INFINITE DIMENSIONAL HOLOMORPHY AND APPLICATIONS
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NORTH-HOLLAND MATH E MATICS STU D I ES
12
Notas de Matematica (54) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester
Infinite Dimensional Holomorphy and Applications Edited b y
MARIO C. MATOS Universidade Estadusl de Campinas.Brazil
1977
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK OXFORD
@North-Holland Publishing Company - 1977
All rights reserved. N o part ofthis publication may be reproduced,stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission ofthe copyright owner.
North-Holland ISBN: 0 444 85084 8
PUBLISHERS:
NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM, NEW YORK,OXFORD SOLE DISTRIBUTORS FORTHE U.S.A.ANDCANADA:
ELSEVIER/ NORTH HOLLAND, INC. 52 VANDERBILT AVENUE, NEW YORK, N.Y. 10017
Library of Congress Cataloging in Publication Data
I n t e r n a t i o n a l Symposium on I n f i n i t e Dimensional Holomorphy, Universidade E s t a d u a l d e Campinas, 1975. I n f i n i t e dimensional holomorphy and a p p l i c a t i o n s . (Notas d e matem&ica * 54) (North-Holland mathematics s t u d i e s j 12j Includes index. 1. Holomorphic functions--Congresses. 2. Domains of holomorphy--Congresses. I. Matos, M h o Carvalho de. 11. T i t l e . 111. S e r i e s . QAl.N86 no. 54 [QA3jl] 510'.8s [ 5 1 5 ' . 9 ] ISBN 0-444-85084-8 7 7 -2 0010
PRINTED IN THE NETHERLANDS
F 0REWORD
'This book c o n t a i n s t h e P r o c e e d i n g s
of
Syn,sosium on I n f i i i i t e D i m e n s i o n a l Holomorphy
the International held a t
the
Vni-
v e r s i d a d e E s t a d u a l d e Campinas, B r a z i l , d u r i n g August 4 - 3 , 1 9 7 5 . I t c o n t a i n s t w e n t y f i v e o r i g i n a l r e s e a r c h a r t i c l e s COrreS?Ondin[J
t o t h e l e c t u r e s g i v e n a t t h e Synposium a n d some p a p e r s by i n v i t e d
or an-
ICC~U~:CI-S who c o u l d n o t a t t e n d t'lc m e e t i n g f o r o n e r e a s o n oi:!icr.
T!ie
articles include complete p r o o f s
and t h e y cover tho
c u r r e n t m o s t a c t i v e r e s e a r c h l i n e s o f I n i f i n i t e D i m e n s i o n a l HC 10ilorphy a n d i t s a p ? l i c a t i o n s . The m e e t i n g r e c e i v e d s u p n o r t from t h e
ds P e s q u i s a s (CNPq)
I'
I
'Conselho l,-zional
"Coordenaqao do A p e r f e i q o a n e n t o d o P a r s o a l
cle l J i v e l S u ; 2 e r i o r ( C A P E S )
' "Fundaqao d e Amparo 2 P e s q u i s a do
7:s-
cado d e S a o P a u l o (FAPESP) ' I I " F i n a n c i a d o r a d e E s t u d o s e ? r o - j e t o s ( P I i J E P ) " , a n d " U n i v e r s i d a d e E s t a d u a l d e Campinas (UNICAMP) m
i
m
'.
o r q a n i z i n q committee f o r t h i s m e e t i n g w a s formeci
J. A . B a r r o s o , G .
I. Katz, IT.
C . Matos ( c h 3 i r m a n ) ,
L.
by
IJadihin
a n d 9. P i s a n e l l i . T h e r e Tiere p a r t i c i p a n t s from Brazil
Chile
France,
the
Sermany, I r e l a n d
Yugoslavia.
V
following Sweden
countries U . S.A.
and
VI
FOREWORD
I would l i k e t o r e g i s t e r my t h a n k s t o t h e f o l l o w i n g p e r -
s o n s : P r o f e s s o r U b i r a t a n D'Ambrosio, d i r e c t o r
of Mathematics
of
UNICAMP,
of
the Institute
whose s u p p o r t made t h e m e e t i n g pos-
s i b l e ; M i s s E l d a M o r t a r i , who t y p e d
t h i s volume: M r .
R o d r i g u e s Q u e i r o z , who p r o v i d e d a l l technical
Gilbert0
f a c i l i t i e s for t h e
t y p i n g of t h e m a i i u s c r i p t .
Pilsrio C . Matos
TABLE OF CONTENTS
R i c h a r d M. Aron
Approximation o f d i f f e r e n t i a b l e
i
f u n c t i o n s on a Banach s p a c e . Volker Aurich
F o n c t i o n s meromorphes s u r C
A
.
Jorge Alberco Barroso,
On holomorphy v e r s u s l i n e a r i t y
M6rio C . Matos
in classifying locally
and Leopoldo Nachbin
spaces.
Paul Berner
T o p o l o g i e s on s p a c e s o f h o l o
convex
31
morphic f u n c t i o n s
of
-
certain 75
surjective l i m i t s . I<. -D.
Bierstedt
and P.. Eleise
Nuclearity
19
and
the
Schwartz
p r o p e r t v i n t h e t h e o r v of
ho-
l o m o r p h i c f u n c t i o n s on m e t r i z 93
a b l e l o c a l l y convex s p a c e s . P h i l i p J . Boland
D u a l i t y and s p a c e s of holomor-
131
phic functions. A holomorphic c h a r a c t e r i z a t i o n
S o 0 Bong Chae
o f Banach s p a c e s w i t h b a s e s . Holomorphic f u n c t i o n s on strong
Se6n Dineen
d u a l s o f F r g c h e t-Monte1 spaces. Thomas A . W .
139
Dwyer
D i f f e r e n t i a l e q u a t i o n s of
147
in-
f i n i t e order i n vector -valued 167
h o l o m o r p h i c Fock s p a c e s . J . Globevnik
On t h e r a n g e of a n a l y t i c f u n c t i o n s i n t o a Banach s p a c e .
201
Erik Grusell
w-spaces and a-convex s p a c e s .
211
Michel Hervg
Some p r o p e r t i e s o f t h e
of a n a l y t i c maps.
VII
images 217
VIII
TABLE OF CONTENTS
Bengt Josefson
Some remarks on Banach polynomials on co (A)
.
G. Katz C. 0. Kiselman
Paul Kr6e
Pierre Lelong
Jorge Mujica Philippe Noverraz Domingos Pisanelli
K. Rusek
and J. Siciak Martin Schottenloher
valued 23 1
Domains of existence in infinite dimension.
2 39
Geometric aspects ofthe theory of bounds for entire functions in normed spaces.
249
Holomorphie et thgorie des distributions en dimension infinie.
277
Sur l'application exponentiel1.e dans l'espace des fonctions entigres.
297
Xolomorphic germs on infinite dimensional spaces.
313
On aparticular case of surjective limit.
323
The connected finite dimensional Lie sub - groupes of the group Gh(n , C ) .
333
Maximal analytic extensions of Riemann domains over topological vector spaces.
347
Polynomial approximationon canpact sets.
379
Cm.
Martin Schottenloher
'I
James 0. Stevenson
Holomorphy of composition.
397
L. Waelbroeck
The nuclearity of
425
Index of Terms and Concepts
w
= T
for domains in
@(U).
393
437
Infinite Dimensional Holomorphy and Applications, MatoS ( e d . ) @ North-Holland Publishing Company, 1977
APPROXIMATION OF DIFFERENTIABLE FUNCTIONS ON A BANACH SPACE
By R I C H A R D M . A R C "
(*)
The study of approximation of holomorphic and differentia ble functions defined on Banach spaces has received considerable attention in recent years. For example, in the corrrplex case, there has been work done on polynomial approximation of analytic fun2 tions, defined on Runqe or polynomially convex sets in infinite dimensional spaces (see, for example, ]12,5,9,11]). In the
real
case, there has been interest in the general problem of approxL mating, in one of several topologies, certain classes of diffeK entiable functions by smoother ones, such as polynomials or real analytic functions. In this note, we will outline some new
re-
sults relating to approximation of differentiable functions with respect to four topologies. In Section 1 , we examine the approximation of differentiable
............................................................... (*)
Research partially supported by the Instituto de Matemst& cat Universidade Federal do Rio de Janeiro, Brasil, Conse lho Nacional de Desenvohimento Cientlfico e Tecnologico
.
(CNPq) and Financiadora de Estudos e Projetos (FINEP)
1
2
R . ARON
f u n c t i o n s , i n t h e f i n e t o p o l o g y , d e f i n e d on C(K) where K
is
a
compact H a u s d o r f f s p a c e . Our r e s u l t g e n e r a l i z e s s l i g h t l y work of @5]
Wells
[17].
and W u l b e r t
I n S e c t i o n s 2-4,
o u r main i n t e r
e s t i s w i t h p o l y n o m i a l a p p r o x i m a t i o n of k - c o n t i n u o u s l y d i f f e r e n _ t i a b l e f u n c t i o n s d e f i n e d on a Banach s p a c e E l w i t h r e s p e c t t o p o l o g i e s s t u d i e s by P r o l l a and G u e r r e i r o Bombal 131 , L l a v o n a [ S ] ,
Restrep [14],
to
L e s m e s [71,
;13],
and o t h e r s . I n these
sec
t i o n s , t h e t o p o l o g i e s w e c o n s i d e r w i l l b e g e n e r a t e d by t h e f o l lowing seminorms: p ( f ) = sup where X
and Y
c
; a j f ( x )( y ) I : x E XI y E Y),
a r e s u b s e t s of
E
and
Y r a n g e o v e r t h e compact s u b s e t s o f
studied i n [3,8,12,13].
E
j E N , j 5 k.
,
and k w e g e t t h e t o p o l o g y -cC
I n S e c t i o n 2, w e show
If
that E
( G r o t h e n d i e c k ) a p p r o x i m a t i o n p r o p e r t y i f and o n l y i f
X
has k
the
( C ( E ) ,$)
h a s t h e a p p r o x i m a t i o n p r o p e r t y f o r s o m e , and h e n c e f o r a l l , k,l. If
X r a n g e s o v e r t h e compact s u b s e t s o f
bounded s u b s e t s of both X
E
, we
and
g e t t h e t o p o l o g y 'T U ( c f
Y
over
the
1171) ,and i f
and Y are a l l o w e d t o r a n g e o v e r t h e bounded s u b s e t s o f
E l w e g e t t h e topology
k=1in
E
k
T~
which h a s b e e n s t u d i e d i n t h e
I n S e c t i o n s 3 and 4 , w e s t u d y t h e c o m p l e t i o n of
c14].
t h e f i n i t e t y p e p o l y n o m i a l s on E w i t h r e s p e c t t o For
case
k
U
and
T
k
~
denotes
the
space of k - continuously Frechet d i f f e r e n t i a b l e F - v a l u e d
map
E
and F r e a l Banach s p a c e s , C (E;F)
T~
p i n g s on E . W e r e c a l l t h a t t h e s p a c e o f f i n i t e t y p e p o l y n o m i a l s
from
E to F
,
63f(E;F) , i s t h e vector s p a c e generated by pf("E:F)
( n E N ) , w h e r e Qf("E;F)
i s t h e subspace of
@("E;F)
(of a l l
c o n t i n u o u s n-homogeneous F-valued p o l y n o m i a l s o n E ) s p a n n e d @' (0"
b : I $ E E ' , b e F ) . When F
k i s t h e scalar f i e l d , C ( E ; F )
will
.
APPROXIMATION OF DIFFERENTIABLE FUNCTION?
be d e n o t e d
Ck(E)
, Qf (E;F)
by
Pf ( E ) ,
3
e t c . Throughout, w e
re-
s t r i c t o u r a t t e n t i o n t o s c a l a r v a l u e d f u n c t i o n s d e f i n e d on E l although t h e obvious extensions t o t h e vector-valued case t o f u n c t i o n s d e f i n e d on open s u b s e t s of
E
and
a l s o hold.
Much of t h e work o u t l i n e d h e r e was done w h i l e t h e a u t h o r was v i s i t i n g t h e I n s t i t u t o d e Matemstica of t h e U n i v e r s i d a d e F e d e r a l do Rio d e J a n e i r o , and i t i s a p l e a s u r e t o t h a n k t h e i n -
s t i t u t e f o r i t s h o s p i t a l i t y . I n addition, t h e author is
espe-
c i a l l y g r a t e f u l t o P r o f e s s o r J O ~ OBosco P r o l l a f o r many
fruit
f u l d i s c u s s i o n s c o n c e r n i n g t h i s a r e a of a p p r o x i m a t i o n t h e o r y The author i s a l s o g r a t e f u l t o Sean Dineen, P a u l Berner,and
.
Ray
Ryan f o r some h e l p f u l c o n v e r s a t i o n s . S E C T I O N 1.
I n [15], function
f
wells showed t h a t a c o n t i n u o u s l y d i f f e r e n t i a b l e co =
d e f i n e d on
( 5 C ( N (,J{m)))
which h a s a u n i f o r m
l y c o n t i n u o u s d e r i v a t i v e must have unbounded s u p p o r t (or be ide; t i c a l l y z e r o ) . I n t h i s s e c t i o n , w e o u t l i n e an extension o f t h i s
r e s u l t t o f u n c t i o n s d e f i n e d on C ( K ) f o r some compact H a u s d o r f f s p a c e K . A s a consequence, w e show t h a t i t i s i m p o s s i b l e to a2 p r o x i m a t e a n a r b i t r a r y C 2 f u n c t i o n by a C 3 f u n c t i o n i n the f i ne t o p o l o g y of o r d e r
2 ( c f [6,10,17]). I n c o n t r a s t t o t h i s , i n
S e c t i o n 3 , w e n o t e t h a t f o r d i s p e r s e d compact s e t s K approximate a r b i t r a r y nomials i n space
-rk
PROPOSITION
1.1:
dohmly con-tinuoun,
K
let
one can
Ck f u n c t i o n s on C(K) by f i n i t e type p l y
. Throughout
C ( X ) , where
I
t h i s s e c t i o n , E w i l l b e t h e Banach
i s compact, Hausdorff and i n f i n i t e .
G E c ~ ( E ) ,G
the ~
p
Gt
3 0. z d ;ZG:
id unbounded.
E
+
E'
i d
un&
R. ARON
4
The p r o o f of t h i s r e s u l t i s m o d e l l e d on t h a t o f Wells and r e q u i r e s t h e f o l l o w i n g lemmas.
11
h
11
y
:
[O,l]
h E E, xl,...
Let
LEMMA 1 . 2 .
= Ih(xi)
*
( i = 2 , . . .,k
I
=
E
b u c h Xhat
,t
E
x1
E
extend g
to K
11 Define
11
g
*
B : iO,l]
=
and
E
u? ,
,
path
y ( t ) ( x j ) = h(xi)
y ( l ) (x,)=-h(xl).
b e open s u b s e t s o f K s u c h
q n T = + .W e
and
3
t o be 1 o n
and -1
on
(where 1
is
i n s u c h a way a t h a t
= Sup
{I
g(X)
I
:
x
E
K) = 1.
B ( t ) = (1- t ) l + t g
by
E
the constant function) verify that y
11
and V 2
V2,
d e f i n e a f u n c t i o n g on
5 , and
y(t)
L e t V1
x2
E V1,
11
[0,1]), y ( 0 ) = h , a n d
PROOF OF LEMMA 1 . 2 : that
Then t h e h e e x i n t o a
(i=l,...,k).
E
dixed pVintb, buch t h a t
Xk E K
and s e t
,
y ( t ) = B(t)h. I t is easy
has t h e r e q u i r e d p r o p e r t i e s .
to
Q.E.D.
The p r o o f of Lemma 1 . 3 i s t r i v i a l , and w e p r o c e e d now t o s k e t c h a p r o o f of p r o p o s i t i o n 1.1. I f t h e p r o p o s i t i o n i s f a l s e , then there i s a
1 1 1. 1,
11
f
E
= l / n , w e have
Let
and
s u c h t h a t G ( 0 ) = 1, G ( f ) = 0
G E C1(E)
dG
11
i s u n i f o r m l y c o n t i n u o u s . Thus
iG(f)
-
Bo = { Xl r . . . , p+ ~ 1} C K
iG(g)
11
for
< 1 / 2 whenever IIf - g l l
for some <
E.
be a r b i t r a r y , and l e t ho=O E E
.
APPROXIMATION O F DIFFERENTIABLE FUNCTIONS
g1 E E, g1 (XI
Let
wise,
.
E
i G ( h o ) (9,)
and
= 0 , s e t hl = gl. 0 t h ~
Z G (h,) (9,)
If
are o f o p p o s i t e s i g n . Ap-
iG(ho) (-gl)
p l y i n g Lemmas 1.2 and 1 . 3 , t h e r e i s hl
and a s u b s e t B I C B o
E E
c o n s i s t i n g o f a l l b u t p o s s i b l y o n e p o i n t xi
11
hl
[I
card
=
E
I
= Ihl(xi)
( i E B1)
and l e t g 2
S = 2"-l,
+
be such t h a t g2(xi)
h l ) ( h l ) = 0 , s e t h 2 = hl.
and e i t h e r
+
iG(ho
or
hl) (9,)
such t h a t
iG(ho) (hl) = 0 . Let S C B l ,
11
(i E S), and
(i E S), g 2 ( x i ) = - h l ( x i ) iG(ho
and
E
E
5
g211 =
Otherwise, iG(ho
+
hl)
If
E.
+
dG(ho (
- g2)
hl(xi)
=
hl) ( h l )
a r e o f oph2 E E
p o s i t e s i g n . Applying Lemmas 1 . 2 and 1 . 3 , t h e r e e x i s t s
11
such t h a t
11
h2
=
,
E
f o r a l l i i n a s e t B2
zG(ho
+
of c a r d i n a l i t y a t l e a s t
.
ing, we g e t functions
-,
of c a r d i n a l i t y 3
+ hl + 3
iG(ho
n i t e sequence
II
fj
hi
/I
hi) ( h i + l )
=
E
(i = 1,. . . , n )
,
2 1) , and = 0 ( i= 0 ,
... , n - l ) .
3
-
fj-1
1
=
II
,
I1 h j II
=
since
n
I
11
( x E B n , i ,j
. .. +
Coptinu
C hi ( j = O , l , . . . , n ) , w e see t h a t t h e f i i=O ( f . ) s a t i s f i e s t h e following conditions:
f j =
/I f n 11
Therefore,
E
2I-l.
E and sets B.3B23.. -JJHBi
,hn ho,hl,.. 2n-i+l , such t h a t
h i ( x ) = h . ( x ) and o f modulus
Thus , s e t t i n g
h l ) ( h 2 ) = 0 , and h 2 ( x i ) = h l ( x i )
G(fn)
-
G(fo)
I
=
E
,
/ f n ( x )I = 1
for
x E B~
,
6
AWN
l?.
denotes
all
= 1, which
is
by t h e Lagrange mean v a l u e theorem (where [ a , b ] p o i n t s on t h e l i n e segment from a n
z
-<
1/2
E
j-1 We c o n c l u d e t h a t I G ( f n ) I a contradiction.
COROLLARY 1 . 4 .
to b ) ,
2
= 1/2.
1/2,
although
L e t p b e a pooitive,conLLnuoud,
[10,17II).
(cf.
lF(f)
,F e
- G(f)
/I
Q.E.D.
bounded d u n c t i o n o n E b u c h t h a t p ( f ) F E C2(E)
IIfn
Then thefie exiot.:, ma
C3(E).
and
< p(f)
ad
0
.+
-
lla2F(f)
i2G(f)
11
/I
Let th&
< p ( f ) , 6otl a l l f E E. C3
~un&on
2.
Suppose t h a t s u c h a G e x i s t s ; w i t h o u t l o s s o f g e n e r a l
PROOF: ity,
w e may s u p p o s e t h a t
be a
Cm
I
IF(0)
-
G(O)
I
= a > 0.
Let $ : R
-+
R
f u n c t i o n such t h a t
$ (t)=
0
if
It1 5 a/2
1
if
It/ ? a
and c o n s i d e r t h e f u n c t i o n
H = $
0
(F
,
-
: E -+ R . Then&(f) =
G)
- G) ( f j i ( F - G) ( f ) and z 2 H ( f ) = $"[F - G ) ( f i ( i ( F - G ) ( f ) ) * @'((F - G ) ( f ) ) z 2 ( F - G ) ( f ) . Hence 1122H(f) 11 2 MC + [ $ " ( ( F - G I ( f ) ) l l l i ( F - G I (f) 11'
$@
+
.
G E C 3 ( ~ ) buch
T h a t i d , it ib i m p o o n i b l e t o a p p t o x i m a t e F b y a
i n ,the d i n e t o p o l o g y o d o a d e t
f / [ +m
where
I
M = sup { 11i2(F
C = Sup { I $ ' ( t )
for a l l
f
For IIflI
5 r,
E
E,
I
-
G) ( f )
11
.
f E El <
m
and
I $ " ( t )I: t E i . For saner>OIl(F-G)(f)I
r , so t h a t $ " ( (F - G ) ( f ) ) = 0 for [If 11 > r . I
an a p p l i c a t i o n of t h e Lagrange mean v a l u e theorem
I)<
shows t h a t I[ i ( F - G ) ( f ) - Ilii (F-G)(O) 1) + r M , so t h a t sup(11~2H(f))l;fEEl
7
APPROXIMATION OF DIFFERENTIABLE FUNCTIONS t h e uniform c o n t i n u i t y of
aH.
By P r o p o s i t i o n 1.1, H m u s t e i t h e r have unbounded s u p p o r t or
H
5
But s i n c e p ( f )
0.
)If
)I
G
does n o t e x i s t . Q.E.D.
+ m ,
SO H Z
0 a s IIf
-t
11
-+ m
G) ( f ) I + O
, I (F -
as
0 , However, H ( 0 ) = 1, a c o n t r a d i c t i o n , s o that
W e remark t h a t i f
K
i s compact, m e t r i z a b l e and uncount-
a b l e , t h e n t h e above P r o p o s i t i o n and C o r o l l a r y a r e t r i v i a l and
i n f a c t much weaker
[16,17] ) . I n f a c t , as Wulbert n o t e s , dens C(K)
ample, Thus,
t h a t what i s a l r e a d y known (see, f o r e x -
if
i s uncountable, dens C ( K )
K
G E C2(C(K))
F E C2 ( K ) )
F E C1(C(K)),
such t h a t
IF(€)
-
G(f)l
= card X.
> d e n s C ( K ) ,so that there
F E C'(C(K))
does n o t exist a n o n - t r i v i a l Consequently, i f
'
'
+
0
w i t h bounded support.
,
t h e r e dct.s not e x i s t
as
IIflI
.
+
SECTION 2 .
Throughout t h i s and t h e s u b s e q u e n t s e c t i o n s , E w i l l den o t e a r e a l Banach s p a c e . I n t h i s s e c t i o n , w e o u t l i n e a
proof
of t h e following result.
PROPOSITION 2 . 1 :
k
k
(C ( E ) , T ~ hao ) the
borne ( h e n c e doh a L L ) k 1. 1
apphoximation p a p m y doh
id a n d o n L q .id E hao t h e appoxhns
t i o n phopehtq. W e n o t e t h a t t h i s r e s u l t complements a n d , i n a sense,com p l e t e s t h e analogy f i r s t n o t i c e d by Llavona Guerreiro
[13
]
[8]
w i t h t h e holomorphic s i t u a t i o n
p l e x c a s e , one p r o v e s t h a t p e r t y i f and o n l y if E i n g t h e fact that the
E
(H (E)
, ):T
and Prolla and [2
1.
I n the c~m-
h a s t h e approximation pro-
h a s t h e a p p r o x i m a t i o n p r o p e r t y by
prcduct
(H(E)
,
T):
E
F 2 (H(E;F),
0
TC)
usfor
8
R. ARON
any complex Banach s p a c e F and t h e n u s i n g a
characterization
of t h e a p p r o x i m a t i o n p r o p e r t y i n terms of t h e
E
(see,
-product
. However,
i n t h e r e a l case, a s i m i l a r product (Ck ( E ) , Tk ~ )a r e unknown. Thus, f o r m u l a and t h e c o m p l e t e n e s s of
f o r example
[2] )
o u r approach t o p r o v i n g t h e a s s e r t i o n must b e c o m p l e t e l y
dif-
f e r e n t ; i t w i l l , i n f a c t , be more d i r e c t t h a n t h e E-product a2 gument. I t s h o u l d be n o t e d t h a t i n 1 4 1 s t u d i e d t h e topology
,
Bombal and Llavona have
on s p a c e s o f Hadamard d i f f e r e n t i a b l e
T:
f u n c t i o n s , and have proved t h e c o m p l e t e n e s s of t h i s s p a c e
as
w e l l as v a r i o u s a p p r o x i m a t i o n r e s u l t s . Using t h i s , Bombal
131
o b t a i n e d P r o p o s i t i o n 2 . 1 by a d i f f e r e n t method.
