXAMPLE 12.2
E, g E V,
Let
V
U = V-{g}.
be a non-empty connected open subset of
If
gb-holomorphic extension of
V
there exists
dim E
U
2
2
then
V
(Proposition 12.1), and hence
is a proper #b-holomorphic prolongation of
Therefore
U
EXAMPLE 12.3
is a proper
U
by
U.
is not a domain of gb-holomorphy. Every non-empty connected open subset
U
of
C
134
CHAPTER 12
is a domain of holomorphy that is, a domain of Wb-holomorphy.
U
F o r suppose that
has a proper #b-prolongation.
exist non-empty connected open sets, W c U
n
V $?! U,
V,
g € W(V)
f I W= gIw.
ed component of
n
U
V
Then, if
containing
g E V n aU
awe,
n
then the function
belongs to
W(U),
and there is no
f I W= glw,
since if such a
But then since
Wo.
which is impossible, as
5
€
g(5)
g
such that
W,
there exists is the connect-
Wo
V
we have
W,
(see the proof o f Proposition 12.2).
@
in
and
f E W(U)
and for every
such that
V
Then there
n aU
f: x E
1 uk- t 6
x-5 such that
g E #(V)
and
g
aWo#
If we choose
existed, we would have
Go
n
f = g
is continuous at
E 6. Therefore
U
5
,
is a domain
of holomorphy.
This example is the inspiration for the following proposit i o n . PROPOSITION 12.2 open subset of f E W,(U)
Suppose that
E,
for which
U
is a non-empty connected
such that for every lim If(x)l
3:
f E
au
there exists
a.
-5
Then PROOF: Then set
U
W.
taining
U
x€u is a domain of #b-holomorphy.
Suppose that
U
is not a domain of #b-holomorphy.
has a proper #b-holomorphic prolongation Let W,
Wo let
be the connected component of a € W
and
b € V\U.
Let
y
U
V
n
by a
V
con-
be a path in
135
DOMAINS OF gb-HOLOMORPHY
V
from
a
to
V,
into
[O,l]
b;
y.
and
fore the set of points in
(y)
(y) n aWo element;
is a closed subset is connected.
There-
corresponding to points in
to E [O,l]
that is, there exists and
(y)
Let
It follows that this set has a least
is closed.
awo,
E
[O,l]
= b.
y(1)
(y) n aWo
Now
and it is non-empty since
V,
y(to)
is a continuous mapping from
y(0) = a
with
denote the image of of
y
thus
awo
y(t)
for all
such that
t E [O,to).
Let
Y(t0> = 5 . E aU.
We claim that
If this is not s o there are two
possibilities: 1)
5 E U.
contained in
U
n
V.
By the choice of
~ ( 5 ) contains points in Wo is,
w0
+
and then
w0 u ~ ( 1 5 Wo U V ( 5 )
and containing since 2)
Wo
~ u t~
U.
Then
5,
( 5
that
Wo;
may be chosen to be connected,
is a connected set, contained in as a proper subset.
5
#
5
of
this implies that
and points outside
is a connected component of
Wo
V(5)
Then there is a neighbourhood
U U aU =
f?,
U
n
V,
This is impossible U
n
V.
implying that
5 $ Go,
which is absurd. Therefore
5 E au n ( y ) n a w 0
f E Wb(U)
then exists
such that
.
BY hypothesis, there
lim If(x)l
=
00.
X 4
XEu If V
is a proper gb-holomorphic prolongation of
there exists
,
g E Wb(V)
such that
and since
5 E V
n
U
glw = f l W . Then
aWo,
by
W,
C H A P T E R 12
136
Q.E.D.
which is absurd. D E F I N I T I O N 12.4
x E E\X
if every
S
X
A subset
RE'MARK 12.6:
n
S
We note that every complex affine variety S
a)
S = E
i s a complex vector subspace of
If U
b)
5 E
i s said to be subconvex
X = Q.
of codimension 1 is of the form T
E
is contained in a complex affine variety
codimension 1, with
of
of
+ T,
where
which passes through
of codimension 1.
4;
S
E
and
of codimension 1
and does not intersect
is neces-
U
sarily closed, since it would otherwise be dense in as
E E and
Tl
is a non-empty open subconvex subset of
the complex affine variety
E\U,
T-
E,
and
is open, this is impossible.
U
P R O P O S I T I O N 12.3
Every open convex subset
U
of
E
is sub-
convex. PROOF:
If
=
U
this is clear.
@
If U
#
Q
and
g E E\U,
then by the Hahn-Banach theorem applied to the space
E,
considered as a real vector spaces there exists a real vector subspace (5+T)
n
T
of
U = @.
E,
of real codimension 1, such that
S =
Let
+
(TniT).
Then
S
is a complex
affine variety of complex codimension 1, 5 E S ,
s n u = @ . REMARK 12.7 For if CO]
and
Q.E.D.
Every subset of
X c d:
and
g !$ X ,
is a subspace of
d:
C,
convex o r not, is subconvex.
we may take
S = 5
o f codimension 1.
+
{O},
since
137
DOMAINS OF #b-HOLOMORPHY
PROPOSITION 12.4 E
U
of
is a domain of #b-holomorphy.
PROOF:
U
Every subconvex connected open subset
g E
Let
Since
aU.
U
is open,
g $! U,
and s o ,
being subconvex, there exists a complex affine variety
S = q
+ T
of codimension 1, such that
By Remark 12.6 (b) there exists
cp
S,
E E',
Thus
= cp(5)
and if
= A};
cp(x) f 1.
Since
and
T, is closed.
the topological dual of
T = {x E E : cp(x) = 01.
Cp(5) = cp(q).
and hence
5 E S
E,
S
n
U = 0.
Therefore for which
E S, 5-7 E T, and
SO
S = {x E E : x-7 E T} = (x E E : cp(x) x E U
then
x $! S,
=
which implies that
It follows that the mapping
belongs to
#,(U),
and
lim If(x)l
=
m.
Therefore, by Propo-
X-G XE u sition 12.2,
U
is a domain of Wb-holomorphy.
Q.E.D.
By Proposition 12.3, we deduce: COROLLARY 12.1
Every convex connected open subset of
a domain of #b-holomorphy.
E
is
This page intentionally left blank
13
CHAPTER
THE CARTAN-THULLEN THEOREM FOR DOMAINS OF ab-HOLOMORPHY
DEFINITION 13.1
Let
and l e t
The g b - h u l l of
X c U.
U
E,
be a non-empty open s u b s e t of X
i s the subset o f
U
f i n e d by:
13.1
PROPOSITION
(2)
3.tb(U)
(4)
If
(5) If
i s closed i n
2
X c Y
C U,
U, V
a r e non-empty
then
f o r every
U
X c U.
c?
)'(b'
open s u b s e t s of
E
with
A
U c V,
(6)
(8)
,sup
k
If
then
X
c ?
I f ( t ) l = sup If(x)l x€x
i s the closure of
X
for e v e r y
in
A
=x PROOF:
W e s h a l l prove ( 2 ) and ( 6 ) only.
139
X c U.
f o r every
U
then
f
E #,(u).
de-
CHAPTER 13
140
thus
?
is the intersection of a family of closed sub-
) ' ( b # sets of U.
can be described as the largest of the sub-
( 6 ) iwb(u) sets T of U
with the property:
A
X c X w b ( ' >
But as
'
the reverse inequality also holds.
On the other hand, if sup If(t)l tET
f E wb(U),
If(x)l form
5;
c
T
and therefore
REMARK 13.1:
for every
xE x
If(t)l
Let
x E X.
f E Wb(U),
2
XE
x
w,(u),
f E
for every Q.E.D.
wb(u> '
and suppose that
x E
]iHb(u)
which is valid in
.
Thus, an inequality of the
X,
is valid in
This is the fundamental property of the ab(U)-hull
x
of
?
) ' ( b S ' of a subset
u.
PROPOSITION 13.2
If X
is a subset of
E,
then
?#b
A
contained in the closed convex balanced hull
E.
s C
If(x)l
By Definition 13.1 we h .ve that
for every
(fl 4 C
for every
SUP I f ( x )
L C
is such that
I
= sup If(x)
t E T,
then for every
Tc U
X,
of
is
X
in
THE CARTAN-THULLEN T H E O R E M FOR DOMAINS OF Ub-HOLOMORPHY
141
We shall employ the following formulation of the
PROOF:
Hahn-Banach theorem:
(*) Then E
E
Let t
X c E
be a real normed space,
t E X.
and
belongs to the closed convex balanced hull of
if and only if cp(t)
sup cp(x)
5
X
in
for every continuous
XE x linear form
ER
Let cp E
let $:
on
cp
that is,
the topological dual of
-
@(x) = cp(x)
by
sider the mapping
ez: C -+ C .
Also,
x
'
SUP xEx
( * ) that of
E A.
I,
c sup J e
Define Then
H
E E', E.
The mapping
Con-
m
e
and the entire function
is of bounded type, since if
Therefore, if
for every
cp E
A c E
then
t E
which implies that
cp(X)
t
H
M = sup IIxII, x€A
is bounded, and
for every
em
E C.
e@
and
E, being the composition of the
continuous linear function
cp(t)
x E E.
icp(ix),
el : x E E I+
is an entire function on
XE x
ER.
is a continuous complex-linear form on
@
E,
denote the real space associated with
(ER)',
E M Q:
E.
?
Wb(E)
ecp(t)
(ER)',
g
'
Ie@(t)/ s
sup ecp(x). xE x
Hence
and it follows from
belongs to the closed convex balanced hull,
2,
Q.E.D.
X.
COROLLARY 13.1
If X
is bounded (respectively, precompact),
n
X
is also bounded (respectively, precompact). #b(E) More generally, if X is bounded (respectively, precompact), then
and
X
is a subset of the non-empty open subset
U of
E,
n
then
X
wb(u)
is also bounded (respectively, precompact).
CHAPTER 13
142
PROOF:
These statements all follow from the fact that
xc 2
c f
W b ( ' > precompact)
C
gb(E) when X
?,
since
?
is bounded (respectively,
is bounded (respectively, precompact). Q.E.D.
REMARK 13.2:
U,
in
The
Slb(U)-hull
X
of a subset
E,
but is not necessarily closed in
compact.
U
of
is closed
X
even if
is
The following example illustrates this phenomenon.
EXAMPLE 13.1
n
E = c ,
Let
n
2
2, g E Cn
n
U = 6: - [ 5 } .
and
Let X = { z E C
X
Then
n
is compact.
If
f E W(U)
has a holomorphic extension to
f
then, by Proposition 12.1,
f E 3L(E).
By the maximum
norm theorem (Proposition 10.2) we then have I:
sup If(x)
I
zE
for every
cn, 1) z-511 ?;
s:
If(z)l
1.
X€X z
E Cn
but since W b ( ' > ' closed in E = en.
#
Therefore every A
X
DEFINITION 13.2
0 < I(z-gll c 1
with
belongs to A
U, 4 @
?3Lb(u).
Thus
A non-empty open subset
U of E
be gb-holomorphically convex if for every subset which is U-bounded, its ab(U)-hull,
2
is not
X
)'(b'
wb(u>
'
is said to X
of
U
is also U-bound-
ed. We recall that to say dist(X,aU) > 0, and
LEMMA 13.1 :mf
If
E #,(U;P(%;F))
X
Xc U
is bounded in E.
f E ab(U;F), and
is U-bounded means that
dmf
then f o r every
E Hb(U;Ss(%;F)).
m E IN,
THE CARTAN-THTJLLEN
E
t h e n , by P r o p o s i t i o n 5.4,
f E Wb(U;F)
If
PROOF:
THEOREM FOR DOMAINS O F Hb-HOLOMORPHY
dmf E
and
W(U;P(%;F))
2rnf
Thus w e have o n l y t o p r o v e t h a t
o n the U - b o u n d e d Let
s o that the set Gp(X)
rn
E
subsets o f
=
dmf
a r e bounded
be a U - b o u n d e d
p = d/2.
Let
subset of
U,
W e c l a i m that
u
Gp(x) i s U - b o u n d e d . It i s clear that XE x i s bounded, and t h a t g p ( X ) c U. Let z E ap(X), and
u E aU.
ap(X)
W e have
dist(u,X)
Therefore SO
X
= d > 0.
dist(X,aU)
and
m E N.
U.
and l e t
N,
irnf g
f o r every
#(U;Ss(%;F))
143
dist(z,U)
dist($,(x),au) Since
2
2
d
> p
2
Idist(u,X)-dist(z,X)(
2
2
d-p
= p,
p. i s U-bounded,
gp(X)
dist(z,X).
and
f E Zb(U;F),
and s o by t h e C a u c h y i n e q u a l i t i e s ,
f o r every
E
N.
x € X,
m
m E W.
Also,
Therefore
and s o
f o r every
by P r o p o s i t i o n
1.3, we have
and
144
CHAPTER 13
for every
m E N.
LEMMA 13.2
Let
Q.E.D. X
r > 0).
r = dist(X,aU)
(thus
and for every
t E $b(U)
Taylor series of
PROOF:
Let
Lemma 13.1,
U,
be a U-bounded subset of
f
at
fe r (X)
f
E Wb(U)
the radius of convergence of the t
0<
g R,
Then f o r every
and let
is greater than or equal to
r.
< 1; then, f r o m the proof of f E slb(U),
is U-bounded, and so i f
Applying the Cauchy inequalities, we have
Thus 1 *m
( * > l s d f(x)(y)
1
s
+
~lyll~ f o r every x E
x,
y E E, m
c
( 0 r)
By Lemma 13.1, that, for every
zmfE
y E E,
is of bounded type.
Wb(U;P(%)),
and this implies
the function
F o r if
T
is U-bounded, then
Now Remark 13.1 applied to the function E Wb(U)
s h o w s that inequality
Therefore
( * ) is valid in
2mf( 2
) (y) E
wb(u)
'
R.
THE CARTAN-THULLEN THEOREM FOR DOMAINS OF Wb-HOLOMORPHY
x E X
for every
wb(u)
’
y E E
and
Now fix a point
m E IN.
Therefore
p
and let
of convergence of the Taylor series of
f
145
be the radius
at
t.
Applying
the Cauchy-Hadamard formula to the above,
p
Therefore p 2 r. REMARK
re
2
for every
E R,
Hence
Q.E.D.
13.3:
If P E P ( % ; F ) ,
it is clear that P/W E Wb(W;F)
for every non-empty open subset
P E P(E;F), set
0 < 0 < 1.
then
of
W
PIw E Wb(W;F)
E.
Hence, if
for every non-empty open
W c E.
LEMMcl 13.3
Let
5 E E, and
E
F
and
Pm E 6‘(%;F)
F
be normed spaces, for every
m E N.
complete,
If the series
0
f(x)
=
C
Pm(x-s)
has radius of convergence
r > 0,
then
IIkO
E ab(Br(% ) ;F)* PROOF:
It is easy to see that it is sufficient to consider
the case
5 = 0. Let U = Br(0),
and for each m E m be the mapping defined by fm = C Pk/U,
fm: U + F m fm(x) = c pk(x) k=O for every m E N.
[N,
let
that is
k=O
for
x E U.
BY Remark 13.3,
fm E
w~(u;F)
B y the definition of the radius of conver-
146
CHAPTER 13
gence, the sequence
gp(0)
for every
is contained in
rf
I
p
‘
converges uniformly to
fm’ mEw
E [O,r),
gp(0)
f
in
and since every U-bounded set p E [O,r),
for some
Wb(U;F)
is a Cauchy sequence in
it follows that
with the natural
&IN
topology.
Since
Wb(U;F)
that
F
is complete, we have, by Proposition 8.1,
is complete, and hence
g E Zb(U;F).
the natural topology to some ‘fJmEN and f
U
converges pointwise in
E Wb(U;F).
converges in
EfmIrnem
to
In particular, Therefore
g.
g = f,
Q.E.D.
THE CARTAN-THULLEN T H E O m M (part I): E.
connected open subset of
Let
U
be a non-empty
Then the following are equi-
valent : a)
U
b)
F o r every U-bounded subset
is a domain of gb-holomorphy.
dist(X,aU)
on
of
U,
= dist(2 Wb(U)
c)
U
d)
F o r every sequence
is Zb-holomorphically convex.
5 E aU
a point
X
Eg I nEN ,
Let
f
U
Wb(U)
=
sup If(gn)I
which converges to which is unbounded
m,
n€[N
PROOF: a) a b). dist(X,au)
there exists
that is,
in
‘5 n’nEN
Since
X c
?
’
W&J)
it is clear that
dist(:
2
= r
If
t E
?
V = B,(t)
Wb(’) ’ the set is a non-empty connected open subset of E. De-
noting by
W
tains
U, V
t,
dist(X,aU)
7
0.
the connected component of and
W
U
n V
which con-
are all non-empty connected open sets,
147
THE CARTAN-THULLEN T H E O W M FOR DOMAINS OF Hb-HOLOMORPHY
Uc U
and
n
V.
Now if
f E gb(U),
cr > 0, depending on
number
there exists a real
BD(t) c
f, such that
U,
and
W
f(x)
1 c m ! Arnf(t)(x-t)
=
m= 0 uniformly in
If p
BD(t).
is the radius of convergence of
this Taylor series, then by Lemma 13.2,
p ;z r ,
and by
Lemma 1 3 . 3 , the function CD
go: x E B ( t ) w go(x) = P
is an element of g
V
to
definition of and hence
m=O
go
we have
flW = g l w
*
p 2 r
3~~(~~(t)). Since
to obtain a function
1 7 imf(t)(x-t)
C
g = g o l v E W,(V).
= g(x)
f(x)
g E gb(V)
this means that
W,
by
morphy. words,
wI'
is a #b-holomorphic prolongation of
V
Therefore
U
Wb(u>
,au) 2
3ib(U) r
=:
U,
'
dist(X,aU).
*
X
is bounded, and so by the Corollary 13.1,
X
in other
and so
b)
Let
U
is a domain of Ub-holo-
must be a subset of
f o r every
dist(?
c).
there
f E Wb(U)
= gIw. But if V # U,
which is impossible since
Br(t) c U
x E B,(t),
.
such that
V
From the
for every
Thus, we have shown that for every exists
we may restrict
be a U-bounded set.
Then, in particular, A
bounded.
is
x3b(u)
Now by b) dist(X,aU)
(u),au),
= dist(% b
and s o since
dist(X,aU) > 0, we have also
dist(j;d b
( u ) ,au)> 0.
148
CHAPTER
13
A
U
Hence c)
d).
Suppose d) is false. of points in
Cg,,]n
X.
is U-bounded f o r every U-bounded set X ) ' ( b W is holomorphically convex.
Therefore
such that every
U
Then there is a sequence
which converges to a point is bounde on the set
f E 3ib(U)
[sn
5 E
aU,
: n E N].
I t follows that the mapping
W,(U).
is a seminorm on H
f(x) E C
f t Wb(U)w
Since the linear form
is continuous f o r every
x E U,
and
p
is the
supremum of a sequence of the modulus o f those linear forms p
By Proposition 8.1 we have that
is lower semicontinuous.
Wb(U)
with the natural topology is a Fre'chet space.
In par-
ticular, it is barreled, which implies that every lower semicontinuous seminorm is continuous. tinuous seminorm on
Wb(U)
Therefore
X
It
and a real number
such that
("1 f o r every
P(f) f E Wb(U).
we may replace
Thus
is a con-
for the natural topology.
follows that there exist a U-bounded set C > 0
p
f
by
*
CPX(f) Since fm
= c SUP xE x Wb(U)
(m f N)
If(4I
is an algebra over
in ( * ) :
C
THE CARTAN-THULLEN THEOFG3M FOR DOMAINS OF Wb-HOLOMORPHY
for every
f E W,(U),
m E N.
f E Hb(U),
m
149
Therefore
(**I for every
E
m
Letting
!N.
tend to
we
m,
obtain
PO) f E Wb(U).
for every
PJf)
It follows from the definition of
p
that
for every the
f
E Wb(U),
Wb(U)-hull,
gn
hypothesis,
n
E
and s o , by the definition of
N,
5, E
Wb(u) -+ 5 E a U as
n
A
,aU) = 0. #b(’> which contradicts c).
Therefore
dist(X
d)
a).
3
such that for every fIW
which
r:
-
[O,l] -+ V
glw.
f
V
?
E Ub(U)
Choose
be the first point on
aWo
Then there exist non-
W,
and
r(0) =
‘5 n’nEN
image of as
‘
n
-+
tn’nEm
n -+
60,
m.
r,
W”C
U
r(l) =
Then
$ U,
V, V
g E W,(V)
b E V\U, a,
n
for
and let b.
Let
5 E V
n
aU
5
r
which lies in the image o f
(see the proof of Proposition 12.3). N o w let
with
there exists
a E W
be a path with
is not U-bounded,
Wb(’)
and
But by
IN.
which implies that
-t m ,
Suppose that a) is false.
empty connected open sets
n E
for every
n
aWo
be a sequence of points belonging to the
and strictly preceding
5,
such that
5,
-+
5
This can be accomplished by choosing a sequence
c [O,t),
where
and taking We then have
gn
r(t)
= 5,
such that
tn -+ T
as
= r(tn).
In E
Wo c U
for every
n E lN
and so by
.
150
CHAPTER 13
SUP I f({,)/ = m e HOWn€IN ever, by o u r initial assumption, there exists g E Wb(V) such
f E gb(U)
d) there exists
that
fIW
5 E aWo
,
-
such that
g I w . This implies that
,
f
and since
we have
Ig(5)I
which is absurd, since
is finite, and
lim f(5,)=-. n-tm
Q.E.D. DEFINITION 13.3
Let
f E ab(U),
g E aU.
and
be a non-empty connected open set,
U
5
is said to be Sfb-regular at
f
if there exists a pair of non-empty connected open sets V, W,
n
W c U
such that
and there exists
5
Conversely,
5 E V
V,
Ub(V)
g
(which implies that such that
au
if every point o f
= f(W*
f
W c U
which
g
n
= f
Sb(U)
V in
and
V
#
is a gb-singular point of
f.
V, W
of
U,
E
for
W.
f E #,(U)
will denote the set of all
domain of existence if
aU.
Sb(U)
f
separable, and let
U
which are
is said to be a gb-
U
@.
THE CARTAN-THULLEN THEOFLEM (part 11):
E.
g E Wb(V)
there is no
Wb-singular at every point of
of
if
f
is said to be gb-singular o n
This means that for all non-empty open subsets with
U),
is said to be a gb-singular point for
no such pair of sets exist. aU
g(w
#
V
Suppose that
E
is
be a non-empty connected open subset
Then the following are equivalent:
a)
U
is a domain of
b)
U
is a Wb-domain of existence.
#b-holomorphy.
THE CARTAN-THULLEN THEOREM FOR DOMAINS OF kib-HOLOMORPHY
c)
The complement
CSb(U)
of
Sb(U)
in
#,(U)
151
is of
#,(u).
first category in
In order to prove this theorem, we need the following propositions: PROPOSITION 13.3 (Montelts Theorem).
Ef,lnEN
and
is a sequence in
sup Ifn(x)I < xEX,n€[N cular, if sup
-
Wb(U)
If E
is separable,
such that
for every U-bounded set
< -),
Ifn(x)I
X
(in parti-
then there is a subsequence
x€U,nEN
'
which converges uniformly on every compact subfn' nEN set of u to a function f E w,(u).
of
LEMMA 13.4 (Ascoli). and
Ef,3"
Let
M
be a separable metric spaces
a sequence of complex functions on
that the sequence
M.
Suppose
is equicontinuous, and that
sup Ifn(x)( < = for every x E M. Then there is a subsenEN quence of [fnInEN which converges uniformly on every compact subset of
M
to a continuous function on
PROOF OF PROPOSITION 13.3: separable metric space. is equicontinuous in 0
< r < dist(5 ,aU).
U.
Since
E
M.
is separable,
We claim that the sequence
T o see this, let
g E U
By Corollary 4.2 we have
U
'
is a
fn' ng" r E R,
and
CHAPTER 13
152
II x-5 ll r-llx-5 I1
I fn( x) -fn(s ) I
Therefore shows that
is equicontinuous.
Efn’nEN <
sup Ifn(x)I
n E IN, which
Also, by hypothesis,
and so by Lemma 1 3 . 4
x E U,
f o r every
Q
for every
s C .--___
n€N } which converges uniformly nk k€N on every compact subset of U to a continuous function f. (f
there is a subsequence
f
W(U)
is holomorphic, since
is closed
compact-open topology, and since every U-bounded set REMARK 13.4:
SUP
X€X,kEIN it follows that
X,
Montelcs Theorem can be rephrased as follows:
every subset of
wb(U)
which is bounded in the natural to-
pology is relatively compact in the compact-open topology Is this true f o r other topologies, such as
QUESTION:
PROOF OF THE CARTAN-THULLEN THEOREM c)
3
If cSb(U)
b).
since
gb(U)
pology, b) a) of
3
#
Let
V
such that
the subalgebra o f
(part 11):
is of first category in
Wb(U)
then,
and thus
@,
U
is a ab-domain of existence.
W
and
W c U
be non-empty connected open subsets
n
#,(U)
V
and
V
#
U.
ab(U,V,W)
denotes
consisting of all functions
f E Hb(U)
for which there exists a (necessarily unique)
g E Wb(V)
such that
#b,m(U,V,W) of all
Tu13
is obvious.
c).
E
9
is a complete metric space in the natural to-
Sb(U)
a)
T~
To.
f =
in
g
W.
be the convex subset of
f E Wb(U,V,W)
satisfies .the relation
F o r each
gb(U,V,W)
m E N,
consisting
for which the corresponding
Igl
5
m
in
V.
let
g E Wb(V)
THE CARTAN-THULLEN TNEOREM FOR DOMAINS OF #b-HOLOMORPHY
We claim that
#b,m(U,V,W)
Wb(U).
is closed in
Since
with the natural topology is metrizable, it suffices
Wb(U)
to show that the limit o f a convergent sequence in 51
b ,m
belongs to
Wb,,,(U,V,W).
ab,m(U,V,W),
wb(V)
let
E N,
j
be a sequence in in
f . -+ f J
W,(U)
in
W.
(gjl s m
Since
gj -+ g
fj = gj
pointwise in
j E IN,
f = g
in
lgl s m
in
Wb,m(U,V,W)
V,
and since
W.
Since
V.
Igjl < m
Therefore
W,(U)
is closed in
Wb(U)
Wb,m(U,V,W)
in
#b,m(U,V,W)
is nowhere dense in
W,(U).
vector space
H
for every and rn E
CWb,,(U,V,W)
Wb(U),
is dense in
Wb(U).