W e prove t h e r e s u l t f o r k E N , only k minor m o d i f i c a t i o n s b e i n g n e c e s s a r y f o r k = m . If (Ck(E) , T ~
PROOF OF PROPOSITION 2 . 1 :
1,define n : $(E)
h a s t h e a p p r o x i m a t i o n p r o p e r t y f o r some k by
TI
duced sup
c
of
E'
( f ) = i f ( 0 ) , Note t h a t i f w e r e g a r d E I C Ck ( E ) with the i"
k
topology, then
T~
~ ; J ( I T ~ (XI )
where K
fore,
+
(y)
I
TI
is a continuous p r o j e c t i o n , s i n c e
: (x,y) E K x L, j
5 kl
= sup(
and L a r e compact s u b s e t s o f
E
lif(0)(y) I
: y E L1
and L1 = K
u
1
I
L.There
E' w i t h t h e i n d u c e d t o p o l o g y i s a complemented s u b s p a c e
Ck ( E ) and hence h a s t h e a p p r o x i m a t i o n p r o p e r t y .
However,
since t h e induced t o p o l o g y on E' i s j u s t t h e compact open t o pology, i t follows t h a t E has t h e approximation p r o p e r t y . To p r o v e t h e c o n v e r s e , l e t
17-1 C (Ck ( E l , . Given
and l e t K , L be compact s u b s e t s o f
E
t h a t t h e r e is a
(x,y)
with
IIx'
-
x
11
[ d J f C x )( y )
6 > 0 such t h a t i f < 6
-
,
I[y'
-
y
11
< 6
:jf(x') (y') 1 <
E
T
,
&
k
~
b) e precarrpact > 0,
E
we claim
and x ' , y ' E E
K
x
L
,
j
5 k)
then
(f F a
("1.
)
9
APPROXIMATION OF DIFFERENTIABLE FUNCTIONS
I n f a c t , i f t h i s i s f a l s e , t h e n f o r some and (y;)
s e q u e n c e s (x;)
& such
s p e c t i v e l y , and ( f n ) i n
II yn - YA I/
< l/n,
- , K2
K1 = (x,)
and
, (x,)
in E
,
L1 =
5
in K
and (y,)
exist
k, there
- i Jf n ( x J
w,
( y ~1 )2
E
and L2 =
re-
and L
t h a t f o r a l l n E N,//xn-x;
/dJfn(xn)(yn)
=
j
. are
/ / < lh, Now ,
compact
i n E , so t h a t t h e seminorm p d e f i n e d by
{I
p ( f ) = sup
k
is
-
and
~ / 2 ,c o n t r a d i c t i n g t h e p r e c o m p a c t n e s s o f
fm)
n
,
zf.
Thus
- x 11
< 6.
holds.
(*)
Let
T E El 0 E
&IT(E)
of functions i n
s i o n a l space T ( E ) . S i n c e k
C (T(E))
, which
(y)
KUL, llTx
& restricted
;k / T ( E )
t o t h e f i n i t e dimen-
i s a precompact s u b s e t of
has t h e approximation property, t h e r e e x i s t s a
such t h a t f o r a l l /&(XI
E
@ ( f )= f o T. C o n s i d e r now the far&
c o n t i n u o u s l i n e a r mapping of f i n i t e r a n k
sup I
x
such t h a t f o r a l l
I$ : Ck ( E ) -+ Ck ( E ) by
Define ly
d’f(xA) (yA)l : n E N 1
c o n t i n u o u s . However, f o r i n f i n i t e l y many m
T~
p (fn
-
d J f ( x n ) (y,)
g E
& I T (El
- ;iJ+(g)(x) ( y ~/ :
$ : C k ( T ( E ) ) + Ck(T(E))
I
(x,y) E T(K)
F i n a l l y , t h e mapping taking f E $(El
x
T(L) , j
into $(f
i s f i n i t e r a n k , l i n e a r and c o n t i n u o u s , and i f f E&,
k~ <
(**I.
E
) o T
E
(x,y) E K x L,
then
- 2’ ( $ ( f I T ( E ) 1 o TI ( X I ( y ) I( - 21(f o T) (x) (y) I + 1 dJ(f o 3’) (XI (y) - 21 (f IT I
/ a J f(x) (y)
d J f ( x ) (y)
+ / d J ( f / T ( E )o TI (x) (y) By ( * ) and ( * * ) , t h e middle t e r m i s 0 p r o p e r t y f o r any
k.
.
o T) (XI (y)
/ +
- 2 J ( $ ( f / T ( E )o T) (XI ( y ) ] .
t h e f i r s t and t h i r d terms a r e < E w h i l e k k Thus, ( C ( E ) , T ~ )h a s t h e a p p r o x i m a t i o n
Q.E.D.
10
R. ARON We remark t h a t t h e p r o o f o f t h e above i m p l i c a t i o n m e r e l y
employs t h e f a c t t h a t i f E h a s t h e a p p r o x i m a t i o n property,then k e v e r y e l e m e n t o f C (El can be approximated i n t h e T: t o p o l o g y
1:12]),
by f i n i t e t y p e p o l y n o m i a l s ( c f
a function i n
and t h a t t h e p a s s a g e of
t o a polynomial which a p p r o x i m a t e s it can
Ck(E)
be accomplished i n a c o n t i n u o u s , l i n e a r manner.
SECTION 3 . I n t h i s and t h e n e x t s e c t i o n , w e w i l l s t a t e , m o s t l y
With
o u t p r o o f , some r e s u l t s on a p p r o x i m a t i o n by f i n i t e t y p e p o l y n g k k and ' I ~ W e w i l l f r e q u e n t l y find mials f o r t h e topologies ' I ~
.
n e c e s s a r y t h e f o l l o w i n g h y p o t h e s i s on t h e Banach s p a c e E
,
the
f i r s t two c o n d i t i o n s of which a r e f o r m a l l y s t r o n g e r thars the a s sumption t h a t E' h a s t h e bounded a p p r o x i m a t i o n p r o p e r t y . For some c o n s t a n t
I $ ~ , . . , ,E+ E~' (k E N) , t h e r e e x i s t s a sequence
For e a c h
(n,)
E
El 0 E
C > 0 , t h e following holds:
satisfying
(i)
I[
(ii)
9, o
nn
11 5
+
11.
(iii) n n ( x )
-+
( n E N).
C I$,
x
in
E'
( i= 1,.
. . , k ) , and
(x E E ) .
C o n d i t i o n (+) i s c e r t a i n l y a weaker a s s u m p t i o n t h a n f o l l o w i n g , which i s found i n There e x i s t s a sequence linear projections satisfying
in
9 5 p n - + 9
E'
1141 ( c f 171 1 . (P,l) of f i n i t e r a n k Pn(x) ($ E
-+
s a t i s f i e s (+)
.
continuous
x ( x E E ) and
(++I
E).
I n f a c t , c o n d i t i o n !++) i m p l i e s t h a t E '
L1
the
i s separable,
while
11
APPROXIMATION OF DIFFERENTIABLE FUNCTIONS
k
k
c0mpLete.
(C ( E l , T ~ ) LO
PROPOSITION 3.1.
k = l , t h e proof f o r k > 1 being
W e s k e t c h t h e proof f o r
similar. L e t (fa)
(C1( E ) ,
be a Cauchy n e t i n
A
1
T ~ ) .
is
It
e a s y t o see t h a t t h e r e a r e c o n t i n u o u s f u n c t i o n s f : E + R
g : E
f = lim f
such t h a t
+ E'
x
d f = g . I n f a c t , i f t h i s f a i l s , then there
and a s e q u e n c e (h,)
E E,
If(x+hn) - f ( x ) -g(x)(hn)I > and e a c h
1)
h,
[I
n E N,
sup{Ilsfa(z)
u [x,
that
a,B >
-
fa(x)
-
zfB(z)
11
:
z
un
E
f B ( x + hn)
-
-
fa(x)
f B ( x + hn)
a0 E A, so t h a t
some
-
f B ( x )1 5
a o . Also, f o r a > scane
a b e any i n d e x l a r g e r t h a n b o t h
. Therefore,
0
I f ( x + hn)
< If(x +
+ I f a (X < 3
-
f(x1
-
hn)
+ hn)
II hn I1
E
-
[ f a ( x t hn)
such t h a t n 2 n
-
f(x)
-
for
fa(X)
-
fa(x)
cto
and
all
E.
al,
there i s
<
/ih,[I f o r
E
I -<
-
afa (XI
(h,)
Ckc ( E ; F )
fa(x)I +
I
+ I;fa(x) (hn) -g(x)(hJ
I<
Q.E.D.
t o b e t h e subspace of dC(E;F)
of a l l f u n c t i o n s g s u c h t h a t f o r e a c h
--
1
d f a ( x ) (h,)
f a ( x + h,)
-
for
I
W e d e f i n e t h e space
iJg(x) E
Noting
,
n 2 no
g ( x ) (h,)
-
/ / h n 11
E
a c o n t r a d i c t i o n , which e s t a b l i s h e s t h e r e s u l t .
-
I 5
[/g(x) - dfa(x)([<
a1 & A ,
Letting
all
fB(x)
( f a ( x + h n )-f,(x) -f(x+\h)-f(x)l(EllhJl
a
0
-
[ x , x + h, 1 1 .
for a l l
n
0 , such t h a t
-+
x + h, J i s compact, i t f o l l o w s t h a t
n
l f a ( x + hn)
-
+
i n E , hn
3 IIhn]I ~ (nEN).Now, f o r e a c h a , B E A
hn)
lfa(x
is differell
-
t i a b l e with d e r i v a t i v e E > 0,
u n i f o r m l y on
E . I t s u f f i c e s t o show t h a t f
compact s u b s e t s o f
exist
g = l i m ;fa
and
a
and
j
5 k and
x
P f ( J E ; F ) ; a s b e f o r e , when F is t h e s c a l a r
E E,
field,
R. ARON
12
Ckc ( E ) .
Ck,(E;F) i s d e n o t e d by
k
tine t o verify that
Cc(E)
f o r e v e r y Banach s p a c e E ; for
,
E = C(K)
1 1 i s complete. N o t e t h a t Cc(E) = C (El k k i n [I] , w e n o t e t h a t C c ( E ) = C ( E )
i s a d i s p e r s e d c o m p a c t , Hausdorff s p a c e , k k Cc(co) = C (CJ When E i s a r e a l s e p a r a b l e
where K
.
and i n p a r t i c u l a r
H i l b e r t s p a c e , Lesmes
in
Using P r o p o s i t i o n 3.1, i t i s r o u
[7]
proved t h a t
Qf ( E )
1
is
T~
dense
k
,k
2. I n
C ( E ) and n o t e d t h a t t h e r e s u l t f a i l s f o r C ( E )
f a c t , we have t h e f o l l o w i n g .
PROPOSITION 3.2. k
1,
Let
in
Qf(E)
T~k
Assuming (++) F,
k
,T
(Cc(E)
k
~
E )
P r o p o s i t i o n 3.2
k
(Cc(E)
suming ( + ) ,
,
T
,
Then
P r o l l a h a s n o t e d t h a t f o r any
all
6011
Banach space
k F = C c ( E ; F ) . Using t h e v e c t o r v a l u e d form o f
(namely, t h a t
he c o n c l u d e s t h a t that
batiddy C o n d i t i o n ( + ) . denne i n Ckc ( E ) .
E
k
Cc(E)
pf(E;F)
is
T~ U
k
k
dense i n Cc(E;F)),
(Cc(E;F) ,
0 F i s dense i n
T
k
~
,)
so
k ~ h) a s t h e a p p r o x i m a t i o n p r o p e r t y . I n f a c t , a s -
o n e can show t h i s d i r e c t l y by n o t i n g , a s i n
s i t i o n 2.1, t h a t t h e p a s s a g e o f a f u n c t i o n i n
Prop2 fi-
Ck(E) t o a
n i t e t y p e p o l y n o m i a l a p p r o x i m a t i n g i t c a n b e done i n a c o n t i n u o u s l i n e a r manner. Summarizing, w e have t h e f o l l o w i n g .
PROPOSITION 3.3.
76 E'
han p t a p e t t y
(+)
,
then
k
(Cc(E)
k han t h e a p p h o x i m a t i a n p t o p e t t y . Conwehnely, id ( C c ( E )
,
T
k
k
I
TU)
~
)ha6
t h e a p p t o x i m a t i o n p t o p e t t y , t h e n no d o e n i t n complemented n u b -
Apace
E'.
SECTION 4. In
114
1 , Restrepo,
c o n s i d e r i n g a r e f l e x i v e Banach s p a c e
E which s a t i s f i e s c o n d i t i o n ( + + ) , s t u d i e d t h e
completion
of
APPROXIMATIUlJ OF DIFFERENTIABLE FUNCTIONS
t h e f i n i t e type polynomials
13
1
f o r t h e topology
Qf(E)
Tb
g E C 1 ( E ) i s weakly r:ontinuous on bounded
found t h a t i f
. He sets
and u n i f o r m l y ( o n e ) d i f f e r e n t i a b l e on bounded s e t s , t h e n g is 1
a
' I ~l
(+)
,
i m i t of e l e m e n t s of
we discuss the A function
Qf ( E ) .
H e r e , under t h e assumption
c o m p l e t i o n of
T:
Qf ( E )
for a l l
1. 0 .
k
and F is said
g between two Banach s p a c e s E
t o be weakly c o n t i n u o u s on bounded s e t s i f f o r a l l M > 0 , and e a c h
I] x 1 1 -< M I t h e r e e x i s t 6 > 0 and $l,...,I$kE E' y E E l 1) y 11 < M , and / @ i ( x- y ) 1 < 6 (i=l,...lk), - g ( y ) I < E . The f u n c t i o n i s u n i f o r m l y weakly m-
lg(x)
t i n u o u s on bounded sets i f , i n t h e above d e f i n i t i o n ,
o1
.. .
can be chosen i n d e p e n d e n t of
11
x,
x
11
6
and
< M . Finally,
g i s u n i f o r m l y d i f f e r e n t i a b l e of o r d e r n on bounded s e t s for a l l all
M > 0
and a l l
E
+
E
> 0, there is
6 > 0
if
such t h a t f o r
1lxllLM1 IIh11(6,
x , h E E , Ig(x
If
> 0,
x E El
s u c h t h a t if then
E
h)
-
g(x)
-
i g ( x ) (h)
-
... - 'ng(x) n.
(h)l 'E)I
h
Iln
.
is r e f l e x i v e , t h e n t h e weak compactness of t h e b a l l of E
i m p l i e s t h a t a f u n c t i o n which i s weakly c o n t i n u o u s on
sets i s u n i f o r m l y weakly c o n t i n u o u s on bounded s e t s .
bounded Further,
w e have t h e f o l l o w i n g . PROPOSITION 4 . 1 .
d u c h 2haA: g
Zg,..
bounded
16 g
Zng
id
betd.
[ 1 4 , Theorem
(cf
.,in-'g id
61).
Let
g : E
g
be
c o n t i n u v u s o n baunded
ohdeh n,then
06
be.td.
W e d o n o t know i f t h e c o n v e r s e h o l d s , even i n
n = 1. That i s , i f
F
ahe u n i d o h m l y loeakly c o n t i n u o u d on
u n i ~ o h m l yd i d d e h e n t i a b l e
u n i d o h m l y Ltleakly
-f
and d g a r e u n i f o r m l y weakly
bounded s e t s , i s g u n i f o r m l y d i f f e r e n t i a b l e
of
the
case
continuous on
order
1 on
I<. ARON
14
bounded s e t s ? Note t h a t t h e s p a c e CEU(E;F), of functions g:E+ F such t h a t
, z g , .. . ,z k g
g
a r e u n i f o r m l y weakly c o n t i n u o u s
bounded s e t s , i s c o m p l e t e when endowed w i t h t h e
w e d o n o t know if
{ g E Ck(E;F) : g
topology;
' kI ~
i s u n i f o r m l y weakly
t i n u o u s and u n i f o r m l y d i f f e r e n t i a b l e of o r d e r <- k I k when endowed w i t h t h e T~ topology.
t i o n (+I. zjg(x)
( c f r14
4.2.
PROPOSITION
Then t h e b e t
- (x Pf
E E,
('3)
E
,
Theorem 81 )
5 k) i n t h e
j
con-
is canplete
.
dunction~
06
on
Let E n a t i n d y c o n d i g E Ckw ( E ) Auch t h a t completion
T !
Qf(E).
06
W e p r o v e P r o p o s i t i o n 4.2 i n t h e c a s e k = O w i t h o u t assum
t h a t i s , w e show t h a t f o r any Banach
ing c o n d i t i o n (+);
space
i s u n i f o r m l y weakly c o n t i n u o u s on boundedsets,
E, i f g : E + R
t h e n g i s a uniform l i m i t (on bounded s e t s ) of f i n i t e ( c f [14
polynomials
o f P r o p o s i t i o n 4.2
to
F
.
31 1 . The p r o o f of
the general case
makes u s e of f o l l o w i n g .
L e t T be a ( n o t n e c e n n a h i l y l i n e a k l mapping
LEMMA 4 . 3 .
E
, Theorem
compact o u b n e t n
To show P r o p o s i t i o n 4 . 2 i n t h e case
k=O, Let
be u n i f o r m l y weakly c o n t i n u o u s on bounded s e t s , l e t
bounded s u b s e t of E
. . . ,$,
then
E E
Ig(x)
with
-
xl,...,xn sup
I
,
,
and l e t
w e have
g(y)
I
<
E.
I$ ( x ) = ( $ , ( x )
R k f where ist
dhom
16 T i d u n i 6 o h m l y w e a k l y c o n t i n u o u n on bounded oets,
t h e n T mapb b a l l o i n E t o h e l a t i v e l y
I$,,
type
{I
that i f
> 0 . For some
I $ I ~ ( x - y) I< 6
-
g :E +R B
,.
.. f I $ k ( x ) . By
I$ f ( xI )
I
: i =
be
a
6 > 0
and
...
(x,y E B, i=l, ,k),
Consider t h e s u b s e t {$(XI : x
such t h a t f o r a l l
E B
$i(x)
E
F.
06
E B}
in
compactness, t h e r e exx E B , t h e r e i s some x
l,... ,k) < 6 / 2 .
j Now,defining
APPROXIMATION OF DIFFERENTIABLE FUNCTIONS
U j = {(Y1r.-.ryk) E Rk : Iyi
.. ,nr let
j = 1,.
-
oi(xj) I < 6
15
(i= l,...,k)}
for
.
cl,.. ,cn be continuous functions from Rk to
R such that k rn), cj(y) 2 0 (y E R r j = l , n E cj(y) = 1 (y = $ ( X,I for some x
...
i) ii)
E
j=1
iii) spt c
j
c Uj
B) ,
(j=l,...,n).
Define
Finally, there exists a polynomial p : Rk
-f
R
such that
REFERENCES
[ 1 1 R. M. ARON
-
Compact polynomials and compact differen-
tiable mappings between Banach spacesfto appear.
[ 2
1
R.
M. ARON'AND R. M. SCHOTTENLOHER
-
Compact holanorphic
mappings on Banach spaces and the
approximation
property, to appear in J. Funct. Anal.
[
3
] F. BOMBAL GORDON
-
Differentiable function spaces
with
the approximation property, to appear.
[
4
] F. BOMBAL GORDON AND J. L. G. LLAVONA
-
La propiedad
de
aproximaci6n en espacios de functions diferencia bles, to appear.
R. ARON
16
151
S. DINEEN
-
Runge domains in Banach spaces, Pr0C.R.I.A.
,
71, Sect. A, nQ 7(1971).
[ 6 ] J. KURZWEIL
-
On approximation in real Banach spacesfst:
dia Math. 14 (1954), 214 - 231.
[ 7]
J. LESMES
-
On the approximation of continuously differ+
tiable functions in Hilbert spaces,Rev. Colanbiana de Matemdticas 8, (1974), 2.17 - 223.
[ 8 ] J. L. G. LLAVONA
-
Aproximacicn de funciones diferencia-
bles, Thesis, Universidad Complutense, Madrid.
[
9
1
-
C. MATYSZCZYK
Approximation of analytic and continuous
mappings by polynomials in Frechet spaces, to a2 pear in Studia Math.
[ 101 N. MOULIS
-
Approximation de functions differentiables sur
certains espaces de Banach, Ann. Inst. Fourier 21 (1971), 293
[ 113 Ph. NOVERRAZ
-
- 345.
Pseudo- convexite, convexite polynomiale,
et domaines d'holomorphie en dimension infinie, Mathematics Studies 3, North Holland (1973).
[ 121 J. B. PROLLA
-
On polynomial algebras of continuously
dif
ferentiable functions, Rend. dell'Accad. dei Lincei,
to appear.
[ 131 J. B. PROLLA AND C. S. GUERREIRO
-
An extension of Nachbin's
theorem to differentiable functions
on
Banach
spaces with the approximation property, to appear.
[ 1 4 1 G. RESTREPO
-
An infinite dimensional version of a theo-
rem of Bernstein, Proc. A.M.S. 23 (19691, 193-198.
17
APPROXIMATION OF DIi~PEI'SZiJ'rIABLEFUNCTIONS
-
[ 1 5 ] J. WELLS
Differentiable functions on co, Bull,
A.M.S.
75 (1969), 117 - 118.
[ 1 6 1 J. H. M. WHITFIELD
-
Differentiable functions with bound
ed nonempty support on Banach spaces, Bul1.A.M.S. 72 (1966), 145
[ 1 7 1 D. WULBERT
-
- 146.
Approximation by Ck
- functions in
approxima
tion Theory, Proc. of Intern. Symp., G. G. Lorentz (ed.) , Acd. Press (1973), 217
- 239.
School of Mathematics, 39 Trinity College,
Dublin 2 , Ireland.
ADDED IN PROOF:
4.1 all
The questions raised after Proposition
have affirmative answers. In addition, W. B. Johnson has pointed out that condition
(+)
is equivalent to E' having the
bounded
approximation property. A much fuller investigation of the completion of spaces of polynomials, containing the material cussed in Sections 3 and 4 of this paper as well as
dis-
the above
points, is contained in a joint paper by the author and
J. B.
Prolla, "Polynomial Approximation of Differentiable Functionson Banach Spaces", to appear.
This Page Intentionally Left Blank
Infinite Dimensional Holomorphy and Applications, M a t 5 (ed.) @ North-Holland Publishing Company, 1977
PONCTIONS MEROMORPEIES SUR CA
par V O L K E R A l l R l C H
INTRODUCTION Toute fonction holomorphe sur un domaine 6ta16
p:X+
C
A
013 A est un ensemble arhitraire se factorise 5 travers un doma&
ne de dimension finie ( 5 savoir son domaine d'existence).
Cela
reste vrai pour toute fonction mdromorphe. En utilisant des r6sultats en dimensions
finies on obtient que toute fonction mg
romorphe sur un domaine dta16 au-dessus de CA est le
quotient
de deux fonctions holomorphes et se prolonne 5 l'enveloppe d'hg lomorphie. Donc il suffit d'btudier les fonctions
m6romorphes
sur un domaine de Stein. On sait qu'un domaine de Stein au-dessus de C h est isomorphe 5
S x CA-'
03 'Y C A est fini et
un domaine de Stein 6ta 6 au-dessus de trons que l'espace S x CA-'
%CS
est la limite
x
C
Y
S
([l] , [3] 1 . Nous d&oF
des fonctions m6romorphes
nductive des n(S
est
CO-'),
4, 3
Y
sur fini.
Un thbort5me analogue pour les donn6es de Cousin n'est pas vrai. Dans [ 4 ] DINEEN a ddmontrg que s u r tout ouvert de C m
il existe
une donnde de Cousin I non rdsoluble. Cependant, une
propri6tb
de factorisation pour certaines classes de donn6es de CousinpeL met d'6noncer des conditions ndcessaires et suffisantes 19
pour
q u ' u n e donnbe d e Cousin s u r un domaine d e S t e i n s o i t r 6 s o l u b l e . La n o t a t i o n e s t p o u r l a p l u p a r t l a m e m e q u e c e l l e
dam
[I]. II d b s i g n e t o u j o u r s un ensemble e t F i n A e s t l ' e n s e m b l e d e s sousensembles f i n i s d e A .
S i p:X
-+
C A e s t un domaine 6 t a l b
ou simplement
0
1. PROPRIETES
FONDAMENTALES DES FONCTIONS 14t?R3"HEs SUR @'
Ux
e s t l e f a i s c e a u d e s f o n c t i o n s holomorphes.
S o i t p:X * CA un domaine Q t a l 6 . Pour t o u t o u v e r t
s o i t M ( U ) l'anneau t o t a l des f r a c t i o n s de @ ( U ) .
U C X
M e s t un
prG-
f a i s c e a u s u r X. L e f a i s c e a u a s s o c i 6 e s t a p p e l g l e f a i s c e a u f o n c t i o n s mbromorphes s u r X e t n o t 6
Vx ou
f o n c t i o n m6romorphe s u r X I c ' e s t - G - d i r e
simplement
@.
toute section m
f i , f i ) iEI
p e u t Ztre r e p r e s e n t g e p a r une f a m i l l e ( U i
e s t un recouvrement o u v e r t d e X e t f i , f i
E
@(Ui)
oC
des
Toute
n(X)
E
(Ui)
t e l s que
l'in
Uin
t b r i e u r d e { f i = 0) s o i t v i d e e t f
= f . f . sur U i j 1 1 j. p e u v e n t Ztre c h o i s i s comme d e s polydisques I n v e r s e m e n t ,
Les
t e l l e f a m i l l e d b t e r m i n e une f o n c t i o n mbromorphe. Evidemment,
Ux
Ui
p e u t Gtre c o n s i d b r 6 comme un s o u s f a i s c e a u d e x E X
qxIx est
le corps des f r a c t i o n s de
A chaque m E % ( X I
qx.Pour
une
chaque
ox,,.
on a s s o c i e une f o n c t i o n :
{y E X : rn E axry1. Parce q u e x E D~ s i e t s e u l e m e n t Y s i i l s e x i s t e n t une f a m i l l e ( U i r f i r f i ) i E I r e p r e s e n t a n t m e t un
S o i t Dm:=
i E I t e l s que
fi
# 0 l a v a l e u r F m ( x ):= f i ( x ) / f i ( x ) e s t b i e n dE
f i n i e . A i n s i on o b t i e n t une f o n c t i o n Fm:Dm
-+
C.
C o m e en d i m e n s i o n s f i n i e s on v e r i f i e q u e l ' e s p a c e de phe.
btalg
e s t s 6 p a r b . Donc on a le p r i n c i p e du prolonqement mbromog
FONCTI ONS FEROMORPHES
Soient u c
11. PROPOSITION
s i Re4 4 e c . t i o n 4 m e t !en
v o k t non v i d e
un c o t p 4
h i
eLle4
V
4 i
O0Alo
e6.t
[7
E Fin A,
e s t i r r g d u c t i b l e dans E
%?o
~ c x ) re.
et tout x
E I
mic44
entap
DEM.:
Soit ( V j l g j l s )
E
I
cX
V
u u v ~ t t ,e 6 . t
connexe.
ou bien en u t i l i s a n t que
0A 0(I: @
que
= l i m ind 0 EFinA
l o
si e t
I0
seulement
*
ui
4 o ~ ut n
ui [ e n gehmes [ f i ]
( L J ~ ~ ~I ~ ~ ? ~
p o l y d i h q u e e t que p u ~ r et
L Ti:
no-cent p t t -
eiix.
j j FJ
une f a r n i l l e r e p r g s e n t a n t m. Pour cha-
q u e x E X c h o i s i s j (x) E: J t e l q u e
x
E
,
V.
11s e x i s t e n t
p o l y d i s q u e Wx o u v e r t d e c e n t r e x e t u n e f o n c t i o n h j
t e l l e que
@@ (11) (I:
1 0
e x i n t e u ~ l ed u m i t e e
4 e p 4 Z s e n f a n t m t e e e e q u e chuque tout i
'&..(v)
oO@lo est i r r g d u c t i b l e d a n s
1.3 LEMMA s o i t m
b u n t ~ g u R e odunb u n
Fn
e 4 t un anneuu z ( u c t o k i e l .
est factoriel pour t o u t
si q
Fm e t
ZguRen. V a n c
40n.t
Ou b i e n comme d a n s
E
E %(XI.
n 4 v n . t ZguRe4 e n u n p o i n t ~ L e e n h, v n t e g u -
e t neuRemen2
1 . 2 PROPOSITION
e t que q
u n o u v e t t c o n n e x e ~t m,n
U . S i Re4 I ; o n c R i o n 4
dUnb
DEM. :
x
21
h J. ( X ) - l X
s o i t le plus grand
diviseur
.
(XI
E
commun
qu'il
1.f x-I Y
s o i e n t p r e m i e r s e n t r e e u x e n t o u t p o i n t y E Ux.
Evi-
d e m e n t ( U x lf
lux,FX lux)
Nous d i r o n s que x
x
de
tel que
e x i s t e un p o l y d i s c o u v e r t U
mais m - l
U(WJ
-
n ' o n t p a s d e d i v i s e u r s communs, d o n c on s a i t (1.5-1 , p . 1 4 9 )
et
un
E QylX
X
de c e n t r e x , U x C W x l
reprssente m E X
.
e s t un pu^Re d e m
e t que x e s t un
E
%!
p o i n t d'indZte4rn
V. AURICH
22
1.4 PROPOSITION
Soit m
comme d a m 1.3.
ALohb
(i)
x
e.6t
(ii)
x
E
E
R(X). (Uirfir?i)iEIb a i t
o n equivaLe.nce. enthe
u n p6Pe. d e m.
ui c n t a a i n e
+
fi(x)
D
e t Fi(x) = 0.
(iii) Pouh t o u t e b u i t e (xnInEm d a n b Dm x
On
DeM.:
Fm(xn) t e n d V e h A
a equivalence
Q U ~c o n w e h g e
weM
m.
enthe
u n point d ' i n d E t e h m i n a t i o n d e m.
(iv)
x
ebt
(v)
x
E
(vi)
Pou4 t o u t v o i d i n a g e V d e x O n a F,(V
ui entaraine
fi(x) = 0
=
-
fi(x).
C o m e en d i m e n s i o w finies en utilisant
1.5 COROLLAIRE
choibie
I51
Dm)
= C.
6.2.3.
L'enbtmble deb p o i n t b d'i.ndetehmination
et
l ' e n b e m b l e deb p 6 l e b e t deb p o i n t b d ' i n d Z t e t m i n a t i o n b o u t
deb
endembleb anaLi4tique.b.
C o m e dans [7!
p . 23 on prouve
1.6 COROLLAIRE
Pout t o u t m
E
m ( X ) Dm e b t u n o u v e h t
1.7 COROLLAIRE
Pouh t o u t m
E m(X)
Fm
ebt
connexe.
une ( o n c t i o n holmoh
phe. A
Pour une fonction holomorphe f sur une varidte q: Y-C et x
E
Y on dgfinit depx f:= l'intersection Ae tous les sousen-
sembles
0
de A tels que f depend au voisinaae Ae x
des variables qj,j
E 0.
depx f est un
connexe depx f ne dipend pas de x
E
seulement
ensemble fini. Si 4!
est
Y, et sa valeur constante
sera not6e dep f . (voir rl]). Pour Q, E Finh et U ouvert dans X nous d6finissons %'(U)
/zn'
:=
{m
E a ( U ) :
dep, Pm c 0
pour chanue
est un faisceau. 11 est le faisceau associg au
x
E
U}.
prefaisceau
23
FONCTIONS MEROMORPHES
u
+
l'anneau total ctes fractions de
1.8 PROPOSITION
m(U) =
u{ %'(U)
8@ (u) .
: @ E
Fin A ) pouh
05
tout
v e h t U c o n n e x e d a n b X. DEM.:
Parce que Dm est connexe depx Fm est constant sur Dm,
2. LE PROBLEMME DE POINCARE ET L'ENVELOPPE DE Mf?ROMORPMIE 2.1 DEFINITION domaine q:Y
+
Soient p:X CAI A C A,
+
C A un domaine et m E m(X).
est appel6 un domaine d'existence de
m s'ils existent une fonction m6romorphe n phisme p : X
+
Un
E
M ( Y ) et un
mor-
Y tels que les conditions suivantes soient satis-
faites: (i)
m = n o p
(ii)
Etant donn6s un domaine q':Y'
un morphisme u ' : X me
$:Y'
+
+
Y' tels que m
=
+
CA '
, n'
E
'&?(Y')et
n'o 1-1' i l existe un morphis
Y tel que 1-1 = $ o 1.1'.
Le domaine d'existence d'une fonction mdrornorphe est unique ii un isomorphisme prgs (s'il existe). 2.2 PROPOSITION
T o u t e d o n c t i o n mhhomohphe m b ~ uhn
p : j ~t * a d m e t un d o m a i n e d ' e x i a t e n c e pm:Xm j ~ F , M . : Choisis x E
X. m induit un germe q
E
C
den Fm
Tc@ (XI ,@ I?
xm
domaine
= dep Fm.
soit la con2osante connexe de q dans l'espace 6tal6 de Elle est un domaine &a16
au-dessus de
C'.
ncQ.
Dans la
n i k e usuelle on dgmontre cru'elle satisfait (i) et (ii)
ma(voir
p.ex. [ 8 ] ) . 2 . 3 REMAROUE
11 est connu ffu'un domaine d'existence d'une f o F
tion mGromorphe en dimensions finies est pseudoconvexe
(
[2] ,
p . 86, consgquence du "Rontinuitatssatz" cle Hartoss-Kneser dans
24
V . AURICH
,
161 )
donc il e s t un domaine d e S t e i n .
2 . 4 THGOR&TZ
T o u t e donction mZtomotphe n u t
(Poincarg)
m a i n e z t a l z au - d e n n u n d e C A e h t l e q u o t i e n t d e deux
un do-
donctionn
hotomotrphen. D ~ M . : Appliquer 2 . 2 ,
7.4.6.
;tale
n e ptrolonge a l ' e n w e l o p p e d'hvLomotphie.
CA
Consgquence immgdiate de 2 . 4 .
DEM.:
2.6.
au
[5]
T o u t e 1;onction mztomotphe h u t un domaine
2 . 5 . THEOREME au - d e n n u h d e
2.3 e t
L ' e n v e l o p p e de m e t o m o t p h i e d l u n domaine Z t a l Q
COROLLAIR~
-
d e b b u h de
a l'enweloppe d ' h o l o m o t p h i e .
e n t ;gale
CA
3 . LES FONCTIONS MROMORPHES SUR UN DOMAINE DE STEIN
3 . 1 LEMME
T o u t e 1 ; o n c t i o n metomotphe m n u t un domaine p : X
admet un t e p t r e n e n t a n t ( U i l ~ i , G i ) i
t e l q u e c h a q u e Ui
I
-f
CA
noit u n
m e t dep Gi c d e p Fm p o u t c h a q u e i E I. E n p l u n o n p e u t o b t e n i t q u e [ g i I x e t L G i I x h o i e n t pdydinc ouvett
e t q u e d e p gi
dep F
CI
ptemietrn enthe tux en t o u t p o i n t DEFT.:
N :=X-Dm
$J
: = d e p Fm. C h o i s i s un r e p r g s e n t a n t
de m c o m e dans 1.3. S o i t p/Ui
+
Pi
E
a
1
U
c
Ui-N
de c e n t r e
(Uirfilfi)
i E I . On p e u t s u p p o s e r q u e
p ( U . ) s o i t topo1ogique.Choisis
ouvert
x
ui.
x E
a.
e t un p o l y d i s q u e
E Ui-N
a := n A - ) z ) .
On d e f i n i t
pour
-1 et Gi(X) := ( ? T @o P ( X ) r a ) g i ( x ) := fi o (pIui) -1 o ( P IUi) ( T o ~ p ( x ) I a ) . A c a u s e du p r i n c i p e d u p r o l o n g e m e n t
ui
analytique l ' i n t e r i e u r de U gi
/ gi
= Fm = fi
/ fi
( U i l g i l ~ , ) d g t e r m i n e m.
isi
= O}
e s t v i d e . Parce que
on o b t i e n t s u r
Ui
giZi
=
figi
sur donc
En p r o c g d a n t m a i n t e n a n t d a n s l a manis-
re de 1 . 3 on dgmontre la deuxisme p a r t i e du lemme.
FOIWTI ONS ME ROMORP HE S
E t a n t donne6 u n damuine q:Y
3 . 2 PROPOSITION
F i n A e A u n p u t ? y d i h q u e u u v e t l t P t CAL-',
Y x P
-f
a!'
t e e que
Lu p t l o j e c t i o n
Y i n d u i t un ibomohphibme a * d e
hut
@(U)
un isomorphisme de
%(Y)
a*:
phisme
:=
(Ui)
0
.
n'(X).C h o i s i s
E
Pour
x
- -1 ( x ) n gi(u
fii(x) :=
I1 r e s t e
/n2'(X).
+
m
jective. Soit
sur @@(UxP)
vi
E
soient
et
U i ) .hi
{fii
p h e s e t l ' i n t g r i e u r de
5,
,
m@(X).
m ( Y ) huh
donc
0
+
U induit
i n d u i t un monomog
5 d s m o n t r e r que a* est
(UifgilG,) c o m e dans
hi ( x ) : = gi ( a-1 ( x ) f l
Dm = a (D,)
Pour x
E
Par consequent F
vi
V.
-1 F m ( o ( x ) n Ui) nuit6 h . h
1 1
7
n a (D,)
s . (x)= 1
m se f a c t o r i s e 5 t r a v e r s o (D,).
-1 ( x ) /I U j ) = h . / h j ( x ) , donc p a r c o n t i g./i.(a 1 1 3 =h.G. sur Vir) V Cela prouve que ( V i , h . , 6 . ) r e p r g =
1'
1 1
S o i e n t q:Y
d i n q u e o u v e t l t dann
-+
a!'
d-@e t
1
% s u r Y. Evidemment
dep f
c@. Aboth
DEM.:
Soient a = (a',a")
que l e segment
a * ( $ ) = m.
E
A u n en4emb8e
a n a l y t i g u e duns Y
A = o ( A ) x P ou a cbt
[a,bl
1
u n d o n i u i n e , Q,
Fin A ,
q u i d o i t 1ucnLernen.t d 4 6 i n i n b u b l e pah line donct-ion
E A
et
:= { ( a ' , a "
f
poey-
un
P
teLet?
v e r t s U1,...,Un
fi
E
Ui+l
-1 ( O ( A 0
0
# $
00 (ui)
P
que
b = (a',b'') E a-l(a(a)).Parce
+ t(b" - a")): t
avec l e s propriEt6s suivantes: a
E [ O J ] )
E
U1,
est OU-
b E Un,
e t a(Ui) = U ( U ~ + ~pour ) t o u t i, i l s e x i s t e n t des A 0 ui = fyl(o). P a r c o n s g q u e n t A Q ui =
t e l s que
ui)
x
p h o j e c t i o n de Y X P + Y .
compact il e x i s t e un r e c o u v r e m e n t f i n i p a r d e s p o l y d i s q u e s
ui f l
donc
x P,
on o b t i e n t hi/hi ( x )=gi/Gi (0-l( x ) CI u i ) =
s e n t c une f o n c t i o n mGromorphe 3 . 3 LEIWE
et
l'ensemble
D'aprgs l e l e m m e 2 . 3 ci-dessous N = o ( N )
x P.
2.1.
ui)
d e s p 6 l e s e t d e s p o i n t s d ' i n d s t e r m i n a t i o n d e m = { x E X: 0 s i x E Ui}.
s 5
s o n t b i e n d s f i n i e s e t holomor
est vide. S o i t N:=
= O}
E
d e X :=
G
Pour t o u t p o l y d i s q u e U Ca!' la p r o j e c t i o n U x P
DEM.:
Vi
25
fl ui
pour t o u t
i
.
Cel'a e n t r a i n e
V. AURICH
26
a
-1
(o(AnUi))
n
UiflUi+, = AflUiflUi+l=
5
-1
(Ar)Ui+l))flUi
(
Parce que U i R Ui+l ast cylindrique et non vide
0 'i+l.
u ( A nui) n u ( U i / l Uitl) = u (A QUi+l) fl a (Uin Uitl)
que a(Ui) = U ( U ~ " U ~ + ~= )0 on obtient o(A Cela implique a' 3 . 4 REMARQUE
donc b
u(Af)IJn)
E
p:X
E
X A o i t ibomohphe a S
e n t x,:=
s
e t o@:x
c'-'
En
obssrvant
Ui) = a(Ar)Uitl).
A.
Fin A e t un d o m a i n e d e S t e i n
t e l b yue
x
E
.
u n d o m a i n e d e S t e i n . On b a i t ([l],
-+ C A d o i t
yu'il~e x i b t e n t Y
[3])
0
-+
x CA-'.
x, la
POU4
q:S
0 E Fin A ,
+
Q:
Y
@ 3'4,AoL
ptojecti.on.
3 . 2 et 3 . 4 entrahe le corollaire suivant.
3.5 COROLLAIRE @ E
Fin A,
p:X
# PY,
+
CA d o i t
un d o m a i n e d e S t e i n . Pouh
tout
'M(X,)
a # i n d u i t un i b o m o h p h i d m e a t d e
huh
'nz@(XI . Pour Y
C # C - @ 'E
X, sur X,
Fin A on a des morphismes canoniques a, "
tels que
3.6 COROLLAIRE
4.
I
,'
(a,
)*)
soit un systgme inductive.
S u h u n d o m a i n e d e S t e i n p:X
Limite inductive de DEM.:
@(X,)
(
(
fl(XQ)
,' (0,
)*)
de
I
Y c 0
-+
CA @(X)
c 0'
E
ebt
La
Fin A .
Consgquence de 3 . 5 et 1.8.
LES PRORLEMES DE COUSIN S U R UN DOMAINE DE STEIN p:X
+
CA
soit un domaine 6tal6. On a les suites
de faisceaux 0 0
($* ( @ * I
-b
-+
o - + mv a/@
0
wy@*
0
**
-+
b*
-+
w * -+
-+
-+
est le faisceau des fonctions holomorphes
exactes
(fonctions
mgromorphes) ne s'annulant dans aucun point de X (dans aucun 02 vert non vide de X)
. Une section dans r (X,
(dans
FONCTIONS MEROMORPHES
e s t a p p e l g e u n e donnze de Couhin 1 ( C o u s i n 1 I ) s u r
r(X,%*/@*)) X.
E l l e p e u t stre r e p r e s e n t g e p a r une f a m i l l e ( U i , m i )
-
Oii
I
e s t un r e c o u v r e m e n t de X p a r d e s p o l y d i s q u e s o u v e r t s e t
(Ui)iEI
mi
27
m
j
0 (ui f~ u j )
E
(mi/m
0*(ui
E
j
0 u j ) ) . Inversement,
une
t e l l e f a m i l l e d g t e r m i n e t o u j o u r s une s e c t i o n . Une donnge de Cousin I (11) e s t d i t e tZ4oLubt.e
si elle est
u(fl*). Nous a p p e l o n s une donnce
c o n t e n u e d a n s l ' i m a g e de
de
Cousin de d i m e n s i o n a i n i e s i e l l e admet un r e p r s s e n t a n t ( U i , m i )
t e l que l e s Ui
s o i e n t des polydisques e t U I d e p F m , , i E I } soit 1
f i n i . A c a u s e d e 1 . 8 on a l e l e m e s u i v a n t .
4.1 LEi"IME
Une d o n n g e de C o u n i n k h o L u b L e enA de d i m e n h i o n
6.i
-
nie.
d e s donnges (Uilm. 1 1
de Cousin I (11) q u i a d m e t t e n t
reprcsentant
p o u r t o u t i . Evidemment
0
r (x,w@/@ @ I e t r, (x,w * / u * ) r (x,c~')*/c
X := Y x Cn-'
A.
@ F: F i n
e t cf : X
t e L que Ui
@I
q :Y
-+
@@)*I.
u n domaine c?AtaLz,
6oiA
Y n o i t La p k o j e c t i o n . A e o k n Aoute dunnee
+
de C o u s i n duns r m( Y ,/n7/@ (Ui,mi)iEI
un
l'ensemble
i
Soit
LEFVIE
c
d e p Fn
t c l que
i e I
r Q ( x lWu 4.2
(r@(X,m*/@*)) dgsigne
l ? @ ( X , m/,)
Pour 0 C A ,
7
)
ori
r @( X I % * / @ * I
u (Vi)
x
C
admet un kepkEnentanR
/I-@
e t dep
C
F
9
p o ~ tl o t d i.
"i
DEM.: Cousin I : T : x
+
p r s s e n t a n t (Vi2ni)iEI v e r t s e t ni
E fi@(Vi).
t i o n mEromorpne m
m x
i i
-
m
j
E Vi
i
a''-'
s o i t l a p r o j e c t i o n . 11 e x i s t e un re-
t e l q u e l e s Vi
s o i e n t d e s p o l y d i s q u e s ou-
3 ' a p r z s 3 . 2 chaque n i d g f i n i t une
sur U
i
:= a(Vi) x
a*-'.
fonc-
I1 f a u t d g m o n t r e r q u e
S o i t x E Ui n U Choisis 1' j * o ( x i ) = ~ ( x . =) o(x). L e segmnt 3
s o i t holomorphe s u r Ui f l U
et
{ ( u ( x ) , 7(xi)
x. E V 1 1
+
t e l s que
t ( T ( x . 1 - T ( x ~ ) ) : t E [O,I;} 3
peut S t r e recouvert
28
V.
d ' u n nombre f i n i d e s V k l
vk fl v
~ # ~B .
AURICH
-
d i s o n s Vi
P+a r c e~ q u e
- n nkV
V
L
- m
mi
consgquent
i, O ( V k
)
x
t e l que
1 '
e s t holomorphe s u r
kv+l
e s t holomornhe s u r
fv
Uk
v=l
"=l
j
v =1
= V
I
C
=
r
vklr-ivk
=
v
cA-0.
V
C o u s i n 11: a n a l o g u e .
p:X
-f
Stein
Une d o n n z e d e C o u b i n 7 h u t u n d o n i u i n e d e
4 . 4 THEOF&.IE
C A e n t aiP.noBubLe n i
e t hsuLemer~Z hi. e.kLe.
de. d i m e n s i o n
ebt
64riie.
4 .I. ijous p r o u v o n s
D ~ M .:
=3
tout
Q E F i n A,
:
@ 3 Y
t
.
~ ' a p r e s3. /1 e t 4 . 2
ona pour
u n diagramme coi-xiuta-kive
I
1 11 e n r G s u l t e le t G o r Z m e n a r c e q u e H (x@,@)
= 0,
X@ G t a n t u n e
v a r i g t i de S t e i n . 4.5
S o i t p:X
THfhREPIE
-f
(II
n un d o m a i n s d e S t e i n .
e x i s t e un domuine d e S t e i n q:S chaniofiplze a
n
E
s
x
c'-*.
+
Q , Y c A d i n i , t e L que X l o i t 2
s u p p o s o n s que H ( S x Q",z) =
IN. ALofin u n e d o n n e e d e C o u n i n 1 1
n e u L e m e n R b i eLLe DEN.:
ebt
Analogue 4 . 4 ,
On b a i t q u ' i l
YJ
Aufi
o
pout t o u ~
X s n t fiZhutubLe
b i
de d i n i e n b i o n 6 i n i e . car
H
1
( x Q r@*I
-
2
= H
(x@,z) =
o
(
[s]).
eX
29
FONCTIONS MEROMORPHES
BIBLIOGRAPHIE
V.
AURICII:
The s p e c t r u m as e n v e l o p e of holomorphy o f
a
domain over a n a r b i t r a r y p r o d u c t of complex l i n e s . P r o c e e d i n g s on i n f i n i t e d i m e n s i o n a l holomorphy, p . 1 0 9 , S p r i n g e r L e c t u r e Notes 364. H.
BEHNKE, P. THULLEN: T h e o r i e der F u n k t i o n e n
mehrerer
k o m p l e x e r V e r a n d e r l i c h e n . S p r i n g e r 1970. G.
ZOEURE:
A n a l y t i c f u n c t i o n s and m a n i f o l d s i n
dimensional s p a c e s . North-Holland S . DINEEN:
infinite
1974.
C o u s i n ' s f i r s t problem on c e r t a i n l o c a l l y
con-
v e x t o p o l o g i c a l vector s p a c e s . M a t h e m a t i c s Resear& R e p o r t No.
75-2,
January 1975, U n i v e r s i t y
of
Maryland. L . HORMANDER:
An i n t r o d u c t i o n of complex a n a l y s i s i n
sevg
r a l v a r i a b l e s . Van N o s t r a n d . H.
KNESER: E i n S a t z bber d i e M e r o m o r p h i e b e r e i c h e
analy-
t i s c h e r F u n k t i o n e n von m e h r e r e n V e r a n d e r l i c h e n . Math. Ann. 1 0 6 , p . 648-655. J.P.
RAMIS: Sous-ensembles a n a l y t i q u e s d ' u n e v a r i g t d banac h i q u e complexe. S p r i n g e r 1970.
M.
SCHOTTENLOHER: D a s L e v i p r o b l e m i n u n e n d l i c h d i m e n s i o n a l e n Rxumen m i t S c h a u d e r z e r l e g u n g . Munchen 1974.