(The complement
[N.
of
in other words,
Since
gb(U,V,W)
CG
it fol-
C#,(U,V,W)
c
Cgb(U,V,W)
is
U
is
is a proper
in a topological
of a proper subspaceG is always a dense sub-
T o see this, note first that it suffices to prove
that
G
lies in the closure of
Then
bila + b X E C,
V
But this follows from the fact that
subspace of
H.
in
W,
for every
a domain of Wb-holomorphy, since then
set of
E gb(V)
g
in
it will suffice to prove that
#,(U).
dense in
m.
V
in
f E Wb,m(U,V,W),
We claim next that the complement
c C#b,m(U,V,W),
+
In particular,
V.
lows that
j
it follows from Montelcs Theorem that
uniformly on the compact subsets of
hence
as
has a subsequence which converges to a
jcw
(U,V,W)
be the corresponding element of
gj
fj = gj
such that
for every
[f,} j€[N
Let
and suppose that
j E IN,
F o r each
for
153
X
as
#
0.
X
-+
0, 1 E C,
Hence
Finally, we show that
b
If b E G ,
CG. X
#
0, and
let
CG.
b+Xa E CG
lies in the closure of
CSb(U)
a E
CG.)
is the union of a count-
able family of nowhere dense sets of the form
gb,m(U,V,W).
154
CHAPTER 13
M
Let then
f E Wb(U,V,W)
for some
that
f = g
and let
U
of
n
V
in
from
a point number V’
5 E
n
q E M
s
V
aV.
n
Wo
aW0.
Br(g)
Then
m E IN
g E Hb(V)
Let
be such
r > 0 be the distance
Let
c V
r
is contained in
V,
V‘
sup I g l
U.
Br(C)
and
sufficiently close to
be such that
E CSb(U)
From the proof of Proposition 1 2 . 2 ,
n
aU
f
be the connected component
Wo
sufficiently close to
= Bs(q)
Let
W.
containing
there exists of
W,
V, W.
If
E.
be a countable dense subset of
$ i ! !
g
Now choose
and a rational
so that the ball
#
U
sup I g ( < V‘
and
s m,
and let
t
a.
be a suf-
V
ficiently small positive rational number so that Bt(q) c Wo. Let so
W’ = B t ( q ) f
E
and
g‘
= glw’.
f = g‘
Then
Since the family of sets H
# b , m (U,V’ yW’).
defined in this way is countable,
REMARK
13.5:
the above.
w,(u).
W’,
and
(U,V’ ,W‘)
b,m is the union of a
CSb(U)
countable family of nowhere dense sets, o f first category in
in
Therefore
@Sb(U)
is
Q.E.D.
There are various open questions relating to F o r example, what are the complex Banach spaces
f o r which the Cartan-Thullen theorem holds?
same question with
H(U)
in place of
Hb(U).
Also open is the The answer to
the last question is y e s when the Levi problem has solution, f o r example, when
See Dineen [ L
3.
E
is a Banach space with a Schauder basis.
PART I1
THE LOCALLY CONVEX CASE
This page intentionally left blank
CHAPTER 14 NOTATION AND MULTILINEAR MAPPINGS
Unless stated otherwise, U
locally convex spaces, and
IN, IR
E.
subset of
G
and
E
and
F
will denote complex
will denote a non-empty open denote respectively the sets of
natural numbers, of real numbers and of complex numbers. IN*
{1,2,3, ...).
denotes the set and
SC(E)
SC(F)
denote respectively the sets of conand
tinuous seminorms on
E
DEFINITION 14.1
m E N*.
Let
F.
ra(%;F)
E~ = EXE
all m-linear mappings o f
denotes the set of
x...~ E
times) into F,
(m
the operations of addition and scalar multiplication being defined pointwise. Ca(%;F)
Ca,(%;F)
denotes the subspace of
of all symmetric m-linear mappings. (%;F)
'as
for every
means that
xl,...,x
m E E
and every
set of all permutations of
If A E Sa(%;F), element
As
Thus
of
u E Sm, Sm
{1,2,...,m].
the symmetrization of
.Cas(%;F)
being the
defined by
155
A
is the
14
CHAPTER
156 for
x1,x2
,...,
xm E E .
We d e n o t e by
C(%;F)
v e c t o r subspaces of
and
ga(%;F)
Xs(%;F)
respectively the
gas (%;F)
and
c o n s i s t i n g of
c o n t i n u o u s mappings.
m = 0,
For
we d e f i n e
La(OE;F)
a s v e c t o r s p a c e s , and we s e t
:= L(OE;F) := Ss(OE;F) := F
= A
A
for
E
:= SaS(OE;F) :=
~,(OE;F).
Ak+As
i s a pro-
which maps
S(%;F)
I t i s e a s y t o s e e t h a t t h e mapping j e c t i o n of onto
ea(%;F)
Ss(%;F)
onto
= S(%)
write
Sa(’E;F)
t h e spaces
m
for e v e r y
E
= ga(E;F)
gas
ga(”’E),
= Cs(%).
gs(%;C)
and
and
X(’E;F)
,...,1,
E
(c,
s o t h e mapping
then
A(X1
and
f i n e d as follows:
m = 0,
if
Ax
0
we
If E = C,
Ls(mC;F) a r e
A(1,
...,I),
and
i s an isomorphism.
A E Sa(%;F)
Let
= Sas(%),
For if
= X1...l,
.,.,l) E F
A+-A(l,
DEFINITION 1 4 . 2
,..,X m )
we
m = 1,
= L(E;F).
a l l n a t u r a l l y isomorphic w i t h one a n o t h e r .
hl
(%el
For
gas (mC;F), S(%;F)
Ca(mG;F),
F = C
I n the case
[N.
s ~ ( ? E ; c )=
w r i t e for s i m p l i c i t y ,
L(%;C)
Sas(%;F)
and
x E E.
= A E F.
Axm
m E
If
i s deIN”,
m times
...,x ) . Sa(%;F), x1 ,...,xk E 7----L_\
Axm More g e n e r a l l y , l e t
E N,
m,nl,n2,,..’nk n Axl’.
If
.
n .xkk
A
E
and
= A(x,x,
n = n +n2 1
i s d e f i n e d as f o l l o w s :
m = n > 0,
+...+ If
n
E,
s m.
m = 0,
k E IN”,
Then
nl Axl
.,
n .xkk = A.
NOTATION AND MULTILINEAR MAPPINGS
where
each
xi
is repeated
by
where
~
1
Then
, Y ~ ,E ~E
9
n l Axl
times if n m > n, Axl
...x2
5
n n 1 k (AX1 * * * X k)(Y1,***tYm-,)
...
A
is defined
nk times ,. - J
nl in each casey and Ax1
E Xa(m-%;F)
is symmetric if
and
0,
'(X~Y...YX~Y***,~,*~. ,X~,Y~Y***YY,-,)
nk
xk
times
,-A
=
ni >
ni
ni = 0. And if
omitted if
157
is symmetic, and continuous if
...xn
k
k
is
A
continuous. LEMMA 14.1 (Newtonfs Formula). x1
,...,xk E
E,
k E N*,
m,n E IN
A(x 1+X 2+...+x~)~ =
PROOF: A(xl+
A E Las(%;F),
and
n c m.
' ...
the sum being taken over all n = n,
Let
nl!
Then
n
n!
nk!
nl,...,nk E
ax^ l [N
nk k '
f o r which
+...+nk The case
...+xk)
n
then f o r each
n = 0
is trivial.
If
...+x~,...~x1+...+x (y, ,...,ymen) E Em-n,
= A(xl+
m = n > 0, then
k).
If m > n >
0,
In each case, the given expression may be expanded, using the fact that
A
is multilinear and symmetric, and it is easy to
see that the number of occurrences of.:xA the number of permutations of
xl,.. .,xk,
.
.xF
where
is equal to x1
is re-
158
peated
CHAPTER
nl
times,
this number is
...,xk
... n!
nl!
!nk!
14
i s repeated
nk
times.
t h e lemma is proved.
Since
Q.E.D.
CHAPTER 15
POLYNOMIALS
DEFINITION 15.1
m E IN,
polynomial, where
-+ F
P: E
A mapping
is an m-homogeneous
if there exists
A E
la(%;F)
such
that m P(x) = AX
f o r every
P
T o express this relationship between A
P = A.
A: E
each
A
Em
i s the diagonal mapping,
x E E,
P = AoA.
then
we write
If m z 1, and
...,
= (x,
A(x)
If P
A,
and
A
As = A .
It is easy to see that
-I
x E E.
X)
for
is an m-homogeneous poly-
nomial, we have ~ ( x x )= xmp(x)
We denote by
for
pa(%;F)
E
being defined pointwise.
P(%;F)
case
E E,
E
(c.
the vector space of all m-homo-
geneous polynomials from
Pa(%;F)
x
into
F, the vector operations denotes the subspace of
of continuous m-homogeneous polynomials.
m = 0,
we take
= p(OE;F)
pa(OE;F)
space of constant mappings of
E
into
isomorphic with the vector space P a (%;c)
= pa(%)
Pa(m@;F)
and
and 6'("C;F)
as vector spaces.
p(%;C)
F.
to be the vector
F, which is naturally
When
= p(%).
In the
F =
When
(c,
we write
E =
(c,
are both naturally isomorphic with
F
CHAPTER 15
160
REMARK 15.1:
...,Em
If El,
m E IN*,
convex spaces,
Xa(E1,.
..
,Em;F)
F
and
are complex locally
then the vector spaces
..
L' (El,.
and
,Em;F) of m-linear mappings
Em into and of continuous m-linear mappings of El x...x m Em);F) and respectively, are subspaces of Pa( (El x...x
P(m(E1 x...x
Em);F)
To see this, let m 1 If X1 = (xl,.-.,Xl), x2 = (x2,
respectively.
1
A E ga(%l,...,Em;F).
,..., Xm =
1
(xm
,...,
m xm) E El x . . . x
Em )" + F
B: (El x . . . ~
X1 = X2 =...=
E m , define
B E Ca(m(E1
x...~
Em);F),
and that
...,xm), then = BXm = A(x1,x2, ...,xm)
Xm = X = (xl, 6(X)
for every
...,xrn)
by
It is easy to see that if
F,
...,xm) E
El x...x
(xl,
Em. Thus statements con-
cerning polynomials may also be applied to multilinear mappings. LEMMA 15. 1 (The Polarization Formula). m
E
N*,
and
,...,xm E
x1
1 A(x~,x~,...,x~) =- m! 2
c
E,
If
A E Xas(%;F),
then
C1C2...E
m ;i(e 1x 1+E 2x 2+...+
the summation extending over all possible values of
c1 = r t l , . . . , e m = PROPOSITION 15.1
fl.
The mapping
A E Sa(%;F),-
E Pa(%;F)
EmXm),
2
161
POLYNOMIALS
is linear, and induces an isomorphism between the vector spaces
‘as
(%;F)
and
ba(%;F),
spaces
LS(%;F)
and
PROOF:
The case
m = 0
ping
A €
ea(%;F)
I--*
and between the vector
ra(%;F),
fi
for every
is trivial.
Let
m
m
E
2
1.
Since
As = A ,
the mapping induces a surjective mapping of
onto
Pa(%;F).
The polarization formula shows
Ls(%;F)
that this mapping is injective and maps P (%;F).
Let
aa(E;F)
and
3(E;F)
ly the vector space of all mappings of
E
vector space o f all continuous mappings of
denote respective-
Za(E;F),
m E
[N,
Thus, a mapping
E
and
is called a polynomial from
P: E -+ F
P = P pa(E;F)
P(E;(C) =
into
F.
When
An
Pa(%;F) E
into
of
F.
+
Pl
+...+
such that
’ m
denotes the vector space of polynomials from
F, and P(E;F) nomials.
and the
is a polynomial if there exist k = 0,1,...,m,
Pk E ba(%;F),
F
into
element o f the algebraic sum of the subspaces
E into
denotes the subspace of continuous poly-
F = C,
we write
Pa(E;C) = P,(E)
and
P(E).
PROPOSITION 15.2 of the families
1Y.
onto
Q.E.D.
DEFINITION 15.2
E D
Pa(”E;F).
A
.Cas(%;F)
m
The map-
is easily seen to be
€ ea(%;F)
linear and surjective, from the definition of A
N.
Pa(E;F)
and
(Pa(%;F)jmElN
P(E;F)
are the direct sums respec tive-
162
CHAPTER
PROOF:
We show first that the family of subspaces
;Pa(E;F), m E
of Pk E
15
Pa(%;F),
is linearly independent.
N,
k = 0,1,...,m,
we must show that induction.
Po = P1 =...=
m-1,
m
2
1.
0,
Pm = 0. We prove this by m = 0; we assume
The assertion is trivial if
its truth for
Thus if
and
N,
+...+Pm =
Po + P1
(1)
m E
Pa(%;F)
Now condition (1) implies that
m
c
hm
and
= 0
P,(x)
k=0
m
c
(3)
h
k
Pk(X) = 0
k=0
x E E,
f o r every
1 E
Subtracting ( 3 ) from ( 2 ) , we have
(c.
(hrn-1)Po(X) +...+
(4)
x E E,
f o r every
Am # Xk
for
applied to
X E C.
k = O,l,...,m-1,
To show that m E N,
Pk E Pa(%;F),
so that
then our induction hypothesis
P(E;F)
0.
is the direct sum of the family
P E b(E;F).
k = 0,1,..., m,
We must show that
Pm,l =
Pm = 0.
let
P = p0 + p1
trivial f o r
C
0
(4) yields
And from (1) we have
(a)
1 E
I f we choose
Po = P1 =...=
P(”E;F),
=
( A m -1 m-1 )Pm,1(X)
+...+
Then there exist
m E N
such that
pm.
Pk E P ( % ; F ) ,
k
0,1,..., m.
I
This is
m = 0; we assume the truth of this assertion
POLYNOMIALS
for
F r o m (a) we obtain
m-1.
(b)
x E E,
f o r every
k = O,l,...,m-1.
?,
?, E G
Fix
x,
m-1
)Pm-,(x)
such that
Xm
#
h k for
it follows by the induction hypo-
Po,P1,...,Pm,l
thesis that
( X -1
Since the left hand side of (b) is a con-
tinuous function of
Pm
E C.
rn
+. ..+
= (Xm-l)Po(x)
X"P(x)-P(?,x)
are continuous.
is also continuous.
Hence, from (a),
Q.E.D.
REMARK 15.2:
The preceeding proof shows that the following
is true:
m t N,
for
if
k = O,l,...,m,
Pk E P ( % ; F )
unique
rn E N
The polynomial
REMARK
-1
15.3:
P
#
0
and
P = or
#
and
15.4
a neighbourhood
SUP eCf(x)3 xE v
<
V
f: U
QoP E Pa(E;G).
Then
the degree of
A mapping
bounded if for every
p.
be complex vector spaces,
G
to the product of the degrees of DEFINITION
Po,P1,.
depending o n the context.
E, F
0,
Then there is a
0.
is called the degree o f
m
Q E Pa(E;G).
and Q
#
is conventionally assigned a degree
0 -03,
Let
P E Pa(E;F)
P
..,Pm , with such that P=Po+...+ PmandPm+O.
k = 0,1,...,m,
The non-negative integer
Pa(kE;F)
k = 0,1,...,m.
k,
P E Pa(E;F),
Let
Pk E
if and only if
and unique polynomials
Pk E Pa(%;F),
of either
Prn with
P E P(E;F)
then
for every
DEFINITION 15.3
+...+
P = P
QoP Q
-t
P.
is said to be amply
f E U
and every
5
contained in
of
is less than o r equal
and
F
If
B
E CS(F) U
there is
such that
164
CHAPTER 15
PROPOSITION 15.3
P E bJa(E;F), the following are equi-
For
valent : (1) P
is continuous.
(2)
P
is amply bounded.
(3)
P
is continuous at one point.
(4) P PROOF:
The implications (1)
are clear.
-
U
0
.
Let
(1).
By hypothesis, there is a non-empty
E
and a number
for every
M t 0
such that
x E U. t: E
-I
E,
t(x) =
Then (a) is equivalent to
V
4
M
for every
Y E V = t(U),
is a neighbourhood of zero and
lynomial.
P = Qot,
Since
Q.
to continuity of
If Q =
C
continuity of
-1
Q = Pot P
is a po-
is equivalent
We claim that
m (c)
(4) =
and consider the translation
p{Q(y)]
(b) where
of
,@{P(x)] s M
uo E U,
x-u
5 E E.
and
open subset (a)
= (2) 3 (4) and (1) 9 ( 3 ) * (4)
Thus it suffices to prove
p E CS(F)
Let
amply bounded at one point.
i s
,
Qj
where
Qj E Pa(jE;F),
j
= 0,1,... ,m,
j=O
Qm
#
0,
@oQj
8.Q
then
is bounded on
We prove this by induction.
is bounded on
V
for
The case
j
V
if and only if
= 0,1,...,m. rn = 0
is trivial.
Assuming the truth of (c) for m-1, with m 2 1, let m Q = C Qj Qj E Pa(jE;F), Qm # 0, and suppose Q is
,
j=O bounded on
V. We have m-l
(d)
hmQ(x)-Q(kx)
=
C j=O
(Im-XJ)Qj(x)
165
POLYNOMIALS
for every
x E E
Xm-XJ #
for
0
X E C.
and j
Fixing
1 E C
such that m-1 the polynomial C (hm-XJ)Qj,
= O,...,m-1,
j=O
which has degree at most
m-1,
is such that
m- 1
p[ C
(Xm-Xj)Qj]
j=O
V,
is bounded o n
since
Hence, by the induction hypothesis, bounded on
Therefore
V.
P0Qm(X)
s
is bounded o n
p{XrnQ(x)-Q(Xx)]
@ O Q ~ , . . . , @ O Q ~ - ~are
is also bounded o n
Po&,
PoQ(X)
@E
+
V.
V,
since
m-1 C Qj(X)I* j=O
This completes the proof of (c). A j E ga,(JE;F)
Now let j = O,l,...,m.
be such that
Q. = J
i. J ’
Then m
(c)
=
Q(x)
C j=O
For
j = 1,2,...,m,
E
the origin in
A.xj. J
W . be a balanced neighbourhood of J such that let
,-
j ,times,
w J.
Then for
(xl,
c i = fl for
...,x j ) E
+...+w ,
j
c
v. J
CiXi E v if i=l I t follows from the polarization
(Wj) j
i = l,...,j.
we have
C
o f the origin in
Ej,
is bounded o n the neighbourhood (W .)’ J for j = 1,2 m. Hence each A
is continuous on
Ej.
But
formula that each
by (c),
Q
BOA.
J
,...,
is continuous o n
A.
is continuous on
j
E.
Q.E.D.
E
and so
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CHAPTER 16
TOPOLOGIES ON SPACES OF MULTILINEAR MAPPINGS AND HOMOGENEOUS POLYNOMIALS
DEFINITION 16.1 and £
s
m E N*.
(~;F)
for
Let
E
and
F
be complex seminormed spaces,
Then a seminorm is defined in a natural way on
by defining
A E £ (~;F), s
where the supremum is taken over all
xl, ••• ,xm E E norm of
E
such that
Ilxlll';'
o, ••• ,llxmll
is identically zero, we define
.;. 0;
if the semi-
II All = O.
We then
have
and
For
m
= 0,
seminorm of
is seminormed in a natural way by the F.
The natural seminorm of in a similar way;
Ilpll
for
P(~;F),
P E P(~;F),
= sup{" p(x)// m
mE N*,
is defined
we define
xE
E,
II xii .;.
O}.
Ilxll
If the seminorm of For
m = 0,
P(oE;F)
E
is identically zero, we define Ilpll
= O.
is seminormed in a natural way by the
167
CHAPTER 16
168 seminorm of
F.
Ss(%;F)
then
F
If
is a normed space or a Banach space,
p(%;F)
and
are normed spaces or Banach m E
spaces respectively, f o r every
IN.
We point out that the same symbol,
E, F, Xs(?E;F)
denote the seminorms of PROPOSITION 16.1 and
m E
Let
E
The mapping
[N.
and
F
// 11,
and
is used to
P(”E;F).
be complex seminormed spaces,
iE
A E Ss(%;F)-
P(%;F)
isomorphism of vector spaces, and a homeomorphism.
is an
Further-
more, mm
IIill
(*) for every PROOF:
A E
IIAlI
S~(%;F).
The case
m = 0
1 f o r convenience.
that the mapping
is trivial,
m > 0.
Let
i
A-
ping is a homeomorphism. If the seminorm of
=.
..=
m
being defined to be
We know, by Proposition 15.1
it will follow that this map-
It is easy to see that
IIill
s /)All.
is identically zero, the proposition
Suppose then that the seminorm of
identically zero.
/I Xllj
E
m
is an isomorphism of vector spaces.
If we prove the inequality ( * ) ,
is trivial.
A
m! IlAlI
F o r all
xl,.
..
,x
E E
E
is not
such that
I[xml( = 1, we have
Using the polarization formula (Proposition 2.1),
we have
T O P O L O G I E S ON SPACES O F MJLTILINEAR X A P P I N G S
REMARK 16.1: E P(%;F) each
m E N,
the smallest constant
mm/m!
A E Ss(%;F),
for
Cm
such that
0
2
bt
In fact, f o r
does not preserve the seminorms.
/IAll s C m l i i / / is
A E Ss(%;:F)
In general, the mapping
169
independently of
E
F,
and
This is shown by the following example.
EXAMPLE 16.1
Let
e
be
x = (x1,x2, ...)
quences
the Banach space o f all se-
&I,
of complex numbers such that
m
co
lxnl <
C
with the norm
m,
IIxlI =
C
then
I)Amll = l/m!
and
\liml] = l/mm.
a E CS(E),
For
E
for details.
denotes the space
Ci
If a E CS(E)
-
IIAm]l = mm lIAmll.
Thus
We refer the reader to Part I, Example 1.2 REMARK 16.2:
es(%)
If Am E
lxnl.
n=1
n=1 is defined by
B E CS(F),
E
semi-
normed by
a.
P(mECi;F ) B
denote respectively the vector spaces of continuous
and
symmetric m-linear mappings of
E:
into E
uous m-homogeneous polynomials from A E
Xs(%a;FB),
p
E CEu;FB),
Ci
spaces, and
nl
Let
m E N,
+...+nk
E
and r;
m,
and
F
,
A
and of contin-
F B '
into and
IIAll,,B
ively denote the natural seainorms of REMARK 16.3:
FB
es(mEa;FB) and
Ilpllu,8
and
For respect-
P.
be complex seminorrned vector
A E X(%;F).
If
k E IN*,
nl,...,\
EN
we define
where the supremum is taken over all
xl,...,xk
E E
such that
CHAPTER 16
170
I)xlll f Oy...,~~xk~~ # 0. zero, we set
IIA
(nl,
If the seminorm o f tnk)
11
E
is identically
= 0.
We then have
In particular, we have
where
m
7
0, and the number
is repeated
1
m
times in
the first equation.
DEFINITION 16.2 spaces and
Let
m E N.
seminormed spaces
E
For
and
F
be complex locally convex
a E CS(E)
and
p E CS(F),
Ss(mEa;FB) and P ( % , ; F B )
phic by the natural mapping
AW
i. Therefore
the
are homeomorthe locally
convex spaces
and
with their corresponding locally convex inductive topologies, are homeomorphic by the natural mapping
A H A.
Finally, consider the locally convex spaces
and
with the corresponding projective topologies;
the natural
171
TOPOLOGIES ON SPACES O F MULTILINEAR MAPPINGS
isomorphism spaces.
Aw
2
establishes a homeomorphism between these
The locally convex topologries obtained in this way
are known as the limit topologies o n DEFINITION 16.3
The bounded topologies on
and
Xs(%;F)
P(%;F). and
are defined respectively by the seminorms
P (%;F)
B E CS(F)
where
Xs(%;F)
and
In the case
of
E.
be
B(A)
and
X1,
m =
...,
Xm
(3,
and
X
are bounded subsets
these seminorms are defined to.
respectively.
p(P)
The compact topologies on
Xs(%;F)
and
P(%;F)
are
defined respectively by the seminorms
a
where of
B(P)
E.
f CS(F),
and
X1,
...,Xm
and
X
are compact subsets
m = 0, these seminorms are
Again, for
p(A)
and
respectively. The finite topologies on
Xs(%;F)
and
ff(%;F)
are
defined by families of seminorms defined in the same inanner as above with subsets of
X1,
...,Xm
and
X
ranging over the finite
E.
It is easy to see that in the case of each of these four topologies, the locally convex spaces p (%;F)
Xs(%;F)
are homeomorphic by the natural isomorphism
and
A&;.
172
CHAPTER 16
REMARK 16.3: .Cs(%;F)
The f o u r topologies which we have defined on P(%;F)
and
limit topology 2
2
are related in the following way:
bounded topology
2
compact topology
z
finite topology.
REMARK
16.4: The limit, bounded, compact and finite topolo&es
can be defined in a similar way on the space
rn E N .
.C(%;F),
Each one of these topologies induces the corresponding topology on
.Cs(%;F),
and the natural mappings
are continuous. REMARK
16.5:
...,Em
If El,
we can define on the space m-linear mappings of
El x
are locally convex spaces, m EN*,
...,E,;F)
s(E1,
...x Em
into
bounded, compact and finite topologies.
of continuous
F,
the limit,
Each of these topo-
..,Ern;F)
logies coincides with the topology induced on .C(%l,. by the corresponding topology on
in each case, the natural projection of onto
...
.
63 (m(E1~. .xEm);F),
di ( E ~ , , E ~ ; F )is continuous.
6 ("(EIX..
and
.xEm);F)
CHAPTER 17 FORMAL POWER SERIES
DEFINITION
17.1
A formal power series, o r more simply, a
E
power series, f r o m
into
x E E
series in the variable
where
F
Am E Xas(%;F),
m E IN.
about a point
of
E
is a
of the form
Equivalently, this series
can be written in the form
.
A
Pm = -4m Am and Pm are both referred to as the as the origin coefficient of order m of the series, and 5
where
of the series. from
E
The space
F
into
about
g
Fa[[E]]
of f o r m a l power series
is a vector space which is cano-
nically isomorphic to
dias(%;F),
and to
We denote by
F[[E]]
the subspace of
E
of all formal power series from
are continuous on
TT ba(%;F). =IN
m€ IN
E,
F,[[E]]
into
F
consisting about
g
which
by which we mean that each coefficient
is a continuous mapping,
Then
F[fE]]
morphic to
173
is canonically
iso-
174
17
CHAPTER
TT .Cs(%;F)
P(%;F).
and to
m N
mEIN
5 E E, Pln E Pa(%;F)
LEMMA 1 7 . 1
Let
$ 5 CS(F).
Then
(m E IN)
and
m
lim m3m
for every Bopm = 0
af c
k=O
P,(x-~)]
o
=
5
in some neighbourhood of
x
for every
m
E
if and only if
IN.
m PROOF: x
every
m;
m lim fIr C m+m
p(um) = 0
X = b ,
k
1 E C,
for
m E N.
we obtain
and
1x1
fI(u0)
that there exists a m E N.
=
c
0.
for
0,
6,
then
and if
It follows from a), with 0
2
such that
- @k(=cn + l 1kU k ) } = m
p(un) = lim p ( m-w 0
<
p(u,)
?:
c/hm
Applying the induction hypothesis to a),
and hence
(c,
s
6 >
Suppose that
m
1 E
BOP,, = 0
We prove this by induction.
n 'r 1.
p ( u ~ - ~= )0 ,
lim fp(knun) m-w
for
for
k=O
1 = 0,
for every
(m E N)
um E F
X uk] = 0
for every
=...=
B(U0)
= 0,
the converse is clear.
We claim that if
Taking
Pk(x-g)]
m-tk=O in some neighbourhood of 5 , then
every
a)
C
l i m
We shall prove that if
1x1
< 6.
z
x~-~u,)
k=n+1
This implies that
175
FORMAL POWER S E R I E S
Letting
),
-t
0,
N o w let
we o b t a i n x = g+lt,
p(u,) where
= 0.
1 E C
T h i s proves the claim. and
t E E.
By hypo-
thesis,
for
11 I
h
6,
follows that
where p(P,(t))
6
i s s o m e positive real n u m b e r .
= 0
for every
t E E,
It
m E N. Q.E.D.
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CHAPTER 18
HOLOMORPHIC MAPPINGS
We adopt the Weierstrass point of view, defining holomorphy in terms of power series;
one could equally well
employ the Cauchy-Riemann definition in terms of complex differentiability. DEFINITION 18.1 if for every m E
(N
5 E
f: U -+ F
A mapping
U
there exists a sequence A m E L s ( ” E ; F ) ,
U
with the following property:
V
there exists a neighbourhood
uniformly in
is holomorphic in
B E CS(F)
for every
g
of
in
U
such that
x E V.
We denote by phic mappings o f
U
defined pointwise.
the vector space of all holomor-
#(U;F)
into
F,
the vector operations being F = CI:
In the case
we write
W(U)
for
#(U;CI:) a
Pm = A m ,
A m , or the polynomial coefficient of order is separated, then
that the sequences write
A .
m
of
= f(g), me”
at
f
s.
is called the Taylor
If we assume that
F
and it follows f r o m Leama17.1
and
are uniq.de.
We
CHAPTER 18
= m!im.
Z r n f ( g ) = m!P,
d r n f ( g ) = rn!A,,
E i t h e r of t h e s e mappings i s known a s t h e d i f f e r e n t i a l of order
of
m
at
f
5.
W e can t h e n d e f i n e t h e mappings:
The s e r i e s
i s c a l l e d the Taylor s e r i e s o f
f
5 ,
at
in E
For
[N,
we
write
7
m,fA 5 . The
difference
(f
rn
rnainder o r o r d e r
I n the case
-
r
A E Ls(%;F),
and
E = 6,
rn
F
grnf(g) of
f
A
P(%;F)
and
rn E
for each
A(1,1,
with
can be
for
[N;
...,
1) E F.
a r e i d e n t i f i e d with an element o f
E(rn)(g), at
g.
f
5.
Ss(%;F)
we i d e n t i f y
which we denote by
of o r d e r
at
f
of
of
i s c a l l e d the Taylor r e -
m , f , 5 ) Iu
identified naturally with
dmf(g)
m
i s c a l l e d t h e T a y l o r polynomial o f o r d e r
Thus F
and r e f e r t o a s t h e d e r i v a t i v e The mapping
i s c a l l e d the d e r i v a t i v e of order
m
of
f
in
U.
at
HOLOMORPHIC ! U P P I N G S
3-7 9
The mappings
E 5,(U;Ls(%;F))
f E #(U;F)-dmf
and
f E #(U;F)b2mf E Za(U;P(%;F)) are linear for every
REMARK 18.1:
m E
(N.
Suppose that the mapping
property that for every Am E Xs(%;F),
m E IN,
5
C U
f: U
-t
F
has the
there exists a sequence
such that m
uniformly in a neighbourhood
V
of
5
in
U.
This means
that
-
lim p:f(x) m-1.n uniformly in that
F
V
m C k= 0
1
dkf(g)(x-g)
p E CS(F).
for every
] = 0
Therefore, assuming
is separated, the Taylor series of
presents the function
example, normed spaces. f € #(U;F)
f
in a neighbourhood of
f
holds for every holomorphic mapping if
ping
k
E
snd
at
5. F
f
reThis
are, for
However, it can happen that a map-
cannot be uniformly represented in any
neighbourhood of a point
5 E
U
by its Taylor series at
5.
We shall see an example of this phenomenon later (seepagesm-288)
REMARK 18.2:
The concepts of holomorphicity and of the dif-
ferential mapping are local in nature, in the following sense: a)
U,
If f E #(U;F)
and
V
f E #(V;F),
and
dm(flV) = (dmf)
then
= (Yf)JV for every
m E N.
is non-empty open subset of
lv,
2m(f(v)=
180
CHAPTER 18
If
b)
U
is the union o f a family
open subsets, f 6 M(VX;F)
If
f
1 E A,
f o r every
REMARK 18.3:
U
is a mapping of
f
Let
F
into
rn E IN
5
then
5.
m E
for every f
shall see later (see page 222) bourhood of
,
5
and
E U.
= 0
dm({)
by the uniquexess of the Taylor coefficients.
dmf(g) = 0
Conversely, if
and if
f E #(U;F)
vanishes in a neighbourhood o f
for every
F,
f E #(U;F).
then
be separated,
of non-empty
vanishes in some neigli-
f E #(U;F)
Thus every
then, as we
[N,
is, locally, uniquely
determined by its Taylor series.
18.4:
REMARK
E Am3 mE[N
W e point out that the uniqueness of the sequence
of differentials in Dafinition 5.1
(assuming that
is separated) does not depend o n the continuity of
Am, m E
F [N,
but o n l y on the existence of the limits as stated in the defi-
nition.
REMARK 18.5:
The definition of a holomorphic mapping between
complex topological vector spaces which are not locally convex This definition is as follows:
must be worded very carefully. f: U
-P
F
sequence
is holomorphic if for every
Am E diS(%;F), W
f o r every neighbourhood
neighbourhood
V
of
g
-
m
C
Ak(x-g)
k
zero in
U
there exists a
F
there exists a
such that for every
M 6 N E ew
U
with the following property:
[N,
of
in
there is a corresponding f(x)
rn €
g E
e > 0
for which
for
m
2
M,
and
x E V.
k=0
When
F
is locally convex, this is clearly equivalent to
Definition
5.1.
However, in general, we may not omit the
181
HOLOMORPHIC MAPPINGS
phrase “ f o r every might mapping A.
be e x p e c t e d . f: U + F
= f(g),
aznd
e >
and r e p l a c e
O”,
For,
cW
by
as
i f t h i s were done, e v e r y c o n t i n u o u s
would s a t i s f y the d e f i n i t i o n ,
Am = 0
W,
for
rn > 0.
taking
I t would not be de-
s i r a b l e t o have a d e f i n i t i o n which a l l o w e d e v e r y c o n t i n u o u s mapping t o be holomorphic.
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CHAPTER 19 SEPARATION AND PASSAGE T O THE QUOTIENT
REMARK
19.1: If, in Definition 5.1, the locally convex space
F is not assumed to be separated the sequence (A,]
mE!N
corres-
ponding to f E W(U;F) and f E U is not necessarily unique. Let KEs(%;F)
denote the subspace of Es(%;F)
consisting of the
mappings whose image lies in the closure of (y E F : p ( y )
in the set
Dm E KXs(%;F)
= 0
for every
{O]
in F, that is,
p E CS(F)].
Then if
is a sequence of Taylor coefficients
and (Am] mE N
of f at f , the sequence [Am+Dm] coefficients of f at 5 .
mEN
is also a sequence o f Taylor
In order that the differentials be m
uniquely defined, we are forced to consider d f(f) as an element of the quotient space
ss(%;F)/KXS(%;F),
equivalence class modulo
KXs(%;F).
63 (%;F)/KP(%;F).
an element of
all cases, whether or not
F
that is, as an
Similarly,
is
This procedure applies to is separated.
is the closure o f the origin in
KEs(%;F)
Z"f(5 )
We remark that
Ss(%;F)
for any
one of the four topologies which we have defined on this space. The s a m e is true of REMARK 1 9 . 2 : in which
F
KP(%;F)
in
The general case can also be reduced to thecase is separated in the following way.
the separated space associated to mapping
P(%;F).
IT: F -+
Fs
F.
Let
Fs
be
The canonical linear
is continuous and open.
It is easy to
154
CIIAPTEK 19
f: U
prove that
-I F
fs= ~ o P U : -+ Fs then
dmf(%)
is holomorphic in
is holomorphic.
and
dmiS(g)
:mf($)
Similarly,
19.3:
gs(%;F,).
and
correspond under the cano-
63 (?E;F)/I@(%;F)
nical isomorphism between REMARK
5 E U
correspond under the canonical
;lllfs(% )
and
if and only if
Furthermore, if
Es(nh;F)/Zs(%;i?)
isomorphism between
U
P (%;Fs).
and
This reduction t o separated s p a c e s can be car-
ried one step farther.
Let
nE: E
the canonical linear mappings of
-t
E
Es
and
respective associated separated spaces. a non-empty open subset of
if and only if there
Es
and
nF: F
and F
be
onto their
Then
f: U
-I F s
-t
F
i s a holomorphic mapping
Us = n E ( U )
is
is holomorphic fs: U s
-t
Fs
such that the diagram f
U
commutes.
The mapping
fs
-F
is uniquely determined by
The viewpoints of Remarks 19.1, 1 9 . 2 the following way:
of representing
if
dlnfs
(5 )
E U,
then
Zmf(5)
and and
and 19.3 are related in dmf(g)
and the two ways
correspond under the canonical iso-
morphisms between the vector spaces Xs(%;Fs)
f.
X,(%;F)/a,(%;F),
Ss(mEs;Fs). A similar remark applies to
amfs(g).
These observations show that the study of holomorphic mappings can be reduced to the case in which both, are separated.
E,
or
F,
or
CHAPTER 20
B-HOLOMORPdY AND !i-HOLOYOKPHY
We consider another way of defining a holomorphic mapping, slightly different to Definition 18.1, but not in any essential way. D E F I N I T I O N 20.1
We define the vector space
H(U;F) = where
6
is a completion of
U
o f all mappings of
f: U
all mappings sider
f
-t
F
into
sr(u;i?) n F F.
U
F
and 'F Thus
H(U;F)
, is the vector space
H(U;F)
consists o f
which are holomorphic when we con-
6.
as taking its values in
Clearly,
H(U;F)
independent of the particular choice of completion ever, assuming that
F
depend on the choice of
by
is separated,
I?,
where
dlnf(S)
and
5 E U, m E
IN.
I?.
is How-
:"f(<)
We have
#(u;F) c H(U;F) c w(u;9). If
f E
#(U;F)
or
f E H(U;F),
we say that
f
is #-holo-
morphic or H-holomorphic respectively. REiMARK 20.1:
Definition 7 . 1 expressses the concept of H-ho-
lomorphy in terms of #-holomorphy.
It is also possible to
express 3-holomorphy i n terms of H-holomorphy. simplicity, that
F
is separated
-
S7~ppose,for
this requirement is not
186
CHAPTER 20
essential. that
a(U;F)
d f(g)(xl,...,xln)
xl,...,xm every
Then
5
E U
morphy, w e deina-.ld that every point of H-holomorphy,
:1
F
f o r every
U,
f
and f,
x E E.
such
5 E U
m E W",
o r equivalently, such that
E E,
m E N",
f E H(U;F)
consists of a l l
m
;"f(g)(x)
and
E F for
In the case of kl-holo-
and all its differentials at
take their values in takes its values in
F,
F,
whereas for
w h i l e its differA
entials at each point may take their values in
F.
Whether one uses the notion of #-holoinorphy or of H-holomorphy is often simply a matter of preference or convenience; the theory of holomorphy can be developed using either notion. F r o m the intuitive point of view, H-holomorphy seeins more na-
tural, but from a technical viewpoint, H-holomorphy is preferable.
F o r example, the differentials of a holoinorphic map-
ping may be represented by integrals such as the Cauchy integral, or by infinite series, whose values, in principle, will lie in a completion, F,
P,
of
F, and may not belong to
even when the napping itself takes its values in
If F
F.
is complete in some suitable sense, the notions
of 3-holomorphy and H-holomorphy coincide.
We shall continue
to work with #-holomorphy, while mentioning H-holomorphy f r o m time to time.
CHAPTER 21
ENTIRE MAPPINGS
DEFINITION 21.1
An element o f
mapping.
f: E + F
in
Thus,
#(E;F)
is called an entire
i s entire if
f
is holoaorpbic
E. The following proposition shows that every continuous
polynomial is entire.
PROPOSITION PROOF:
k
E
IN*.
P(E;F) c H(E;F).
21.1
It suffices to s h o w that
P(%;F)
A E dis(%;F).
Accordingly, let
c W(E;F)
f o r every
By Newton's formula
(Lemma 14.1), we have
f o r every
x E E
and
5 E
E.
It follows immediately that
n
P = A
is holomorphic in
If F
REMARK 21.1: n
P = A,
E.
Q.E.D.
is separated,
the proof o f Proposition 21.1 sliows that if
and
If
k 6 IN
A E Xs(%;F),
dmP(g)
= 0
P E P(E;F)
if
m = O,l,...,k
m > k.
has degree
m,
then
k d P = 0
if
k > m.
and
188
CHAPTER 21
LEMMA 21.1
Let
P E P(%;F),
If
be separated.
F
k
t
N,
then
for
dlnP E P ( k - % ;
Ls(%;F))
amp E 6'(k-%;
P(%;F))
rn = 0,1,...,k,
Xs("E;F)
where
and
P(%;F)
and
carry
their limit topologies. A E Xs(%;F)
L3t
PROOF:
let
T: Ek-m
( xl,.
..,x ~ - E~ )Ek-rn,
-t
Xs(%;F)
associates with
T E Xa(k-%; T(x,x,
..
(yl,.
,ym) E Em
k
= (,)Ax
.
k-rn
x = x1 =...=
Therefore
dmP E P a ( k - % ;
for
(X~,...,X~-~) E E
mapping which associates with k
of
(,n)A(~l,...,~k,m,~,...,y) G
E X a (k-?E,
G(x,x,.
P(%;F)),
Now for every that
llAlla,p < 1.
for every
G(xl,
y E E
,xm)l
x,xl,...,xrn E E .
L
'
then
Gs(%;F)).
be defined as fol-
...,
xk - m )
is the
the element
x = x1 =...=
zmP E
Therefore
p E SC(F)
xk-m
It is easy to see that
F.
Therefore,
BrAxk-rn(xl,
,
and that if
.
..,x) =
P(%;F)
-+
k-rn
lows:
It is easy to see that
and that if
G: Ek-rn
Similarly, let
the element
F.
of
if
is the mapping which
T(xl,...,xkmm)
Xs(%;F)),
...,x)
rn = 0,1,..., k ,
For
be defined a s follows:
...,X~,~,Y~,'..,Y,)
k
(,)A(xl,
-;.
P =
with
there exists for
a(x)
Hence
rn
k-m
;E
x ~ - ~ then , p(%;F)).
a E
C S ( E ) such
1, a(xl)
* a ( ~ , ~ )
Axk-rn E = C s ( % a ; F B ) ,
and
189
ENTIRE M A P P I N G S
f o r every
x E E.
This also holds f o r
m = 0.
Therefore the
(k-m) -homogeneous polynomial
gS(mEu;FB)
is continuous, where
11 /Iu
carries the seminorm
9 B
c
x E E is continuous, the a k-m Ax E XS(mEcL;F4) (k-m)-homogeneous polynomial x E E I-
Since the mapping
continuous.
gs(%;Fp)
x
E
I---,
Since the inclusion mapping of
is
s ~ ( ~ E ~ ; Finto ~ )
is continuous, where the second space carries the
inductive topology,
x E E
I-
Since this is true for every (k-m)-homogeneous polynomial
Ax
k-m
E Xs(%;FB)
p E CS(F), x E E
continuous for the limit topology on dmP
is continuous, and similarly
+-
it follows that the
AX^'^ E
dZs(%;F).
;Imp
is continuous.
Ss(%;F)
is
Therefore
is continuous.
Q.E.D. REMARK 2 1 . 3 :
This leinrna is also valid for the bounded,
compact and finite topologies on
Ss(%;F)
and
P(%;F),
since each of these topologies is weaker than the limit topology.
.
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CIIAPTER 22
SOME ELEMENTARY PROPERTIES OF HOLOMORPHIC MAPPINGS
#(U;F)
PROPOSITION 22.1
F
is strengthened and the topology o f
F
topology of
Let
j(E)
T
weakened;
U
and
T
into j(F),
F
are unchanged.
j = 1,2,
Let
spaces
E
E . and J
and
F
j = 1,2,
Fj ,
El,
a non-empty open subset o f Let
U1
and
El,
and of
c W(U2;F2).
denote
U2
U
respect-
T
j(E)
and
T~(F)2 T~(F). then
'r j ( F ) .
If U
is
i s also open in
U
E2 '
considered as an open subset of
E2, respectively. Furthermore, if
F
and
respectively denote the
with the topologies
T~(E)2; T ~ ( E ) and
Suppose that
be locally convex
E
topologies on the complex vector spaces ively.
if the
remains separated, the differentials of a
holomorphic mapping of PROOF:
E
increases if the topology o f
Then clearly
f E W(U1;F1)
#(U1;F1) c
and
F2
is se-
parated, it follows from the uniqueness of the Taylor series that the differentials we consider ping of
U
dm€
and
as a mapping of
f
into
Gmf U
are the same whether
into
F1,
or as a map-
Q.E.D.
F2.
REMARK 22.1: I t is easy to see that the first part of this
proposition is also true for H-holomorphic mappings: second part takes the following form: completions o f
F1
and
F2
Let
respectively.
I?,
and
The
F2
be
Then the continuous
CHAPTER 2 2
192
F1
identity mapping of
n: F1
linear mapping dmf(5),
;"f(g)
nodmf(5) when
f
-t
into
i2.
extends to a continuous
F2
f E H(U;F1),
Then if
5 E
are its differentials at
and
and then
U,
are the differentials of
.oG"f(g)
is considered as an element o f
f
5 ,
at
H(U;F2).
Proposition 2 2 . 1 is a special case of Propositions 22.2
and 2 2 . 3 .
These are themselves special cases of propo-
sitions concerning composition o f holomorphic mappings. Let
PROPOSITCOA 22.2
convex spaces, let
E, F
and
f E 3S(U;F)
be complex locally
G
continuous affine mapping; that is, A E G G
and
B E L(F;G).
Then
g E H(F;G)
and let g(x)
be a
= 4 + B(x),
gof E W ( U ; G ) ,
where
and i f
and
F
are separated, dm(g o f)( 5
) = Bo dmf ({ )
and
= Ba;tmf(g)
for
; I m ( g O f ) ( g )
PROOF:
m E PJ,
E U.
The proof follows easily from the definition of a Q.E.D.
holomorphic mapping and its differentials. PROPOSITION 22.3 spaces, and let f E #(E;F)
= A + B(x),
Let
V
E, F
and
G
be complex locally convex
be a non-empty open subset of
where
A E F
and
U = f-l(V)
and
are separated, then
B E C(E;F),
is non-empty, then
g o f
d"(gof)(f)
= dmg[f(5)]oB
;tm(gof)(5)
= ~ m g [ f ( g ) ] o B m for
m
if
If
F.
is a continuous affine mapping, so that
and if G
5
f(x) =
g E W(V;G),
E #(U;G).
If
and m E IN,
P E
U,
F
193
SOME ELEMENTARY PROPERTIES OF HOLOMORPHIC MAPPINGS
where
Bm: Em
Bm(xl,. PROOF:
m
is defined by
F
-b
..
,Xm )
...,B(xm)),
= (B(xl),
..
,xm E E.
xl,.
The proof f o l l o w s easily from the definition of a ho-
lomorphic mapping and its differentials.
Q.E.D.
The proofs of the following propositions are immediate.
E
Let
PROPOSITION 22.4
F
and
A’
,)
F =
locally convex spaces, and let
F
A€ A E,
empty open subset of
U + F
the mappings
f
fX E #(U;Fx)
f o r every
*
h’
E‘
of
es(?E;F)
f: U
then
x’
parated for every corresponds to
and
) , E A.
A,
-t
is a non-
U
has as its components
f 6 #(U;F)
if and only if
Furtheremore, if
5 E
F
is se-
k
dmf(g)
U,
under the canonical isomorphism
es(%;Fk).
with
F
be complex
@,
If
x’
then f o r every
(dmfh(g))XEA
A #
E A,
Similarly,
Gmf(5)
cor-
k€ A
responds to
Am (d fk(g))ktA.
PROPOSITION 2 2 . 5 and
F
Let
E
be a complex locally convex space,
a complex vector space.
Let
a family of locally convex topologies on denote the locally convex space the locally convex topology
F,
and let
(F,T~(F)). If
sup ‘rk(F) on
A€ n
#
A
{Tl(F)lkEA,
F,
T
@,
be
Fl
denotes
then
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CHAPTER 23
HOLOMORPHY, CONTINUITY AND AMPLE BOUNDEDNESS
PROPOSITION 2 3 . 1
# ( u ; F ) c H ( U ; F ) c c(u;F),
U
is the vector space of continuous mappings of PROOF:
We show first that
5 E U.
and let
of
f
5.
at
g(U;F) c C(U;F).
f
Since
Since
sm E C(E;F),
where
V
)
F.
F.
f E g(U;F),
B E CS(F)
5
U
in
such that
x E V.
uniformly in
p E CS(F),
U.
E C(V;Fg)
Therefore
f E C(U;F).
and s o
H(U;F) c C(U;F),
f/V
let
be a completion
Then, from what we have already proved,
H(U;F) c that
o
of
it follows from ( * ) that
f o r every
To see that of
=
V
has the induced topology from 0
Let
is holomorphic, for every
lim p[f(x)-s,(x)] m-rm
f E C(U;F
into
Consider the Taylor polynomials
there exists a neighbourhood
("1
C(U;F)
where
~(u;F)c
c(u;?).
H(U;F) c C(U;F).
REMARK 23.1:
Since
U
H(U;F) c F
,
it follows
Q.E.D.
The assumption of continuity o f the Taylor coef-
ficients in Definition 18.1 implies that every holomorphic mapping is continuous.
The next proposition gives a very use-
1 95
196
CHAPTER
23
ful characterization of holomorphy.
PROPOSITION 2 3 . 2
In order that
f: U
it is necessary and sufficient that
-t
f
€ o r every
in
U
Am E Las(?E;F), @
be either continuous there exists a
with the following property:
V
there exists a neighbourhood
of
5
such that
uniformly in PROOF: 23.1
E CS(F)
m € IN,
be holomorphic,
E U
o r amply bounded, and that, for every
sequence
F
x
E
V.
Necessity follows from Definition 18.1 and Proposition
and the fact that continuity implies ample boundedness. To prove that the stated condition is sufficient, supis amply bounded, and that for each 5 € U
f: U -+ F
pose that
there is a sequence
Am E
We must s h o w that Am m k sm(x) = C Ak(x-{) ,
Las(%;F)
with the given property. m E N.
is continuous, f o r every x E E.
sm E P a ( E ; F ) , .
Then
Let
F o r every
k=0
B E
CS(F)
V
let
= 0 uniformly i n lim @[f(x)-sm(x)] m-Ko Then there exists C > 0 such that every
m E
[N,
x E V.
x E V
m E IN.
Thus
Proposition 1 5 . 3 , A m = sm - S m - 1
sm
f
such that
sup p[f(x)]<=.
s C
sup pfs,(x)]
v
5 ,
is amply bounded at
is continuous for every
is continuous f o r every
Finally, if
and
U
B[f(x)-sm(x)]
I t follows that sm
in
XE V
XE
every
5
b e a neighbourhood of
m E
is continuous, then
for
<
m
for
and so, by m E iN.
Hence
[N.
f
ed, and so the stated condition is sufficient.
is amply bound-
Q.E.D.
CHAPTER 24
BOUNDING SETS
REMARK 24.1:
cp E E L ,
E i
Let
b E F
and
m E N,
then
cpm-b E P a ( % ; F )
If
E.
denote the algebraic dual of
is de-
fined by [prn*b](x) = [cp(x)lmb belongs to
W'rren
pm*b E P(?E;F).
E'
the topological dual of
m = 0, we define
For
constant mapping
,
x E E
x E E.
for
+- b
m
E F.
0
cp .b
.b
E,
we have
to be the
is the simplest
possible example o f an m-homogeneous polynomial. Given that
F
pm E E' and
bm E F
(m E N*),
and assuming
is separated, we seek conditions under which the
series
converges in mapping,
If this is
f,
so,
for every
F
of
E
into
I
and defines a holomorphic
F, where
then (1) is the Taylor series o f
Consider the set
If
x E E
is finite, then
I
of all
f E P(E);F)
197
m E IN*
f
at
f o r which bm
c #(E;F).
0.
#
0.
Suppose that I
198
CHAPTER 24
cpm
and
IIbm/l = 1
is normed we may assume that
m E I, since
f o r every / \ b m1\/ \m
F
If
is infinite.
and
p ',
can he replaced by
bm
This motivates t h e
respectively.
bm/l\b,/\
condition o n the sequence
(bm]mEN* in the next proposition.
F,
We remark that in the case of a general separated space
F
even when given
bm
is metrizable it is not necessarily true that,
E F,
bm
f
there is a sequence
m E I
0
for
A,
E C,
F
in a separated space
f r o m zero if there exist ~(a,) 2 6
m E
for every
PROPOSITION 24.1
plete, let
p ',
e
Let
E',
I = ( m E fN* : bm f
F
{Xmbm]mEIis
We recall that a sequence
is said to be bounded away
a E CS(F)
6 > 0
and
such that
[N.
be separated and sequentially com-
and
O}
is infinite,
IN*
m E I, such that
bounded and bounded away f r o m zero.
{
Ic
where
e
bm
F
for
be infinite.
bounded and bounded a w a y from zero.
m
e
N,
and let
Suppose that
bm3
IrlE I
is
Then in order that the
series
x E E,
converges f o r every
and defines an entire mapping,
i t is necessary and sufficient that
x E E,
for every
and that the sequence
continuous, that is, that topology PROOF:
o(E',E),
exists
C
5:
cpm
-b
0
as
m +
m
is equi"JmeI
converges to zero in the weak
and is equicontinuous.