Habilitationsschrift
This Page Intentionally Left Blank
Infinite Dimensional Holomorphy and Applications, Matos (ed.) 0 North-Holland Publishing Company, 1977
ON HOLOMORPHY VERSUS LINEARITY
I N CLASSIFYING LOCALLY CONVEX SPACES
By
J O R G E ALBERT0 BARROSO , MARIO C . MATOS
and
LEOPOLDO NACHBIN
1.
INTRODUCTION
I n t h e l i n e a r t h e o r y of l o c a l l y convex s p a c e s , c l a s s i c a l t o study bornological , b a r r e l e d , i n f r a b a r r e l e d
it is and
Mackey s p a c e s . I n t h e holomorphic approach , t h e corresponding concepts have been i n t r o d u c e d r e c e n t l y a s holomorphically bornologi c a l , ho lomorphica 1l y b a r r e l e d , ho lomorph i c a l l y i n f r ab a r r e l e d and holomorphically Mackey s p a c e s , t h a t a r e more restrid_ ed c l a s s e s t h a n t h e corresponding l i n e a r ones. I n t h i s reasonably self-contained,
e x p o s i t o r y paper, w e p r e s e n t some
basic
r e s u l t s i n such a s t u d y . L e t us i n t r o d u c e t h e following a b b r e v i a t i o n s f o r
pro
p e r t i e s of a complex l o c a l l y convex s p a c e : B = B a i r e , S =Silva,
sm
=
semirnetrizable, hba = holomorphically b a r r e l e d , hbo = ho-
lomor ph i ca 1l y bo r no 1o g i ca 1, h i b = ho 1om0rph i ca 1l y i n f mbarreled ,
31
BARROSO, MATOS
32
hM = h o l o m o r p h i c a l l y Mackey
. We
& NACHBIN
have t h e f o l l o w i n g i m p l i c a t i o n s
f o r t h e named p r o p e r t i e s :
B\hba S s h b o > h i b
-3hm
sm t h a t c o r r e s p o n d t o c l a s s i c a l o n e s d e a l i n g w i t h c o n t i n u o u s linear mappings, i n p l a c e o f h o l o m o r p h i c mappings. An i n t e r e s t i n g h i a h l i g h t i s t h e holomorphic Banach-Steinhaus theorem
on
a
F r 6 c h e t s p a c e , t h a t c o n t a i n s a s a p a r t i c u l a r c a s e t h e classical l i n e a r Banach-Steinhaus
theorem o n s u c h a s p a c e .
W e s h a l l u s e f r e e l y t h e n o t a t i o n and t e r m i n o l o g y [8];
see a l s o t h e r e f e r e n c e s g i v e n t h e r e . L e t u s make a
r e v i e w o f what w i l l b e needed h e r e . U n l e s s s t a t e d
of brief
,
otherwise
w e s h a l l a d h e r e t o t h e f o l l o w i n g c o n v e n t i o n s . E and F
denote
complex l o c a l l y convex s p a c e s ; and U i s a nonvoid open
subset
o f E . The s e t of a l l c o n t i n u o u s seminorms o n E i s d e n o t e d CS(E)
. We
by
d e n o t e by Ea t h e v e c t o r s p a c e E seminormed by a .
We
r e p r e s e n t by wF t h e weakened s p a c e F , t h a t i s , t h e v e c t o r space F endowed w i t h t h e weak t o p o l o g y o ( F , F ' ) d e f i n e d on F by F'
.
I f I i s a s e t and F i s a seminormed s p a c e , w e d e n o t e by l m ( I ; F ) t h e seminormed s p a c e o f a l l bounded mappings o f I i n t o F ;
and
by c o ( I ; F ) t h e seminormed s u b s p a c e o f a l l mappings o f I i n t o F t e n d i n g t o 0 a t i n f i n i t y . A mapping f : U i f 6 o f i s l o c a l l y bounded f o r e v e r y 6
+
E
F i s amply bounded
CS(F) : more g e n e r a l -
l y , a c o l l e c t i o n o f mappings o f U i n t o F i s amply bounded the collection B o W e d e n o t e by
8 (U;F)
if
i s l o c a l l y bounded f o r e v e r y B E C S ( F ) .
t h e v e c t o r s p a c e o f a l l holomorphic
map
019 HOLOMORPHY
VERSUS L I N E A R I T Y
p i n g s o f U i n t o F ; and by H ( U ; F )
33
t h e v e c t o r s p a c e of a l l
map-
p i n g s o f U i n t o F which are h o l o m o r p h i c when c o n s i d e r e d a s map p i n g s of U i n t o a f i x e d c o m p l e t i o n t h e a d j e c t i v e holomorphic r e f e r s t o f: U
I
f
-t
E
o f F. U n s p e c i f i e d u s e
%,
of
not t o H. W e say t h a t
F i s a l g e b r a i c a l l y holomorphic i f t h e r e s t r i c t i o n
(U f l
S)
i s h o l o m o r p h i c , f o r e v e r y f i n i t e d i m e n s i o n a l vector
s p a c e S o f E m e e t i n g U , where S c a r r i e s i t s n a t u r a l t o p o l o g y .
To t h e
On f u n c t i o n s p a c e s from U i n t o F , w e r e p r e s e n t by
pology of u n i f o r m c o n v e r g e n c e o n compact subsets; and by
to-
zof
t h a t f o r f i n i t e d i m e n s i o n a l compact s u b s e t s o n l y . When F = C it i s not included i n t h e notation €or function spaces;
z(U)s t a n d s 2.
for
,
thus
@(U;C).
HOLOMORPHICALLY BORNOLOGICAL SPACES
DEFINITION 1.
A given E
i n a "hoPomohphicaLLy b o h n o P o g i c a l
bpacel' id, d o h e v e h y U a n d e v e h y F, we h a v e t h a t e a c h m a p p i n g f: U
-t
FbePangb t o
E ( U ; F ) LA l a n d u l w a y n o n l y id) f
b t r a i c a L l y h o l u m o t p h i c , avid f 4e.t
i 4
ii6
alge-
bounded o n evehy compact
4ub-
o h u.
R e m a r k 4 b e l l o w m o t i v a t e s t h e above d e f i n i t i o n , b u t w e need the
f o l l o w i n g p r e l i m i n a r y m a t e r i a l which i s known.
LEMMA 2 .
F o h a g i v e n E, t h e 6 o l L o w L n g c o n d i . t i o n h
ahe e q u i v a -
Len.t: (Ib) .to
F o h e v e h y F, me h a v e . t h a t e a c h m a p p i n g f : E &C(E;F)
-f
F belong4
id l a n d a L w a y a v n L y id) f io L i n e a h , a n d f ia bound
34
BARROSO, MATOS & NACHBIN
ed o n e v e h y b o u n d e d 4 u b 4 e t a d E .
Fox evehy F , w e h a v e t h a t e a c h m a p p i n g f : E
(Ic) t o
&,(E;F)
.id
( a n d a t w a y s o n R g id) f
ed o n e v e h y cornpacR 4 u b 4 e . t
a i s bounded o n
E a c h seminohm
(2c)
id)
a i h bounded o n
PROOF.
F
Linea4, and f
belongs i d
bound -
E.
E a c h beminohm a o n E i c l c o n t i n u o u d id
[2b)
id)
06
i 4
-f
( a n d aLways
onLy
evehy bounded 4ub4e.t o d E. c1
on E
i 4
continuoun
id ( a n d a t w a y h
ontg
evehy c o m p u c t s u b s e t o d E .
W e s h a l l p r o v e t h e f o l l o w i n g imp1 i c a t i o n s
. This
( l c ) =>
(lb)
( l b ) =>
(2b). L e t
i s clear. c1
b e a seminorm o n E t h a t i s bounded
on e v e r y bounded s u b s e t o f E . P u t F = E a . f = I: E
-f
The i d e n t i t y mapping
F i s l i n e a r , and f is bounded on e v e r y bounded sub-
s e t of E . By ( l b ) , f i s c o n t i n u o u s . T h u s , a i s c o n t i n u o u s . ( 2 b ) =>
( 2 c ) . L e t a b e a seminorm on E t h a t i s bounded
on e v e r y compact s u b s e t o f E . W e c l a i m t h a t a i s bounded
on
e v e r y bounded s u b s e t X of E . I n f a c t , l e t xm E X ( m E IN) b e ag b i t r a r y . F o r any Am E C (m E E?) s u c h t h a t A m
A mxm
-+
0, a s m
-f
m.
Then,
c1
+
0 , w e have t h a t
i s bounded on {Amxm: m
E IN}
,
sirce
t h i s s u b s e t t o g e t h e r w i t h 0 i s compact: t h a t i s , { A m a(m);mElN}
i s bounded. W e deduce t h a t { a ( x m );m E IN} i s a l s o bounded, since (A,)
i s a r b i t r a r y . Thus a ( X ) i s bounded, b e c a u s e
o n e v e r y d e n u m e r a b l e subset of X . By ( 2 b ) ,
c1
c1
i s bounded
is c o n t i n u o u s .
35
ON HDLOIORPHY V E W U S LINEARITY
( 2 c ) --->
.
(lc) L e t f : E
+
F b e l i n e a r and bounded o n ev-
e r y compact s u b s e t o f E . I f 6 E CS(F) , t h e n 6 o f i s a
semi-
norm on E t h a t i s bounded o n e v e r y compact subset o f E .
By
( 2 c ) I (3 o f i s c o n t i n u o u s . Thus f is c o n t i n u o u s , s i n c e 6 i s a g
bitrary
. The p r o o f c a n a l s o b e c a r r i e d o n w i t h t h e same r e a s o g
i n g , by r e v e r s i n g the a r r o w s . QED The f o l l o w i n g d e f i n i t i o n i s c l a s s i c a l , p a r t i c u l a r l y i n terms o f ( l b ) or ( 2 b ) .
DEFINITION 3 . isdieb
A g i v e n E is a " b o a n o E u g i c a L space"
t h e tqu.ivatent conditions
REMARK 4 .
D e f i n i t i o n 1 was
06
.id it
Lemma 2 .
formulated i n analogy t o D e f i n i -
t i o n 3 t h r o u g h ( l c ) I r a t h e r t h a n ( l b ) , o f Lemma 2 . The
i s t h a t each f
E
sat -
reason
(U;F) i s a l w a y s bounded o n e v e r y compact
subset o f U; w h e r e a s it may o c c u r t h a t some f bounded o n some bounded s u b s e t o f E (see [ 7 ] a s a consequence of t h e J o s e f s o n
-
,
i s un-
E
(E)
p.28)
. Actually,
Nissenzweig theorem
[5]
,
it i s known t h a t , i f E i s a n i n f i n i t e d i m e n s i o n a l normed
IIlO],
s p a c e , a n d X C E h a s a non v o i d i n t e r i o r , t h e r e i s some f E
8 (E)
which i s unbounded o n X (see [ 5 ] ) .
PROPOSITION 5 .
A h o l o m a a p h i c a t L y b o a n o t o g i c a L d p a c e E 0 &no
a bohnoLogicaL s p a c e .
PROOF.
I t s u f f i c e s t o compare D e f i n i t i o n s 1 and 3 , by
using
( l c ) o f Lemma 2 , and by r e m a r k i n g t h a t a l i n e a r mapping i s a l g e b r a i c a l l y h o l o m o r p h i c . QED
36
BARROSO, MATOS
PROPOSITION 6 .
&
NACHBIN
A bemimethizabf?e Apace E i n a ho~omohphicak!.ty
b o h n o l o g i c a l 6 pa c e..
PROOF.
Let f: U
-f
F b e a l g e b r a i c a l l y h o l o m o r p h i c , and bound-
ed on e v e r y compact s u b s e t o f U. S i n c e E i s s e m i m e t r i z a b l e , i t f o l l o w s t h a t f i s amply bounded. Hence f E
%$ (U;F), b e c a u s e
f
i s a l g e b r a i c a l l y h o l o m o r p h i c and amply bounded. QED
REMARK 7 .
P r o p o s i t i o n s 5 and 6 imply t h e known f a c t t h a t
a
semimetrizable space E i s a bornological space. The f o l l o w i n g i s a by now known d e f i n i t i o n .
E
=
'm
U
Em
E IN
and t h a t E c a h h i e o t h e i n d u c t i v e l i m i t t o p o l o g y .
REMARK 9 .
A S i l v a s p a c e i s known t o b e e s s e n t i a l l y t h e
t h i n g a s t h e d u a l of a F r z c h e t - S c h w a r t z s p a c e , o r FS-space s h o r t ; t h u s i t i s a l s o known a s a DFS-space.
More
same for
explicitly,
t h e s t r o n g d u a l s p a c e of a F r s c h e t - S c h w a r t z s p a c e a S i l v a space; t h e s t r o n g d u a l s p a c e of a S i l v a space i s a Frzchet-Schwartz s p a c e ; and b o t h S i l v a s p a c e s and F r g c h e t - S c h w a r t z s p a c e s reflexive.
are
ON tIOLOPlORPIIY VERSUS LINEARITY
A S i l v a space
PROPOSITION 1 0 .
E
ih
37
a holomuhphicaLLy bohno-
logical Apace. The p r o o f w i l l r e s t o n t h e f o l l o w i n g lemma.
L e t E b e a c o m p e e x w e c t o h h p a c e , Em a c o m p l e x Loco..&
LEMMA 11.
l y c a n v e x s p a c e , p,
: Em
+
E u Eiiieaa m a p p i n g , and
a c o m p a c t Lineah t n a p p i n g s u c h t h a t p,
= P,,~
o a m f o r m E IN.
A s ~ u m et h a t
und endow E w i t h t h e i n d u c t i v e L i m i t t o p v e o g y . l e t U c E a p e n . P u t Um = p,-1 ( U )
,
and ahhume t h a t Uo
and Um ahe n o n v o i d d o & m s p a c e and f : U
then f
F,
8Ej (u,;F)
E
f o ,p
->
E IN.
i(uh
E
16 F
be
i s non-void; hence
i n a conipeex k o c u l l y
% (U;F) id and o n l y id
f
conwex E
m
evehy m E IN.
As-
N e c e s s i t y b e i n g c l e a r , l e t us p r o v e s u f f i c i e n c y .
PROOF.
U
sume t h a t f m E Z ( U m ; F ) f o r e v e r y m
€ IN.
W e claim t h a t f is a l
g e b r a i c a l l y h o l o m o r p h i c . I n f a c t , l e t S b e a f i n i t e dimensional vector subspace of E, w i t h U 0 S
= S . Thus,
p,
E
1
a
and
i s a v e c t o r s p a c e isomorphism b e t w e e n Sm
and S . W e h a v e p m ( U m S i n c e f,
E IN
o f Em, o f same d i m e n s i o n a s S , s u c h t h a t
v e c t o r s u b s p a c e S, p,(S,)
# g. T h e r e a r e m
(umnsm)
Sm) = U E
n S.
%(urn n s,;
8 ( U fl S ; F) b e c a u s e p,
I n p a r t i c u l a r , U, /I Sm# g.
F) , i t f o i i o w s t h a t f
is a homeomorphism b e t w e e n S,
j
(uns)
and S ,
where Sm and S c a r r y t h e i r n a t u r a l t o p o l o g i e s . Thus, t h e f i r s t c l a i m i s t r u e . I d e n e x t c l a i m t h a t f is amply bounded. I t
is
enough t o t r e a t F a s b e i n g seminormed. W e may assume t h a t 0 E U,
38
BARROSO, MATOS
% . NACHBIN
and i t s u f f i c e s t o show t h a t f i s l o c a l l y bounded a t 0 . S i n c e f o i s l o c a l l y bounded a t 0 , choose a convex neighborhood V, 0 in U
0
of
s u c h t h a t a o ( V o ) h a s a compact c l o s u r e i n El c o n t a i n e d h e n c e p o ( V o ) c U , and s u c h t h a t
i n U1,
f o r some M
E
IR. A s s u m e t h a t , f o r some m E IN, w e h a v e d e f i n e d
a convex neighborhood Vm o f 0 i n Urn s u c h t h a t om(Vm) h a s a co_m p a c t c l o s u r e i n Em+l c o n t a i n e d i n
hence pm(Vm)
C U,
and
such t h a t
t h i s i s i n d e e d t h e case f o r m = 0 , by ( 1 ) .S i n c e fm+l i s l o c a l l y bounded a t t h e c l o s u r e o f u m ( V m ) i n Em+l, h e n c e
uniformly
c o n t i n u o u s t h e r e , and s u c h a c l o s u r e i n convex, u s e ( 2 ) choose a convex neighborhood Vm+l
of t h a t c l o s u r e , hence of 0 ,
s u c h t h a t U ~ + ~ ( V , + ~h )a s a compact c l o s u r e i n Em+2 con
in
tained i n
hence P , + ~ ( V , + ~ ) t U , and s u c h t h a t SUP {
II
fm+l(X)
w e also have p m ( V m ) c pm+ fVm+l). letting
P r o c e e d i n g i n t h i s way and
w e g e t a neighborhood V o f 0 i n U s u c h t h a t every x f
to
E
and f,
f(x)11
< M
V . Hence t h e s e c o n d claim i s t r u e . I t f o l l o w s
E %;(U;F).
REMARK 1 2 .
11
for
that
QED
I t i s known t h a t Lemma 11 i s t r u e i f w e r e p l a c e f
b e i n g holomorphic by them b e i n g c o n t i n u o u s ; b u t Lemma 11
39
ON HOLOMORPHY VERSUS L I N E A R I T Y
i s f a l s e i f w e r e p l a c e f and f m b e i n g h o l o m o r p h i c by them
b e i n g amply bounded, a s w e see even when E = C(N)
REMARK 1 3 .
and F = C.
I t c a n b e s e e n t h a t , i n Lemma 11, E i s
neces-
s a r i l y a S i l v a space. PROOF OF PROPOSITION 1 0 . n i t i o n 8. L e t f : U
+
C o n s i d e r t h e s e q u e n c e (Em) o f D e f i -
F b e a l g e b r a i c a l l y h o l o m o r p h i c , and bound
e d on e v e r y compact subset o f U . S e t Urn = U
g,
t h a t Uo # g, h e n c e Um #
n Em;
w e may assume
f o r a l l m E p7. Then f
Um i s a l g e -
b r a i c a l l y h o l o m o r p h i c , and bounded o n e v e r y compact subset Um.
By P r o p o s i t i o n 6 , f
Um i s holoinorphic f o r e v e r y
of
m E IN.
By Lemma 11, f i s h o l o m o r p h i c . QED
REMARK 1 4 .
i f Ei
Lemma 11 is a r e m i n i c e n s e o f t h e known f a c t t h a t ,
( i E I ) i s any f a m i l y o f l o c a l l y convex s p a c e s , E
vector space, pi:
Ei
-f
E ( i E I) i s a l i n e a r mapping,
endowed w i t h t h e i n d u c t i v e l i m i t t o p o l o g y , and F i s a convex s p a c e , t h e n a l i n e a r mapping f: E and o n l y i f f o p i :
Ei
+
+
a
is E
is
locally
F is c o n t i n u o u s
if
F is c o n t i n u o u s for e v e r y i E I.Lemia
11 may b r e a k down i n o b s e n c e o f c o m p a c t n e s s
( s e e Example
18
below) or d e n u m e r a b i l i t y (see Example 20 below) c o n d i t i o n s .
REMARK 1 5 .
P r o p o s i t i o n 10 i s a r e m i n i s c e n s e of
the
known
f a c t t h a t any i n d u c t i v e l i m i t o f b o r n o l o g i c a l s p a c e s i s a b o r n o l o g i c a l s p a c e . A d e n u m e r a b l e i n d u c t i v e l i m i t whose connecti?g mappings u a r e n o t compact (see Example 1 8 below)
,
o r a non-dg
numerable i n d u c t i v e l i m i t w i t h compact c o n n e c t i n g mappings
(see Example 20 below)
,
cs
of holomorphically bornological spaces
may f a i l t o b e a h o l o m o r p h i c a l l y b o r n o l o g i c a l s p a c e .
40
BARROSO, MATOS
PROPOSITION 1 6 .
16 E . i b a h o l o m o t p h i c a C l y buanologicai? b p a c e ,
@ (U;F)
then F
i h
& NACHBIN
id
carnpLete doh. t h e campact-open t o p d o g y -r0
campeete.
i h
d
PROOF. f: U
@ (U;F) b e t h e v e c t o r s p a c e o f a l l mappings
Let
+
F which are a l g e b r a i c a l l y h o l o m o r p h i c a l l y h o l o m o r p h i c d
and bounded o n t h e compact s u b s e t s o f U . Then p l e t e f o r t h e compact-open t o p o l o g y
T
~
(U;F) i s com-
s, i n c e F i s c o m p l e t e .
Since E i s a holomorphically bornological space, then a ( U ; F ) = N
= % ( U ; F) a l g e b r a i c a l l y and t o p o l o g i c a l l y . QED
I t i s known t h a t , i f F i s a b o r n o l o g i c a l s p a c e
REMARK 1 7 .
& (E;F)
then ogy
,
,
i s c o m p l e t e f o r t h e s t r o n g , o r compact-open,tapol-
i f F i s c o m p l e t e . P r o p o s i t i o n 1 6 c o r r e s p o n d s t o t h e sec-
ond h a l f o f t h i s r e m a r k .
L e t Xo b e a s e p a r a t e d i n f i n i t e d i m e n s i o n a l mmplex
EXAMPLE 1 8 .
l o c a l l y convex s p a c e . I t i s known t h a t a n %-bounding o f Xo
subset
( t h a t i s , a s u b s e t of Xo on which e v e r y member o f f $ ( X o )
i s bounded) h a s a n empty i n t e r i o r [ 5 ] . T h e r e f o r e , i f Xo i s me-
t r i z a b l e , t h e r e i s a s e q u e n c e y,
E
(m = 1,2,
e(Xo)
t h a t , g i v e n any neighborhood V o f 0 i n Xo
,
where Xm = C ( m = 1,2,...).
i f x = (xm) mEN
E E.
C
' m=O m
I
Define f : E
If we let
-+
such
t h e n some gm is un-
bounded on V . C o n s i d e r t h e t o p o l o g i c a l d i r e c t sum E =
. ..)
C by
ON HOLOMORPHY VERSUS L I N E A R I T Y
Em =
41
Xo @. ..@Xm
and c o n s i d e r it a s a v e c t o r s u b s p a c e o f E l t h e n f l E m E
8$
(Em)
f o r m E IN. N o t i c e t h a t e a c h Em i s m e t r i z a b l e , and e v e n normable i f Xo i s normable. We claim t h a t f i s n o t l o c a l l y bounded a t 0 . I n f a c t , i f V i s a n e i g h b o r h o o d o f 0 i n Xo and
1,2,,..,
xo
E~
m =
> 0 €or
d e f i n e W a s t h e set o f a l l x = (xmImElNE E s u c h
E V and
that
.
lxml 5 ~ ~ ( m = 1 , 2 , .. ) . IT w e c h o o s e k s o t h a t gk
is
unbounded on V , t h e n f i s unbounded on t h e s e t o f a l l x E E w i t h xo E V, xk = ck, and xm = 0 f o r m
1, m
#
k; t h u s f
is
S i n c e a l l s u c h W form a b a s i s o f n e i g h b o r h o o d s
unbounded o n W .
o f 0 i n E l o u r c l a i m i s p r o v e d . Hence f f
&:(El
. This
shows
t h a t Lemma 11 b r e a k s down i f t h e sm a r e assumed t o b e l i n e a r and c o n t i n u o u s , b u t n o t compact, a l t h o u g h d e n u m e r a b i l i t y o f the f a m i l y i s p r e s e r v e d . Such an example a l s o shows t h a t a
denu-
merable i n d u c t i v e l i m i t E of holomorphically bornologicalspaces Em (m E IN) may f a i l t o b e a h o l o m o r p h i c a l l y b o r n o l o g i c a l
space.
I n f a c t , f i s a l g e b r a i c a l l y h o l o m o r p h i c on E ; and it i s bounded on t h e compact s u b s e t s o f E l s i n c e e a c h s u c h s u b s e t i s cont a i n e d i n some Em. However, f
8 (E) .