Sufficiency:
such that
cp,(x)
F o r each
Irp,(x)l
5
0
such that
1/2
for B(b,)
x E E
there exists
m z Mx.
s C
If
B E
for every
Mx E I
CS(F),
m.
there
Hence
BOUNDING SETS
I m + (b,)
sr rcpm(~)irn-b,~=
icpm(x)
~
199 for ~ m z Mx.
2
x € E.
fore the series 1) converges for every that
C
f(x) =
There-
We must show
[ c p , ( ~ ) ] ~ * bdefines ~ an entire mapping of E
m€ I
into
F.
Let
\cp,(s)
I
1/3
that
Icpm(g)Irn
5
h l 3 ,EN*
E E.
There exists
rn 5 M,
for
s D(l/3)"
M E I
such that
D
and so there exists f o r every
m
E
2
such
1
Since
N*.
is equicontinuous, there exists a neighbourhood
of the origin i n for every
m E
E
such that if
t
E V
then
Iv,(t)
I
V
s l/3
We claim that the family
IN*.
is uniformly absolutely summable i n prove that for every
6 E
V.
T o show this, we
p
the value of
CS(F),
on the s u m
of any finite subset of the family 2) is uniformly bounded for
t € V.
Given such a finite subset, let
greatest value of
m
which occurs.
m
0
be the
Then the value of
@
on
the s u m o f the elements of this finite subset is at most
3CD,
for every
t
E V,
where
@(bm)
L
C
for all
rn E IN*.
There-
CHAPTER 24
200
fore the family 2) is uniformly absolutely summable in
V,
and hence
That is,
C Ph(t) E N
=
f(s+t)
V,
uniformly in
where
It is clear from what we have shown that this series converges.
Ph is a continuous h-homogeneous poly-
We shall prove that h,
nomial for every
5.
morphic at
f
is holo-
Consider the mapping h
E E
(tl'"*,th)
from which it follows that
'-
C
( ~ ) ~ ~ m ( ~ ) ] m - h ~ m ( t ~ ) " * ~ m (Et h F; )bm
mEI mg:h this series is absolutely summable in metric h-linear mapping from
hood
V
have
Icpm(t)
of the origin in
I
5
l/3
into
to E E,
Ph. And if
polynomial is
Eh
E
and defines a sym-
F,
whose associated
F
there exists a neighbourt E to
such that for
f o r every
m E IN*.
+ V,
Thus the series de-
fining
Ph
converges uniformly in a neighbourhood of
and s o
Ph
is continuous.
Necessity:
Suppose that
uniqueness of the Taylor series,
f E #(E;F). C
(cp,)
f
at zero.
Since the sequence
bounded away from zero, there exists
to
Then by the
mbm
mEI series of
we
'
is the Taylor
bJ mE I
p E CS(F)
is
such that
201
BOUNDING SETS
@ ( b m )2 1
for
origin in
E
m E I.
V
There is a neighbourhood
of the
such that
m-m
x E V.
uniformly for
m
-
lim e{f(x)
c
k bk =
Cqk(x)]
o
kE I ,k = O
It follows, by the Cauchy condition
for convergence, that
x E V,
uniformly for
and hence lim Jqm(x)lm =
o
m-tw
uniformly for
x E V.
that
Iqm(x)Im
s 1
I~,(x)l
s 1
equicontinuous.
for
x E E.
@[[q,(x)]%,] [q,(x)]” Ax,
-t
0
as
x 6 E,
for every
-t
REMARK 24.2:
tion
Therefore
m
[q,(~)]~b,
p E CS(F) as
m
-t
-,
-t
00,
-t
0
such
that is, [q ]
m
is
C [qm( x)] mbm mEI
as
m
for
-t a ,
as above, we then have f o r every
x E E,
x E E.
for every
and hence
Replacing
x
by
we find that
for every
m
0
x E V,
and
x E V.
x E E,
With -t
M
2
Furthermore, since the series
converges for every every
m
m 2 M,
for
M E IN”
Therefore there exists
q m -t 0
‘Tm’mEN
and
1 E
CC.
x E E.
If the space
Therefore
-t
0
as
Q.E.D. E
is barrelled, then the condi-
in the weak t o p o l o g y
is equicontinuous.
qm(x)
u(E‘,E)
implies that
202
D E F I N I T I O N 24.1
in
24
CHAPTER
E
A subset
if every
X
E #(E) =
f
E
of
is said to be #-bounding is bounded on
X.
For
is H-bounding in
E.
A sub-
#(E;C)
E
example, a compact subset of
set o f a #-bounding set is H-bounding, and the closure o f a #-bounding set is #-bounding.
E
8-bounding subset of E M A R K 24.3: space
F,
f
every
E
every
is bounded.
X c E
If
E' c # ( E ) ,
Since
is #-bounding in
E,
is bounded o n
#(E;F)
(I E F ' ,
from the fact that for every
$of
then for every
X.
This follows
E #(E)
is bounded
hence
f(X)
is bounded in the weak topology
and therefore
f(X)
is bounded in the original topology o f F.
on
X;
Conversely, if every separated space
F
F,
24.4:
E
If
bounded subset of
E
,
then b
#
is bounded on
and it follows that
REMARK
of
[O}
b E F,
see this, choose f*bE #(E;F)
#
f E #(E;F)
E
X
Thus
f E
E.
To
H(E),
is bounded in
f(X)*b
is bounded in
f(X)
for some
is #-bounding in
Then f o r every
0.
X.
X
is bounded on
U(F,F'),
IT.
is a semi-Monte1 space, so that every is relatively compact, then the subsets
which are #-bounding in
E
are precisely the bounded
sets. LEMMA 24.1
Suppose that the family
(am,n]m,nEIN
o f complex
numbers is unbounded, and that
c
<
m
for every
n E
[N.
m€IN Then there exist sequences
n0 < nl <...<
nk <...
in
m [N
0
< ml <. ,.< mh <. such that
..
and
20 3
BOUNDING SETS
P R O P O S I T I O N 24.2
X
Let
equicontinuous sequence in tfie weak topology
PROOF:
in
[Vrn3 m € N
E‘
which converges to 0
Then there is an
Suppose the conclusion is false.
c%3
in
mEN
u ( E ’ ,E),
in the weak topology
X
is #-bounding i n
E’
which converges to 0
such that
E.
Passing to a subsequence if 6 > 0
necessary, we may assume that there exists f o r every
Let
m E N
rn
3
,
am ,n m nEN
0
as
m
+
f o r every
m
c
Iarn,nl <
m
n,
n
0
< nl <...<
nk <
...
in N
indeed,
we have
f o r every
there exist
I t is
Furthermore, since
ri
mE” Applying Lemma 24.1,
m.
m,n
is unbounded;
n.
such that
for which
2Vm 2 @m = 7
is unbounded for every
c am,nLEN -t
where
t
easy to see that
epm(xn)
xrn E X
there exists
a = r&Jxn)l m,n
Then for every
U(E‘ , E ) ,
equicontinuous sequence
where
E.
be #-boundillg in
rno < ml <
such that
E
LN.
...<
m
<
...
and
204
Let
f =
but
f
Imh.
C [,$ KIN mh
'm'
B y Proposition 24.1 we have
'
is unbounded on
If E'
COROLLARY 24.1
'
24
CHAPTER
U ( E ' ,E)
and hence is unboundd on X. XnklkElN
Q.E.D.
contains an equicontinuous sequence
qm
such that mEN
converges to
but does not converge to
B(E',E),
E
then
f E #(E),
0
0
in the weak topology
in the strong topology
contains a bounded set which is not
#-bounding. REMARK
24.5:
Josefson and Nissenzweig, in 1973, r63], [112]
proved the following theorem which provided the solution to a n important and long-standing problem in the theory of normed spaces. PROPOSITION 24.3
If E
is a normed space of infinite dimen-
sion, there exists a sequence
lcp,l
= 1
m E N
for every
weak topology
u(E' ,E)
and
{ v , , , ] ~ ~in cp,
+ 0 as
E'
m
such that -t
in the
.
This result enables us to prove the following proposition. PROPOSITION
24.4
nite dimension.
Let
E
be a complex normed space of infi-
Then every #-bounding set in
E
has empty
interior. PROOF: E
By Proposition 24.3, there is a sequence
which converges weakly to zero, and
m E IN.
Let
cp: E
-t
CC
be the mapping
Ilcp,l
= 1
in for every
BOUNDING SETS
cp E a(E).
By Proposition 24.1, ed on
gr(0),
for
for some
r > 1.
Therefore
IIqmII
Ml/m r
5
rn
r > 1.
s M/rm
E N",
m
m E IN";
Then
E W(E)
Ilcpmll
hence
r;
but this implies that
m E IN",
which is absurd.
have shown that for every closed ball f
is unbound-
Suppose this were not so.
for every
for every
there exists
cp
We claim that
Using the Cauchy inequalities we have
for every
j m l < r s M'
205
Thus we r > 1,
with
Er(0)
gr(0).
which is unbounded on
It
follows easily that the same is true of the closed balls
-
(0) where r1 > 0, and hence, by translation, it follows 1 that for every 5 E E and every r > 0 there exists
Br
f
E W(E)
which is unbounded on
LEMMA 24.2 let
X
Let
E
be a complex locally convex space, and
be a subset of
E
which is bounded and not precompad.
Then there exist a sequence
6 > 0 x0
such that, if
,...,xm '
PROOF:
Since in
Fix
6,
then
X
X,
Q.E.D.
Er(f).
Sm
U(U-X,+~)
EXmlmEN
in
x, u
is the subspace of 2
6
for every
€ CS(E),
E
and
generated by
u E Srn
,
m E N.
is not precompact, there is a sequence
a E CS(E)
0 < 6 < y/2.
mensional subspace of
and
y > 0, such that
We claim that if
S
E,
p 6 IN
there exists
is a finite disuch that
206
C I U P T E R 24
a(u-t )
b
2
P
u E S.
f o r every p E N,
then f o r every a(up-tp) < 6 .
Since
Suppose this were not so;
there exists (a(tp))
u
P
E S
such that
is bounded,
[U(up)]
P€I N also bounded, and s o , since po < p1 <...<
exist
)
a(u-t
5
for
+
~ ( u - u)
ph
in IN
u E
and
y/2
ph
sufficiently large, which contradicts 1).
h
is
such that
S
+ 6 <
-t ) < ( y / 2 - b ) ph
U(u
ph
"
is finite dimensional, there
S
...
ph <
pE
Thus our
claim is established. We n o w define the sequence
= t o , and suppose
'
recursively. Let mc" have been chosen t o satisfy
xO,...,x m the desired condition. Let S be the subspace o f
x
x0
rated by a(u-t Pm )
2
,...,xm .
b
Then there exists u E S.
for every
'
the sequence
(Dineen-Hirschowitz)
continuous sequence i n
E'
verges i n the weak topology set i n
E
PROOF:
Let
We define
such that
x m+l
--
tp,
.
.
Then
Q.E.D.
If every equi-
contains a subsequence which con,E)
D(E'
then every #-bounding
is precornpact. X
be a subset o f
we must show that
X
E
which is not precompact;
is not #-bounding i n
E.
#-bounding set is bounded, we may assume that
'
c X , u E C S ( E ) and 6 7 0 mcN By the Hahn-Banach theorem, there exist Let
gene-
€ LN
has the stated property.
niED
P R O P O S I T I O N 24.5
p,,
E
such that
cpm/Sm = 0 ,
f o r every
x E E.
(p,(~,+~)
Since every
X
is bounded.
be as in Lemma 24.2. (p,
= 1, and
Therefore the sequence
E E'
,
I(pm(x)l
'
'm'
mcN
m E N, s ~1 a a ( x )
is equi-
207
BOUNDING SETS
m o < ml <...<
continuous and s o , by hypothesis, there exist
...
< mh < to
cp
in
and
LN
a ( E ’ ,E).
in
(xh) = 0
have
such that
JI,
=
- rp.
5
k,
and so
Let for
converges rcpmh3hEN Since mk 2 k , we
cp E E‘
h
cp(xh)
= 0
for every
cp mk
h E IN. Therefore
for every
h E LN.
converges to
f o r every
0
in
h E IN.
#-bounding in FIEMARK
Thus
24.6:
is equicontinuous, and c’h’h€N
u(E’,E),
but
X
Therefore, by Proposition 24.2,
E.
$h
is not
Q.E.D.
In general, a precompact subset of
E.
be #-bounding in
Clearly, if
E
E
need not
has the property that
every closed precompact subset is compact, then every precompact subset is a-bounding in true if
E
is complete.
completeness:
if
E
In particular, this is
E.
We are thus led to a new concept of
satisfies the hypothesis of Proposition
24.5 and if every closed precompact subset of E then the a-bounding subsets of
E
is compact,
are precisely the precompact
sets. EXAMPLE
24.1 (see Dineen).
m E N,
let
em
Let
be the element of
E
E = La ( N )
and f o r each
whose m-th coordinate
is 1, with every other coordinate equal to 0. a-bounding in
,
E, but is not precompact.
Then [em}
Evidently,
not satisfy the conditions of Proposition 24.5.
m€ IN
E
is
does
This page intentionally left blank
CHAPTER 25
THE CAUCHY INTEGRAL AND THE CAUCHY INEQUALITIES
In this chapter we shall be considering integrals o f functions from subsets of
C
F.
into
Although in principle
these integrals will take their values in a completion, F,
of
the equations we derive show that the integrals in
question lie in PROPOSITION 25.1 f
E #(U;F)
,
5 E
F. (The Cauchy Integral). x E U
U,
+ Ax E U for every 1
PROOF: C
E,(c)
#,
Suppose that C
V
E
V
for some
p
7
5
p.
and
a!,
)A)
1
Let
be separated,
(1-1)s +
such that Then
is an open subset of
6 > 0 and
F
7
(c,
g E #(V;F).
E B6(S)c
Then
r
This can be proved in exactly the same way as the classical case of
F = C,
F = C
or by reducing the case of general
F
to that
by means of the Hahn-Banach theorem.
Now let
g(1)
= f[ (l-X)C+hx]
.
Then
and is holomorphic, in the open subset
V
g of
is defined,
C
consisting
210
CHAPTER 2 5
e
of all
6:
gp(0) c
1 E B (0) c
P
+
(1-X)g
for which
Xx E U.
By hypothesis,
and hence
V,
f
that is,
PROPOSITION 2 5 . 2
(The Cauchy Integral). m E
that
f+Xx E U
1 dmf(5)xm m?
=
-
PROOF:
Suppose that
with centre ed in
V.
5
p.
S
p
be separat-
> 0
such
Then
__
2ni
V c CC
and radii
Then, if
and
[N
1x1
E a!,
for every
F
Let
is open, and the closed annulus 0 < r s R,
R,
and
r
is contain-
g € W(V;F),
This can be proved exactly as in the classical case or by reducing the case of general
F
to that of
F = C,
F = C
by
means o f the Hahn-Banach theorem. NOW
let
.
g ( x ) = f('+xx)
holomorphic, in the open set such that
c V.
g+Xx E
U
Therefore, if
X
and 0
<
0
#
Then
V C CC
0.
s p,
g
is defined, and is
consisting o f all
By hypothesis, we have
),
gp(0)-{O]
E
(c
c
THE CAUCHY INTEGRAL AND THE CAUCHY I N E Q U A L I T I E S
211
that i s ,
f
(-
I 'IX(=E
For
1
pn =
3
Then for
E
t
n E IN,
E,
let
n
;"f(s)
and
c
sn(t) =
P,(t-g).
k=0
n
n
c
=
dX
PJX)
2 n i P,(x).
k=0
T h e r e f o r e , for
n
5
m,
we have
I
r
and hence, f o r e v e r y
But for e v e r y
E
B
CS(F)
E
CS(F),
t h e r e i s a neighbourhood
U'
of
5
l i m @ [ f ( y ) - s n ( y ) ] = 0 u n i f o r m l y on U'.Theren+= f o r e , i f we choose c > 0 s u f f i c i e n t l y s m a l l and l e t n + a , in
U
such t h a t
we o b t a i n
CHAPTER 25
2 12
B E CS(F),
Since this is true for every
we have
f
REMARK 25.1:
m = 0
The case
valent to Proposition 25.1;
of Proposition 2 5 . 2
is equi-
this can be seen by performing a
change of variable.
If a E CS(E),
FtEMARK 25.2: A
E x~,(%;F),
A
$ Ss(%a;Fa);
we define
5 E U
PROOF:
and
f
J J A J I ~to , ~ be
E H(U;F),
a E CS(E),
g a , p ( g ) c U.
and
if p,(%;F).
a
#
Let
F
5 E CS(F),
0,
be sep > 0,
Then
taking
x E E
such that
then
PROPOSITION 25.4 E CS(F),
+m
(The Cauchy Inequalities).
We apply Proposition 25.2,
a ( x ) = 1;
m E IN
a similar definition applies to P
PROPOSITION 25.3 parated,
5 E CS(F),
p
>
Let 0,
5
F
E U
be separated, and
f E #(U;F),
5a ,B ( g ) c U.
Then
a E CS(E),
213
THE CAUCHY INTEGRAL AND THE CAUCHY INEQUALITIES
pose that so
a =
U
Sup-
m = 0, the inequality is trivial,
For
0.
a =
Since
E.
must be
is trivial.
this reduces to Proposition 25.3.
0,
m z 1.
let
that
#
If a
PROOF:
0
8.f
If
Suppose that
@of
B'
( 5 ) c U , it follows a *P is unbounded, the inequality
and
is bounded on
E.
We shall
prove that in this case
for every Then to
x E E.
= dmf(g)(x).
g("'(0)
g,
g E #((C;F)
Define
8 E CS(F),
1
= pra
This holds for every
p
> 0, and
obtain the desired result.
REMARK 25.3:
g(X) = f(s+xx)*
Hence, applying Proposition 25.3
we have, for every
Brh ;"f(g)(x)i
by
The case
a =
so,
letting
p -+
*
we
Q.E.D. 0
in this proposition is closely
related to Liouvillets Theorem. REMARX
25.4:
If
a #
0, the constant
l/pm
in the Cauchy
inequality is the best possible constant which is independent of
E
then
and
F.
1 3 z"f(S)
If a = 0 , sition 2 5 . 4 ,
F o r , if
= P,
P E P(%;F),
5 E E, f(x)
= P(x-g),
and
there is no minimal universal constant in Proposince any strictly positive constant could be
substituted for
l/pm.
CHAPTER 25
214
REMARK 25.5:
25.4,
U n d e r the conditions of Propositions 25.3
and
w e h a v e respectively
and
In the first case,
mm/m!pm
constant, while in the case universal constant.
is the least possible universal a = 0,
there i s n o smallest
CHAPTER 26
THE TAYLOR REMAINDER
PROPOSITION 26.1 separated,
f
E
(The Taylor Remainder Formula).
5 E
H(U;F),
(1-1)s + Ax E U
1x1
1 E C,
for
x E U
U,
s p.
and
p > 1
Let
F
such that
Then
f
where
r
PROOF:
m,f,5
(x) =
m 1 C I<= 0
B y Proposition
By Proposition 25.2,
^k d f(S)(x-%),
for every
25.1,
with
x-5
in place o f
r
Using the identity
valid f o r
h E
(c,
X f 0,
f 1, we obtain r
x,
be
m E IN.
216
CHAPTER 26
COROLLARY 26.1
p E CS(F),
PROOF: that
Under the conditions of Proposition 26.1, if
then
Apply the usual inequalitites for integrals, and note IX-11
p-1
2
X
Q.E.D.
is the closed convex balanced hull of the set
X E C, 1x1 =
:
{f[(l-X)5+AxI
p}.
V = (f[(l-X)5+Xx]
PROOF:
Let
6
be such that
E F’
= P.
Under the conditions of Proposition 26.1 we have
COROLLARY 26.2
where
1x1
X E C,
for
:
E C,
1x1
= p],
I
sup{ 1 @ ( y ) : y 6 V] s 1.
and let
Then, by
Proposition 26.1,
Therefore
-
f(x)
(x)
T,,~
V,
lanced hull of
belongs to the closed convex bathat is, to
Q.E.D.
X.
--___-
Prn(P-1)
P”(P-1)
We have the following converse of Proposition 26.1: PROPOSITION 26.2 order that f
F
Let
E #(U;F)
f
f: U
be separated, and
-t
In
F.
it is necessary and sufficient that
satisfies: is amply bounded in
a)
f
b)
For every
a E U
For every each
for which
f E U
m E IN,
(1-x)e
b E E,
the mapping
is continuous in V = [ X E
1 I-+ f(a+Ab) c)
and
U.
there exist
Am E gas (YE;F)
such that, for every
+ Xx E U
if
: a+hb
1 E 6,
1x1
x E U 5
p,
and
E U]
.
for p
> 1
we have
THE TAYLOR REMAINDER
m
f o r every PROOF:
E
[N.
1x1
C,
s 1.
M
We r e c a l l t h a t a s u b s e t
(l-h)s+Xx E M
-balanced i f
X E
and 2 6 . 1 .
N e c e s s i t y follows from P r o p o s i t i o n s 2 3 . 2 Sufficiency:
5
217
f o r every
x E M
V
is
and e v e r y
p E CS(F),
B y condition a ) , given
e x i s t s an open 5 - b a l a n c e d neighbourhood
E
of
g
of
there
in
U
such t h a t
( 1 , ~E) C x E
I t f o l l o w s from t h e c o n t i n u i t y of t h e mapping +I
(1-X)5+Xx € E ,
and t h e f a c t t h a t
a number
p > 1,
(1-),)4;+Xx
E
such t h a t
F o r each
V.
t h a t t h e mapping
[X €
tinuous i n
:
1x1
The r i g h t hand s i d e of ly in
W
as
m -+
-.
holomorphic.
U.
U
5
= p},
X E
W,
x € W
W
of (c,
5
in
1x1
s p
and
U,
implies
we h a v e , by c o n d i t i o n b ) ,
= f[(l-X)$+Xx]
i s con-
and h e n c e , f r o m c ) ,
t h i s i n e q u a l i t y t e n d s t o z e r o uniformT h e r e f o r e , by P r o p o s i t i o n 2 3 . 2 ,
f
is
Q.E.D.
PROPOSITION 26.3 and l e t
E
f[s+X(x<)]
-I
(c
x
i s an open s - b a l a n c e d
V
s e t t h a t t h e r e e x i s t s a neighbourhood
-I
Let
F
be separated,
E W(U),
f
5 € U,
be t h e l a r g e s t open 5 - b a l a n c e d se-t; c o n t a i n e d i n
Then f o r e v e r y compact s u b s e t
K
of
U
5
and every
CEIAPTER 261
218
P E SC(F)
V
there exist a neighbourhood c; 8 ,
c
PI: f(x)-Tm,f ,g
(x)]
real numbers
< c
em
x E V,
for every
m E N.
The mapping
PROOF:
is defined and continuous in the open subset consisting of all
(x,x)
compact set
D X K
D x K.
is bounded on
V
of
lies in
K
in
13,
of
C
x E D
If
K C U
then since
W.
W
(l-X)S+Xx E U.
f o r which
is the closed unit disc in
hood
and
U,
< 1, such that
0, 0 <
5.
in
K
of
the
5'
Therefore the above mapping
It follows that there is a neighbour-
U
and : x E
sup(P[f((l-X)g+Xx]
p > 1
v, x
such that
E 0,
1x1
h 03
= €4 <
- *
The desired inequality now follows from Corollary 26.1, with 8' = l / p
C = B/(p-1).
and
D E F I N I T I O N 26.1
Z
Let
X
Q.E.D.
be a set of mappings o f
is said to be amply bounded in
and
g E
U
if for every
there exists a neighbourhood
U
U
V
of
g
into
F.
p E CS(F) in
U
such that
REMARK 26.1:
In practice, Proposition 26.3 is often applied
i n the following form: tion, if
{Am}
under the conditions of the Proposi-
is a sequence of complex numbers such that mE N
lim sup
IXm/l/m
?;
1,
then the sequence
THE TAYLOR FlEMAINDER
is amply bounded in REMARK 26.2:
F
.
X c #(U;F)
s 1,
xm(
f -7
is amply bounded in
f o r every
g
This result can be generalized as follows:
is separated,
lim sup ll,/'/m
U
219
is amply bounded,
5 E U
If
and
then the family of mappings
m,f,g
)/ug
f E
9
X ,
m E IN
'
m E N.
To see this, we apply Corollary 26.1 with noting that
a [ (l-X)Z+Xx-f]
=
1
p = r/a(x-g),
a(x4) = r
when
1x1
= p.
This page intentionally left blank
CHAPTER
27
COMPACT AND LOCAL CONVERGENCE OF THE TAYLOR SERIES
5
We recall that if
largest 5-balanced subset o f
u5 =
u
Ex E
:
Let
us F
K
in
0<
Thus
U
x
for every
E
1x1
(c,
1).
K
V
of of
and every p E CS(F)
U
5
K
and 5 E U .
f E #(U;F)
in
U
such that
x E V.
By Proposition 26.3 there is a neighbourhood U,
5
Open*
and
real numbers
c, 8,
where
c
2
0
V
and
< 1, such that
The proposition f o l l o w s immediately. PROPOSITION 27.2 E U.
denotes the
s
.be separated,
there exists a neighbourhood
PROOF:
U.
is
Then f o r every compact subset
uniformly in
then
( ~ - x ) ~ + xEx u
It is easy to see that PROPOSITION 2'7.1
U,
Let
F
be separated,
Q.E.D. f E #(U;F)
Then m
f(x) =
c m= 0
-&2 % ( 5 ) ( x - g ) 221
and
of
2 22
27
CHAPTER
uniformly on every compact subset of
U
g
.
This follows immediately from Proposition 27.1.
PROOF:
Q.E.D.
COROLLARY
27.1
(Unique Determination o f a Holomorphic Map-
ping by its Taylor Series). f , g E W(U;F).
and then
f = g
in
If
U
5
PROPOSITION 27.3
F
Let
= dmg(4;)
d”f(4;)
% E U
be separated,
m E
for every
. 4; E U,
Let
F
be separated,
and suppose also that
f
is locally bounded.
uniformly o n some neighbourhood in
U
f E #(U;F),
Then
of every compact sub-
.
set o f
U
PROOF:
It suffices to show that the Taylor series o f
5
5
converges uniformly on a neighbourhood of each x E Us
We claim that if x
in
U
N
and
x
f
E
at
Ug.
W
there exists a neighbourhood
of
> 1 such that the set
p
U,
is contained in
and
f(T)
is bounded in
F.
T o see this, consider the continuous mapping G:
(X,t) E CxE If
( A ( i 11
(l-x)%+kt E E.
I---
x E U
5:’
the compact set
is contained in
there is a neighbourhood is bounded in
F.
Since
U
4;
U‘ G-’(U’
.
M = {(l-X)s+),x :
Since
M
of
)
f
in
X
E CC,
is locally bounded,
U
such that
f(U’)
is a neighbourhood of
223
COMPACT AND LOCAL CONVERGENCE O F THE TAYLOR SERIES
gl(0) x {x]
in
U
1x1
5
in p
and p
and
(I:
> 1
x E
there exist a neighbourhood (l-X)S+kx E U'
such that
t E W.
if
and
Since
V
series of
REMARK
t E W,
27.1:
series of
f
p E CS(F),
at
5
If
f E P(E;F)
8,
it follows that the Taylor
converges to
at any point o f
and E
f(t)
E,
whether or not
example shows that
f
f
is separated, the Taylor
F
contains only a finite
is locally bounded.
is not seminormable then the identity mapping
REMARK 27.2: every
f
uniform-
A simple
need not be locally bounded:
not locally bounded, but
t E W.
uniformly in
number of non-zero terms, and hence converges to ly in
(I:,
then
is independent of f
X E
x
of
This establishes o u r claim.
It follows from Corollary 26.1 that if m E LN
W
If E
I: E -+ E
is
I E P(E;E).
There are two interesting situations in which
f E #(U;F)
is locally bounded;
finite dimensional, the other where
F
one is where
E
is
is seminormed.
We have the following partial converse to Proposition
27.3: PROPOSITION 27.4 f E #(U;F).
converges t o f
Let
E
be seminormable,
If the Taylor series of f
f
F separated, and
at a point
uniformly in some neighbourhood o f
is locally bounded at
5.
< 5,
of
U
then
CHAPTER 27
224
PROOF:
E
Since of
V
bourhood
is seminormable, there is a bounded neigh-
5
U
in
such that
m
uniformly in
x E V.
Continuous polynomials are bounded on
bounded sets, and hence
p E CS(F).
Therefore, since
bounded on V.
27.3:
REMARK
is bounded on
B o f
V
for every
is independent o f
8,
f
is
Q.E.D.
E
The requirement that
Proposition 27.4 is essential, for if the identity
V
I: E
-I
E
be seminormable in
E
is not seminormable,
is not locally bounded.
We now present two examples of a holomorphic mapping
g
whose Taylor series at a point
% .In Example
ly in any neighbourhood of
are Fdchet-Monte1 spaces if
E
27.2
EXAMPLE
g E W(C),
Let
If I
I
is a normed space and
27.1 Let
f
5.
c > 0
E E
j)
F
a Fsechet-Monte1 space.
E = F = CI
by
does not converge uniform-
Suppose this were not s o . there exists a finite subset
j€I
J
of
5
converges uniformly in
and
F
f E E
at any point
5 = (5
and
is not a polynomial, the Taylor
g
ly in any neighbourhood of
I
E
is countable, and i n Example
f E #(E;F)
and define
Then, for some
27.1
I be a non-empty set, and let
is infinite, and
series of
does not converge uniform-
such that the Taylor series of
v =
(x
I
f
at
.
COMPACT AND LOCAL CONVERGENCE OF THE TAYLOR SERIES
for
j E J].
p(Y) = l Y k l
k E I\J
Let
Y = (Yj)j~~ E F.
9
Taylor series of
5 ,
at
f
dimension.
Let
am = 0
for
m
F = CR,
xk E C .
n,
m
2
5
n,
which means that
contrary to our hypothesis.
E
be a complex normed space of infinite
Then, by Proposition 24.4, there exists g E #(E)
which is unbounded on some bounded subset of gn E # ( E ) ,
n E N,
and let
by
gn(x)
f E #(E;F)
Suppose that for some
= g(nx),
E.
x E E.
We define Now let
be given by
r > 0
the Taylor series of
the origin converges uniformly in the open ball, radius
r.
to the
a = m m!
where
C,
for every
is a polynomial,
EXAMPLE 27.2
E
1
But this implies that g
xk
such that
5
8
Then, applying
n E N
lam(xk-5k)ml
be defined by
we find that the series
is uniformly convergent for Hence there exists
p E SC(F)
and let
225
f
V,
at of
Then
converges uniformly to
gn
in
V
for every
n E N.
This
nV
for every
n E
Since
implies that
converges uniformly to
g
in
[N.
CHAPTER 27
226
every continuous polynomial is bounded onthe bounded set n 6 N,
for every
n E N.
every set of
E,
it follows that
Therefore
is bounded on
g
nV
nV
for
is bounded on every bounded sub-
g
which contradicts o u r choice o f
Therefore
g.
f
is not locally bounded at the origin. REMARK 27.4:
U,
in
Given
5 E U, a neighbourhood
E,
and a separated space f E #(U;F)
exists
F =
mension o f
E
If E
V
#
F
r~ E
(Ea ) ‘
f
there exists
V.
a l E CS(E)
such that
such that
cp
B
(0) c V.
Since E
Ul,l
a 2 E CS(E)
c E IR+.
is false f o r all
2
g
E
whose Taylor series at the origin does not c o n -
Choose
cUl
S
is separated, then
{O]
o f the origin in
is not seminormable, there exists a2
does not con-
i s not seminormable (hence the di-
is infinite) and
verge uniformly in PROOF:
4,
A solution to this problem when U = E ,
for every neighbourhood f E #(E;F)
g
implies a general solution.
(r:
PROPOSITION 27.5
V.
of
one can ask whether there
whose Taylor series at
verge uniformly in
C = 0 and
F,
V
<
is not a polynomial, and
b E F,
= (garQ)*b E W(Eu ;F) C #(E;F),
Hence there exists g E w(C)
Choose
(Eal)’.
such that
b
0.
so that
Then
and the Taylor series of
2
at the origin is co
C
amprn *b, where
am = -iiiT g (m)(o).
m= 0
If this series converges uniformly in V , p E CS(F) that
such that
@(b)
= 1,
then, choosing
there exists
n E IN
such
f
22 7
COMPACT AND LOCAL CONVERGENCE OF THE TAYLOR SERIES
Theref ore
I
Icp(x)
Since
g
m E IN.
x E V,
for
is not a polynomial,
am
and
m
# o
for infinitely many
In particular, there exists B
Since
Ul,1
(0) c V,
with respect to
this implies that
E
Let
empty open subset of
E.
the Taylor series of
f
The dimension of a)
3
for which am f 0.
rp
is continuous
b).
Q.E.D. be a non-
U
The following are equivalent:
at each point
E
f E #(U;F),
and every
F,
uniformly in some neighbourhood of
PROOF:
n
be separated, and let
F o r every separated space
b)
2
such that amfO.
a l , which is a contradiction.
PROPOSITION 27.6
a)
m
n
2r
g
U
of
converges
5.
is finite.
Suppose first that
E
is not normable.
I be a base of neighbourhood of the origin in
Let each
V E I
fv
there exists, by Proposition 27.5,
E.
For
E #(E)
whose Taylor series at the origin does not converge uniformly in
V.
Let
F =
I,
(I:
x E E
and define
f E Sf(E;F) by
f(x) = (fV(X))
I-
E
F a
VE I
Then, as in Example 2 7 . 1 ,
the Taylor series of
f
at the
origin does not converge uniformly in any neighbourhood of the origin. Now suppose that mension, and let
F = CN.
E
is normable and has infinite diExample 27.2
shows that there
CHAPTER 27
228
exists
f E #(E;F)
whose Taylor series at the origin does
not converge in any neighbourhood of the origin. b)
a)
3
follows from Proposition 2 7 . 3 ,
since every finite
Q.E.D.
dimensional space is locally compact.
REMARK 27.3:
The proof of this proposition relies on the
Josefson-Nissenzweig Theorem (Proposition normed.
24.3) when E
is
The following result shows that this theorem is an
essential part of the proof:
If E a) of
f b)
is a normed space, the following are equivalent:
F o r every
at
5
Every
F , and every
f
E #(U;F),
the Taylor series
g.
converges uniformly in some neighbourhood of f E #(E)
is bounded on every bounded subset o f
E. T o see this suppose a) holds.
Then, by Proposition 27.6,
is finite dimensional and hence every bounded subset of is relatively compact.
on every bounded subset of f E #(E;F)
In particular,
f
E
Therefore a) implies b).
Conversely, suppose that every
every
E
E.
f
E
#(E)
is bounded
F,
It follows that for every
is bounded on every bounded subset of
E.
is bounded on every bounded neighbourhood
o f the origin, and it follows from the Cauchy integral formula
that the Taylor series of every
f E #(E;F)
converges uniform-
ly on some neighbourhood of the origin, from which a) follows. Without the Josefson-Nissenzweig Theorem, one would be unable to show that for every infinite dimensional complex normed space
E
there exists
on every bounded ball.
f
E
#(E)
which is not bounded
CHAPTER 28
THE MULTIPLE CAUCHY INTEGRAL AND THE CAUCHY INEQUALITIES
(The Multiple Cauchy Integral).
PROPOSITION 28.1
...,nk E
nl,
that j
5 E
f E #(U;F),
separated,
m = nl
N,
+...+hkxk
5 + hLxl
= l,,..,k.
+...+
k
nk
and
E U
xl,.
E,
> 0, such
P 1 , ...,pk
for every
..,xk E
be
lhjl
E 6,
P j
Then n
..%!
1
nl!.
-
N",
E
U,
F
Let
dmf(g)xl
f(5+hlX1+.
1
(2ni)
k
...xkn .
l
x1
-
-
.
.+hkXk)
nl+l
*/hjl"Pj
k
nk+ 1
dh l...dXk
"'h,
1sj s k REMARK 28.1: The proof of Proposition 28.1 is similar to the
proof of Proposition 25.2, the single integral in the latter case being replaced by a multiple integral.
Alternatively,
Proposition 28.1 can be obtained by repeated application of Proposition 25.2.
Proposition 25.2
Proposition 28.1 in which where
nl =...=
k = m,
COROLLARY 28.1 m E N*,
x1
,...,xm E
g + xlxl +...+ j = l,...,m.
Let
xmxmE Then
k
is the extreme case of
= 1. The other extreme case,
nm = 1
is as follows:
F
be separated,
E
and
u
p1
,...,pm
for every
xj
5
f E a(U;F),
E U,
> 0 such that E C,
IxjI
4
p j
9
9
2 30
CHAPTER 2 8
Xx = XIXl and if
+...+
IX,I
...,nk) E
n = (nl,
If we write
REMARK 2 8 . 2 :
'kXk
= pl,
Xn+l
'
nl+l
= X,
...,I
hkl
= pk
...Iknk+l ,
i s written
k
N
,
d ( n ) = k,
dX = dX,.
1x1
= p ,
..dx,
,
then
integral formula given in Proposition 2 8 . 1 becomes f
a form similar to Proposition 2 5 . 2 . REMARK 2 8 . 3 :
sition 2 8 . 1 ,
Cauchy inequalities can be derived from Propoo r from Remark 2 8 . 2 ,
in the same way as the
Cauchy inequalities of Chapter 2 5 were derived from Proposition 2 2 . 2 . REMARK 2 8 . 4 : A E Ss(%;F),
If we apply Corollary 28.1 to
5 =
with
and
0
U = E,
f =
i,
where
we obtain a new po-
larization formula:
A(xl,
...,xm) =
f1
1
m!(2ni)m
m xm )
2(X1Xl+...+h
.$ jl=l i
(XI*
-__ 2
dX1. .dX,
*Xm)
1s j 4 m
where we have taken
p1
=...=
P,
= 1;
in this case, it can
be shown easily by a change o f variable that any choice of
t
THE MULTIPLE CAUCHY INTEGRAL AND THE CAUCHY INEQUALITIES
> 0 gives the same value for the integral.
p l , ...,pm
231
Like
the original polarization formula (Lemma 15.1), we can use this formula to obtain an estimate f o r a(xj)
r;
1
REMARK 28.5:
to
f =
i,
j = l,...,m,
for
BIA(xl,
...,xm)];
then
More generally, if we apply Proposition 28.1 where
A E Xs(%;F),
m E N",
5
= 0
U = E,
and
we obtain another new polarization formula:
n
k k -
. -
if
1 k
m! ( m i )
!Ixjl=l 1sjsk
n +1 1
1,
nk+ 1
"'1,
dX l...dXk
.
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CHAPTER 29
DIFFERENTIALLY STABLE SPACES
DEFINITION 29.1
F
F
Let
be a complex locally convex space.
is said to be differentially stable if for every
every non-empty open subset recall that
F
H(U;F) = # ( U ; F )
U
of
n Fu.
E
and
E, #(U;F) = H(U;F). Thus if
F
We
is separated,
is differentially stable if and only if, for every
f E H(U;F),
we have
5 E
xl,
and
U,
dmf(g)(xl,
...,xm E
E.
...,xm) E
F
for every
m E N*,
Bearing in mind the proof of the
Cauchy integral formula (Proposition 2 5 . 2 ) , it suffices to
E = C,
consider the case f(m)(0)
E F
for every
U = B1(0),
and to show that
f E #(U;F).
It is easy to see that a complete space is differentially stable.
The following proposition describes two more ge-
neral conditions, either of which guarantees differential stability. PROPOSITION 29.1
F
is differentially stable in each of the
following cases: 1)
F
2)
The closed convex balanced hull of every compact sub-
is sequentially complete.
set of
F
PROOF:
It suffices to consider the case where
is compact.
233
F
is separated.
CHAPTER 29
234
Let
f
that
E H(U;F),
5 E
(l-X)g+Xx E U
x E E.
and
U,
1x1
X E C,
for
s p.
> 0
p
Choose
such
Then, by Propo-
sition 25.2, --1
f = % 12 \
dmf({)xm
for every
d?,
~
m!
Xm+1
m E N.
-IXI=P
Since
f(C+Xx)
E F
5
for every
,
A , x,
conditions 1) and 2)
each imply that the integral which appears in this equation takes its values in f
Hence, by the polarization formula,
F.
E SI(U;F).
Q.E.D.
PROPOSITION 29.2 the space
WF
F
is differentially stable if and only if
= (F, a(F,F'))
is differentially stable.
In order to prove this proposition we need the following result from the theory of weakly holomorphic mappings. LEMMA 29.1
Let
and only if PROOF:
@of
f: U + F.
E = C,
and
E H(U)
f o r every
f
E
H(U;F)
suppose that f
F
@of E H ( U )
is separated, and let
is continuous at
then f o r
z E Bp (g )
5.
If
for every
5 E U.
g E z(U)
r
-
1
I t-i: I =P r
and hence
t-z
dt,
T o prove
JI E F',
We show first
and
we have
g(z)
if
JI E F'.
The necessity of this condition is obvious.
its sufficiency, suppose that
that
Then
gp(5) c
U,
235
DIFFERENTIALLY STABLE SPACES
Theref ore
g = $of,
Applying t h i s t o
$ E F',
Since t h i s holds f o r every
f o r every
p E CS(F).
f
,
we have
i t follows that
It i s e a s y t o s e e t h a t
i s bounded
f
@ [ f ( z ) - f ( r ) ] -+
on e v e r y compact s e t , and hence Therefore
(I E F'
where
0
as
z-5.
i s continuous.
We have shown t h a t
r
f o r every
$ E F'.
Since
is s e p a r a t e d , i t f o l l o w s t h a t
F f
where
for
z E Bp(5).
I t-g 1
= p
i s uniform f o r
We have
and
It-51
z
E = p
Bp (
5).
and
Furthermore, z
E
Er(g),
t h e convergence
where
0
< r < p.
23 6
CHAPTER 2 9
I t follows that
,
uniformly on every compact subset of
W(U;i),
f E
Therefore
follows that
and since
f E H(U,WF).
$ o f
F
Suppose that
it
is differential-
Therefore
d:
and let
Jr E F’, and
f o r every
H(U)
f E H(U;V) = W(U;F).
f E w(u,WF).
F,
into
Q.E.D.
so,
Since the topology
F, #(U;F) c s J ( U , W F ) ,
is weaker than the topology of
and hence
U
be a non-empty open subset of
Then
by Lemma 29.1,
u(F,F’)
U
Let
maps
f
f E H(U;F).
PROOF OF PROPOSITION 29.2:
ly stable.
Bp(<).
is differentially
WF
stable. Conversely, suppose that f E H(U;F),
and let C.
Then
(+of)(”)(g)
2) (+of)(”)(S)
Since
= Since =
$of
F.
f o r every
for every
a holomorphic mapping of
U,
f(m)(5)
U
=
which implies that
m
of
into
$
be
f E H(U;F)
with
$,
m E N. f E #(U;WF)
m E N, f
at
where
5,
with (I,
fp)
(g) E F
taking
f
WF.
fp)(g)E F
and let
then
is the composition of
$[fp)(g )]
Therefore
g E U,
Let
is the composition of
$~f(~)(g)~ $ o f
a non-empty open subset of
(p)‘,
E
If
is the differential o r order
5 E
i s
f E H(U;WF) = #(U;WF).
a completion of 1)
U
where
is differentially stable,
WF
F
for every
m E IN,
is differentially stable.
Q.E.D.
as
CHAPTER
30
LIMITS O F HOLOMORPHIC MAPPINGS
30.1
PROPOSITION
is closed in
H(U;F)
C(U;F)
in the com-
pact-open topology. We recall that the compact-open topology on
PROOF:
C(U;F)
is defined by the family of seminorms g where
@
E CS(F)
E
C(U;F)
K c U
and
consider the case where the closure of pology.
Fixing
Am: Em + F
g
e
for each
+
Xmxm
Am(xl,
E
for
U
..., m ) X
A .
m E IN"
f(5).
>
0
I:
pj,
1
if
such that j
f( 5
=
f
belong to
We define a mapping
as follows:
,...,p,
X j E a,
=
Let
in the compact-open to-
C(U;F)
let
U,
I t suffices to
is separated.
in
p1
choose real numbers
is compact.
F
H(U;F)
SUP erg(x)i xE K
I---
= 1
..,xm E
and let
..+hmxm) dkl..
1%j % m
We claim that: 1)
The value of
choice of 2)
pl,...,pm
Each
Am
Am(xl,
...,
xm)
is independent of the
satisfying the stated condition. is m-linear and symmetric.
237
E,
5 + Xlxl +...+
,...,m,
+h lxl+.
m
xl,.
.dXm
238
CHAPTER 30
3)
If x E U ,
(l-X)g+AxE
U
and
p
> 1
1x1
X E @,
for
is a real number such that
s p,
then
,-
f E H(U;F),
If
then
1
A m = -mT
dmf(S),
and these
properties follow from the Cauchy integral formula (Proposition 25.2) and the Taylor remainder formula (Proposition 26.1). Using the same methods as were used to prove Proposition 4 . 6 , Part I, it follows that l), 2 ) and 3 ) hold when to the closure of
H(U;F)
Proposition 26.2 that REMAFtK 30.1:
f
in
C(U;F).
f E H(U;F).
f
belongs
It now follows from
Q.E.D.
In the proof of this proposition, the fact that
is contixuous implies that each
Am
is continuous.
It
then follows from 3 ) , using the methods of proof of Proposition 26.2,
that
f E H(U;F),
without appealing to Proposi-
tion 26.3. FLEMARK 30.2:
H(U;F)
The proof of Proposition 30.1 also shows that
is closed in
C(U;F)
in the topology of uniform con-
vergence on the finite dimensional compact subsets of
U.
(If E
#
is separated and has infinite dimension, and F
{O},
this topology is strictly weaker than the compact-open topology on
H(u;F)).
REMARK 3 0 . 3 :
It is possible to avoid the use of multiple
integrals in the proof of Proposition 30.1. is closed in
C(U;F)
Hence
H(U;F)
in the topology o f uniform convergence
on one dimensional compact subsets of
U.
LIMITS O F HOLOMORPHIC MAPPINGS
30.4:
REMARK
If
C(U;F)
239
is complete in the compact-open to-
pology (the case most frequently encountered is that i n which E
is semimetrizable and
is complete) then
F
complete i n the compact-open topology.
H(U;F)
is
C(U;F)
However,
need
not be complete in the compact-open topology, even when F = C. Furthermore, even when
H(U;F)
need not be complete in this topology
F = C.
EXAMPLE 30.1
E
Let
a!('),
be
power of the continuum. convex topology.
I has at least the
where
We place on
E
the finest locally
This is the locally convex inductive limit
topology corresponding to the decomposition
S of
E.
There exists a 2-homogeneous polynomial,
P,
from
pages
42-44). There exist unique
cij
E C,
c
ij
--
c. ji
'
E I, such that P(x) = x = (x.)
where E
+
G
E E.
iEI be defined by
PJ E 6 ( 2E) c H ( U )
compact subsets of
,
E.
c
cij xi x j '
J c I, let
F o r each finite
PJ(X) = Then
E
which is not continuous f o r this topology (see r l l ] ,
C
PJ:
where
ranges over the set of finite dimensional vector subspaces
into
i , j
E = US,
c
c
and
PJ
i,jEJ
However,
ij
x
+ P P
i
x
j
.
uniformly o n the H(U)i
not complete in the compact-open topology, where non-empty open subset o f
E.
H(U)
thus U
is
is any
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CHAPTER 31
UNIQUENESS O F HOLOMORPHIC CONTINUATION
PROPOSITION 31.1 U
Let
is connected. a)
f
f E #(U;F),
F
where
is separated and
Then
U
is identically zero in
if and only if
identically zero in some non-empty open subset of b)
is identically zero in
f
exists
5 E U
PROOF:
We note first that if
such that
of
U
V
for every
m €
W.
= 0 for every
E U
m E !N,
then
subset
U
f
of
s
U.
m E N.
is identically zero in a
zmf
then
is also identical-
This follows immediately
from the uniqueness of the Taylor series of Moreover, if
is
if and only if there
U
f
V
non-empty open subset ly zero in
dmf(g)
f
is such that
f
at each point.
= 0 for every
dmf(g)
is identically zero in the largest g-balanced U,
which is necessarily open. f
from the fact that the Taylor series of at every point of
This follows
at
5
converges
9
The necessity of the condition given in a) is obvious. To prove its sufficiency, suppose that in some non-empty open subset of points in
U
U.
at which the value of
is a non-empty open subset of
U.
241
f
Let f
is identically zero
V
be the set of
is not zero.
Then
V
It follows from our first
CHAPTER 31
242
remark above that m E N.
$mf
for every
N o w , by the Cauchy integral formula, it is easy to
rn E IN*
see that for each t E U +-+ ;Imf(t)(x)
x E E,
and
is continuous.
7,
tically zero in the closure,
T.
is also identically zero in
the mapping
7c
V,
of
V
in
U.
throughout
is iden-
Hence
dmf
By o u r second remark,
V
which implies that
7 in
is closed.
V = U,
connected, we conclude that
zmf
Therefore
then identically zero in a neighbourhood of fore
V
is identically zero in
that is,
f
There-
U.
Since f
is
is
U
vanishes
U.
The necessity of the condition given in b) follows from the first remark. such that
f
g E U
Conversely, suppose there exists
dmf(g) = 0
m E IN.
for every
By the second remark,
5,
is then identically zero in some neighbourhood of
hence, by a),
f
PROPOSITION 31.2
is identically zero in f E #(U;F),
Let
U.
where
and
Q.E.D.
U
is connected.
Then
For every non-empty open subset
a)
vector subspaces of
F
generated by
V
f(V)
of
U
the closed
and
respect-
f(U)
ively are equal,
If F
b)
is separated, then f o r every
vector subspaces of and
f(U)
PROOF: of
F
a)
F
generated by
g E U
[dmf(g)xm
:
the closed
xEE, m@N}
respectively are equal. Let
Su
generated by
and f(U)
and
be the canonical mapping of convex quotient space
be theclosed vector subspaces
Sv
F/SV
F
.
f(V)
respectively.
Let
nV
onto the separated locally Then
nVof E #(U;F/SV),
and
24 3
UNIQUENESS OF HOLOMORPHIC CONTINCTATION
nVof
is identically zero in
V.
Hence, by Proposition 31.1,
nvof
is identically zero in
U.
Therefore
S c Sv. U
implies that
c f(U).
Therefore
ed by =U
T
Let
b)
Therefore E Su
dmf(S)xm
c Su,
.
x E E}
since
f(V) c
F
f o r every
and hence, since
U,
= 0
dm(nUof)(g)xm
nu[dmf(g)xT
generat-
The holomorphic mapping
is identically zero in
is separated,
x E E.
v
Sv.
: m E IN,
{dmf(g)xm
F/SU
=
S
be the closed vector subspace of
g
U + F/SU
of:
Su
Moreover,
f(U) c S v , which
m E N,
for every
= 0, which implies that
m E N,
x E E.
Hence
T c Su.
.
5
Now consider the canonical mapping w : F + F/T We have 5 5 that w g o f E 31(U;F/Tg), and dm(wgof)(g)xm = w,[:dm€(f)x m ] = 0 m E N,
for every
6
Hence, by Proposition 31.1,
is identically zero in
ugof
c T
x E E.
.
REMARK
Therefore
31.1:
tion 31.2.
U,
Su c T
which implies that
f(U) c
Q.E.D.
%'
We have used Proposition 31.1 to prove ProposiIt is easy to see that Proposition 31.1 follows
from Proposition 31.2. When applying part b) of Proposition 31.2,
REMARK 31.2:
often uses the fact that, for each space of
F
generated by
dmf($)xm,
g E U, the closed subm € IN,
des with the closed subspace generated by dmf(5 ) (xl,.
.
,xm),
m E N*,
xl,.
one
.., xm E
E.
x € E,
f(g)
coinci-
and
This can be
proved easily using the polarization formula.
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CHAPTER 32
HOLOMORPHY AND F I N I T E HOLOMORPHY
D E F I N I T I O N 32.1
f: U
is finitely holomorphic in
F
-t
for every finite dimensional subspace U
n
S f
REMARK 32.1:
(provided
f: U -+ F
A mapping
E
E
of
if,
for which
f/UnS E #(UnS;F).
we have that
@,
S
U
is finitely holomorphic if
is separated) and only if, the following condi-
tion is satisfied:
for every
xl,. ..,x n E E,
V
if
N*
and
denotes the open subset of
..,A,)
(Al,.
siting of all
g E U, n E
such that
g + Xlxl
(Cn
con-
+...+Xnxn E U,
then the mapping
V.
is holomorphic in D E F I N I T I O N 32.2
Let
and suppose that subspace
S
of
F
E
f: U
-t
F
be finitely holomorphic in U,
For each finite dimensional
is separated. such that
U
n
S f 0
let
fS = f
j uns
then dmfs(5) for every
5
E .Es(mS;F) and
E U fl S,
mensional subspace of
c U fl T ,
and
rn E
E
dmfT(I)ISm
(N.
zmfs(5) E
If T
such that
= dmfS(5),
P(mS;F)
is another finite diS c T,
then
dmfT(g)l,
=
g E U n S c
2mfs($) for
246
CHAPTER 32
m E N.
every
= fS(g).