?j
Thus, E i s n o t a h o l o -
morphically bornological space. Actually,
@ ( E l i s n o t complete,
dir even s e q u e n t i a l l y c o m p l e t e , f o r t h e compact-open t o p o l o g y
To. To
see t h i s , it i s enough t o
ntroduce t h e truncated f u n s
d e f i n e d by
tion f k E %(E)
k
c
fk(X) =
gm x 0 ) x m
m=l for k = 1 , 2 , .
s e t of E as k
.; -+
since f k
m,
+
f u n i f o r m l y on e v e r y compact
but f $ %(E),
not s e q u e n t i a l l y complete f o r
w e conclude t h a t % ( E l
TO.A c t u a l l y ,
subis
i f w e look a t
E
42
BARROSO, MATOS
NACHBIN
&
as E = X
Q: (IN
x
0
,
w e see t h a t E is a b o r n o l o g i c a l s p a c e , a s t h e C a r t e s i a n produ c t o f two b o r n o l o g i c a l s p a c e s . Hence, a b o r n o l o g i c a l s p a c e may f a i l t o b e a h o l o m o r p h i c a l l y b o r n o l o g i c a l s p a c e . W e
also
see t h a t a C a r t e s i a n p r o d u c t o f two h o l o m o r p h i c a l l y b o r n o l o g i c a l s p a c e s may f a i l t o b e a h o l o m o r p h i c a l l y b o r n o l o q i c a l s p a c e .
We s h a l l need t h e f o l l o w i n g lemma i n Example 20 below.
l e t E be
LEMMA 1 9 .
dimenbion
i b
at
a compRex v e c t o f i
leUbt
b p U C e UJhobc?
(algebfiaicl
equaR t o t h e c o n t i n u u m . Endow E w i t h it.5
LatLgebt RocaRRy c o n v e x t o p o l o g y . 7 h e n t h e h e i n a 2-homogeneoub p o l y n o m i a l P: E
+
C w h i c h i6 n o t c o n t i n u o u s .
,
L e t B b e a b a s i s f o r E . S i n c e B is i n f i n i t e
FIRST PROOF.
t h e r e i s a set S o f f u n c t i o n s s : B
+
IN s u c h t h a t S h a s t h e pow_
er o f t h e continuum; and s u c h t h a t , f o r e v e r y f u r c t i o n t:3 t h e r e i s some s E S f o r which s
5
+
B+,
c t i s f a l s e f o r a l l c E IR+.
I n f a c t , f i x a n i n f i n i t e d e n u m e r a b l e s u b s e t I o f B , and c a l l S t h e set of a l l functions s: B Then S h a s t h e power o f e v e r y t: B
-t
+
IN v a n i s h i n g o f f t h a t s u b s e t .
18, that
IR+ , w e c a n f i n d s
i s , o f t h e continuum. For E S
such t h a t s
5
c t is f a l s e
on I , h e n c e on B , f o r a l l c E B+ . S i n c e t h e power o f B
is a t
l e a s t e q u a l t o t h e continuum, t h e r e i s a s u r j e c t i v e mapping b E B
+
sb E S . D e f i n e
f o r bl,b2 E B;
then
43
ON HOLONORPHY VERSUS LINEARITY
r(b2,bl)
= r(bl,b2)
W e c l a i m t h a t t h e r e is no t: B t ( b l ) . t ( b 2 ) f o r a l l bl,b2 choose s E S s u c h t h a t s
bl
E B.
Sbl
-+
(b21
2
0.
lR+ s u c h t h a t
r(bl,b2)
5
I n f a c t , i f t did e x i s t , we a u l d
5 c t is
f a l s e f o r a l l c E IR+.
Let
.
so t h a t s = sb Then s ( b 2 ) = sb ( b 2 ) 5 r ( b l , b 2 ) 5 1 1 t ( b l ) . t ( b 2 ) f o r a l l b 2 E B; t h u s s 5 c t i f c = t ( b l l , a c o n t r a E B
d i c t i o n . Now, d e f i n e t h e s y m m e t r i c b i l i n e a r form A: E L
f o r x1,x2 E E l where b
x
E E
*
+
C
by
i s t h e l i n e a r form o n E which t o every
a s s o c i a t e s i t s b-component by B ,
i s f i n i t e . L e t t h e 2-homogeneous
i f b E B ; t h e above sum
p o l y n o m i a l P: E
+
Q: b e g i v e n
by P(x) = A ( x , x ) f o r x E E . W e c l a i m t h a t P i s n o t c o n t i n u o u s . O t h e r w i s e , A would b e c o n t i n u o u s t o o , t h a t i s , w e would h a v e a seminorm
c1
I 5 a(x,) .a(x2)
for all
B and x2 = b 2
B , w e would
on E such t h a t ]A(x1,x2)
x1,x2 E E . Then, l e t t i n g x1 = bl get r(bl,b2)
E
5 cl(bl) .ci(b2) f o r a l l b l , b 2
E B,
E
a contradiction.
QED
SECOND PROOF.
L e t X b e a n i n f i n i t e d i m e n s i o n a l complex vector
s p a c e , and Y b e i t s ( a l g e b r a i c ) d u a l s p a c e . A s s u m e f i r s t l y t h a t E = X x Y . L e t P: E
-+
C b e t h e 2-homogeneous p o l y n o m i a l defined
by P ( x , y ) = y ( x ) f o r a l l x E X I y E Y . W e claim t h a t P is
not
c o n t i n u o u s i f E i s g i v e n i t s l a r g e s t l o c a l l y convex t o p o l o g y . I n f a c t , assume t h a t P i s c o n t i n u o u s . Now, t h e l a r g e s t l o c a l l y convex t o p o l o g y o n E i s t h e C a r t e s i a n p r o d u c t o f t h e l a r g e s t l o c a l l y convex t o p o l o g i e s o n X and Y ; and P i s a b i l i n e a r form
44
EARROSO, XATOS & NACHBIN
on X and P on Y s u c h t h a t
on X x Y . Then, t h e r e a r e seminorms
LY
IP(x,y) I
y E Y . Once t h e seminorm a
5 a(x) .B(y)
for a l l x
E X,
i s g i v e n , and X i s i n f i n i t e d i m e n s i o n a l , t h e r e i s a l i n e a r form
b on X which i s n o t c o n t i n u o u s f o r a. However I b ( x ) I < c . a ( x ) f o r a l l x E X, where c =
=
IP(x,b)l
6 ( b ) , showing t h a t b i s conti
nuous f o r a , a c o n t r a d i c t i o n . Hence, P i s n o t c o n t i n u o u s . Comi n g back t o any E, i n o r d e r t o f i n i s h t h e p r o o f , w e a r g u e t h a t i t i s enough t o prove t h e lemma when t h e dimension o f
is
E
e q u a l t o t h e continuum; i n f a c t , t h e g e n e r a l c a s e r e d u c e s
to
t h i s o n e b e c a u s e E i s a d i r e c t sum o f t w o v e c t o r s u b s p a c e s , one of which h a s dimension e q u a l t o t h e continuum.
NOW,
if
the
above X h a s a n i n f i n i t e denumerable dimension, t h e c o r r e s p n i i i n g Y h a s dimension e q u a l t o t h e continuum; h e n c e X x Y h a s dimen-
s i o n e q u a l t o t h e continuum t o o . QED
EXAMPLE 20.
L e t E b e a complex v e c t o r s p a c e whose dimension
i s a t l e a s t e q u a l t o t h e continuum. Endow E w i t h i t s l a r g e s t l o c a l l y convex t o p o l o g y ; E i s t h e i n d u c t i v e l i m i t o f i t s
i t e d i m e n s i o n a l v e c t o r s u b s p a c e s . By Lemma 1 9 , l e t f : E
fin+
C be
a 2-homogeneous polynomial which i s n o t c o n t i n u o u s . For e v e r y f i n i t e d i m e n s i o n a l v e c t o r s u b s p a c e X o f E , it is c l e a r t h a t f IX
E
%(XI.
However, f E B ( E ) .This shows t h a t Lemma 11 breaks
down i n a b s e n c e of d e n u m e r a b i l i t y o f t h e f a m i l y , a l t h o u g h comp a c t n e s s o f t h e c o n n e c t i n g mappings o i s p r e s e r v e d . Such example a l s o shows t h a t a non-denumerable i n d u c t i v e l i m i t
an of
h o l o m o r p h i c a l l y b o r n o l o g i c a l s p a c e s may f a i l t o b e a holomorpk i c a l l y b o r n o l o g i c a l s p a c e , even i f t h e c o n n e c t i n g mappings a r e compact.
u
45
ON HOLOMORPH Y VE RS U S L I N E A R I T Y
REMARK 2 1 .
I n Example 1 8 , i f Xo i s n o t normable, t h e n
every
bounded s u b s e t of X h a s an empty i n t e r i o r . I n t h i s c a s e , gm may be chosen t o b e a c o n t i n u o u s l i n e a r form. Thus, f
a
is
2-homogeneous p o l y n o m i a l . T h i s i s t o b e compared t o Example 20, where f i s a l s o a 2-homogeneous p o l y n o m i a l . The f o l l o w i n g
ex-
ample is t h e n i n o r d e r ( b u t X c o u l d n o t b e a m e t r i z a b l e l o c a l l y convex s p a c e which is n o t a normable s p a c e , i n i t ) .
EXAMPLE 22.
W e now show t h a t E may f a i l t o b e a h o l o m o r p h i c a l
l y b o r n o l o g i c a l s p a c e , and y e t have t h e f o l l o w i n g p r o p e r t y : f o r e v e r y U and e v e r y F, w e h a v e t h a t e a c h polynomial f : E is continuous i f
p a c t s u b s e t of
+
F
( a n d always o n l y i f ) f i s bounded o n e v e r y m
u.
I n f a c t , l e t X be a n i n f i n i t e d i m e n s i o n a l o q
p l e x normed s p a c e Y = C ( N )
and E = X x Y . L e t f : E
+
F be a pg
l y n o m i a l t h a t i s bounded on e v e r y compact s u b s e t of U .
It
is
enough t o t r e a t F as b e i n g seminormed, and U a s b e i n g V
x W
,
where V C X I W
cY
a r e open and non-void.
W e w r i t e , f o r x E X,
Y E y,
f(x,y) =
c
ga(x)ya
a
u n i q u e l y , where a i s any s e q u e n c e of p o s i t i v e i n t e g e r s a l l b u t f i n i t e l y many o f which are z e r o , and each g a : X nomial. I f y
F i s a poly-
-
X -+ F by by f ( x ) = f ( x , y ) for Y' Y Hence x E X . Each ga i s a f i n i t e l i n e a r c o m b i n a t i o n of t h e f Y' ga i s bounded on e v e r y compact s u b s e t of V . S i n c e X i s a normed E
Y,
define f
+
s p a c e , t h e n ga i s c o n t i n u o u s on X , h e n c e bounded on e v e r y born+ ed subset of X. S e t
ca = SUP{II g c l ( x )II and
E
b e a sequence
Em
> 0
:
I I X l I I
1)
( m E IN) s u c h t h a t
BARROSO, MATOS
46
11 5 11 f(x,y) 11 5
then
1 and Iyml
x
5
E~
&
NACHDIN
f o r e v e r y m E IN imply t h a t
1, t h a t i s , f is bounded on a neighborhood f o r 0
i n E . Hence, f i s c o n t i n u o u s . We now show i n Example 26 below t h a t i t is n o t
t o u s e only F
= C i n Definition
emugh
1; see however P r o p o s i t i o n s 5 4
and 76 below. To t h i s e n d , w e s h a l l need t h e f o l l o w i n g r e s u l t s .
A*(il,..
.,im)= A ( e i , .. .,e i 1
d o t ill
...,imE
I , w h e h e m E IN. T h e n A,
E
m
m c o ( I 1;
i n pahtieu-
v a n i h h e b o d d a denurnetabee A u b o e t 0 6 Im.
h h , A,
F o r m = 0 , t h e lemma i s t r u e , s u b j e c t t o t h e conven
PROOF.
t i o n that co(Io) i s reduced t o 0 . L e t m > 1. I n case m sumed t h a t t h e lemma i s t r u e for m-1 x1
, .. .,xm E
(1)
A(xlf
E,
..
2, a s
t o a r g u e by i n d u c t i o n . I f
then -
.,Xm)
c
2
-
A*(il,..
i l l. . . , i m E I
.,im)
xli
. . . xm i
m
where t h e series i s c o n v e r g e n t by p a r t i a l summation o v e r
all
f i n i t e subsets of Im t h a t a r e C a r t e s i a n p r o d u c t s . W e must prove t h a t , f o r every
E
. ..,im) I 1.
IA, (ill
> 0 , t h e s e t of t h e ( i l l. E
. .,i,)
E I
m
f o r which
h a s t o b e f i n i t e . Assume t h a t t h i s s e t
is
ON HOLOMORPHY VERSUS LINEARITY
i n f i n i t e f o r some
. Let
E
t h e n ( i l n , ...,imn) € Imbe
d i s t i n c t , and s u c h t h a t \ A * ( i l n , . . . , i m n ) l F o r each f i x e d h
=
1,.
47
.. , m ,
E
pairwise
f o r n=1,2,.
w e must h a v e t h a t ihn+
.. .
as n
meaning t h a t e v e r y f i n i t e s u b s e t of I c o n t a i n s ihn f o r
,
+
only
f i n i t e l y many v a l u e s of n; t h i s i s c l e a r i f m = 1, and i f m
1. 2
t h i s f o l l o w s from t h e a s s u m p t i o n t h a t t h e l e m m a h o l d s for m-1. By p a s s i n g t o s u b s e q u e n c e s , w e may assume, f o r e a c h f i x e d h = 1, . . . , m ,
2
t h a t t h e ihna r e p a i r w i s e d i s t i n c t . I n case m
2,
i n view of t h e a s s u m p t i o n t h a t t h e lemma h o l d s f o r l , . . . , m - 1 , and by p a s s i n g t o s u b s e q u e n c e s , we may a l s o assume i n d u c t i v e l y that
1
-
m 'V
for n
2
2n
kl...km
2, where summation is o v e r a l l k l ,
..., km E
{l,*..,d ,
o n e a t l e a s t b u t n o t a l l of them b e i n g e q u a l t o n. I n case m = 1, t h e above s t e p of t h e r e a s o n i n g i s t o b e a b o l i s h e d . i n e xl,.
.. , xm tkl..
f o r kl,
€
E i n d u c t i v e l y as f o l l o w s . S e t
.km
-
... ,km -- 1,2,..., 'n
and -
-
' tkl..
.km
where summation i s over a l l k l , . . . , k m Then r e q u i r e :
Deg
E
{l,...,n~ f o r n
2
1.
BRRROSO, MATOS
48
1)
f o r h = 1,
2,
tn...n
3)
xhi
...,m
&
NACHBIN
and k = 1,2,
...
h a s the same argument as s
= 0 f o r h = 1,
...,m
~ f o-r n~ 2 2 ,
and t h e r e m a i n i n g i E I .
W e have
It,.
, we
and, by u s i n g ( 2 )
bnI ? proving t h a t sn
-+
..nl
> E/n,
C l/h
h=l m,
E
r
get
n E
2
Is11
n
-
C
h= 2
,
1/2h
a g a i n s t s n * A(xl , . . . , x m ) a s n *
m,
b y (1).
QED
S e t E = co(I). Let
DEFINITION 2 4 .
z
b e t h e t o p o L u g y on
E
d e i i n e d b y d h e BULL supkernurn n o t m x E E
wheheas L e t
+
IjxII
= sup \ X i \
iEI
E
IR
be t h e t o p o l o g y o n E d c { i n c d b y t h e 6 a m i d y
t h e dcnurnembl e hupkemum s ~ m i v i o t r n s
06
19
ON HOLOPIORPHY VERSUS LINEARITY
The f o l l o w i n g r e s u l t i s due t o J o s e f s o n [ 4 ] .
1 6 E = c o ( I ) and U c E
LEMMA 2 5 .
z
hence dotr
, then
t h a t we endow E w i t h
in n a n v a i d a n d a p e n d o h 8 ,
%(u)
i n t h e name r r e g a h d L e n n
8 oh
c.
06
t h e duct
I n t h e f o l l o w i n g , an i n d e x J d e n o t e s t h a t w e a r e
PROOF.
t a k i n g a c o n c e p t w i t h r e s p e c t t o t h e seminorm o n E d e f i n e d b y t h e denumerable s u b s e t J of I; w h e r e a s l a c k o f t h a t i n d e x means t h a t w e a r e u s i n g the f u l l supremum norm (see D e f i n i t i o n 2 4 ) . I t i s enough t o c o n s i d e r f E
f E
z
%(U)
(m
E
for
8 . Fix
5
for
%(U) U.
E
7 and c o n c l u d e t h a t
P u t Am = d m f ( S ) E d l ( m E )
for
T h e r e is E > 0 s u c h t h a t BE ( 5 ) c U and
IN).
u n i f o r m l y f o r x E B E ( 5 ) . Moreover,, (1) h o l d s t r u e p o i n t w i s e l y o n t h e l a r g e s t 6-balanced
subset U
5
o f U . From t h e Cauchy-Hadg
mard f o r m u l a , it f o l l o w s t h a t
i s bounded. By Lemma 2 3 , t h e r e i s a d e n u m e r a b l e s u b s e t J o f such t h a t
)I
,
nuous f o r
11
Am
=
I/
Am
11
I
a n d , i n p a r t i c u l a r , Am i s c o n t L
f o r a l l m E IN. I t f o l l o w s t h a t
( m = 1,Z , . . . I i s bounded. , S i n c e U i s o p e n f o r
l a r g e enough and
and
Em*
11
Am
11
E
J
8 , we
may assume t h a t J
i s s u f f i c i e n t l y s m a l l so t h a t BJE
(5) C
is U
( m F IN) is bounded. Then, (1) h o l d s n o t only
50
BARROSO, PIATOS 8r IJACHBIN
p o i n t w i s e l y on B J E ( C ) C Us b u t a l s o uniformly on B JE/~(') proving t h e claim. QED
L e t E b e a complex v e c t o r s p a c e . A s s u m e t h a t
EXAMPLE 2 6 .
and
z
8
a r e two l o c a l l y convex t o p o l o g i e s on E such t h a t :
1.
4
Cz
2.
3
and
d # 'L:
and
z
;
have t h e same compact s u b s e t s o f E ,
hence
t h e same bounded s u b s e t s of E . 3.
f o r every nonvoid s u b s e t U C E
,
hence f o r
then
that w e endow E with 4. with
i s t h e same r e g a r d l e s s of t h e f a c t
Pf&(LJ)
d
or
,
t h a t i s open f o r
z.
E is holomorphically b o r n o l o g i c a l when i t i s endowed
T. Then, i f E is endowed w i t h
& ,
w e c l a i m t h a t E is not
b o r n o l o g i c a l , hence n o t holomorphically b o r n o l o g i c a l . f o r every nonvoid s u b s e t U C E t h a t i s open f o r tion f: U
-+
%(U)
C belongs to
8 , each
ping I : ( E ,
$C
s)
+
funs
i f f is a l g e b r a i c a l l y holomor-
p h i c , and f i s bounded on every compact s u b s e t of U . E endowed w i t h
However,
In
fact,
is n o t b o r n o l o g i c a l s i n c e t h e i d e n t i t y (El
c) i s
map-
l i n e a r , and i t maps bounded s u b s e t s
i n t o bounded subsets, but i t is n o t continuous. Now l e t U C E b e nonvoid and open f o r
8 ,
and l e t f : U
-t
C be a l g e b r a i c a l l y
holomorphic and bounded on every s u b s e t of U which i s compact for
8 . Since
U i s open f o r
,
and f is a l g e b r a i c a l l y holomog
p h i c and bounded on every s u b s e t of U which i s compact f o r then f f
E
E
%(U)
%(U)
i f E i s endowed w i t h '1:. I t follows t h a t
i f E is endowed w i t h
8 . An
i n s t a n c e of t h i s s i t u a -
,
ON HOLOMORPHY VERSUS L I N E A R I T Y
51
3
t i o n is t h e f o l l o w i n g . Take a nondenumerable s e t I , and u s e and
z of
D e f i n i t i o n 2 4 o n E = c o ( I ) . Then, a l l t h e a b o v e f o u r
c o n d i t i o n s c a n b e checked; t h e t h i r d c o n d i t i o n f o l l o w s from Lemma 25.
REMARK 2 1 .
Example 2 6 a l s o shows t h a t i t i s n o t enough t o u s e
o n l y F = Q: i n D e f i n i t i o n 3 v i a (lb) o r ( l c ) o f Lemma 2 . However, i n t h i s case there a r e simpler c l a s s i c a l conditions.
3.
HOLOMORPHICALLY BARRELED SPACES
A g i v e n E i a a "haComohphicaLLy b a h t e l e d
DEFINITION 28.
n p a c e " ib,
60.t
tian
%(U;F) i b ampLy bounded id ( a n d alwayn o n t y id)
C
ewehy U and e v e h q F, w e h a v e t h a t each c o t l e c -
in baunded o n ewehy 6 i n i . t e REMARK 2 9 .
dimenaionat compact aubaet ad
x
U.
I t w i l l f o l l o w from P r o p o s i t i o n 38 below t h a t , i n
D e f i n i t i o n 28 and i n s i m i l a r s i t u a t i o n s , it i s e q u i v a l e n t t o c o n s i d e r o n l y t h e a f f i n e o n e d i m e n s i o n a l compact s u b s e t s o f U . REMARK 30.
x
C E ( U ; F ) i s amply bounded i f and o n l y i f
35
i s e q u i c o n t i n u o u s and bounded a t e v e r y p o i n t o f U . Thus DefinL t i o n 28 may b e r e p h r a s e d by r e q u i r i n g t h a t
is equicontinuous
i f it i s bounded on e v e r y f i n i t e d i m e n s i o n a l compact s u b s e t o f U;
t h e n t h e r e i s no "and a l w a y s o n l y i f " .
REMARK 33.
below m o t i v a t e s t h e above d e f i n i t i o n , b u t w e need
t h e f o l l o w i n g p r e l i m i n a r y m a t e r i a l which is known.
L E W 31.
Foh a
given E, t h e da&Lowing canditionn a t e equiva-
BARROSO, llATOS
52
& NACHBIN
Lent: F o t e w e t y F, w e h a v e t h a t each c o L L e c t i o n %
( lp)
i n ampLy b o u n d e d , ways o n l y id) X ( IC)
F
O
04
c
L(E;F)
e q u i w a l e n t L y e q u i c o n t i n u o u n , iQ ( a n d
i n bounded a t evehy p o i n t
04
aL-
E.
35 C L( E ; F ) i n
e~ w e t y F, w e hawe t h a t e a c h c o L L e c t i o n
amply b o u n d e d , o t e q u i v a L e n t C y e q u i c o n t i n u o u n , iQ ( a n d aLwayn
o n L y id)
i n b o u n d e d o n e w e t y Q i n i R e d i m e n n i o n a l c o m p a c t nub-
net ad E
W e s h a l l prove t h e following implications
PROOF.
(lp) =>
( 2 ) . L e t cx b e a seminorm o n E t h a t i s
t h e c o l l e c t i o n of t h e continuous linear
semicontinuous. C a l l
h r m s f on E such t h a t I f ( x ) I 5 a ( x ) f o r a l l x
x
E E.
Since
uous, by ( l p ) (2)
=>
By
E E.
theorem, w e h a v e a ( x ) = s u p { I f ( x ) 1 ;
Hahn-Banach
lower-
the
f F Z } for all
i s bounded a t e v e r y p o i n t of E , i t i s W C O n t i E
. It
f o l l o w s t h a t cx i s c o n t i n u o u s .
.
(lc) L e t
5 C 6, ( E ; F )
b e bounded o n e v e r y
f i n i t e d i m e n s i o n a l compact subset o f E , h e n c e a t e v e r y p o i n t of E. I f 6 E C S ( F ) , then a ( x ) = sup{P[f(x)];
f E
x
1
d e f i n e s a l o w e r s e m i c o n t i n u o u s seminorm cx o n E . By ( 2 ) tinuous. I t follows t h a t
(lc)
=>
(lp)
5 is
. Let x c
for x E E
,
a i s mg
equicontinuous a s B is arbitrary. ( E ; F ) b e bounded a t e v e r y
ON HOLOHORPHY VERSUS LINEARITY
5
p o i n t o f E . Thus
53
is 'bounded o n e v e r y f i n i t e d i m e n s i o n a l s i g
p l e x , h e n c e o n e v e r y f i n i t e d i m e n s i o n a l compact s u b s e t , o f E . BY
(IC)
, 5 is equicontinuous
.
The p r o o f c a n also b e c a r r i e d o n w i t h t h e same r e a s o r i n g , by r e v e r s i n g t h e a r r o w s . QED The f o l l o w i n g d e f i n i t i o n is c l a s s i c a l i n terms o f
particularly
(lb) o r ( 2 ) . A g i v e n E is a " b a h h e l e d n p a c e "
DEFINITION 32.
t h e e q u i v a l e n t condi.tionh REMARK 33.
,
06
.id it saLL5,$Le~
Lemma 3 1 .
D e f i n i t i o n 28 was f o r m u l a t e d i n a n a l o g y t o D e f i n i -
t i o n 32 t r o u g h ( l c ) , r a t h e r t h a n ( l p ) , o f Lemma 31. The r e a s o n
i s t h a t , by a c l a s s i c a l example, it c a n o c c u r t h a t a s e q u e n c e f,
E
(C)
(m
E IN)
i s bounded a t e v e r y p o i n t o f C , and y e t i t
f a i l s t o b e bounded on some compact subset of C , t h a t i s ,
it
i s n o t l o c a l l y bounded.