=
f,(g)
we have
These compatibility relations imply the existence
of unique
Am E Xas(%;F)
f o r every
m E N,
5.
containing
Pm
both
m = 0
In particular, when
and
Pm E Pa(%;F)
and
such that
S
and every finite dimensional subspace *
Am
,
Pm = Am
It is easy to see that
and that
are finitely continuous, that is,
P
ml S and
A
are continuous for the induced topology
ml S m
for every finite dimensional subspace
= Am,
amf(g)
Let then
f
for every
PROPOSITION 32.1 f
imf(g) = Pm
m E IN,
+ F
f: U
bounded in
p
f(a+hb)
if and only
and continuous o r amply
U
F
be separated, and let
a E U
(1 E 6:
b E E,
and
: a+Xb
E U]
(l-X){+AxE
of
a.
U
Theref ore
for
X
E V.
Let
for
V = (1 E J ' : (l-X)g+Xx E U ] ,
= f[(l-X)g+Xx]
U.
then the mapping
is holomorphic, by Remark 32.1, in the open
F
> 1 such that
Let
U
be finitely holomorphic and amply bounded in
We note first that if
subset
U.
The necessity of the stated condition is obvious.
f: U + F
F-
5 E
U.
T o prove its sufficiency, let
X
S((U;F),
dmf(g) = amf(g),
is holomorphic in
is finitely holomorphic in
PROOF:
. Clearly, if f E
is finitely holomorphic, and
imf(g) = :"f(5)
if
and
g.
containing
S
S,
on
Then
5:
E U,
1x1
A E C,
and let
x E U,
and
s p.
g(X) =
g E W(V;F),
and
gp(0) c V.
HOLOMORPHY AND FINITE HOLOMORPHY
~ ( ~ ' ( = 0 a"f(g)(x-s)", )
f o r every
rn E N.
But
f o r every
rn E IN.
Therefore,
holornorphic i n
247
and hence
by Proposition 26.2,
f
is
Q.E.D.
U.
REMARK 3 2 . 2 :
Each polynomial
P E P,(E;F)
l o m o r p h i c in
E.
need n o t be c o n t i n u o u s (even
when
P E Ca(
morphic
.
1
However,
E;F)),
P
and hence
P
i s f i n i t e l y ho-
i s n o t necessarily h o l o -
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33
CHAPTER
THE MAXIMUM SEMINORM THEOREM
PROPOSITION 33.1 ~
E
U.
If t E X
that
(l-A)~
and
X
Let
F
is a non-empty subset of
+A t E U
E X
(l-),,)~+At
implies
A E
for
Ilf(OII s PROOF:
Let
placed by
f E ~(U;F)
be seminormed,
t
E
X.
with the property
AE
for
IA I
e,
U
and
s 1,
IAI = 1,
0;,
then
x E X}.
sup{llf(x)11
25.2
Applying Proposition
with
x
re-
we have
t-~,
f(~)
=
2~i
f[(l-~)~+AtJ
\
d)"
'I)" 1=1 and hence "f(S)" ~
sup
11.1=1
Ilfr(l-A)~+AtJII
sup{llf(x)"
S
: x EX},
tEX where
II
·11
denotes the seminorm considered in
COROLLARY 33.1 Let
V
Let
be a closed
F
S -balanced oV
such that the boundary
Ilf(~ PROPOSITION f E ~(U;F).
33.2
)11 Let
f E ~(U)
be seminormed,
s F
neighbourhood of
of
V
in
sup{llf(x)11
U
II fll: 249
and in
~
is non-empty.
~
E U. U Then
: x E oV}.
be seminormed,
If the mapping
Q.E.D.
F.
U
x E U t--
conncected and
II f(x)1I
E
(R
has a
CHAPTER 3 3
250
local maximum at a point
5
in a neighbourhood of in
then
U,
of
l/f(S)il
and
is constant
IIfIl
is the minimum of /If11
U.
PROOF:
E = F =
When
this proposition is simply the clas-
(c
sical Maximum Modulus Theorem, which states that if
+ /f(x)l E
If\:x E U
is constant in
4, E U,
f
-
be arbitrary, and let
E
If I : x E U then
has a local maximum in
If(x)
I
E IR
G1(0) c V.
X
E V.
U.
is constant in
g
that
(1-),)g+),x
Define
Since
hence
E U)
F =
To see this, let
U,
and let
g E W(V;C)
by
g(X)
= g(0).
g(1)
constant in
Since
Us.
U
Then and
C,
is constant in
V
which
S1(0),
and so
it follows from Proposition 31.1 that
f
are arbitrary.
F
$ E F'
exists Let
I[f({)I(
V
such that
[lf(x)ll
for every
(I$I(
maximum at
for every
x E V.
g.
Thus
in
U,
U. E
By the Hahn-Banach Theorem, there
be a neighbourhood of 2.
is
is constant in
Finally, let us consider the general case, where and
and
f
is a neighbourhood o f
S
for
0, it follows
f ( x ) = f(S),
Therefore
be
g
= f[ (l-X)g+Xx]
( g ( has a local maximum at
g
U
9'
is an open subset of
In particular,
0.
f
We claim
(c.
is constant in the connected component of
contains
then
has a local maximum at
the largest 5-balanced subset of
V = [), E C :
U,
U.
N o w let
that if
IR
$of
L
1
5
and
in
x E V.
E W(U)
U
$ff(f)]
= I[f(s)I(.
such that
Then
and
IJrofl
has a local
From the cases already considered, it follows
THE MAXIMUM SEMINORM THEOREM
that
i s constant i n
$of
m ( s 11
I I ~ ( )CI I =
ilf(S)li =
Hence
s \If(x)jl
I/fll
c m i
iif(x)i~.
5
x E V,
x E U.
x E U,
if
Ilf(r)II
and
b
Q.E.D.
Under t h e c o n d i t i o n s o f t h e P r o p o s i t i o n , i f
has a maximum a t a p o i n t o f
then
U,
/If11
i s constant
U.
in
EXAMPLE 33.1 norm.
for
1.
=
f o r every
ilf(x)II
f o r every
COROLLARY 3 3 . 2
Therefore,
U.
251
Then
f:
1,
and l e t
E E -+ ( 1 , z )
z
Ilf(z)II =
F
F = C2
with t h e supremum
1.1.
Thus
(If11
E,
i s holomorphic i n
I l f ( z ) l / = s u p { l , l z l ] = 1.
s 1 we have
zI
>
E = C,
Let
and
However, f o r
i s constant in aneighborhood
of the o r i g i n , but I/fll i s not constant i n E ; Ilf(0)ll i s the minimum of
11 fll
i n E. Example 33.1 shows t h a t , i n c o n t r a s t t o t h e case F = C,
we cannot, i n g e n e r a l , conclude i n P r o p o s i t i o n 3 0 . 2 i s constant i n a neighbourhood i s constant i n
another c o n d i t i o n on
nected,
such t h a t
Y E F,
Y
F
Let
lowing property:
5,
but only t h a t I ( f ( (
F: be a normed space, l e t
f € W(U;F).
and l e t
of
f
This conclusion i s v a l i d if we impose
V.
PROPOSITION 33.3
V
that
q E F,
if
II$I/
= 1,
f 7,
))YII
$(q) = L
s 1.
(Iqll = 1, and
Then, if
has a l o c a l maximum a t a p o i n t
F
Suppose t h a t
g
be con-
has t h e f o l -
there e x i s t s
IQ(y)I < 1
IIflI: in
U
for every
x E U -llf(x)/j U,
f
Q E F’
E IR
i s constant i n
U .
PROOF:
If
[If(g)ll =
0,
the r e s u l t i s t r i v i a l .
Suppose t h a t
CHAPTER 33
252
IIf(g)II
>
0.
assume that
# f(5).
Ilf(x)ll
f
//f(s)\l= 1.
Let
= 1,
$CfG)I y
Multiplying
5
V
x E V,
Therefore
I$ofl
be such that
Y E F,
when
x E V.
f o r every
has a local maximum in
x E V
Thus f o r
Ilf(x)ll
r~
tinuation,
33.1:
f
V,
f(x) =
U
in $of
= 1,
such that
E U(U)
and,
U,
and it follows
$of
is constant in
= $ff(g)]
+[f(x)]
1, which implies that
is constant in
REMARK
we have
//$I/
IlYII s 1,
Then
f r o m the proof of Proposition 33.2 that U.
5
be a neighbourhood of
l\f(g)/I = 1
for every
$ E F’
I$(Y)I < 1
and
Let
by a suitable constant, we may
f(g).
= 1, while Therefore
f
and so, by uniqueness o f holomorphic con-
must be constant in
U.
This proposition applies when
space, and in particular when
F = CC.
Q.E.D. F
is a Hilbert
CHAPTER 34
PROJECTIVE AND INDUCTIVE LIMITS AND HOLOMORPHY
PROPOSITION 3 4 . 1
Let
{F.] 1
ly convex s p a c e s , l e t i E I,
each
let
F
pi:
F
be a f a m i l y o f complex l o c a l -
i€I
be a complex v e c t o r s p a c e , and f o r -I Fi
be a l i n e a r mapping.
Let
F
be g i v e n t h e p r o j e c t i v e l i m i t t o p o l o g y d e f i n e d by t h e map-
i €
Pi,
pings
I.
If
i s a non-empty
U
E,
complex l o c a l l y convex s p a c e into
F,
then
f E H(U;F),
If
C o n v e r s e l y , suppose t h a t let
ii
i s a mapping o f
f
i f and o n l y i f
U
E H(U;Fi)
piof
i E I.
f o r every PROOF:
f E H(U;F)
and
open s u b s e t of a
i t i s obvious t h a t piof
be a c o m p l e t i o n o f
E H(U;Fi).
Fi.
Pi
E H(U;Fi).
For each
G =
If
o f
Fi
i E I,
and
i€ I
Fi,
= i€I g : U -t G
Then
then
6
i s a completion o f
Now d e f i n e
p:
F + G
p : Y E F+.r ( p i ( Y ) ) S = p(F).
Then
Since t h e topology o f
logy o f
Define
by
g E H(U;G).
and l e t
G.
S
F
g(U) c S ,
i EI
by
E G,
and hence
g E H(U;S).
i s t h e i n v e r s e image o f t h e topo-
u n d e r t h e mapping
p,
253
and
g = pof,
it follows
254
CHAPTER 34
that
f 6 H(U;F).
Q.E.D.
In the proof of Proposition 34.1 we have made
REMARK 34.1:
if
use o f the following fact:
p:
F
surjective, and the topology of
F
P
pof
of the topology of
then
G,
-t
G
is linear and
is the inverse image under
E
H(U;F)
if and only if
f 6 B(U;F).
COROLLARY
34.1
If F
is a complex locally convex space,
COROLLARY
34.2
If F
is a complex locally convex space,
and if
denotes the space
WF
,
u (F,F‘ )
F
with the weak topology
then
H ( u ; W ) = {f:
u + F
:
$ o f
E W(u)
for every
$ E F’].
Corollary 34.1 follows immediately from Proposition 34.1, and Corollary 34.2 follows from Corollary 34.1. O u r next example shows that it is not, in general,
possible t o replace the symbol Thus Proposition valid if EXAMPLE
H
by
a
in Corollary 34.2.
34.1 and Corollary 34.1 are not, in general,
H
i s replaced by
34.1
Suppose that
Then, by Proposition 29.2,
W. F
is not differentially stable.
WF
is not differentially stable.
It follows that there exists a non-empty open subset such that
H(U;WF)
be false for
#(U;WF).
C
PROPOSITION 34.2
Let
#
a(U;WF).
‘Em’ me”
locally convex spaces, let
E
U
of
Hence Corollary 34.2 must
be a sequence of complex be a complex vector space,
25 5
PROJECTIVE AND INDUCTIVE LIMITS AND HOLOMORPHY
m E IN
f o r each
om:
let Pm
--
Em
-i
Pm+loum
E
and let
let
Ern+ 1
p,:
Em
E
-t
be a compact linear mapping such that
f o r every
E
m
Uo
and
E,
let
REMARK
Urn =
If
a mapping of
only if
fop,
34.2:
m E N,
-l(U), Pm
If U
is an open
and suppose that
is a complex locally convex space,
F
U
m E IN.
p,,
into
F,
f E #(U;F)
then
E H ( u ~ ; F ) for every
if and
rn E N.
T o say that the linear mapping
Om: Em
is compact means that there exists a neighbourhood the origin in in
Em+l.
Pm(Em),
me N be given the locally convex inductive limit to-
is non-empty. f
u
Suppose that E =
IN.
p o l o g y defined by the mappings
subset of
be a linear mapping, and
Em
such that
Gm(Vm)
+
Vm
Em+l of
is relatively compact
The compatibility condition in the statement of
the proposition concerning the mappings
Urn
and
p,
states
that the diagram
commutes f o r every implies that
Urn
m E N.
The condition
#
= p,l(U)
for every
Q
m E N.
This fol-
lows by induction from the relation:
To prove (*),
let
xm = p
-1
(x),
= x E U, which implies that and hence
um(xm) E P,,~
(u).
where
x E U.
Then
P~+~[U~(X~) = ]pm(xm)
pm(xm) =
= x E U,
25 6
CHAPTER 34
PROOF OF PROPOSITION 34.2:
#(u;F),
f E
If
it is clear
that
fop,
E B(Um;F)
for every
m E IN. Conversely, suppose
that
fop,
E B(U,,,;F)
for every
m E IN.
is finitely holomorphic.
f
E
Suppose that
Em
for which
E.
contains a subspace
and
S.
Sm
such that
i
zi = um-l~um-2
E Em
for every
yi = pm(zi). nerated by
{zl,
of
S Sm
is separated. with
...,nk].
Choose
# 0
m E N
we have
,...
is linearly independent, and
Sm
denotes the subspace o f
then
S.
p,
Furthermore,
p,
Therefore
if
U flS
#
Em
ge-
is an isomorphism of the
p,
is continuous
is also a homeomorphism
S.
pm(UmnSs,)
N
Then
Furthermore, we have
and s o
ni E
be
i = 1 ,k. Then i i and { zl,., , z k ] is linearly inde-
zk],
and
[ nl ,
is a
(yl,...,yk]
there exists
un (xn ) ,
...,
Sm
Let
p,
.
i,
Hence, if
vector spaces and
0 . . . 0
{yl,...,yk]
pendent, since
S.
with
i = l,...,k
and similarly, for each
such that
yi = pni(xni).
greater than the maximum of
Let
Sm
i = l,.,.,k
For each
xni E E ni
be a finite
We claim that there exists m E N
topological isomorphism of a basis for
S
is separated, and let
dimensional subspace of
z
We show first that
@.
Now, since
257
PROJECTIVE AND INDUCTIVE LIMITS AND HOLOMORPHY
fop,
E #(U,;F),
U
(fopm)/UmnSm
(foPm)/UmlSm = (f/UlS)o (pm/Um1Sm),
But 3
we have that
n
S
is bijective.
and it follows that
W(umnsm;~).
E
and
pm/UmlSm: UmlSm
Therefore
f
is holomorphic in
U
n
S.
Hence
f
is finitely holomorphic. We complete the proof by showing that bounded in where
may assume
0 6 U,
It is not difficult to show that we
and that it suffices to prove that
locally bounded at
m E N.
Then
Em+l.
such that
om(Vm)
-t
o,(V,)
is relatively compact
is bounded, and so, if
neighbourhood of the origin in
Em+l,
Wm+l
there exists
is a
1 > 0
such that
that i s ,
-1
is a neighbourhood of the origin in
om (Wm)
Therefore
om
0.
Em.
is continuous.
We now assume bounded at
is
Em+l is continuous for Vm be a neighbourhood of
om: Em
T o see this, let
the origin in Em in
f
0.
We show first that every
is amply
F o r this, it suffices to consider the case
U.
is semnormed.
F
f
Let
the origin in
E,
-1 U h = p, (U’);
each
0 E U,
U‘ c U
and prove that
is locally
be a closed neighbourhood o f
and consider f o r each Uh
f
m E
(N
the set
is a closed neighbourhood of the
CHAPTER 34
25 8
origin in
E m , and
U k c Urn= p,l(U).
,
Since
fm = f o p
Let
m
m f IN.
zero, there is a neighbourhood and real numbers
Since
uo
Let
Vo = V b
and
ao(Vo)
_
oo(Vo)
_
Eo
Vb
of zero contained in
c
n vh
I
Uo(V’A)
. Then
c ao(Ub)
_
E
so
~ , +a,(~,) ~
fm+laam
,
is compact in
m f IN
such that -1
P,+~(U’),
am(Vm) and
f Vm} = M m < M.
we then have
suPEllfm+l(Y)ll
= sup{llf,(x)ll
: y
E
El
we have a convex neigh-
Em
u,’,,+~= : x
Vo c Tfo,
Also, since
oo(Vo)
c
sup[l)fm(x))/
fm =
El ; hence
_
of the origin in
Vm
compact in
and hence
9
is compact in E1’
which is closed in
U;,
C
~,(v0),and
bourhood
u;
such that
such that
NOW suppose that f o r
Since
is locally bounded at
c a o ( U b ) c U; c U1 = p;’(U).
_oo(Vo)
M
fo
is compact, there exists a convex neighbourhood
of the origin in
V’L
_
and
Mo
Also,
Om(Vm)3
: x E Vm}
= Mm <
= Mt
is
259
P R O J E C T I V E AND I N D U C T I V E L I M I T S AND HOLOMORPHY
Choose
> 0 such that Mm+l =
E
< M.
M,+E
Since
compact, there exists a convex neighbourhood origin in Since
Em+l
Em+l
-
and
of the
is compact in E
m+2 '
there exists a closed neighbourhood
origin in
-
(0))
um+l(v;,+l
Vk+l(0)
is uniformly continuous on the compact set
fm+l
GmTm),
such that
is
'm+l
=
V>+l(0)
of the
such that
_U __
_
(Y+{VL+~(O)O)
c
u;+~
9
where
~ ~ m ( V m )
[V;+,(O)] Let
0
is the interior of
Vm+l(0) = [Vk,,(O)
..
<7q-
Now, since
Um(Vm)
yl,.
,yh E
n
v/;+l(0)
V''+l(0)]o.
in
Em+l.
Then there exist
such that
-___
there exists
tx
is compact, and is contained in
E CS(Em+l)
such that
%+I
'
CHAPTER 3 +
260
'
am+l[B~,l(a~rV~))J
Therefore
am+l[B~,l(~(Vm))J Vm+l
= a m-t- I[B
l(~-(V»)J. m m
~,
V =
the origin in
U
p
mEIN
m
and
U,
We define
Then
(V ). m
Then
is a neighbourhood of
V
that
Em
{Em} mEIN'
is a vector subspace of
clusion mapping
Q.E.D.
A locally convex space
if' there exists a sequence
Em
E m+
~
Therefore
x E V} < M.
sup{" f(x)11
is locally bounded at the origin.
DEFINITION 34.1
E
and hence
is relatively compact.
Let
f
c am+l(V~+l)'
E
is a Silva space
of Banach spaces such E,
Em
C
E m+
l,
is compact for every
l
the inmEN,
= U
Em' and the topology of E is the locally convex mEN inductive limit topology corresponding to this decomposition.
REMARK 34.3:
Silva spaces,apart from being of interest in
themselves, appear frequently in holomorphy, even in finite dimensions. n E IN*,
For example, if
the space
~(K)
K
is a compact subset of
of holomorphic germs on
its natural topology, is a Silva space. open subset of
R
n,
the space
e'(u)
with compact support is a Silva space. open sUbset of
~n,
~'
(u) = r~(U)J'
If
U
K
~n,
with
is a non-empty
of distributions on If
U
U
is a non-empty
is a Silva space.
We point out that the hypotheses of Proposition 34.1 characterise the class of Silva spaces, by virtue of the fo1-
261
PROJECTIVE AND I N D U C T I V E LIMITS AND HOLOMORPHY
If E
lowing result:
is a vector space,
quence o f locally convex spaces, ping and
Urn:
prn = P r n + l o U r n E
Em
-+
Ern+l
f o r every
p,:
Em
a se-
-t
E
a linear map-
a compact linear mapping, with rn E N,
if
E =
(J
p,(E,),
and if
m€ N
carries the corresponding locally convex inductive limit
topology, then FtEMARK
34.4:
E
is a Silva space.
Proposition 21.1 is not, in general, valid if
the assumption that the mappings or if the sequence
family of spaces.
(Emjmm
are compact is dropped,
is replaced by an uncountable
This page intentionally left blank
CHAPTER 35
TOPOLOGIES ON
DEFINITION 35.1 K
Let
be a seminorm on
be a compact subset o f
if there exists
V
p
of
U
with
E CS(F)
@
K c V
H(U;F)
U.
p
and let
g(U;F),
is said to be ported by
K
such that for every open subset
C(V) > 0,
there exists a real number
with the property that p(f) s C ( V ) sup @{f(x)] XE
f € #(U;F).
for every
v
The Nachbin topology, denoted by
is the locally
T (u’
convex topology on #(U;F)
#(U;F)
defined by the seminorms on
which are ported by compact subsets of
EXAMPLE 3 5 . 1
Let
K c U
be compact and
@
U.
E CS(F).
Then
the seminorm
is ported by REMARK 35.1
and
p
over
K. As
K
CS(F),
ranges over the compact subsets of the seminorms
define the compact-open topology, follows from Example 3 5 . 1 that EXAMPLE 3 5 . 2
Suppose that
F
T
‘r0 s
f E #(U;F)i~ 7 W
on , on
is separated.
U,
sup pff(x)]
xE K #(U;F). Thus, it #(U;F). Let
K
be a
264
CHAPTER
compact subset of
35
B E CS(F),
U,
and let
'
quence o f non-negative real numbers such that m
-t
L
Let
m.
E.
be a bounded subset of
be a se-
mEN c
-t
m
as
0
Then the mapping
m
K.
defines a seminorm which is ported by f
E #(U;F).
V = B
Then there exists
(K) =
U $1
p
Choose
1x1
5
p.
u
Bu,l(x) c U
a E CS(E)
x+Xt E V
such that
such that
sup e{f(x)} = M < m. xE v for t E L, x E K ,
and
*K
> 0
To see this, let
X E
C,
Then, by the Cauchy integral formula, /-
and hence
Y s - : - ~ for
1 Am B[--Tm. d f(x)(t)]
x E K,
t E M,
m E IN.
P
Theref ore
tiL for every
m E
[N.
For
m
and hence the series which defines
lows easily that that
p
with
K c V c U.
BU,l
is ported by
( K ) c V,
such that
1 E C,
p
Let
V
x+Xt E B u r l ( K )
s 1/2,
converges.
It fol-
We now show
a
€
CS(E)
E,
such that
r > 0, depending only on
f o r every
Therefore
U(U;F).
be an open subset of
Then there exists
and there exists
(11 4 r .
p(f)
is a seminorm on K.
cl/"/p m
sufficiently large,
x E K,
t E L
and
V
TOPOLOGIES ON
U(U;F)
265
tE L f E H(u;F).
for every
cl / m
m C
Since
(--!--)"
is independent of
m= 1 f,
we may define cl
m
c
C(V) =
/m
m
m
m= 0
Therefore
sup @{f(y)]
p ( f ) S C(V)
.
(y-)
for every
f E U(U;F),
YE v and so
p
is ported by
35.1
PROPOSITION
K
let on
E CS(F)
is ported by
for every
there exists
f
E
E CS(F)
V
5
K
f E U,
let
and
A seminorm
U.
If
C(V)
> 0
V
of
con-
U
with the property that
SI(U;F).
#(U;F)
is an open subset of
U
K,
p
be
by Definition
Then
V.
p
and let
containing
be the largest 9-balanced subset o f
open and contains
K,
which is ported by
be a seminorm associated with V
p
if and only if there exists
T o prove that this condition is necessary, let
a seminorm on
35.1.
be separated,
such that for every open subset K,
taining
@
F
be a compact {-balanced subset of
U(U;F)
PROOF:
Let
K.
and hence there exists
K,
let
V
is
f
C(Vg) > 0
such that a)
p(f)
5
c(vg)
sup p[f(x)] xE v<
f o r every
By Proposition 27.2, the Taylor series of
f
f
E H(u;F).
at
9
converges
26 6
to
CHAPTER
f
v5
in
V
f
.
Hence, for every
35 f E 5J(U;F) and every
we have
Taking this in conjunction with a), we have
f o r every
f E 5J(U;F).
Since
V
depe.nds only on
g
V,
this
proves the necessity o f the given condition. To prove that this is sufficient, suppose that there exists
p E CS(F)
containing
for every
K
such that f o r every open subset
there is a constant
f E #(U;F).
there exists an open set
W,
(l-),)S+Xx E V
and
C(W) > 0
f o r every
if
x E W
K
Since
A E
of
U
such that
is compact and 5-balanced,
K c W c V, (c,
1x1
and p > 1, such that
s p.
There exists
such that
f E #(U;F).
B y the Cauchy integral formula ( P r o -
position 25.2) it follows that
and hence
> 0
C(V)
V
TOPOLOGIES ON f E W(U;F).
f o r every
this shows that PROPOSITION 35.2 that for every
Since
Let
fX XE
p E CS(F)
is independent of f,
C(W)
K.
is ported by
p
267
W(U;F)
*
Q.E.D.
be a net in
Suppose
#(U;F).
K
and every compact subset
there exists an open subset
V
of
containing
U
K
of
U
such
that lim sup p{f(x)-f
x
f E #(U;F).
for some
PROOF:
Let
p
x (x)]
x€V
Then
be a
7
lim f
x
x
a seminorm corresponding to for every open subset
= f
in the topology
-continuous seminorm on
W
K c U.
i s ported by some compact set
p
o
=
of
W
Let
7
W
.
Then
#(U).
B E CS(F)
be
in Definition 35.2, so that
p
U
K,
containing
there exists
> 0 for which
C(W)
a)
p(f)
4
f o r every
sup p[f(x)]
C(W)
XE w
f
By hypothesis, there exists an open subset K
taining
such that for every
E #(U;F).
V
of
U
e > 0 there exists
con),,
€ I\,
with b)
Sup
S: C
P{f(X)-f),(X)}
for
5
1,-
X€V
Combining a) and b), we find that
1 z- l o 7
W
.
Therefore
< eC(V)
converges to
f
when
in the topology
Q.E.D.
.
PROPOSITION 3 5 . 3 lanced. at
EfA€A
df-fx )
f
5 E
Let
Then f o r every converges to
f
U,
and suppose that
f E #(U;F)
U
is 5-ba-
the Taylor series of
in the topology
7
W
.
f
CHAPTER 35
26 8
PROOF:
Since
of
K
such that
U,
lim sup
{f(X)-7
converges to
f
p(%;F)
V
Let
F
U
of
containing
(x)) = 0 .
the Taylor series of
f
at
s
Q.E.D.
in the topology
PROPOSITION 35.4 Then
and hence, by Pro-
arid every compact subset
m,f,f
x€V
Therefore, by Proposition 35.2,
N.
= U,
there exists an open subset
m+=
m E
s
p g CS(F)
position 27.1, f o r every
K
U
is 5-balanced,
U
be separated,
imf E # ( U ; Pi(%;F)),
f E #(U;F),
Pi(%;F)
where
and denotes
with the limit topology (see Definition 16.2).
Fur-
thermore, if m
m
is the Taylor series of
f
5,
at
then
Z
ZmPk+,,,(x-{)
is
k=0
the Taylor series o f PROOF:
dmf
f .
at
We s h a l l prove the proposition for the bounded topo(see Definition 16.3).
logy on P(%;F) choose
a E CS(E)
B (g) c U a9 1
such that
Given and
h
uniformly in
B
a,l
(5 )
when
Tg: x E E F * x-f E E , Then Let
lim e{f(x) h-m L
-
h Z Qk(x)] k=O
h
=.
-t
and
&k = pk"Tf
= 0
be a bounded subset of
Let
E,
,
uniformly in and let
TOPOLOGIES ON
#(U;F)
Y E L,
and
Applying the Cauchy integral formula far x E
269
B'
a ,r/3
(f),
y E L
and
we have
Theref ore
for every
uniformly in
imf E
E
x E
a ,1/3
(5).
-
B (g) a ,1/3
W(U; b ( % ; F ) )
EIence
as
h
+
Therefore
m.
and m
c
;mPk(x-5)
k= 0
is the Taylor series of the bounded topology.
imf
at
g
,
where
P (%;F)
carries
It follows f r o m Remark 21.1 that this
series can be written as
The p r o o f for the limit topology on
b(%;F)
follows
CHAPTER 35
270
Q.E.D.
easily from the case we have considered. DEFINITION 35.2
P(%;F).
T~(%;F)
Let
denote the limit topology on
denotes the locally convex topology o n
T&
Jj(U;F)
defined by the family o f seminorms
K
where
ranges over the compact subsets of
ranges over the and
T
i(%;F)-continuous
seminorms on
q
53 (%;F)
,
m E IN.
35.2
REMARK
We note that this definition is valid, since
Pi(%;F)
where pology
35.5
Let
K
T
subset of
U
c s Tw
with the to-
seminorm on
K,
containing
-
*U0J
Since the seminorm there exists
w(u;F).
on
be a compact subset o f
Ti(%;F)-continuous
q
B E CS(F)
U,
P(?E;F).
a,
choose
m E N
If V E CS(E)
and
q
a
is an open such that
(K) c V.
i s continuous f o r the limit topology,
such that
the canonical linear mapping of
YB
p(%;F)
7 i(%;F).
PROPOSITION PROOF:
denotes the space
m E N,
for every
P~(%;F)) c C(U; P~(%;F))
imf E # ( U ;
and
and
U,
is a seminorm on
q = yBonB,
P(%;F)
P(%;FB)
into
where
n
0
is
P(%;FB),
which is continuous for
the inductive limit topology corresponding to the decomposition
T O P O L O G I E S ON
f E #(U;F)
If
is such that
# (U;F)
zmf(x)
271
E
;FB)
for every
0
x E K,
then there exists
On the other hand, if
#
P(?Ea
;FP)
for some
r
>
0
f E #(U;F) x E K,
such that
i s such that
then 1) is trivial, since in
0
this case
Applying the Cauchy inequality we have
and hence, by l),
for every
f E #(U;F).
f
is
T
W
-continuous.
E
Therefore the seminorm #(U;F)I-
Q.E.D.
Zmf(x) @
sup q[Zmf(x)] x€K
This page intentionally left blank
CHAPTER 36
BOUNDED SUBSETS OF
PROPOSITION 36.1 set
X
of
Let
F
U,
be separated.
E CS(F),
13
and every bounded subset
numbers
C,
c
Am
1
PROOF:
m E
every compact sUbset B
of
E,
K
there exist real
such that
0
~
sup{I3[-, m. d f(t)(x)J for every
In order that a sub-
be To-bounded, it is necessary and suf-
~(U;F)
ficient that for every of
~(U;F)
f EX, t E K, x E B} :s; C. c
m
f\T
Taking
m = 0,
ly sufficient.
we see that this condition is evident-
Conversely, let
first the case where
B
X
be T -bounded. o
is compact.
Consider
r > 0
Choose
such that
L= {t+".x: tE K, xE B, ". E ([:,I".I:s; r}cU. By the Cauchy inequalities we have 1
Am
sup{I3[-, m. d f(t)(x)J :s;
~
sup{l3[ f(y)J
: fE X, tE K, xE B}:s; :
f
EX, y E L} = C. c
m
r
where
C = sup{l3[f(y)J
f EX, y E L}
and
c
= 1/r.
We now consider the case of a general bounded set We argue by contradiction; thus suppose that, taking
273
C
B.
=k
274 and tk
CHAPTER 36 c = k2
E K
xk E L
and
mk E N ,
there exist
fk E
X,
+
as
such that
^mk d fk(tk)(xk)]
1
T, [p
k = 1,2,...,
for
2 mk
> k*(k )
for every
k.
k'
Let
yk = xk/k.
J =
The set k +
U [yl,y2,...)
[O)
is compact, since
yk
0
and we have
m ,
for
Then
k = 1,2,...,
contradicting the first part of the p r o o f . Q.E.D.
PROPOSITION 36.2 #(U;F)
is
PROOF:
Since
T
W
If
E
W
every
;r T o ,
x
Conversely, suppose that
f o r every
p E
CS(F),
We claim that i f
(*)
f o r every
of
U
d
K C U
W
T
0
containing
of
-bounded.
-bounded set is To-bounded.
is ~ ~ - b o u n d e d .Then
K
of
U.
is compact, then
p E CS(F)
be a metric on
T
is
and every compact subset
K
sup~~~f(x)l Let
x
-bounded if and only if T
x
is metrizable then a subset
there exists an open subset such that : f E X,
E
V
E V] <
+m.
which defines the topology of
E.
BOUNDED SUBSETS O F
( * ) is false, there exists
If
n
E
xn E U
there exist
N"
p E CS(F) and
275
#(U;F)
fn
such that, for each
E X
for which
K U [
But this is impossible, since
is compact.
nEm* Thus ( * ) is proved, and it follows that X is
T
W
-bounded. Q.E.D.
REMARK
36.1:
E
If
is not metrizable, we can no longer
assert that every r -bounded subset of (see [ 9 REMARK
I,
Let us recall that a subset 7, o f # ( U ; F )
there is a open subset
T
W
W
-bounded
36.3
V
U
of
is said
E U and every p E CS(F)
to be amply bounded if, for every
PROPOSITION
T
53-55).
pages
36.2:
is
;#(U;F)
containing
5
Every amply bounded su.bset of
such that
#(U;F)
is
-bounded.
PROOF:
Let
be a seminorm on
p
a compact set
K
associated with open subset of
c p
U
U,
and let
#(U;F)
B E CS(F)
by Definition 22.1. containing
K,
which is ported by be a seminorm Thus, if
there exists
V
is an
C(V) > 0
such that
Let
x
c
#(U;F)
be amply bounded.
I t follows easily from
Definition 3 6 . 1 that there exists an open subset containing
K,
and a number
p > 0,
such that
V'
of
U
CHAPTER 36
276 b)
s
p[f(x)]
p
f E X,
for every
v’.
x E
From a) and b), we have
c(v‘)*P
p(f) s
x
Therefore
f E
for every
is T -bounded.
X.
Q.E.D.
W
PROPOSITION 36.4
If
sets of
the amply bounded subsets of
#(U;F),
E
is metrizable, the ‘ro-bounded sub-
the ‘rW-bounded subsets of
amply bounded.
x
Let
Then, for every compact C > 0
exists a number fE
x
and
x E K.
bounded
be a K
c U
bounded
K,
s c
p[f(x)]
V
C1 > 0
such that
x
of
f E
REMARK
36.3: There are situations in which the
subsets of
x E V.
W(U;F)
Hence
A subset
there exists a number
fE
s Cl
T
W
for Q.E.D.
-bounded
are not amply bounded (see the work o f
X
of
pointwise bounded if, for every
every
contain-
is amply bounded.
Barroso, Matos and Nachbin in [ 101 ) DEFINITION 36.1
36.2, this
U
B[f(x)]
every
there
f o r every
A s in the proof of Proposition
and a number
x,
#(U;F).
p E CS(F),
and every
such that
set is
subset of
implies that there exists an open subset ing
and
are the same.
#(U;F)
I t suffices to show that every
PROOF:
#(U;F)
C > 0
.
#(U;F)
is said to be
p E CS(F)
and every
such that
s C
p[f(x)]
x E U, for
x.
PROPOSITION 36.5
Let
F
following are equivalent:
be separated.
For
x
c #(U;F)
the
BOUNDED SUBSETS OF
277
1)
x
is amply bounded in
2)
X
is -ro-bounded in
3)
X
is pointwise bounded and equicontinuous in 1)
PROOF:
36.2, Let
B(U;F)
x
is
4; E U
number
T
w
#(U;F)
and equicontinuous in
be amply bounded.
-bounded, and hence @ E CS(F).
and
c > 0
x
Let
2).
3
SI(U;F).
T
0
U.
By Proposition
-bounded in
There exist
U.
#(U;F).
a E CS(E),
and a
such that
and m
c)
1 c m ! Zmf(5)(x-5)
converges uniformly to
E
in
m=O
(5)
relative to
8.
Theref ore
PC m
s
c
c
Cm
ZG
cc/l-e
m= 1
if
U(X-5)
5
E < 1.
2)
* 3)
3)
s 1).
tinuous in
U.
number
0
a)
c
7
p{f(g)]
Hence
X
is equicontinuous at
5.
i s trivial.
Let
x
be pointwise bounded and equicon-
E U
If
and
p E CS(F),
there exists a
such that s. c
Furthermore, since
for every
x
f E
x.
is equicontinuous, there is a neigh-
27 8
CHAPTER 36
V
bourhood b)
of
5
in s
p[f(x)-f($)}
such that
U
X,
f E
f o r every
1
x E V.
Combining a) and b) , we have s
p[f(x)]
x
and hence
EXAMPLE 36.1
Then
E
Let
amply bounded in
V,
c > 0
such that
Since
E
Q.E.D.
denote the space
B
Let and
B
#(ED);
there exist
-bounded.
T
fl(hxo) =...= for every
Therefore
B
However,
V
of the origin in
...,fn E
fl,f2,
E’
Eu.
B
E’
B
.
is not
fl(Xo)
=...=
...,fn.
fl,
fn(xo)
for every
h E C;
lf(hxo)l
is not amply bounded.
is not equicontinuous in
E.
f E B
+
Since
which
Hence there exists
= 0, and
fn(Xxo) = 0 but
F o r such
and a number
has infinite dimension, there exists
such that
Xx, E V
with the weak
in fact,
is not a linear combination of
Then
E
be the closed unit ball of
is
for every neighbourhood a set
x E V,
be a complex normed space of infinite
Eu
u(E,E’ ).
B c #(Eu),
x0 E E
f E X,
for every
is amply bounded.
dimension, and let topology
C+I
1 E C,
f(xo)
#
0.
that is,
1x1
-+
m
as
B
is T,-bounded,
m.
BIBLIOGRAPHY
Note: -
The following list of books and papers is far from comIt contains the references quoted in the text andofhers
plete.
that aim to arouse the reader's interest in many topics not dealt with in this book.
Fairly complete bibliographies
be found in [L] and in [ 1021 ject.
References [A],
.
can
[ Q] was a landmark in the sub-
[B], [C],
[D], [El, [F], [GI indicate
proceedings of Meetings on infinite dimensional Holomorphy and other areas, with numerous bibliographical quotations. already-published (and more advanced) [I] , [: Jl [0] , [R]
,
[K]
,
The
[L] ,
[MI ,
, [S] ,
[TI and the soon-to-appear (and on the same level as this book) [HI, [N] , [PI, all of which have different
objectives from this book, are recommended as important reading material. The reader's attention should also be drawn to [109], in which motivations suggested by different branches of Mathematics point to the necessity and importance of the study o f holomorphic mappings between spaces o f infinite dimension.
In [U],
special recommendation is made o f the reading of J. Horvith'e study.
In the same volume, the papers by H. Upmeier and J.-F.
Colombeau present suggestive aspects of the relationship between infinite dimensional Holomorphy and other branches of Mathematics and Mathematical Physics.
In [27] there appears
a proof of the Josefson-Nissenzweig theorem which is more accessible than the others. [A] Proceedings on Infinite Dimensional Holomorphy (Ed.: P.L. Hayden and P.J.
Suffridge).
in Mathematics
364 (1974).
Springer-Verlag Lecture Notes
[B] Infinite Dimensional Holomorphy arid Applications (Ed. : M.C.
Matos) North-Holland Pub. Co. (1977).
279
B I BL IOGRAPHY
280
[C] Advances in Holomorphy (Ed.: J.A.
Barroso) North-Holland
Pub. Co. (1979).
[D] Functional Analysj.~,Xololnorphy and Approximation Theory (Ed.: S. Machado) Springer-Verlag Lecture Notes in Math.
843 (1981). [E] Functional Analysis, Holomorphy and Approximation Theory (Ed.: J.A. Barroso) North-Holland Pub. Co. (1982). [F] Functional Analysis, Holomorphy and Approximation Theory (Ed.:
G.I. Zapat") Marcel Dekker, Inc. (1983).
[G] Functional Analysis, Holomorphy and Approximation Theory (Ed.:
G.I. Zapata) North-Holland Pub. Co. (1984).
[HI Chae, S.B.,
Holomorphy and Calculus in normed spaces.
Marcel Dekker, Inc.
[I] Colombeau, J.-F.,
To appear.
Differential Calculus and Holomorphy.
North-Holland Pub. Co. (1982). [J] Colombeau, J.-F.,
New Generalized Functions and Multi-
plication of Distributions. [K]
North-Holland Pub. Co. (1984).
Coeurg, G., Analytic functions and manifolds in infinite dimensional spaces.
North-Holland Pub. Co. (1974).
[L] Dineen, S., Complex Analysis in Locally Convex Spaces. North-Holland Pub. Co. (1981). [MI
Franzoni, T. and E. Vesentini, variant distances.
[N] Isidro, J.M.
Holomorphic maps and in-
North-Holland Pub. Co. (1980).
and L.L. Stachb,
groups in Banach spaces.
Holomorphic automorphism
North-Holland Pub. Co.
[ O ] Mazet, P., Analytic sets in locally convex spaces.
T o appear. North-
Holland Pub. Co. (1984).
[PI Mujica, J., Complex Analysis in Banach spaces. Holland Pub. Co.
North-
T o appear.
[Q] Nachbin, L., Topology on spaces of holomorphic mappings. Springer-Verlag (1969).
BIBLIOGRAPHY [R]
Noverraz, P.,
Pseudo-convexit&.
281
Convexit6 polynomial et
domains dtholomorphie en dimension infinie.
IS]
North-
(1973).
Holland Pub.
Co.
Ramis, J.P.,
Sous ensembles analytiques drune vari&t&
banachique complexe.
Springer-Verlag Erg. der Math. 53,
(1970) [TI
Upmeier, H., Jordan C"-Algebras and Symmetric Banach Manifolds.
[U]
Advances in Mathematics and Its Applications (Ed.: Barroso) North-Holland Pub. C o .
[l]
(1984).
North-Holland Pub. C o .
J.A.
T o appear.
Abuabara, T., A version of the Paley-Wiener-Schwartz theorem in infinite dimensions.
Advances in Holomorphy
(Ed.: J.A. Barroso) North-Holland Pub. Co. (1979), 1-29. I21
Alexander, H., Analytic functions on Banach spaces. Thesis, University of California, Berkeley (1968).
[3]
Ansemil, J.M.
and S. Ponte,
Topologies associated with
W(U),
the compact-open topology on
[4]
Aragona, J . , Holomorphically significant properties of spaces of holomorphic germs.
[5]
preprint (1980).
Advances in Holomorphy
(Ed.: J . A .
Barroso) North-Holland Pub. Co.
Aron, R.M.
and P.D. Berner,
orem f o r analytic mappings.
(1979), 31-46.
A Hahn-Banach extension theBull. SOC. Math. France,
106 (1978), 3-24. [6]
Aron, R.M.
and Carlos Herves,
Weakly sequentially con-
tinuous analytic functions on a Banach space.
Functional
Analysis, Holomorphy and Approximation Theory (Ed.:
G.I.
Zapata) North-Holland Pub. Co. (l984), 23-38.
[7]
Aron, R.M.
and M. Schottenloher,
Compact holomorphic
mappings on Banach spaces and the approximation property. J. Funct. Analysis, 21, 1 (1976), 7-30.
BIBLIOGRAPHY Barroso, J.A., Topologias nos espasos de aplicapzes holomorfas entre espagos localmente convexos. Bras. de Cigncias, Barroso, J.A.
Anais Acad.
43 (1971), 527-546.
and L. Nachbin,
Sur certaines proprie'te's
bornologiques des espaces titapplications holomorphes. Troisi6me Colloque s u r 1fAnalyse Fonctionelle, Li6ge 1970, C.B.R.M.,
Vander (1971), 47-55.
Barroso, J.A.,
M.C.
Matos and L. Nachbin,
sets of holomorphic mappings.
O n bounded
Proc. Infinite Dimensional
Holomorphy (Ed.: T.L. Hayden and T.J. Suffridge) SpringerVerlag Lecture Notes in Math. 364 Barroso, J.A.,
M.C.
(1974), 31-74.
Matos and L. Nachbin,
On holomorphy
versus linearity in classifying locally convex spaces. Infinite Dimensional Holomorphy and Applications (Ed.: M.C.
Matos) North-Holland Pub. Co.
Barroso, J.A. and L. Nachbin,
(1977), 31-74.
Some topological properties
of spaces of holomorphic mappings in infinitely many variables.
Advances in Holomorphy (Ed.: J.A. Barroso)
North-Holland (1979), 67-91. Barroso, J.A.
and L. Nachbin,
A direct sum is holomor-
phically bornological with the topology induced by a Cartesian product. Bierstedt, K.-D.
T o appear in Portugaliae Math.
and R. Meise,
Nuclearity and the Schwartz
property in the theory of holomorphic functions on metrizable locally convex spaces.
Infinite Dimensional Holo-
morphy and Applications ( E d . M.C.
Matos) North-Holland
Pub. CO. (1977), 93-129. Bierstedt, K.-D.
and R. Meise,
Aspects of inductive
limits in spaces of germs of holomorphic functions on locally convex spaces and applications to a study of Advances in Holomorphy (Ed. J.A. Barroso) (W(U) ,'r',,). North-Holland Pub. Co. (1979),111-178.
283
BIBLIOGRAPHY
1161 Boland, J.P.,
Some spaces of entire and nuclearly entire
functions o n a Banach space. I.
Jour. ffir die Reine und
Angew. Math. 270 (1974), 38-60. [l7] Boland, J.P.,
Some spaces of entire and nuclearly entire
functions o n a Banach spaces. 11. und Angew. Math. 271
[18] Bremermann, H . I . ,
Jodr. ffir die Reine
(1974), 8-27.
Complex Convexity.
T.A.M.S.
82 (1956),
17-51 [l9] Bourbaki, N. Topologie Ge'drale, Chapitre X. Hermann
(1949) [20] Cartan, H. and P. Thullen,
Zur Theorie der Singularitaten
der Funktionen meherer Veranderlichen: Konvergenzbereiche. [21] Chae, S.B.,
Replaritats iind
Math. Annalen 106 (1932), 617-647.
Holomorphic Germs on Banach Spaces.
Ann.
Inst. Fourier 21, 3 (1971), 107-141. [22]
Coeure', G.,
Fonctions plurisousharmoniques s u r les es-
paces vectoriels topologiques et applications a llktude des fonctions analytiques.
Ann. Inst. Fourier 20 (1970),
361-432. [23] Colombeau, J.-F.,
On some various notions of infinite
dimensional holomorphy. Holomorphy (Ed.:
Proceedings Infinite Dimensional
T.L. Hayden and T.J.
Suffridge) Springer-
Verlag Lecture Notes in Math. 364 (1974), 145-149.
[ 241 Colombeau, J.-F.,
Hololnorphic mappings with a given
asymptotic expansion at a boundary point.
J. Math. Anal.
and Appl., 72, 1 (1979), 274-282.
[ 2 5 ] Colombeau, J.-F.
and B. Perrot,
The Fourier-Bore1 trans-
form in infinite many variables and applications.
Func-
tional Analysis, Holomorphy and Approximation Theory (Ed.:
S. Machado) Springer-Verlag Lecture Notes in Math-
ematics 843 (1981), 163-186.
BIBLIOGRAPHY
284
[ 261 Colombeau, J.-F. and Mario Matos,
Convolution equatioLw
in infinite cliineiis::>ixs: Brief Survey, new results and proofs.
Functional Analysis, Holomorphy and Approxima-
tion Theory (Ed.:
J.A.
Barroso) North-Holland Pub. C o .
(1982), 131-178.
[27] Diestel, J.,
Sequence and Series in Banach Spaces.
Springer-Verlag Graduate Texts in Mathematics no 92 (1984).
[28] Dieudoud, J.,
Foundations of Modern Analysis. Academic
Press (1960).
[29] Dineen, S.,
The Cartan-ThulLen Theorein for Banach Spaces.
Ann. Sc. Norm. Sup. Pisa, [3O] Dineen, S.,
24, 4 (1970), 667-676.
Holomorphy Types on a Banach Spaces. Studia
Math. 39 (1972), 241-288.
[31] Dineen, S.,
Bounding subsets of a Banach Space.
Math.
Annalen 192 (1971), 61-70.
[32] Dineen, S.,
Holomorphic functions on (Co,Xb)-Modules.
Math. Annalen 196 (1972), 106-116.
[ 331 Dineen, S., Space.
Unbounded holomo rphic f.mctinns
J. London Math, SOC.
[34] Dineen, S., spaces, I.
011
a Banach
4, 3 (1972), 461-465.
Holomorphic functions on locally convex Locally convex topologies
Fourier, Grenoble, 23
1351 Dineen, S.,
23
#(U).
Ann. Inst,
(1973), 19-54.
Holomorphic germs on compact subsets of
locally convex spaces.
Functional Analysis, Holomorphy
and Approximation Theory (Ed.: S. Machado) SpringerVerlag Lecture Notes in Math. 843 (1981),
[36] Douady, A.,
247-263.
Le p r o b l h e des modules pour les s o u s espaces
analytiques compacts d'un espace analytique donne.
Ann.
Inst. Fourier 16, 1 (1966), 1-95.
[37] Dwyer, T.A.W.,
Fourier-Bore1 duality and bilinear realiz-
ations of control systems. Proceedings 1976 Ames (NASA) Conference on Geometric Methods in Control (Ed.: R. Hermann and C. Martin), Math. Sci. Press, M4
(1977), 405-436.
BIBLIOGRAPHY [38] Dwyer, T.A.W.,
285
Differential equations of infinite order
in vector-valued holomorphic Fock spaces.
Infinite Di-
mensional Holomorphy and Applications (Ed. : M.C.
Matos)
North-Holland Pub. Co. (1977), 167-200.
[39] Fantappie', L., I funzionali analitici. Memorie della R. Academia Nazionale dei Lincei, 6, 3, 11 (1930),453-683 [ 401 Fre'chet, M., Une definition fonctionelle des polynGmes. Nou. Ann. Math. 9 (1909), 145-162.
[ 411 Fre'chet , M.,
L e s transformations ponc ttinlles abstraites.
C.R. Acad. Sc. Paris, 180 (1925), 1816-1817.
[42] Fre'chet, M., Les polyn&es
abstraits.
Journal Math.
Pures et Appl. (9), 8 (1929), 71-92.
[43] Ggteaux, R., Fonctions dsune infinite' des variables inde'pendantes. Bull. SOC. Math. de France, 47 (1919), 70-96.
[44] Ggteaux, R.,
S 7 n diverses questions de calcul fonctionnel.
Bull. SOC. Math. de France, 50 (1922)* 1-37.
[45] Globevnik, J., On the ranges of analytic mappings in infinite dimensions.
Advances in Ho1ornorp"ly (Ed.:
Barroso) North-Holland Pub. C o . [h6] Greenfield, S.J.
J.A.
(1979), 303-344.
and N. Wallach,
bounded domains in Banach spaces.
Automorphism groups of T.A.M.S.,
166 (1972),
45-57. [ 4 7 ] Grothendieck, A., Sur certains espaces de fonctions holomorphes.
Jour. ffir die Reine und Angew. Math. 192
(1953), Part I, 35-44, Part 11, 77-95.
[48] Gruman, L . and C.O. Kiselman., Le probleme de Levi dans les espaces de Banach 5 base. C.R. Acad. Sc. Paris 274 (1972), 821-824.
[49] Gunning, R. and H. Rossi., complex variables.
Analytic functions of several
Prentice Hall (1965).
BIBLIOGRAPHY
286 [ 503 Gupta, C.P.,
Malgrange theorem for nuclearly entire
functions of bounded type on a Banach space.
Notas de
Matematica 37, IMPA, Rio de Janeiro (1969).
[: 511 Harris, L.,
Schwarzfs lemma and the maximum principle in
infinite dimensional spaces.
Doctoral Dissertation,
Cornell University (1969). 1521 Harris, L . ,
Schwarz-Pick systems of pseudo-metrics for
domains in normed linear spaces. (Ed.: J.A. Barroso).
Advances in Holomorphy
North-Holland Pub. Co. (1979),
345-406. 1531 Hayden,
and T . J .
T.L.
Suffridge,
Fixed points of holo-
morphic mappings in Banach spaces,
Proc. Am. Math. S O C . ,
60 (1976), 95-105[ 541 Her&,
M.,
Analytic continuation on Banach spaces.
Several Complex Variables I1 (Ed.: J. Horvgth) SpringerVerlag Lecture Notes in Mathematics 185 (1971),
[55] Hewier, Y.,
63-75.
On the Weierstrass problem in Banach spaces.
Proceedings on Infinite Dimensional Holomorphy (Ed.: T.L. Hayden and T.J.
Suffridge) Springer-Verlag Lecture Notes
in Math., 364 (1974), 157-167.
1561 Hilbert, D.,
Wesen und Zieleiner Analysis der unendlich
vielenunabhhngigen Variabeln.
Pend. del Circolo Math. di
Palermo 27 (1909), 59-74.
[57] Hille, E., Methods in Classical and Functional Analysis. Addison-Wesley Pub. Co. (1972).
[58] Hirschowitz,
Sur les suites de fonctions analytiques.
A.,
Ann. Inst. Fourier, 20, 2 (1970), 403-413.