A h o l o m o h p h i c n k ' l y b u h h e l e d Apace
PROPOSITION 3 4 .
is
alno
u
batixeled b p a c e .
PROOF.
I t s u f f i c e s t o compare D e f i n i t i o n s 28 and 32, b y u s i n g
( E ; F ) C I f e ( E ; F ) . QBl
( l c ) of Lemma 31, and by r e m a r k i n g t h a t
Foa a g i w Q n E t o b e a h o l o m o t p h i c a l k ' y baa
PROPOSITION 35.
aei'ed s p a c e , i t LA n e e e s s a h y and h u h d i c i e n t t h a t , we h a v e t l i a t e a c h c o l l e c t i o n (and always o n l y
compact s u b s e t
06
C%&(U)
i6
,504
-
e v e k y U,
LocaLL'y b o u n d e d
id
i d ) & LA bounded o n evetry d i n i t e d i m e n b i o n d U.
BARROSO, IUlTOS
54
& NACHBIN
N e c e s s i t y b e i n g c l e a r , l e t us p r o v e s u f f i c i e n c y . L e t
PROOF.
cd
(U;F) b e bounded on e v e r y f i n i t e d i m e n s i o n a l compact
s u b s e t of U . Given any R E CS(F) , l e t
3
be t h e c o l l e c t i o n
of
t h e l i n e a r forms $ on F s u c h t h a t I $ ( y ) I < @ ( y )f o r a l l y E F. By t h e Hahn-Banach t h e o r e m , w e h a v e t h a t B ( y ) = s u p { f o r a l l y E F. S i n c e t h e c o l l e c t i o n $ E
8
and f E 5
,
a.
REMARK 3 6 .
8
of a l l $ o f , where
i s bounded on e v e r y f i n i t e d i m e n s i o n a l com-
pact subset of U, t h e r e r e s u l t s t h a t f o r every
7 3 o
I $ ( y ) I;$€ 1
I t follows t h a t
5
J
S i s l o c a l l y bounded,
is amply bounded. QED
I t i s known t h a t i t i s enough t o t a k e F = C i n ( l p )
o r ( l c ) o f Lemma 31, when u s i n g them i n D e f i n i t i o n 3 2 . F o r t h e c a s e of ( l c ) , P r o p o s i t i o n 35 c o r r e s p o n d s t o t h i s r e m a r k .
PROPOSITION 3 7 .
A R a i h e Apace E
i b
a h m f o m v 4 p h ~ c a k ? f ybahheted
bpUCe.
PROOF.
I t i s enough t o t r e a t F a s b e i n g a seminormed s p a c e .
W e s t a r t w i t h two c l a s s i c a l r e m a r k s . I f X i s a nonvoid B a i r e s p a c e , and
2 is
a pointwise
bounded s e t o f c o n t i n u o u s mappings o f X t o F, t h e r e i s a t least a p o i n t o f X where I f p: E
+
ai s
l o c a l l y bounded.
F i s a n m-homogeneous p o l y n o m i a l ( m E IN) and
i n f a c t , b y t h e maximum p r i n c i p l e , w e may r e p l a c e / A / 5 1 by
ON HOLOMORPHY VERSUS L I N E A R I T Y
1x1
= 1, and t h e n e q u a l i t y i s c l e a r v i a
X
+
55
1/X,
by m-homogene
i t y . In particular
11
p(b)11
5
supCII p(a+Ab) 1 1
Now, l e t
sc
;
X
\ X I 5 1).
E C,
(U;F) b e bounded on e v e r y a f f i n e
one
d i m e n s i o n a l compact subset of U, which i s t h e c a s e i f bounded o n e v e r y f i n i t e Fix 5 that
E
is
d i m e n s i o n a l compact s u b s e t o f U .
U. Take a b a l a n c e d o p e n neighborhood V of 0 i n E s u c h
5 +
V C U . By t h e Cauchy i n t e g r a l , t h e s e t
i s p o i n t w i s e bounded on V , b e c a u s e
X
i s bounded o n e v e r y a f -
f i n e o n e d i m e n s i o n a l compact subset 1 5 +
AX
:
E
1x1
< 11
o f U , where x E V . By t h e f i r s t remark a b o v e , t h e r e i s a a E V where
/u i s
l o c a l l y bounded, s i n c e V i s a nonvoid B a i r e s p a c e .
L e t W b e a b a l a n c e d n e i g h b o r h o o d of 0 i n E s u c h t h a t a
and
@ is
bounded on a
+
W . By t h e s e c o n d remark a b o v e ,
+
W
/u.
bounded on W . Then T a y l o r s e r i e s e x p a n s i o n a t 5 shows t h a t i s bounded on 5
PROPOSITION 3 8 .
+
W/2. Hence
C V
is
x
i s l o c a l l y bounded. OED
C %%(U;F) is b o u n d e d o n evetry b i n i t e d i m en
b i o n a e c o m p a c t n u b s e t 0 6 U id and o n l y id
32 i n b o u n d e d o n e x
e h y a d d i n e o n e d i m e n b i o n a l compact n u b o e t a d U. PROOF.
Only s u f f i c i e n c y r e q u i r e s j u s t i f i c a t i o n . I t i s enough
t o r e s t r i c t a t t e n t i o n t o t h e case when E i s f i n i t e d i m e n s i o n a l , h e n c e a B a i r e s p a c e . Then, a n i n s p e c t i o n o f t h e p r o o f o f Prop o s i t i o n 35 g i v e s t h e argument f o r t h e p r e s e n t p r o o f . QED
BARROSO, MATOS
56
REMARK 3 9 .
ti XACHBIN
P r o p o s i t i o n s 34 and 37 imply t h e known f a c t t h a t
a B a i r e s p a c e E i s a b a r r e l e d s p a c e . P r o p o s i t i o n s 37 and
38
c o n t a i n as a p a r t i c u l a r case t h e f o l l o w i n g g e n e r a l i z a t i o n t h e c l a s s i c a l Banach-Steinhaus
PROPOSITION 4 0 . i4
of
Theorem.
( H a t o m a h p h i c B a n a c h - S t Q i n h a u n Theohem)
.
16 E
a F h e c h e t n p a c e , each c o l l e c t i o n 5 C Z ( U ; F ) i n e q u i c o n f i E
uciun id nubnet
bounded o n eveay addine one d i m e n n i o n d compact
id
U.
06
PROPOSITION 4 1 .
a h o L o r n o h p h i c a L L y bct4aeLed
A SiCva npace
npace. The p r o o f w i l l r e s t on t h e f o l l o w i n g lemma.
LEMMA 4 2 .
1n t h e notation
06
ampLy bouvtded id a n d o n L g
.id
ampLy b o u n d e d d o h eveaty m
E IN.
PROOF.
Lemma I T , t h e n
x m5
o p,
C
X
in
C &(U;F)
urn;^)
i 4
N e c e s s i t y b e i n g c l e a r , l e t us p r o v e s u f f i c i e n c y . I t i s
enough t o t r e a t F a s b e i n g a seminormed s p a c e . S i n c e e a c h
is p o i n t w i s e bounded, it f o l l o w s t h a t too. Consider g: U
x E U and f E
-f
X,
i s p o i n t w i s e bounded
l w ( 2 & ; F ) d e f i n e d by g ( x ) ( f ) = f ( x ) f o r
x . Since
each
bounded, w e see t h a t g o pm: Um
xmC % (U,;F) -f
L"(5;F)
is l o c a l l y
i s holomorphic
for
e v e r y m E IN. By Lemma 11, w e c o n c l u d e t h a t g i s h o l o m o r p h i c . Thus, q i s l o c a l l y bounded, t h a t i s , 2 i s l o c a l l y bounded. OED
REMARK 4 3 .
Lemma 4 2 may b e p r o v e d d i r e c t l y , by a r e a s o n i n g
q u i t e c l o s e t o t h a t o f t h e p r o o f o f Lemma 11, s e e Lemma 3,
ON HOLOMORPHY VERSUS L I N E A R I T Y
[l]
. Notice
4 2 when
57
t h a t Lemma 11 i s n o t t h e p a r t i c u l a r c a s e of Lemma
3E i s r e d u c e d t o o n e e l e m e n t , a s t h e n Lemma 4 2 i s t r i -
vial.
PROOF OF P R O P O S I T I O N 4 1 .
C o n s i d e r t h e s e q u e n c e (Ern) o f Def-
i n i t i o n 8, and u s e n o t a t i o n o f Lemma 4 2 . L e t 2E C % (U;F) bounded on e v e r y f i n i t e d i m e n s i o n a l compact subset o f U .
Then
m ;F) i s bounded on e v e r y f i n i t e d i m e n s i o n a l compact
% C g ( U
s u b s e t of Urn. By P r o p o s i t i o n 37 ( o r e l s e 4 0 1 , bounded f o r e v e r y m E IN. By Lemma 4 2 ,
E
E mi s amply
i s amply bounded. QED
Lemma 4 2 i s a r e m i n i s c e n s e o f t h e known f a c t t h a t ,
REMARK 4 4 .
i f Ei(i
be
I) i s any f a m i l y of l o c a l l y convex s p a c e s , E
v e c t o r space, pi
: Ei
-t
is
a
E ( i E I ) i s a l i n e a r mapping, E i s el!
dowed w i t h t h e i n d u c t i v e l i m i t t o p o l o g y , and F i s a l o c a l l y convex s p a c e , t h e n a c o l l e c t i o n
35C d ( E , F ) i s amply bounded,
o r e q u i v a l e n t l y e q u i c o n t i n u o u s , i f and o n l y i f
c
E
Z€ o p i
i s amply bounded, o r e q u i v a l e n t l y e q u i c o n t i n u o u s ,
(Ei:F)
f o r every i
Zi
E
I . Lemma 4 2 may b r e a k down i n a b s e n c e o f compact
n e s s (see Example 65 below) o r d e n u m e r a b i l i t y ( s e e Example 6 6 below) c o n d i t i o n s .
REMARK 45.
P r o p o s i t i o n 4 1 i s a r e m i n i s c e n s e o f t h e known
f a c t t h a t any i n d u c t i v e l i m i t o f b a r r e l e d s p a c e s i s a b a r r e l e d s p a c e . A d e n u m e r a b l e i n d u c t i v e l i m i t whose c o n n e c t i n g mappings
u a r e n o t compact (see Example 65 b e l o w ) , o r a non-denumerable i n d u c t i v e l i m i t w i t h compact c o n n e c t i n g mappings u (see Example 6 6 below)
of h o l o m o r p h i c a l l y b a r r e l e d s p a c e s may f a i l t o b e a
holomorph c a l l y b a r r e l e d s p a c e , o r e v e n t o b e a h o l o m o r p h i c a l l y
BARROSO, 1LhTOS
58
&
NACHBIN
i n f r a b a r r e l e d s p a c e i n t h e s e n s e of t h e n e x t s e c t i o n .
4 . HOLOMORPHICALLY INFRABARRELED SPACES A g i v e n E i n a "holomohphicalLy i n , j x a b a ~ ~ f i e f o d
DEFINITION 4 6 .
n p a c e " id, doh ewehy
tAon ~ C % ( U ; F )
Ah
u and
-
amply bounded ih ( a n d a l u ~ a y h o n l y id]
in bounded o n e v e h y compact REMARK 47.
e v e a y F , we h a v e t h a t each c o l l e c
hubhet
0 6 U.
F o r t h e r e a s o n g i v e n i n Remark 30, D e f i n i t i o n 46
may b e r e p h r a s e d by r e q u i r i n g t h a t
is equicontinuous if it
i s bounded o n e v e r y compact s u b s e t of U ; t h e n t h e r e is no "and always o n l y if". Remark 50 below m o t i v a t e s t h e above d e f i n i t i o n , b u t
w e n e e d t h e f o l l o w i n g p r e l i m i n a r y m a t e r i a l which i s known. F o h a g i w e n E , t h e ~ o ~ l o w i ncgo n d i t i o n n a h &
LElllMA 4 8 .
equi-
uaLent: (Ibl
F o h e u e h y F, we h a v e t h a t each c a l e e c t i o n X c L ( ( e ; F )
in amply bounded, ox e q u i w a e e n t l y equicon-t.inuoun, id ( a n d a l wayh o n l y i d 1 5 [Ic)
Fox euehy F, we h a v e t h a t each c o l e e c t i o n X c d : ( E ; F )
in ampLy bounded waqn o n l y id1
(2b)
i n bounded o n eueay bounded h u b h e t od E
ah
e q u i v a l e n t l y e q u i c o n t i n u o u b , Ad ( a n d a L -
g i h bounded on euehy compact
hubnet
06
E.
Each heminohm u o n E i4 c o n t i n u o u b id ( a n d a l w a y b
o n L y id 1 u i n ~ o w e h h e m i c o n t i n u o u h and bounded o n e u e h y bounded nubnet
06
(Zc]
E.
Each heminohm u o n E i n c o n t i n u o u n id ( a n d a l w a y n
59
ON HOLOMORPHY VERSUS L I N E A R I T Y
W e s h a l l prove t h e following implications
. This
-->
(lb)
-
(2b). Let
>
i s c le a r. c1
b e a seminorm on E t h a t i s lowersemi
c o n t i n o u s and bounded on e v e r y bounded subset o f E . C a l l
the
c o l l e c t i o n o f t h e c o n t i n u o u s l i n e a r forms f on E s u c h t h a t
I f ( x ) I -<
a (x) f o r a l l x E E . By t h e Hahn-Banach theorem, w e
have a ( x ) = sup{ ( f ( x )I ; f
1 for a l l x
E
E E.
Since 2 i s
bounded on e v e r y bounded s u b s e t o f E , i t i s e q u i c o n t i n u o u s , by
(lb)
. It
(2b)
follows t h a t =>
i s continuous.
c1
( 2 ~ ) .L e t a b e a seminorm on E t h a t i s lowersemi
c o n t i n u o u s and bounded on e v e r y compact s u b s e t of E; t h e n
is
c1
a l s o bounded on e v e r y bounded subset o f E ( s e e t h e same s t e p i n t h e p r o o f of Lemma 2 ) (2c)
=>
(lc)
. Let
p a c t s u b s e t o f E . If D f E
1 for x
E E
. By
( 2 b ) , a is c o n t i n u o u s .
X c I(E;F) E CS(F),
b e bounded on e v e r y corn
then a ( x )
=
-
:
sup{p[f(x)]
d e f i n e s a lowersemicontinuous seminorm a on
E t h a t i s bounded on e v e r y compact subset of E . By ( 2 c )
continuous. I t follows t h a t
,
ff
is
i s equicontinuous as B i s a r b i -
trary. The p r o o f can b e a l s o c a r r i e d on w i t h t h e same r e a s o n i n g , by r e v e r s i n g t h e a r r o w s . QED
BARROSO, -MATCIS
60
&
NACHBIN
The f o l l o w i n g d e f i n i t i o n i s c l a s s i c a l , p a r t i c u l a r l y i n t e r m s of
( l b ) or ( 2 b )
DEFINITION 49. hatin6ieh
REMARK 5 0 .
.
A given E
i b
i d
a n " i n ~ ~ . a b a ~ ? c hl pe adc e "
t h e e q u i v u f e n t cond.i.tionn
0 6 Lemmu
it
4b.
D e f i n i t i o n 46 w a s f o r m u l a t e d i n analogy t o D e f i n i -
t i o n 4 9 t h r o u g h ( l c ) , r a t h e r t h a n ( l b ) , o f Lemma 4 8 . The reason
i s t h e same g i v e n i n Remark 4 . A h ~ d ~ m o ~ ~ p h i c ai nLddhya b u 4 k e d e d b p a c e
PROPOSITION 5 1 .
&o
an i n 6 ~ . a b a h ~ . e L es~pda c e .
PROOF.
I t s u f f i c e s t o compare D e f i n i t i o n s 4 6 a n d 4 9 , u s i n g
( l c ) o f Lemma 48, a n d b y r e m a r k i n g t h a t
(E;F) C
(E;F).
BED
F O R . a g i v e n E RO b e a h ~ d ~ m ~ h p h i c a il nl dyh a -
PROPOSITION 5 2 .
bahteled hpace,
u,
it
i h
n e c e h a u h y a n d o u 6 d i c i e n R t h a t , doh e u e h y
we have t h a t e a c h coLdectian
X
C (U)~ i b
id ( a n d a d w u y i i a n d y i d ) S i s b o u n d e d a n oh
L o c a l l y bounded
elte4y
compact hubnet
u.
PROOF.
T h e a r g u m e n t i s s i m i l a r t o t h a t of t h e p r o o f o f P r o p o -
s i t i o n 35. OED
REMARK 5 3 .
in
I t i s known t h a t it i s enough t o t a k e F = C
(lb) o r ( l c ) of Lemma 4 8 , when u s i n g t h e m i n D e f i n i t i o n 4 9 . F o r t h e case of remark.
( l c ) , P r o p o s i t i o n 52 c o r r e s p o n d s
to
this
61
ON HOLOMORPHY VERSUS L I N E A R I T Y
Foh E t o b e a h o l o m o h p h i c a L L q b o h n o L a g i c u L
PROPOSITION 5 4 .
n p a c e it i n n e c e n s u h q and s u 6 6 i c i e n t f h c i t E b e a h o L o m o f i p h i c u L
L y indhabahkeled n p a c e , and mofiLcuv?.t t h a t , d o h e v e h q U , w e have
u
t h u t euch 6 u n c t i o n f :
+
c
be1’nvigb
to
$$(u) i d
un-Q!i il;) f i n
aCgebxuicaLLq h a l o m u h p h ~ c , a n d
evehy compact
hubbet
PROOF.
06
( u n d uPu~!qr,
€ i n bounded
OM
U.
L e t us p r o v e n e c e s s i t y , and assume t h a t E is a holomog
phically bornological space. L e t
5C
(U;F) b e bounded o n
i s pointwise
t h e compact s u b s e t s o f U . I t f o l l o w s t h a t
bounded t o o . I t is enough t o t r e a t F a s b e i n g a seminormed space. Consider g: U
x
E
U and f E
2
+
P”(
. Since
X
;F) d e f i n e d by g ( x ) ( f )
= f(x) for
i s bounded on the compact subsets of U, i t
f o l l o w s t h a t g i s bounded on t h e compact subsets o f U .
In
Dar
t i c u l a r , g l (U flS ) i s l o c a l l y bounded f o r e v e r y f i n i t e dimens i o n a l v e c t o r s u b s p a c e S o f E m e e t i n g U ; h e n c e g i s algehraicall y h o l o m o r p h i c . S i n c e E i s a h o l o m o r p h i c a l l y b o r n o l o a i c a l space, t h e n g i s h o l o m o r p h i c , h e n c e l o c a l l y bounded. I t f o l l o w s t h a t
6
i s l o c a l l y bounded. T h i s shows t h a t E i s a h o l o m o r p h i c a l l y
i n f r a b a r r e l e d s p a c e . The rest of n e c e s s i t y i s c l e a r . L e t p r o v e s u f f i c i e n c y , and assume t h a t f : U
-+
us
F is algebraically
h o l o m o r p h i c and bounded on e v e r y compact s u b s e t o f U . F o r any fixed B E CS(F) , l e t on F s u c h t h a t
I$ (y) I
b e t h e c o l l e c t i o n of t h e l i n e a r forms $
5 6 (y) f o r a l l y
E F . Each s u c h )I
0
f is
a l g e b r a i c a l l y h o l o m o r p h i c and bounded on t h e compact s u b s e t s of U;
t h u s i t i s h o l o m o r p h i c . Moreover,
bounded on e v e r y compact s u b s e t of U .
2
Thus
‘7J60
x is
f C %(U)
is
locallybound&
s i n c e E i s a h o l o m o r p h i c a l l y i n f r a b a r r e l e d s p a c e . By t h e Hahn-
BARROSO, PIATCIS
62
& NACHBIN
-Banach t h e o r e m , w e h a v e B ( y ) = $up[ I J , ( y ) I ; J, E
11
for a l l
y E F . I t f o l l o w s t h a t B o f i s l o c a l l y bounded. Thus f i s am p l y bounded. S i n c e f is a l s o a l g e b r a i c a l l y h o l o m o r p h i c ,
it is
h o l o m o r p h i c . Thus E i s a h o l o m o r p h i c a l l y b o r n o l o g i c a l space. QED
REMARK 55.
I t i s known t h a t , f o r E t o b e a b o r n o l o g i c a l s p a c e
i t is n e c e s s a r y and s u f f i c i e n t t h a t E b e a n i n f r a b a r r e l e d s p a c e , and moreover t h a t e a c h f u n c t i o n f : E
-+
C belongs t o E'
i f ( a n d a l w a y s o n l y i f ) f i s l i n e a r , and f i s bounded on e v e r y bounded, o r compact, s u b s e t of E . P r o p o s i t i o n 54 c o r r e s p o n d s t o t h e s e c o n d h a l f o f t h i s remark.
DEFINITION 5 6 .
A g i v e n E had t h e "E.lonRel p h o p e h t y "
id,
doh
e.vehy U and e v e h y F, w e h a v e t h a t each c o l l e c t i o n E c H(U;F) h e l a t i v e l y compact doh
i d
To id
land a l w a y s o n l y id1
X
bounded on e v e h y d i n i t e d i m e n s i o n a l compact h u b b e t 0 6 U, g(x) C F
REMARK 57.
i d
i d
and
h e l a t i v e t y cornpact d o h e v e h y x E U.
The t e r m i n o l o g y i n D e f i n i t i o n 56 comes, o f c o u r s e ,
from t h e c l a s s i c a l Montel theorem s a y i n g t h a t , i f E i s f i n i t e d i m e n s i o n a l and F = C , t h e n for
yo i f and o n l y X
C
@(U)
i s r e l a t i v e l y compact
i s bounded o n e v e r y compact subset o f
U. W e s h o u l d d i s t i n g u i s h between Montel p r o p e r t y o f D e f i n i t i o n
56 and by now c l a s s i c a l Montel p r o p e r t y of E r e q u i r i n g t h a t
ev
e r y bounded s u b s e t o f E b e r e l a t i v e l y compact (see Example 67 below).
PROPOSITION 5 8 .
Foh E t o b e a holamohphicalLy b a h h e l e d h p a c e
ON HOLOMORPHY VERSUS LINEARITY
63
i-t i n n e c e b b a h y a n d n u 6 ~ i c i e n - tthat E b e a h o L o m a t p h i c a 1 1 y i n d k a b a h t e t e d s p a c e , and m 0 5 e o u e t that E had -the M o n t e 1 phope/r -tY*
PROOF.
L e t us p r o v e n e c e s s i t y , and assume t h a t E i s holomor-
p h i c a l l y b a r r e l d . Then, c o m p a r i s o n o f D e f i n i t i o n s 28 and 4 6
let
shows t h a t E i s h o l o m o r p h i c a l l y i n f r a b a r r e l e d . N o r e o v e r ,
x
c H(U;F) c
(U;P)
b e bounded on t h e f i n i t e d i m e n s i o n a l
compact subsets o f U. Then, 5 i s amply bounded, h e n c e e q u i c o n tinuous
. If,
i n addition,
( x ) C F i s r e l a t i v e l y compact
f o r e v e r y x E U , t h e n A s c o l i ' s theorem i m p l i e s t h a t
ZC H (U;F)
i s r e l a t i v e l y compact. Thus E h a s l l o n t e l p r o p e r t y . L e t u s t u r n
t o s u f f i c i e n c y , and assume t h a t E i s h o l o m o r p h i c a l l y i n f r a b a r r e l e d h a v i n g Plontel p r o p e r t y . I f
c a ( U ) i s bounded on ev-
e r y f i n i t e d i m e n s i o n a l compact subset o f U , t h e n
XC
z(U)
r e l a t i v e l y compact f o r
xo by
X i s bounded f o r
t h a t i s , bounded on e v e r y compact
z,
is
Monte1 p r o p e r t y ; i t f o l l o w s t h a t
sub-
s e t o f U . S i n c e E i s h o l o m o r p h i c a l l y i n f r a b a r r e l e d , t h e r e res u l t s t h a t Z&
i s l o c a l l y bounded. B v P r o p o s i t i o n 35, E
is
hg
l o m o r p h i c a l l y b a r r e l d . OED
REMARK 5 9 .
W e may t h i n k o f a v a r i a t i o n of t h e H o n t e l p r o p e r -
t y w i t h j u s t F = C; namely t h a t e a c h compact f o r
zo i f
X
C %(U)
( a n d always o n l y i f ) 2E
is r e l a t i v e l y
i s bounded on e v e r y
f i n i t e d i m e n s i o n a l compact s u b s e t of U. The proof of P r o p o s i t i o n 58 shows t h a t t h e Monte1 p r o p e r t y w i t h a r b i t r a r y F i s e q u i v a l e n t t o s u c h a v a r i a t i o n of i t w i t h j u s t F = C when E i s h o l o m o r p h i c a l l y i n f r a b a r r e l . e d . However , t h e y a r e n o t equivalent by t h e m s e l v e s (see Example 6 8 b e l o w ) .
64
BARROSO, FlATOS & NACHBIN
REMARK 6 0 .
P r o p o s i t i o n 5 8 and Remark 59 c o r r e s p o n d t o t h e
f o l l o w i n g known f a c t s . F o r E t o b e a b a r r e l e d s p a c e it i s c e s s a r y and s u f f i c i e n t t h a t E b e a n i n f r a b a r r e l e d s p a c e ,
neand
moreover t h a t , f o r e v e r y F, w e h a v e t h a t e a c h c o l l e c t i o n
X C & ( E ; F ) i s r e l a t i v e l y compact f o r i s p o i n t w i s e bounded, and
if)
x
2;,
if
(and always o n l y
( x ) C F i s r e l a t i v e l y com-
p a c t f o r e v e r y x E E . I t i s enouqh t o t a k e F = C i n t h e above s t a t e m e n t : t h e two c o n d i t i o n s on 3E. a r e e q u i v a l e n t when E is i n f r a b a r r e l e d ; however , t h e y are n o t e q u i v a l e n t by t h e m s e l v e s .
DEFINITION 6 1 .
4 given E
hub t h e
I ’ i n B k u - M o n t e L pxopektg” i6,
6 o x evetry U a n d e u e k y F , we h a v e t h u t e u c h c o L L e c t i o n C
id) 2
H(U;F) i n ? r e t a t i v P L y c o m p a c t d o h
ToL6
( a n d aLwayn o d g
06
and X ( x ) t F
bounded o n euekg compact n u b n e t
i.4
i n keLatiweLy c o m p a c t
auk
U,
e v e h q x E U.
The t e r m i n o l o g y i n D e f i n i t i o n 6 1 i s m o t i v a t e d
as
i n R e m a r k 5 7 , a n b y c o m p a r i s o n b e t w e e n P r o p o s i t i o n s 58 a n d
63
below. I t i s c l e a r t h a t E h a s t h e i n f r a - M o n t e 1 p r o p e r t y i f
it
REMARK 6 2 .
h a s t h e Montel p r o p e r t y . Except f o r t h a t , w e s h o u l d d i s t i n g u i s h b e t w e e n Montel p r o p e r t y , i n f r a - M o n t e 1 p r o p e r t y and c l a s s i c a l Monte1 p r o p e r t y ( s e e Example 6 7 below)
PROPOSITION 6 3 .
A hoCornohphicaLLy i n d h a b a k h e l e d s p a c e E
had
t h e in6ta-Mantel pfiopektg.
PROOF.
The argument i s a minor m o d i f i c a t i o n o f t h e p r o o f of
t h e c o r r e s p o n d i n g a s s e r t i o n o f P r o p o s i t i o n 58. QED
ON HOLOMORPHY VERSUS L I N E A R I T Y
REMARK 6 4 .
65
W e may t h i n k o f a v a r i a t i o n o f t h e i n f r a - M o n t e 1
p r o p e r t y w i t h j u s t F = @; t i v e l y compact f o r
yo i f
namely t h a t e a c h X c % ( U ) i s r e l a ( a n d always o n l y i f )
x
i s bounded
o n e v e r y compact s u b s e t of U . T h i s amounts t o s a y i n g t h a t e a c h x(U)h a s t h e c l a s s i c a l Monte1 p r o p e r t y f o r
zo. The
infra-Pdog
t e l property with a r b i t r a r y F i s not e q u i v a l e n t t o such a v a r i g t i o n of i t w i t h j u s t F = C ( s e e Example 6 8 b e l o w ) .
EXAMPLE 6 5 .
C o n s i d e r Example 1 8 . C a l l
t i o n of t h e f k f o r a l l k = 1,2,.
3 c %(El the collec-
. . . Then % i s
bounded on
ev-
e r y compact s u b s e t of E . Hence S I E m i s l o c a l l y bounded f o r
m f IN. However, .X i s n o t l o c a l l y bounded a t 0 , b e c a u s e f n o t l o c a l l y bounded a t 0 and f k
+
f pointwisely as k
+
00.
is This
shows t h a t Lemma 4 2 b r e a k s down i f t h e om a r e assumed t o b e l& n e a r c o n t i n u o u s , b u t n o t compact, a l t h o u g h d e n u m e r a b i l i t y o f t h e f a m i l y i s p r e s e r v e d . Such a n example a l s o shows t h a t a denumerable i n d u c t i v e l i m i t E o f h o l o m o r p h i c a l l y b a r r e l e d s p a c e s E m ( m E IN) may f a i l t o b e a h o l o m o r p h i c a l l y i n f r a b a r r e l e d
s p a c e . I n f a c t , i f Xo i s a F r z c h e t s p a c e , t h e n each Em i s a F r g c h e t s p a c e , h e n c e h o l o m o r p h i c a l l y b a r r e l e d (by P r o p o s i t i o n s
37 o r 4 0 ) . However, E i s n o t h o l o m o r p h i c a l l y i n f r a b a r r e l e d .
EXAMPLE 6 6 .
C o n s i d e r Example 2 0 . F i x a b a s i s B f o r E . F o r ev-
e r y f i n i t e s u b s e t I of B , c a l l pI t h e p r o j e c t i o n d e f i n e d by B , o f E o n t o t h e v e c t o r s u b s p a c e of E g e n e r a t e d by I . C a l l
c %(El
t h e c o l l e c t i o n o f t h e f I 5 f o pI f o r a l l s u c h I .
Then 5 i s bounded on e v e r y compact s u b s e t of E . However,
i s n o t l o c a l l y bounded a t 0 , b e c a u s e f i s n o t l o c a l l y bounded a t 0 and f I
+
f p o i n t w i s e l y a s I i n c r e a s e s . T h i s shows
that
BARROSO, MATOS
66
& NACHBIN
Lemma 4 2 b r e a k s down i n a b s e n c e of d e n u m e r a b i l i t y o f t h e fami-
-
l y , a l t h o u g h compactness o f t h e c o n n e c t i n g mappings u i s p r e s e r v e d . Such a n example a l s o shows t h a t a non-denumerable
in-
d u c t i v e l i m i t o f h o l o m o r p h i c a l l y b a r r e l e d s p a c e s may f a i l t o b e a holomorphically i n f r a b a r r e l e d s p a c e , even i f t h e connect i n g mappings u a r e compact.
EXAMPLE 6 7 .
An i n f i n i t e d i m e n s i o n a l Banach s p a c e E h a s t h e
Montel p r o p e r t y , by P r o p o s i t i o n s 3 7 o r 4 0 , and 5 8 . However, E f a i l s t o h a v e t h e c l a s s i c a l Montel p r o p e r t y , by a t h e o r e m
of
R i e s z . C o n v e r s e l y , assume t h a t t h e l o c a l l y convex space E h a s t h e c l a s s i c a l Montel p r o p e r t y . I t may o c c u r t h a t t h e r e i s some
c %(El pact f o r
which i s bounded f o r
zo;t h e n E
To,b u t
i s n o t r e l a t i v e l y com-
does n o t have t h e infra-Monte1 p r o p e r t y .
An i n s t a n c e o f t h i s s i t u a t i o n i s d e s c r i b e d i n Example 6 5 ,
if
Xo i s assumed t o h a v e t h e c l a s s i c a l Montel p r o p e r t y . A n o t h e r i n s t a n c e o f t h e same s i t u a t i o n i s d e s c r i b e d i n Example 6 6 . F i n a l l y , l e t t h e l o c a l l y convex s p a c e E b e m e t r i z a b l e , b u t
not
b a r r e l e d . Then E i s a h o l o m o r p h i c a l l y i n f r a b a r r e l e d s p a c e , by P r o p o s i t i o n s 6 and 5 4 ; t h u s E h a s t h e i n f r a - M o n t e 1 p r o p e r t y
,
by P r o p o s i t i o n 6 3 . However, E d o e s n o t h a v e t h e Monte1 p r o p e r t y , by P r o p o s i t i o n s 3 4 and 58; i n f a c t , E i s h o l o m o r p h i c a l l y i n f r a b a r r e l e d , b u t E i s not holomorphically b a r r e l e d because
i t i s n o t b a r r e l e d . An i n s t a n c e o f t h i s s i t u a t i o n i s E = C (IN) w i t h t h e supremum norm. W e now show i n Example 68 below t h a t i t i s n o t enough t o u s e F = C i n D e f i n i t i o n s 56 and 6 1 .
ON HOLOMORPHY VERSUS L I N E A R I T Y
EXAMPLE 68.
and
z
67
&
L e t E b e a complex v e c t o r s p a c e . Assume t h a t
are two l o c a l l y convex t o p o l o g i e s on E such t h a t condi-
t i o n s 1, 2 and 3 of Example 26 a r e s a t i s f i e d , and moreover:
4.
E i s holomorphically b a r r e l e d when i t i s endowed w i t h
5.
There are a Banach s p a c e F and a c o l l e c t i o n
C
& ( E l & ) ; F ) t h a t is bounded on every compact s u b s e t of E and i s
such t h a t X ( x ) CF i s r e l a t i v e l y compact f o r every x yet
x C L(E,8
E; and
E
;F) is n o t r e l a t i v e l y compact f o r t h e p o i n t w i s e
topology. Then, i f E i s endowed w i t h
C$
, we
sat-
claim t h a t E
i s f i e s D e f i n i t i o n 56 w i t h F = Q: (see Remark 5 9 ) , hence Definit i o n 6 1 w i t h F = C (see R e m a r k 6 4 ) ; b u t E does n o t have t h e i" fra-Monte1 p r o p e r t y of D e f i n i t i o n 6 1 , hence does n o t have
the
,
Monte1 p r o p e r t y of D e f i n i t i o n 5 6 , w i t h a r b i t r a r y F . I n f a c t
l e t U b e nonvoid and open € o r
4 ,
hence
open
.
for
If
2& C a ( ( U , d ) ) i s bounded on every f i n i t e dimensional compact s u b s e t of U , t h e n Z € C @ ( ( U , ' C ) )
is bounded on every
dimensional compact s u b s e t of U . Hence for ( U , Z ) .
I t follows t h a t
5
x
i s l o c a l l y bounded
is equicontinuous f o r ( U , Z ) ,
and a l s o p o i n t w i s e bounded. By A s c o l i ' s theorem, 5 C
i s r e l a t i v e l y compact f o r v e l y compact f o r for
yo. W e t h e n s e e t h a t
for
2,.
F and
T o ;hence E c % (
$ because 86
finite
((U,
z)1 i s
Z C f 8 ( ( U , ~1 )
(
(U,
1
( U , T)1 i s r e l a t i -
closed i n & ( ( U , z 1 )
i s r e l a t i v e l y compact
This proves t h e f i r s t h a l f of t h e claim. Consider now quoted i n c o n d i t i o n 5. T h e n X C
B ( ( E , b);F)
is
bounded on every compact s u b s e t of E , and %(x) C F i s r e l a t i -
68
BARROSO, MATOS & NACHBIN
v e l y compact f o r e v e r y x E E . However, r e l a t i v e l y compact f o r
To,a s
ZC, @ ((E,d) ; F ) i s n o t
c %( (
~) ;,F )ji s n o t r e l a t i v g
c&(( E , 3);F)
l y compact f o r t h e p o i n t w i s e t o p o 1 o g y ; i n f a c t ,
i s n o t r e l a t i v e l y compact f o r t h e p o i n t w i s e t o p o l o g y , &.((El 3 ) ; F ) i s c l o s e d i n
and
% ( ( E , d ) ; F ) f o r t h a t topology.
This
p r o v e s t h e s e c o n d h a l f of t h e c l a i m . An i n s t a n c e of t h i s s i t u a t i o n i s t h e same E = co(I) w i t h t h e t o p o l o g i e s cf
and
ample 2 6 . Then, a l l t h e above f i v e c o n d i t i o n s c a n
be
of Exchecked.
L e t us v e r i f y 5 , as t h e o t h e r f o u r c o n d i t i o n s a r e c l e a r by now.
W e t a k e F = c ( I ) w i t h t h e f u l l supremum norm. F o r e v e r y
denu-
0
merable J C I , l e t f yi = x
i
Then f J
if E
: E + F be d e f i n e d by f J ( x ) = y , where j i E J and y i = 0 i f i E I - J , f o r e v e r y x E E .
J((E,d);F).
a l l such J.
X(K)
Then
t h e c o l l e c t i o n of t h e
Call
fJ
i s compact f o r e v e r y compact
s i n c e K i s b a s e d on a denumerable s u b s e t o f I . Y e t
K C El
X d ( E , & ); F )
i s n o t r e l a t i v e l y compact f o r t h e p o i n t w i s e t o p o l o g y . I n t n e i d e n t i t y mapping I : E
-+
i n the vector space
fact,
$: ( ( E l 5 )
F does n o t belong t o
b u t i t belongs t o t h e c l o s u r e of
for
IF);
f o r t h e pointwise topology
FE of a l l mappings from E
to
F
.
5 . HOLOIIORPHICALLY MACKEY SPACE
A g i v e n E i n a " h o ~ o m o t p h i c a 4 ' L g Mackeg h p a c e
DEFINITION 6 9 .
idl
box C v e h y
U
und e v e t g F , w e l i u v e t h a t each m a p p i n g f:U
b e e o n q n t o H(U;F) id
( a n d ciLwayn o n g g
p h i c , t h a t i n , d,
E a ( U )
tion:
o f
,504
+
"
F
id) f i n w e a k l y h o l o m o t -
e w e t g J, E F ' ;
i n othet
notu -
H(U;F) = H ( U ; criF). Remark 7 2 below m o t i v a t e s t h e above d e f i n i t i o n
,
but
69
ON HOLOMORPHY VERSUS LINEARITY
w e need t h e f o l l o w i n g p r e l i m i n a r y m a t e r i a l which i s known.
Foh
LEMMA 7 0 .
a g i v e n E, t h e ,3ollowing c o n d i t i o n s a4e egui-
vatent:
F o h e v e h y F , we h a v e t h a t e a c h m a p p i n g f : E
[ 1)
t o ~ ( E ; F )id [ a n d a L w a y d o n l y id) f continuouh, t h a t i n , $ o f
.L(E;F)
notation:
=
L
E El
404
i 4
+
F belbngo
l i n e a k , a n d f is weakey
evehy
J, E
F';
i n
otheh
(E;~F).
( 2 ) A l o c a l l y c o ~ v e xt o p o l o g y ;3 o n E i n smrceleh t h a n t h e g i v en t o p o t o g y
z
tinuous L i n e a h
o n E L,3 ( a n d UbUCcyO o n t y dokm4
than
on E
44
2on
The g i v e n t o p o l o g y
ha4 ,3ewck c o n -
t h e gfientent l o c a l l y
t o p o l o g y o n E among t h o h e d e d i n i n y t h e [ 2m)
3
z.
z
The g i v e n topology
129)
id)
4uwe
COMUQX
dual space E ' .
E io maximaL among t h e l o c a l l y
c o n v e x t o p o l o g i e o o n E d e d i n i n g t h e name d u a l n p a c e E ' .
(3)
The g i v e n t o p d o g y
z
on E i 4 t h e topology
04
unidohm
CUM
uehgence o n t h e u(E',E)-compact c u n u e x n u b h e t n 0 6 E l . PROOF.
(2)
(2m)
W e s h a l l prove t h e f o l l o w i n g i m p l i c a t i o n s
=> =>
(2g)
=>
( 3 ) . Call
( 2 m ) . This i s c l e a r .
3
t h e topology o f uniform convergence
o n t h e o ( E ' , E ) - c o m p a c t convex s u b s e t s of E l ; w e may r e s t r i c t a t t e n t i o n t o s u c h s u b s e t s t h a t are a l s o b a l a n c e d . W e c l a i m that
xed . I n
fact, i f V i s a 2-closed
convex b a l a n c e d neighbor-
70
BARROSO,
,
hood of 0 f o r
MATOS & NACHBIN
t h e n i t s p o l a r Vo i n E ' i s u ( E ' , E l -compact,
by t h e Alaoglu-Bourbaki
is
theorem, and a l s o convex. S i n c e V
t h e p o l a r of Vo i n E l t h e n V i s a neighborhood of 0 f o r
3-
This p r o v e s o u r c l a i m . W e n e x t c l a i m t h a t a l i n e a r form $ on E t h a t is continuous f o r
4
Z.
i s a l s o continuous f o r
In fact,
t h e r e i s a a ( E ' , E ) - c o m p a c t convex b a l a n c e d subset K C E ' t h a t 1 9 1 ~ 1)
5
such
1 i f x E KO, where KO i s t h e p o l a r o f K i n E ;
t h u s $ E KOo, where KOo d e n o t e s t h e p o l a r o f KO i n t h e a l g e b r a i c dual space E
*
of E . However, K i s b a l a n c e d ,
convex
and
u ( E ' , E ) - c o m p a c t , hence u ( E ' , E ) - c l o s e d i n E*; t h u s KOo = K
and
showing t h a t
@ E K C E',
z
and
't: =
4.
o u r c l a i m . Hence (2m)
, we
(31
have
z.
T h i s proves
4 d e f i n e t h e same d u a l s p a c e
(1). L e t f : E
=>
i s continuous f o r
+
El.
By
F b e l i n e a r and weakly c o n t i n u o u s .
We have t h e t r a n s p o s e d l i n e a r mapping t f : J, E F'
t h a t i s c o n t i n u o u s from u (F',F) t o u ( E ' , E l
. Let
-f
$ o f E E'
W b e any closed
convex b a l a n c e d neighborhood o f 0 i n F . I t s p o l a r Wo i n F' convex and
0
is
(F' ,F)-compact, by t h e Alaoglu-Bourbaki theorem.
Thus K z tf(Wo) i s convex and o ( E ' , E ) - c o m p a c t . Hence, t h e p l a r V E KO o f K i n E i s a neighborhood o f 0 i n E . Now, x E V implies
l$[f (x)] I < 1 f o r e v e r y J, E Wo,
t h a t i s f ( x ) E Woo
= W , where
i s t h e p o l a r o f Wo i n F . Thus f i s c o n t i n u o u s .
Woo
(1) =>
than
z
(2).
. Put
Let
8
h a v e fewer c o n t i n u o u s l i n e a r forms
F = ( E 1 3 1 . Then t h e i d e n t i t y mapping I : E
weakly c o n t i n u o u s . By (1), i t is c o n t i n u o u s . Thus
3C z
+
F is
.
The proof c a n a l s o b e c a r r i e d on w i t h t h e same r e a s o n i n q , by r e v e r s i n g t h e a r r o w s . OED
-
71
ON HOLOPIORPHY VERSUS LINEARITY
The f o l l o w i n g d e f i n i t i o n i s c l a s s i c a l , p a r t i c u l a r l y
i n terms of ( 2 9 ) . A g i v e n E i n a "Mackey space" id id s a t i n 6 i e n
DEFINITION 7 1 .
d h e e q u i v a l e n t c a n d i t b n b 0 6 lernmu 66. REMARK 7 2 .
D e f i n i t i o n 6 9 was f o r m u l a t e d i n a n a l o g y t o DefinA
t i o n 7 1 through
(1). A holomoxphically Mackey s p a c e is a & b a u
PROPOSITION 7 3 .
Muchey space. I t s u f f i c e s t o compare D e f i n i t i o n 6 9 and 7 1 , b y u s i n q
PROOF.
(1) o f Lemma 7 0 , and by r e m a r k i n g t h a t
d ( E ; F ) C % e ( E ; F ) . OED
A holvma~phically in6xabuxxeled space E i s u
PROPOSITION 7 4 .
halomohphically Muckey space. Let f: U
PROOF. @
o f F
+
F b e weakly h o l o m o r p h i c , t h a t i s ,
86 (U) f o r e v e r y
I t f o l l o w s t h a t f i s alqebrai-
$ E F'.
c a l l y h o l o m o r p h i c i n t h e H-sense
(not necessarily i n t h e
8% -
- s e n s e ) ; i n o t h e r words, w e a r e u s i n g h e r e t h e f a c t t h a t , i f E
i s f i n i t e d i m e n s i o n a l , t h e n it i s a h o l o m o r p h i c a l l y Plackey s p a c e , a s i t i s known. W e n e x t p r o v e t h a t f i s amply bounded. NOW, c l e a r l y f ( K ) i s weakly bounded, h e n c e bounded, i n F
e v e r y compact s u b s e t K o f U .
Thus
ed on a l l comnact s u b s e t s o f U ,
set
3
of F '
. There
results that
%=
{$ o f; $ E
1)
i s bound
f o r e v e r y s t r o n g l y bounded sub_
X
i s l o c a l l y bounded, because
E i s holomorphically i n f r a b a r r e l e d . I t follows t h a t , i f
CS(F) and
9 is
for
l3 E
t h e s e t o f a l l l i n e a r forms @ o n F s a t i s f y i n g
72
1 $. ( y ) I 2
BARROSO, MATOS
& NACHBIN
B ( y ) f o r e v e r y y E F, t h e n
t h e Hahn-Banach
i s l o c a l l y bounded. By
theorem, w e h a v e P ( y ) = s u p { I $ ( y l I ; ii, E
31
for
a l l y E F . Thus 13 o f i s l o c a l l y bounded f o r e v e r y s u c h 6 .
H e n c e f i s amply bounded. I t f o l l o w s t h a t f E H(U;F)
REPIIARK 75.
. OED
I t is known t h a t an i n f r a b a r r e l e d s p a c e i s
a
Mackey s p a c e . P r o p o s i t i o n 7 3 c o r r e s p o n d s t o t h i s r e m a r k .
PROOF.
L e t us p r o v e n e c e s s i t y .
I f E i s a holomorphically bor
n o l o g i c a l s p a c e , t h e n it f o l l o w s from P r o p o s i t i o n s 5 4 and
74
t h a t E i s a h o l o m o r p h i c a l l y Mackey s p a c e . The r e s t of n e c e s s i t y i s c l e a r . L e t us p r o v e s u f f i c i e n c y , and assume t h a t f:U
+
F
i s a l g e b r a i c a l l y h o l o m o r p h i c and bounded on e v e r y compact subs e t of U . Then $ o f i s a l g e b r a i c a l l y h o l o m o r p h i c a n d bounded on e v e r y compact s u b s e t of U , f o r e v e r y t h a t $. o f
REMARK 7 7 .
E
%(U)
)I
E F ' . It follows
f o r e v e r y such $..Hence f E H ( U ; F ) . QED
I t i s known t h a t , f o r E t o b e a b o r n o l o g i c a l s p a c e
it 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 E b e a Mackey space, and
moreover t h a t each f u n c t i o n f : E
-+
C belongs t o E ' i f
(and a l -
, sec
ways o n l y i f ) f i s l i n e a r , and f i s bounded on e v e r y bounded o r compact, s u b s e t o f E . P r o p o s i t i o n 76 c o r r e s p o n d s t o t h e
73
ON HOLOMORPHY VERSUS LINEARITY
ond h a l f of t h i s r e m a r k .
ACKNOWLJEDGEMENTS
6.
The a u t h o r s g r a t e f u l l y acknowledge p a r t i a l f i n a n c i a l s u p p o r t from FAPESP and FINEP.
B IBLIOGFiAP HY
1.
J . A . BARROSO, M .
C . MATOS & L . N A C H B I N , On bounded s e t s
of h o l o m o r p h i c mappings, P r o c e e d i n g s on I n f i n i t e Dimensio n a l Holomorphy ( E d i t o r s : T .L.
Hayden & T . J . Suf f r i d g e ) ,
L e c t u r e Notes i n Mathematics 364 ( 1 9 7 4 ) , 1 2 3 - 1 3 4 . 2.
S . D I N E E N , Holomorphic f u n c t i o n s o n l o c a l l y convex s p a c e s ,
Annales d e 1 ' I n s t i t u t F o u r i e r 2 3 ( 1 9 7 3 )
3.
,
19-54,
153-185.
S . D I N E E N , Holomorphic F u n c t i o n s o n S t r o n g D u a l s o f
Frgchet-Monte1 s p a c e s I n f i n i t e D i m e n s i o n a l Holomorphy and A p p l i c a t i o n s ( E d i t o r : M.C.
M a t o s ) , North-Holland
Mathema-
tics Studies (1977). 4.
B . JOSEFSON, A c o u n t e r e x a m p l e i n t h e Levi problem,
d i n g s on I n f i n i t e Dimensional Holomorphy Hayden & T .
J.
Proceg
(Editors: T. L.