[ 591 Hirschowitz, A., infinie.
Prolongernent analytique en dimension
Ann. Inst. Fourier 22 (1972), 255-292.
[6O] HHrmander, L.,
An introduction to complex analysis in
several variables.
f61] Horvath, J.,
Van Nostrand (1966).
Topological vector spaces and distributions.
Vol. I, Addison-Wesley Pub. Co. (1966).
BIBLIOGRAPHY
[62] Isidro, J.M.
and J.P. Mendez,
287
Topological duality on the
J. Math. Anal. Appl. 67, 1
function space ( # ( U ; F ) , T ~ ) .
(1979)3 239-2480 [63] Josefson, B.,
Weak sequential convergence in the dual of
a Banach space does not imply norm convergence.
Arkiv
far Math., 13 (1975), 79-89. [ 641 Kaup, W.
,
Algebraic characterization of symmetric complex Math. Annalen 228 (1977), 39-64.
Banach manifolds.
[65] Kiselman, C.O.,
Geometric aspects of the theory of bounds
for entire functions in normed spaces.
Infinite Dimen-
sional Holomorphy and Applications (Ed.: M.C.
Matos)
(1977), 249-275.
North-Holland Pub. Co.
[66] Kathe, G., Topological Vector Spaces, Vol. 1, SpringerVerlag (1969).
[67] Kramm, B.,
Analytische Struktur in Spektren ein Zugang
fiber die --dimensionale Holomorphie.
J. Funct. Analysis,
37, 3 (1980)9 249-270. [68] Kre'e, P.,
The'orie de la mesure et holomorphie en dimen-
sion infinie.
Se'minaire Pierre Lelong 1975/76. Springer-
Verlag Lecture Notes in Math., 578 (1977),
44-70.
[69] Kre'e, P., Holonorphie et the'orie des distributions en dimension infinie.
Infinite Dimensional Holomorphy and
Applications (Ed.: M.C.
Matos) North-Holland Pub. Co.
(1977)9 277-296. [70] Lelong, P.,
S u r les fonctions plurisousharmoniques dans
les espaces vectoriels topologiques et une extension du the'oreme de Banach-Steinhaus aux families dfapplications polynomiales.
Troisieme Coll. Anal. Fonct.,
Vander, Belgium (1971),
"711
Lelong, P.,
Lizge 1970,
21-45.
S u r Itapplication exponentielle dans l'espace
des fonctions entieres. and Applications (Ed.:
(1977)9 297-311
Infinite Dimensional Holomorphy
M.C.
Matos) North-Holland Pub. C o .
BlBLIOGRAPHY
288
[72] Lelong, P.,
A class of Frechet complex spaces in which
the bounded sets are C-polar sets.
Functional Analysis,
Holomorphy and Approximation Theory (Ed.: J.A.
Barroso)
North-Holland Pub. Co. (1982), 255-272.
1731 Lelong, P.,
Two equivalent definitions of the density
numbers for a plurisubharmonic function in a topological vector space.
Functional Analysis, Holomorphy and Approx-
imation Theory (Ed.:
G.I.
Zapata) North-Holland Pub. Co.
(1984), 113-132.
[74] Ligocka, E.,
A local factorization of analytic functions
and its applications.
[75] Martineau, A.,
Studia Math.
47 (1973), 239-252.
S u r les fonctionnelles analytiques et la
transformation de Fourier-Borel.
J. Anal. Math., 11
(1963)~1-164.
[76] Martineau, A.,
Oeuvres de And&
Martineau (Ed.:
Hirschowitz) lfditions du CNRS, Paris
f77] Matos, M.C.,
A.
(1977).
O n the Cartan-Thullen theorem for some sub-
algebras of holomorphic functions in a locally convex space.
J. f&r die Reine und Angew. Math. 270
(1974),
7-14. [78] Matos, M.C.,
Convolution equations in spaces of uniform
nuclear entire functions.
Functional Analysis, Holomor-
phy and Approximation Theory (Ed.: G . I .
Zapata) Marcel
Dekker, Inc. (1983), 207-231.
f 791
Matos, M.C.,
O n the Fourier-Bore1 transformation and
spaces of entire functions in a normed space.
Functional
Analysis, Holomorphy and Approximation Theory (Ed.: G.I. Zapata) North-Holland Pub. Co. (1984), 139-169.
[80] Mazet, P.,
G6n6ralization des notions dcanneau noethe'ria
et d'anneau de Cohen-Macauley.
Applications
a
l a ge'orne-
trie de dimension infinie. Fonctions analytiques de Plusieurs Variables et Analyse Complexe. Colloq. Int. d u C.N.R.S., 131-140.
Paris 1972. Agora Math. Gauthier-Villars
(1974),
BIBLIOGRAPHY
[ 8 1 ] M eis e, R.
and D . Vogt,
289
The symmetric t e n s o r a l g e b r a o f
a l o c a l l y convex s p a c e and e n t i r e f u n c t i o n s ( P r e p r i n t ) .
[ 8 2] M eis e, R .
and D . Vogt,
Counterexamples f o r holomorphic
f u n c t i o n s on n u c l e a r F r e c h e t s p a c e . [83] M eis e, R .
and D.
Vogt,
(Preprint).
An i n t e r p r e t a t i o n o f
a s normal t o p o l o g i e s o f sequence s p a c e s .
T
w
and
[84] M i c h a e l , E.A.,
topolo-
11 ( 1 9 5 2 ) .
Memoirs A.M.S.,
L e C a l c u l D i f f g r e n t i e l d a n s l e s e s p a c e s de
[85] M i c h a l , A . D . , Banach.
J.A.
( 1 9 8 2 ) , 273-285.
Co.
Locally multiplicatively-convex
gical algebras.
6
Functional
A n a l y s i s , Holomorphy and Approximation Theory (Ed.: B a r r o s o ) North-Holland Pub.
T
G a u t h i e r - V i l l a r s (1958).
[86] Moraes, L . A . ,
T i p o s de h o l o m o r f i a e a b e r t o s de Runge.
P a r t e s I e 11.
de C i h c i a s 50 ( 3 )
Anais Acad. B r a s .
5-25.
( 1 9 7 8 ) , 277-293 e 5 1 ( I ) ( 1 9 7 8 ) , [87] Moraes, L . A . ,
Holomorphic f u n c t i o n s on holomorphic
i n d u c t i v e l i m i t s and on t h e s t r o n g d u a l s of s t r i c t inductive l i m i t s .
F u n c t i o n a l A n a l y s i s , Holomorphy and
Approximation Theory (Ed.: Pub. C o .
G.I.
Z a p a t a ) North-Holland
( 1 9 8 4 ) , 297-310. Spaces of germs o f holomorphic f u n c t i o n s .
[88] M u j i c a , J . ,
T h e s i s , U n i v e r s i t y o f R o c h e s t e r (1974). A n a l y s i s , Advances i n Math. (Ed. G.C.
Studies i n
Suppl. S t u d i e s , v o l .
4
R o t a ) Academic P r e s s ( 1 9 7 9 ) , 1-41.
[89] Mujica, J.,
A new topology on t h e s p a c e o f germs o f ho-
lomor phic f u n c t i o n s . 1901 Nachbin, L . ,
(Preprint)
.
L e c t u r e s on t h e t h e o r y o f i d s t r i b u t i o n s .
T e x t o s de Matemdtica no 1 5 , U n i v e r s i d a d e F e d e r a l de R e c i f e , Pernambuco, B r a s i l (1964).
R e p r i n t e d by Univer-
s i t y M i c r o f i l m s I n t e r n a t i o n a l , USA ( 1 9 8 0 ) .
[91] Nachbin, L . , given type. Gelbaum).
O n s p a c e s o f holomorphic f u n c t i o n s o f a I r v i n e , 1966. (Ed.: B.R.
Functional Analysis.
Academic P r e s s ( 1 9 6 7 ) , 55-70.
BIBLIOGRAPHY
290
[92]
Nachbin, L.,
On the topolo,T
of the space of all holo-
morphic functions on a given open subset.
Indagationes
Mathematicae, 29 (1967), 366-368.
[93]
Nachbin, L . ,
Sur les espaces vectoriels topologiques
dtapplications continues.
C.R.
Acad. Sc. Paris, 271
(1970)9 596-598 [94]
Nachbin, L.,
Convolution operators in spaces of nuclear-
ly entire functions on a Banach space,
Functional
Analysis and Related Fields, (Ed.: F.E. Browder) Springer-Verlag (1970), 167-171.
[95]
Nachbin, L., spaces.
Concerning holomorphy types for Banach
Intern. Coll. Nuclear Spaces and Ideals in
Operator Algebras.
[96]
Nachbin, L.,
Studia Math., 38 (1970), 407-412.
Concerning spaces of holomorphic mappings.
Publications of Depart. of Math., Rutgers University
(1970) [97] Nachbin, L., Uniformitd dlholomorphie et type exponential. Sdminaire Pierre Lelong 1970. Springer-Verlag Lecture Notes in Math. 205 (1971), 216-224. c98]
Nachbin, L.,
Sur quelques aspects re'cents dcholomorphie
en dimension infinie.
Seminaire Goulaouic-Schwartx,
Paris 1971/72, 18, 1-9.
[99]
Nachbin, L.,
Weak holomorphy, Part I and 11.
Unpublish-
ed manuscript (1972).
[loo]
Nachbin, L. and S. Dineen,
Entire functions of expo-
nential type bounded on the real axis and Fourier transforms of distributions with bounded support.
Israel J.
Math. 13 (1972), 321-326.
[lo13 Nachbin, L.,
On vector-valued versus scalar-valued
lomorphic continuation. 352-354.
Indagationes Math., 35,
ho-
4 (1973),
BIBLIOGRAPHY
[lo21 Nachbin, L . ,
291
Recent developments in infinite dimensional
holomorphy.
B.A.M.S.
[lo31 Nachbin, L.,
79 (1973), 625-640.
Limites et perturbations des applications
holomorphes.
Fonctions analytiques de Plusieurs Va-
riables et Analyse Complexe, Colloq. Ink. du C.N.R.S., Paris 1972. Agora Math, Gauthier-Villars (1974),
131-140. [lob] Nachbin, L., phy.
A glimpse at infinite dimensional holornor-
Proc. on Infinite Dimensional Holomorphy (Ed.: T.L.
Hayden and T . J . in Math.
Suffridge) Springer-Verlag Lecture Notes
364 (1974), 69-79.
[lo51 Nachbin, L., Perturbation of holomorphic mappings revisited. Rend. di Mat., (2), Vol. 8 (1975), 337-344. [106] Nachbin, L.,
Some holomorphically signficant properties
of locally convex spaces.
Funct. Analysis (Ed.:
D.J.
Figueiredo) Marcel Dekker, Inc. (1976). [lo?] Nachbin, L.,
Analogie entre ltholomorphie et la
line'arite'. Se'minaire Paul Kre'e. Partielles en Dimension Infinie,
[I081 Nachbin, L.,
gquations aux De'rive'es
1977/78, nP 1.
Some problems in the application of func-
tional analysis to holornorphy.
Advances in Holomorphy
(Ed. J.A. Barroso) North-Holland Pub. Co. (1979),
577-583 [lo91 Nachbin, L.,
Why holomorphy in infinite dimensions.
L'Enseignement Mathematique 26 (1980), 257-269. [110] Nachbin, L . and M.C.
Matos,
Silva holomorphy types.
Functional Analysis, Holornorphy and Approximation Theory (Ed.:
S.
Machado) Springer-Verlag Lecture Notes in Math.
843 (l98l), 437-487.
[ill] Nachbin, L.
and M.C.
Matos,
Entire functions on locally
convex spaces and convolution operators.
44 (19Sl), 145-181.
Compositio Math.
292
BIBLIOGRAPHY
[112] Nissenzweig, A.,
w*-sequential convergence.
Israel
J. Math. 22 (1975), 266-272. [lls] Noverraz, Ph.,
Fonctions plurisousharmoniques et ana-
lytiques dans les espaces vectoriels topologiques.
Ann.
Inst. Fourier 19, 2 (1969), 419-493. [l14] Noverraz, Ph.,
Sur le theoreme de Cartan-Thullen-Oka en dimension infinie. Seminaire Pierre Lelong 1971/72. Springer-Verlag Lecture Notes in Math. 332 ( 1 9 7 3 ) ,
59-68. [115] Noverraz, Ph.,
Le problame de Levi en dimension infinie.
Bull. Socie'te'Math. de France, Mdmoire
[116] Oka, K.,
Sur les fonctions analytiques de plusieurs
variable, VI.
J.,
46 (1976), 73-82.
Domaines pseudo-convexes.
Tohoku Math.
49 (1942), 15-52.
[117] Oka, K.,
Sur les fonctions analytiques de plusieurs
variables complexes, IX. tique intdrieur.
[I181 Pisanelli, D.,
Domains finis sans point cri-
Japan J. Math. 23 (1953), 97-155. Sur la (LF)-analyticite'.
Fonctionelle et Applications (Ed.:
Analyse
L. Nachbin) Herman
Act. Sc. et Ind. (1975), 215-224.
[ll9] Pisanelli, D., infinie.
Applications analytiques en dimension
Bull. Socidte Math. de France 96 (1976),
181-191. [ 1201 Raboin, P., Le problame du 5 sur un espace de Hilbert. Bull. Societe' Math. de France, 107 (1979), 225-240. l121] Ramis, J.P.,
Sous ensemble analytiques dlune varietd
analytique banachique.
Sdminaire Pierre Lelong 1967/68.
Springer-Verlag Lecture Notes in Math.,
71 (1968),
140-164. [122] Rickart, C.E.,
Analytic functions of an infinite number
of complex variables.
581-597
Duke Math. Journal 36 (1969),
293
BIBLIOGRAPHY [l23] Rickart, C.E.,
Natural functions algebras.
Springer-
Verlag Universitext (1979).
[124] Ryan,
Applications of topological tensor products
R.A.,
to infinite dimensional holomorphy.
Thesis. Trinity
College Dublin (1980). [125] Schottenloher, M.,
rhmen.
Analytische Fortsetzung in Banach-
Dissertation, Munich (1971).
[126] Schottenloher, M.,
c-product and continuation of
analytic mappings.
C.R.
du Colloque dlAnalyse Fonction-
nelle et Applications (Ed.: L. Nachbin) Rio de Janeiro
1972, Hermann (1974), 261-270. [l27] Schottenloher, M.,
The Levi problem for domains spread
over locally convex spaces with a finite dimensional Schauder decomposition.
Ann. Inst. Fourier 26,
4 (1976),
207-237 [128] Sebastigo e Silva, J., se Funcional.
A s funqEes analfticas e a AnB'li-
Port. Math., 9 (1950), 1-130.
[l29] Sebastigo e Silva, J.,
Le calcul differentiel et intd-
gral dans les espaces localement convexes, re'els ou complexes, I.
Atti Acad. Lincei Rend., 20 (1956),
743-750.
[l30] Silva Dias, C.L. da, Espaqos vetoriais topol6gicos e suas aplicaqzes nos espaGos de funcionais analiticos. Bol. SOC. Mat. Sgo Paulo 5 (1950), 1-58.
[131] Soraggi, R.L., germs.
Bounded sets in spaces o f holomorphic
Advances in Holomorphy (Ed.: J.A. Barroso)
North-Holland Pub. Co. (1979), 745-766. [l32] Soraggi, R.L.,
Holomorphic germs on certain locally
convex spaces (1980).
[l33] Stevenson, J.O.,
(Preprint).
Holomorphy of composition.
Infinite
'Dimensional Holomorphy and Applications (Ed.: M.C. North-Holland Pub. C o .
(1977), 397-427.
Matos)
BIBLIOGRAPHY
294
[134] Taylor, A.E.,
O n the properties of analytic functions
in abstract space.
[135] Taylor, A.E.,
Math. Annalen, 115 (1938), 466-484.
Historical notes on analyticity as a
concept in functional analysis. (Ed.: R.C.
Problems in Analysis
Gunning) Princeton Math. Series, Vol. 3 1
(1970)9 325-343
[136] Taylor, A.E.,
Notes on the history o f the uses of
analyticity in operator theory.
Am. Math. Monthly 78
(1971)9 331-342 [ 1 3 7 ] Thorp,,E. and R. Whitley,
The strong maximum modulus
theorem for analytic functions into a Banach space.
P.A.M.S.
18 (1967), 640-646.
[138] Upmeier, H., gebras.
A holomorphic characterization o f C*-al-
Functional Analysis, Holomorphy and Approxima-
tion Theory (Ed,: G . I .
Zapata) North-Holland Pub. C o .
(l984), 427-467.
[139] Vesentini, E.,
Maximum theorems f o r vector-valued
holomorphic functions.
Rend. Sem. Mat. Fis. Milan
40
( 1970)9 23- 55.
[I401 Vigu6, J.P.,
Le groupe des autoaorphismes analytiques
d f u n domaine born6 dlun espace de Banach complexe. Sc. Ec. Norm. Sup.,
[141] Vigue, J.P.,
203-282.
S u r la convexit6 des domaines bornes
circle's homogenes.
1978/79.
4, 9 (1976),
Ann.
Seminaire Pierre Lelong-H. Skoda,
Springer-Verlag Lecture Notes in Math. 822
(1980)9 317-331.
[I423 Volterra, V., funzioni.
Sopra le funzioni che dependone da altre
Nota I. Rend. Accad. Lincei, Series
4, Vol.3
(1887)9 97-105.
[143] Volterra, V., funzioni.
S o p r a le f'unzioni che dependone da altre
Nota 11. Rend. Accad. Lincei, Series
Vol. 3 (1887),
141-146.
4,
295
BIBLIOGRAPHY
[I441 Volterra, V., funzioni. Val.
[I451
Sopra le funzioni che dependone da altre
Nota 111.
Rend. Accad. Lincei, Series
4,
3 (1887), 153-158.
Volterra, V., Nota I.
Sopra le funzioni dependenti da linee.
Rend. Accad. Lincei, Series
4, V o l . 3 (l887),
225-230.
[I461 Volterra, V., Nota 11.
Sopra le funzioni dependenti da linee.
Rend. Accad. Lincei, Series
4, v o l . 3 (1887),
274-281.
c147] Waelbroeck, L.,
The nuclearity of
@(U).
mensional Holomorphy and Applications (Ed.:
Infinite DiM.C.
Matos)
North-Holland Pub. Co. (1977), 425-436.
[I483 Wanderley, A.J.M.,
Germes de aplicagGes holomorfas em
espaFos localmente convexos.
Tese, Universidade Fede-
ral do Rio de Janeiro (1974).
[149]
Z o r n , M.,
Characterization of analytic functions in
Banach spaces.
Annals of Math. 46 (1945), 585-593.
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AN INDEX OF DEFINITIONS
................................ Amply bounded s e t of mappings ........................ B a l a n c e d s e t ......................................... Bounded t o p o l o g i e s on e s ( % ; F ) and P ( % ; F ) ........... C o e f f i c i e n t s o f a power s e r i e s ....................... Compact-open t o p o l o g y on C ( U ; F ) ( ~ ~ - t o p o l o g y........ ) Compact t o p o l o g i e s on a s ( % ; F ) and P ( ? E ; F ) ........... C o n t i n u o u s m-homogeneous p o l y n o m i a l .................. C o n v e r g e n t ( u n i f o r m l y c o n v e r g e n t ) power s e r i e s ....... Degree of a p o l y n o m i a l ............................... D i f f e r e n t i a l l y s t a b l e s p a c e .......................... D i f f e r e n t i a l s o f a holomorphic mapping ............... Domains o f gb-holomorphy ............................. E n t i r e mappings ...................................... Space o f e n t i r e mappings ~ ( E ; F ) ................ E n t i r e mappings of bounded t y p e ...................... Amply bounded mapping
Space o f e n t i r e mappings of bounded t y p e gb(E;F)..
............ ......................... . .................................. ......... .......... ....................................... ............................... ............................. .......................... ................... ....................... .......................
F i n i t e t o p o l o g i e s on
es(%;F)
and p ( % ; F )
F i n i t e l y h o l o m o r p h i c mapping
S p a c e o f f i n i t e l y h o l o m o r p h i c mappings, Wfh(U;F). F o r m a l power s e r i e s
S p a c e of f o r m a l power s e r i e s ,
Fa[[E]]
S p a c e of c o n t i n u o u s f o r m a l power s e r i e s , F[[E]]. G-holomorphic ( G z t e a u x h o l o m o r p h i c ) mapping W-bounding
set
163 218
45 171
17 235
171 4 17 163 233 26
133 28 28
119
119 171 69,245
69 173 173
173 75 202
Wb-domain of e x i s t e n c e
150
8 -holomorphic e x t e n s i o n
131
b
b
-holomorphic p r o l o n g a t i o n
132
g - h o l o m o r p h i c a l l y convex open s e t b Z b - h u l l of a s u b s e t X o f U
142
Wb-regular h o l o m o r p h i c mapping
150
297
139
A N INDEX OF DEFINITIONS
298
Xb-singular
........ ............................... ........
p o i n t of a h o l o m o r p h i c mapping
Holomorphic mapping
S p a c e of h o l o m o r p h i c mappings # ( U ; F )
............... H ~ ( u ; F )...............................
Holomorphic mapping o f bounded t y p e
150
25 9 177 25
,177 81
S p a c e o f h o l o m o r p h i c mappings o f bounded type,
81
Holomorphic mappings b e tween t o p o l o g i c a1 v e c t o r s p a c e s which a r e n o t l o c a l l y c o n v e x
.........
63(%;F) .......... ........................... .......................... . .................... w on ....................... on ....................... di(E1, ...,E m ; F ) ............................ P ( % ; F ) ................................... .............................. ........................................ P ~ ( E ; F ) ............... s e r i e s ...................................... ................... ................ at .......................... a s e r i e s .............. .................................... C ~ ( ’ % ; F ) .............................. P ( % ; F ) ............................... ...................... .................... all E F, ........ ......................................
L i m i t t o p o l o g i e s o n g,(?E;F)
and
L o c a l l y bounded mapping
m-homogeneous p o l y n o m i a l
S p a c e o f m-homogeneous p o l y n o m i a l s , p a ( % ; F )
180 170,171
48 4,159 4e 159
N a c h b i n t o p o l o g y (T
-topology)
N a t u r a l topology
Hb(U;F)
85,263 81
N a t u r a l topology
XI(U;F)
97
N o r m on
2
NO^ on
5 6,160
P o l a r i z a t i o n formula Polynomial
S p a c e of p o l y n o m i a l s , Power
13 161
17
P r o p e r #b-holomorphic e x t e n s i o n
131
P r o p e r #b-holomorphic
132
prolongation
R a d i u s of b o u n d e d n e s s o f a h o l o m o r p h i c mapping
a point
48
Radius o f convergence ( r a d i u s of uniform convergence) of
power
Schauder b a s i s Seminorm on Seminorm on
Seminorm p o r t e d b y a compact
S e p a r a t e l y h o l o m o r p h i c mapping S e t of
mappings from
into
;F(E;F)
S i l v a spaces
S p a c e of bounded h o l o m o r p h i c m a p p i n g s , gg(U;F) S p a c e o f c o n t i n u o u s f o r m a l power s e r i e s , F[[E]]
.... ...
17 108
167 167 85,263
76 11
260
83 173
A N INDEX OF DEFINITIONS
299
m Space o f c o n t i n u o u s m-homogeneous p o l Y n a , 6 ( E;F)
159
Space o f c o n t i n u o u s m - l i n e a r mappings
...
C(E1. Space of
.Em.F). S ( % ; F )
.................... ..........
continuous polynomials. ff(E;F)
Space o f c o n t i n u o u s symmetric m - l i n e a r
......................... # ( E ; F ) ................. f i n i t e l y holomorphic mappings. gfh(U.F) .. f o r m a l power s e r i e s . Fa[[E]] ............ h o l o m o r p h i c mappings. # ( U ; F ) ............
mappings. L ~ ( % ; F ) Space of Space of Space o f Space o f
3 . 156
e n t i r e mappings.
28
69 173 25 y 177
Space o f holomorphic mappings o f bounded t y p e .
...........................
g b ( u . ~ ) .a
b(~;~)
81. 1 1 9
Space of h o l o m o r p h i c mappings which a r e bounded o n e a c h s e t o f a n open c o n t a b l e c o v e r o f U . SI(U;F)
.................................... ..... g a ( ~ l .....E~.F). ................................... pa(^;^) .................... gas(%;^) .. .................................... a ............. of ...... m .................................... ................................. on .......................... .......................... ~ ( u ; F )............ on .......................... on .......................... #(u;F) .......................... ..................................... ....................................
Space o f m-homogeneous
97 159
polynomials. ffa(%;F)
Space of m - l i n e a r mappings. Sa(”E;F)
S p a c e of p o l y n o m i a l s .
Space of symmetric m - l i n e a r mappings. Subconvex s e t
Symmetrization o f
Taylor coefficient
m - l i n e a r mapping
a h o l o m o r p h i c mapping
Taylor polynomial o f o r d e r mapping
1. 155
. .
161
3 155 136 3 155 177
of a h o l o m o r p h i c
178
T a y l o r s e r i e s o f a h o l o m o r p h i c mapping
a t a point Topology
T
Topology
‘ro
Topology
T~
c
270
i n C(U;F)
38. 235
i n # ( U ; F ) i n d u c e d by t h e
compact-open t o p o l o g y on Topology
7-
Topo1oe;y
Tm
T opology
T~
Topology
T~
U-bounded
set
25.1 7 8
W(U;F)
on
84. 263
#(U;F)
84
#(U;F)
84 100
85. 263 81
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AUTHOR INDEX
............................... 2 ............................ 276 Bourbaki, N. ............................. 39 Cartan, H. ............................... 126,139,146,150,152 Cauchy, ............................. 18,31,32,35,209, 210,212,229 Coeure', G. ............................... 100 31 Dieudonne', J. ............................ Dineen, S. ............................... 108,109,154 Fre'chet, M. .............................. 81 Gateaux, R. .............................. 75 Hadarnard, J. ............................. 18 Hartogs, F. .............................. 77 Hille, E. ................................ 31 Hirschowitz, .......................... 206 Josefson, B. ............................. 122,204 Martineau, ............................ 85 Matos, M.C. .............................. 276 151 Montel, P. ............................... .............................. 26,85,100,276 Nachbin, Newton, I. ............................... 157 .......................... Nissenzweig, 102,204 Schauder, J. ............................. 108 Silva, .............................. 2 60 Steinhaus, H. ............................ 66 Taylor, B. ............................... 25 ,215 E. ................................ 116 Thullen, P. .............................. 126,139,146,150,152 Vesentini, E. ............................ 116 Witley, R. ............................... 116
Banach, S.
Barroso, J . A .
A.L.
A.
A.
L.
A.
J.S.
Thorp,
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