S u f f r i d g e ) , L e c t u r e Notes i n Mathematics
364 ( 1 9 7 4 1 , 168-177. 5.
B:
JOSEFSON, Weak s e q u e n t i a l c o n v e r g e n c e i n t h e d u a l of a
Banach s p a c e d o e s n o t imply norm c o n v e r g e n c e , A r k i v f o r Mathematik 1 3 ( 1 9 7 5 ) , 79-89.
BARROSO. MATOS
74
6.
M.
&
NACHBIN
C . Matos, On L o c a l l y Convex S p a c e s w i t h t h e Monte1 P r o
p e r t y , Functional Analysis ( E d i t o r : D.
de Figueiredo)
,
Marcel Dekker ( 1 9 7 6 ) .
I.
L . NACHBIN, T o p o l o g y o n s p a c e s o f h o l o m o r p h i c m a p p i n g s Springer-Verlag
8.
,
(1969).
L. NACHBIN, A g l i m p s e a t I n f i n i t e D i m e n s i o n a l Holomorphy, P r o c e e d i n g s on I n f i n i t e D i m e n s i o n a l Holomorphy ( E d i t o r s : T.L.
Hayden & T . J .
S u f f r i d g e ) , L e c t u r e Notes i n Mathemat-
i c s 3 6 4 ( 1 9 7 4 ) , 69-79. 9.
L. NACHBIN, Some h o l o m o r p h i c a l l y s i g n i f i c a n t p r o p e r t i e s
of l o c a l l y c o n v e x spaces, F u n c t i o n a l A n a l y s i s ( E d i t o r : D . d e F i g u e i r e d o ) , Marcel Dekker ( 1 9 7 6 ) 10.
A . NISSENZWEIG, W*
s e q u e n t i a l convergence, Israel J o u r n a l
of M a t h e m a t i c s 2 2 ( 1 9 7 5 ) , 266-272.
D e p a r t a m e n t o d e Matemstica P u r a Universidade Federal d o R i o d e J a n e i r o Rio d e J a n e i r o
-
R J ZC-32
Brasil
D e p a r t a m e n t o d e Matemztica U n i v e r s i d a d e E s t a d u a l de Campinas Campinas
SP
Brasil
D e p a r t m e n t of M a t h e m a t i c s U n i v e r s i t y of R o c h e s t e r R o c h e s t e r NY 1 4 6 2 7
USA
.
Infinite Dimensional Holomorphy and Applications, Matos (ed.) @ North-Holland Publishing Company, 1977
TOPOLOGIES ON SPACES OF HOLOMORPHIC FUNCTIONS
OF CERTAIN
SURJECTIVE LIMITS
By P A U L B E R N E R
I n t h i s p a p e r w e s t u d y t o p o l o g i e s o n s p a c e s of
holomor-
p h i c f u n c t i o n s d e f i n e d i n an open s u r j e c t i v e l i m i t o f convex s p a c e s , e s p e c i a l l y s u c h s p a c e s a s
a‘
locally
(Schwartz’s
t r i b u t i o n s ) which are open compact c o u n t a b l e s u r j e c t i v e
dis-
limits
o f Dual F r e c h e t N u c l e a r s p a c e s . To do s o w e i n t r o d u c e a n i n d u c t i v e l i m i t t o p o l o g y as f o l l o w s : I f U i s a convex b a l a n c e d s u b s e t o f an open surjective l i m i t E = s u r j limaeA(E,na)
t n a ( H ( n a (U) ) ) where
(see d e f i n i t i o n 2 . 1 ) i s t h e map f
E
then H ( U ) =
H (na(U) )
+
UaEA
f o na
E
H(U).
So w e may d e f i n e a n i n d u c t i v e l i m i t t o p o l o g y on H ( U ) by t h e f o g
mula
(H(U)
,T
I)
E ind limaEA((H(na(U))
, T ~ ,) t n a ) .
If U C E
is
open and c o n n e c t e d b u t n o t convex o r b a l a n c e d , t h e n w e may have that H(U) #
UaEAtna
(H(na (U) ) 1
. For
t h i s reason w e are
t o c o n s i d e r domains s p r e a d o v e r t h e s p a c e s Ea i n s t e a d
sets v a ( U )
(see Theorem 2 . 1 ) i n o r d e r t o o b t a i n a good
t i o n of
on H ( U ) f o r a l l open c o n n e c t e d s e t s U.
T~
forced of
the
defini-
When E i s a n o n - t r i v i a l open c o n p a c t c o u n t a b l e s u r j e c t i v e l i m i t of Q@ s p alc e s , w e show t h a t -rI 75
i s a s t r i c t (LI.”)-Montel
P.
76
s p a c e and c o i n c i d e s w i t h t h e
BERNER
T~~
and
-t6
t o p o l o g i e s . W e t h e n use
?ru
i s quasi-complete,
t h i s f a c t t o show, f o r example, t h a t ?r
ob
but
i s n o t quasi-complete. I n S e c t i o n 1, w e g i v e some p r e l i m i n a r y r e s u l t s c o n c e r n i n g
domains s p r e a d . I n S e c t i o n 2 w e d e f i n e d i r e c t e d s u r j e c t i v e l i m -
i t s and t h e t o p o l o g y
The
-t1.
T~
t o p o l o g y on h o l o m o r p h i c
t i o n s d e f i n e d o n a domain s p r e a d o v e r a
@g# s p a c e
func-
is
stud-
i e d i n S e c t i o n 3 and t h e r e s u l t s a r e a p p l i e d t o g i v e o u r theorem c o n c e r n i n g t h e
-tI
main
t o p o l o g y . S e c t i o n 4 d e a l s w i t h a l l the
v a r i o u s t o p o l o g i e s f o r h o l o m o r p h i c f u n c t i o n s on a compact c o u n t a b l e s u r j e c t i v e limit o?
@g
#
non-trivial
s p a c e s and
w e con-
elude t h i s f i n a l s e c t i o n w i t h a d i s c u s s i o n o f f u r t h e r r e s u l t s .
W e s h a l l u s e t h e s t a n d a r d n o t a t i o n o f i n f i n i t e dimensional holomorphy, and 1 . c . s . w i l l always mean complex H a u s d o r f f locall y convex l i n e a r s p a c e ( s )
.
Some o f t h e s e r e s u l t s a p p e a r e d i n t h e a u t h o r ' s U n i v e r s i t y o f R o c h e s t e r Ph.D t h e s i s (1974). The a u t h o r w i s h e s t o thank D r s . S . Dineen and R .
Aron f o r t h e i r h e l p f u l comments and t o acknow-
l e d g e t h e f i n a n c i a l s u p p o r t of a Department o f E d u c a t i o n ( I r e l a n d ) Post-Doctoral Fellowship.
1.
DOMAINS SPREAD
DEFINITION 1.1
-
PRELIMINARY RESULTS
A c o n n e c t e d Hausdoh56 n p a e e R t o g e t h e t l w i t h a
Lacak homeomotphinm 0 6 R i n t o a 1.c.s. E l $ : R
a domain nptlead ( a w e t
E),
+
E,
in caLLed
and denoted b y [ R , $ , E l a t l j u n t Q .
A c o n n e c t e d non-empty open subset W c Q o f a domain spread
(R,$,E)
i s called a chatlt i f
$IM
: W
+
$(W) i s a homeomorphism.
77
TOPOLOGIES ON SPACES OF HOLOMORPHIC FUNCTIONS
L e t ( R , $ , E ) and ( C , + , F ) b e two domains s p r e a d o v e r 1.c.s.
E and F r e s p e c t i v e l y , and l e t
IT
: E
+
F be a c o n t i n u o u s
open
l i n e a r ( a n d c o n s e q u e n t l y s u r j e c t i v e ) map of E o n t o F . A c o n t i l l uous map J : R If J : R
-+
Z i s c a l l e d a IT-mokphinm i f f
C i s a IT-mohphism, tJ w i l l d e n o t e
-t
tJ : f E H ( C )
f o J
-f
E
+
o J =
the
o c$.
IT
map
H(Q).
S i n c e a Ir-morphism J i s " l o c a l l y t h e same as"
REMARK
the
c o n t i n u o u s l i n e a r map IT, i t i s e a s y t o see t h a t tJ i s w e l l d e f ined. Since
TT
i s open, J i s a l s o o p e n , so by t h e 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 i t f o l l o w s t h a t tJ i s i n j e c t i v e . I f R i s a domain s p r e a d o v e r a n l . c . s . , T*
w i l l d e n o t e , r e s p e c t i v e l y , on H ( R )
,
then
T
~ T,
~
t h e compact-open
l o g y , t h e t o p o l o g y g e n e r a t e d by semi-norms p o r t e d
by
and ,
topocompact
s e t s , and t h e t o p o l o g y g e n e r a t e d by semi-norms p o r t e d by count a b l e c o v e r s (see 1161
, [9]) ,
and
and
T~~
b o r n o l o g i c a l t o p o l o g y a s s o c i a t e d t o -r0 and
w i l l denote T~
respectively.
F o r t h e remainder of t h i s s e c t i o n , E and F fixed 1.c.s.
,
: E
IT
+
the
will
denote
F w i l l b e a c o n t i n u o u s open l i n e a r
map
o f E o n t o F , and ( Q , + , E ) w i l l b e a f i x e d domain s p r e a d o v e r E .
PROOF
If KC
II t ~ ( f ) II K
=
R i s compact, t h e n J ( K ) C C i s compact and
II f
0 J I IK =
II f I I j ( K )
-
I: i s c l e a r from t h i s t h a t
t~ i s To-continuous. Now suppose p i s a -cg-continuous semi-norm on H ( R )
,
it
s u f f i c e s t o show t h a t p o tJ i s r * - c o n t i n u o u s on H ( C ) .Let {VnIn
78
P . BERNER
be any i n c r e a s i n g c o u n t a b l e open c o v e r o f C ,
then
J
-1
n is
(Vn)
an i n c r e a s i n g c o u n t a b l e c o v e r of R s o t h e r e e x i s t s a C > O and N
IN s u c h t h a t p ( h )
E
2
ClI
hlI J-l
for a l l h
(VN,
E
t h i s i m p l i e s t h a t p c t J ( f ) 5 C I l f c J ( I J- l( vn) = CII f all f
E
But
\IvN
for
for the
I t i s e a s y t o show t h a t tJ i s a l s o c o n t i n u o u s
T
topologies.
-cub ,
~
DEFINITION 1 . 2
g
H(R).
H ( C ) so p c t~ i s Tg-continuous on H ( c ) .
E
REMARK
'ob'
and
L(n,R) = {f
. ..flW
H ( I T o $(W))
E
I
H(R)
= g o
T
o
3
a chaht
W
C
R
$ 1 ~ w~ i 1L L d e n o t e t h e n e t
holomoxphic AuncZionh o n R which d u c t o x LocaLey t h x o u g h A IT-morphism,
dactoxization
(doh
J :
R
(C,JI,F), i s c a l l e d a
-t
ad
IT.
06
IT-domain
R ) i f f t J ( I - I ( C ) 13 L ( I T , R ) .
A IT-domain of f a c t o r i z a t i o n , J : 0
tnuL IT-domain
and
-t
C is called t h e m i n i
dactvhizativn [ d o t 0) i f f J i s s u r j e c t i v e
06
and
s a t i s f i e s t h e following universal property: If K : R
-+
( r , r l , F ) i s any o t h e r IT-domain o f f a c t o r i z a t i o n
such t h a t K i s s u r j e c t i v e , t h e n t h e r e e x i s t s a u n i q u e phism,
:
r
E
C such t h a t R o K = J .
Let x
LEMMA 1 . 2
Let f
-+
E
n, LeeZ w be a c h a x t i n R c o n t a i n i n g
H ( R ) . 16 D a f I I v :o d o h euch a
^1
D f ( x ) :d f ( X ) ( a ) , t h e n f
a
PROOF D f
IdF-mor-
E
E
T-'(o),
x , and
whexe
L(.rr,0)
By s h r i n k i n g , w e may assume t h a t $(W) i s convex. S i n c e -1 ( 0 ) , w e have t h a t f o ( $ F o f o r a l l a E IT is local-
a Iw l y c o n s t a n t on e a c h s e t o f t h e form ( $ ( y )
where y
E
W. B u t , by t h e c o n v e x i t y o f
+ $(W) ,
IW
v-lfo))
each s u c h
connected, so t h e function g:zEITO$(W)'fO
($
/I $(W)
-1
set
I
is
TOPOLOGIES ON SPACES O F HOLOMORPHIC FUNCTIONS
i s w e l l d e f i n e d and e v i d e n t l y G-holomorphic.
p i n g , s o g i s a l s o continuous, hence g
$Iw
so f
PROPOSITION 1.1
16 J
f
IW
= g
71 0
0
E
E
79
i s a n open map-
77
H ( n o $ (W))
.
Now
L(n,R).
ization d o h R , t h e n t J t Suppose { J ( f A
PROOF to h
is a net i n t J ( H ( C ) )
which c o n v e r g e s
H ( R ) i n t h e T~ t o p o l o g y . We may assume t h a t t h e r e
E
is a
c h a r t W c Q s u c h t h a t 4 (W) i s a b a l a n c e d n e i g h b o u r h o o d of Let
and a
-
=
Wlh
$(W) ,
E
h=P
($(W))). F o r e a c h x
(@lw)-’(‘/2
WI/~o , < p <
,
c a ) (27ri)-ld<,where hi = t J ( f A ) - h.
+
o u n i f o r m l y on t h e c o n p a c t s e t { ( $
+
112
-1 w e have DahX( x ) : d h A(x)( a ) =
c - 2 h X o ($lw)-l($(x)
Since hA
E
0.
A
1 IW
-1
($(x)+Ca)1 I?l=p},
and $(W) i s a b s o r b i n g , and dhA( x ) ( a ) i s l i n e a r i n a , we conclude t h a t Da t J ( f , ) (x)
-1
$ o J o
a
E
IT-’(o)
all x h
E
( $ ( x ) + Ca) =
and a
E
+
o
XI
2
and a l l a
for a l l x
f ) @(W) , i t f o l l o w s t h a t Da J ( f X )(x) E
n
L(n,R) C t J ( H ( C ) ) ,
: X
IT
W
E
t
DEFINITION 1 . 3 p
Dah(x) f o r a l l x
Let
-1
(0)
.
E
WI,~
and
= o = Dah(x) f o r
Hence, by Lemma
so t J ( I - I ( C ) )
E . Since
E
1.2,
is ro-closed i n H ( R ) .
X and Y be. t w o t o p o l o g i c a l
Y w i L L b e caLLed c o m p a c t l y phopex
A ~ U C ~ AA .
K
doh each
map G Y
c o m p a c t , t h e k e e x i o t a an L C= X c o m p a c t , n a t i b 6 y i n g p ( L ) = K . PROPOSITION 1 . 2
16 J : R
+
(Z,+,F) i n a h u h j e c t i v e
Ti-mvh-
p h i o m and n i n compac-tly phopex, t h e n J i n aLoo c o m p a c t l y pxo-
pex. PROOF
Let
= {W C RI W i s a c h a r t and b o t h I$
a r e i n j e c t i v e ) and l e t $(W)
=
{(W,V)
E
12x321
and
W CV
+ U Z $(V) f o r some o-neighbourhood Ul. S i n c e J
$lm
and is
open
80
P.
and s u r j e c t i v e , [J(CJ)1 K
BERNER
i s a n o p e n c o v e r of C
E~
= Li.
$ o J(Wi)
=
IT
EiC
o $(Wi), ZiCn-l(Li)C=
m
L i c U j Z 1 +Wi) + a
-1 o
u:=l Ri.
2=1 ( + I J ( V i )
COROLLARY 1.1
IT 0 @(Wi)C @(Vi)
.. . , a m
E
n
n-'(O).
-1 ( 0 ) s u c h t h a t
-1( L . )
= K,
16 J
R
1
:
j=l(ci -
t h e i n c L u . ~ i a ntJ : H ( C )
io a 1r-mcr4phiom and n
(C,$,F)
-t
induced
H(C)
OM
i4 cum-
by
cai~cidew . ~i t h t h e c o m p a c t
(H(Q),.ro)
-f
u:=l
h e n c e J i s compactly p r o p e r .
p a c t l y p a u p e a , t h e n t h e hePatice tupvLogy
t o p o ~ O g y ( H ( C ) ,.r0).
OpCM
I n view o f Lemma 1.1, i t s u f f i c e s t o show
PROOF T
+
m ((J a j ) /I $ ( v i ) ) , and i' Then R C R i s compact and J ( R ) = J(ifi) =
f$= ($I-)-'
Let
1'
u
E such t h a t
Since
Ki i s compact s o t h e r e e x i s t a l ,
let R =
&
E
F o r e a c h i = 1 , . . . , n , l e t Li=$(K/lJ(Wi)).
J(Wi).
By h y p o t h e s i s , t h e r e e x i s t s a compact s e t
n(Li)
if
NOW
..., (Wn,Vn)
c C i s compact, t h e n t h e r e e x i s t s (W1,V1),
such t h a t K C U Y S 1
.
- c o n t i n u o u s semi-norm,
p , on H ( C )
that
can b e extended
each
t o a ro-mn-
.
t i n u o u s semi-norm on :-I ( Q ) S i n c e p i s mntinuous,there a r e C > 0 and K C C compact s u c h t h a t p ( f )
5 1)
f
(IK
all f
E
II(C)
.
By Propo-
sition 1.2,
t h e r e i s a g C Q compact s u c h t h a t J ( R ) = K .
semi-nom. h
H(2)
11
fo
2.
JII
t.
=
+
h
1 1 t J ( f ) 1 1 it ,
(IR
is
7
- a n t i n w s on H(2)
o(f) 5 C /If ((J(K)=
hence p can be continuously extended.
SURJECTIVE LIMITS AND THE T~-TOPOLOGY
A 1.c.s. E
DEFINITION 2 . 1
Limit
I(
The
06
i h
caLLed a d i a i e c t e d o u & ; e c t i v e
1 . c . s . { E a I a E A id t h e h e
a n d d o t u L L a,@
E
A ouch t h a t D
2
a ditrected pteoadeh 2 on A
a ,thehe a 4 e c o n t i n u v u b
but-
TOPOLOGIES ON SPACES OF HOLOMORPHIC FUNCTIONS
j e c t i v e map4
TI^
: E
Ecl and
-f
TI
aB
: EB
+
81
TI^
6ati66ying
Ea
=
detehmined nap O TIB, and E ha5 t h e p h o j e c t i v e Limit t o p o L o g y b y t h e map4 { n a I c l E A . We d e n o t e t h i n nituation b y w h i t i n g E = s u r j l i m a E A ( E c l , ~ a , ~ a B ) B > a [ o h E = s u r j l i m c l E A Ea when t h e TI ' s and t h e TI ' s a t e u n d e t n t o o d ) . F u ~ t h e t m o h eiue n a y a di-
aa
c1
kected bu4jective L i m i t E [a)
i6:
o p e n id na i n a n o p e n map, a L L a
i d
TI
i n a n o p e n map aLL a,B
a6
E
E
A,
A [equivaLentLy:
B
2
(b)
compact id r a i n compactLy phopeh, aLL
(c)
c o u n t a b e e id ( A ,-> )
= IN
(d)
n o n - t f i i v i a L id
each a
604
a). A.
c1 E
w i t h i t n uduae o h d e h i n g . E
A, ncl i n n o t a homeomor
phi4m. The s t r o n g d u a l o f a F r e c h e t l l o n t e l s p a c e w i l l b e c a l l e d
a@g#
s p a c e , and i f i t i s a l s o N u c l e a r , a M x s p a c e .
REMARK
S u r j e c t i v e l i m i t s a r e e x t e n s i v e l y s t u d i e d i n [7]
,
where t h e f o l l o w i n g r e s u l t is p r o v e d :
76 F
PROPOSITION 2 . 1
i d
a
b t 4 i C t
inductive Limit 0 6 a 4 e q u e ~
c e 0 6 Fhechet Monted dpacen { F n I n , . t h e n t h e n t 4 o n g d u a l Fi i6 an o p e n compact countabLe n u h j e c t i v e Limit space4 (F,)
06 thc
=#
F,
i.
EXAMPLE 2 . 1
L e t U C I R m b e open and l e t CVn) b e a
fundamen-
t a l s e q u e n c e of r e l a t i v e l y compact open subsets of U
-
06
i n g V n c Vn+l,
n
E
satisfy-
IN, t h e n t h e s p a c e o f d i s t r i b u t i o n s
(U)
i s a n o n - t r i v i a l open compact c o u n t a b l e s u r j e c t i v e l i m i t o f the @$#spaces
c 8 ' (vn)l n . m
EXAMPLE 2 . 2
Z j=,
C
m
x
IIi,o
a: i s a n o n - t r i v i a l open
c o u n t a b l e s u r j e c t i v e l i m i t o f t h e @%#spaces
compact
.{Irn CXII:~ j=o
eln.
82
P.
BERNER
Every d i r e c t e d s u r j e c t i v e l i m i t o f
NOTE
@@
s p a c e s i s nec-
e s s a r i l y open by t h e open mapping t h e o r e m . For t h e remainder o f t h i s s e c t i o n , E = s u r j l i m
CXEA
( E a l n a , ~ , B ) B > a w i l l b e a f i x e d open d i r e c t e d s u r j e c t i v e l i m i t . -
7 6 ( Q , $ , E ) io a domain 4 p t e a d o v e h E l
DEFINITIOIJ 2 . 2 An = { a
E
$(w)
c h a h t W C Q. nuch t h a t
A1 3 a
$(w) +
nil(o)1 .
By d e f i n i t i o n of t h e t o p o l o g y of a d i r e c t e d s u r j e c t i v e
REIIARK
l i m i t , e v e r y neighbourhood i n E c o n t a i n s a -1
satisfying V = V a E An,
=
Let
B
E
+ n,
and
A,
4
(0)f o r
>
a
=3
some a
3
E
An
E
A.
neighbourhood
V
I t i s obvious t h a t :
. Hence
cofinal i n
(AQ,,)is
(A,?).
The f o l l o w i n g r e s u l t i s proved i n 14-1:
L e t ( Q , $ , E ) b e a domain sphead O u c h a n open
THEOPW.1 2 . 1
hected huhjectivc? l i m i t E = s u r j limuEA(Ea,n (1) F v h each a doh
51, J,
:
Q
E +
A ~ ,t h e
miniinaL na-doiiiain
Then:
at71aB Ba '
ol;
~actohizafion
(Ral$alEa)l e x i n t n .
( 2 ) F o h each a I B ~ A Q 4uch t h a t
i L y u n i y u e ) n,gmotphidm, 6 u h t h e t m o h e , JClB: Q B t o h i z a t i o n doh Q
1
di-
-+
Jaa
aa
i 4
4> : .QR
a , thetre e x i n t n +
Qal
a
(nece44ah-
4 u c h t h a t J~ =
t h e minimal n
aB
J,~OJ@;
-domain 0 6
6ac-
8'
( 3 ) Q ha4 t h e ptojectiwe l i m i t t o p o l o g y d e t e h m i n e d b y t h e mapn
With t h e n o t a t i o n o f Theorem 2 . 1 w e make t h e f o l l o w i n g d e f i n i ti on : DEFINITION 2 . 3
We d e d i n e t h e t o p o L o g y - c ~o n H ( Q )
to
be
the
( L v c a L l y c o n v e x ) i n d u c t i w e L i m i t t o p o L o g y o n H ( Q ) detehmined
TOPOLOGIES ON SPACES OF HOLOMORPHIC FUNCTIONS
Let Q be a domain nphead o v e h E . Then o n
PROPOSITION 2 . 2 H(0) :
T~
2
In particular PROOF
83
i s a Hausdorff topology.
T~
BY Lemma 1.1, 'J
€or each a
E
: ( H ( R ~ r) T o )
( H ( R ) , T ~ )i s
-f
continwus
An, h e n c e t h e r e s u l t . 14 E
THEOREM 2 . 2
a l n o a compact d i h e c t e d n u a j e c t i v e L i m i t
ih
and R LA a d o m a i n n p h e a d o v e h E , t h e n ( H ( R ) , T ~ ) i n a n t h i c t i n d u c t i u e L i m i t o 4 c l o n ed n ubn pacen t a J ( H
PROOF
Suppose a,B
(nu) IacAR
An and B 1. a . I f K C Ea i s compact, t h e n
E
s i n c e va i s compactly p r o p e r , t h e r e e x i s t s a s u c h t h a t va ( 8 ) = K.
T
B
(R)
C
R
C E,
compact
E D i s compact and v a B ( r B (R)) = K
s o C o r o l l a r y 1.1 a p p l i e s t o JaB : R B
Ra showing t h a t
-t
s t r i c t . P r o p o s i t i o n 1.1 and Theorem 2 . 1 ( p a r t 2 )
is
T~
show
that
a J ( I I ( Q a ) )i s c l o s e d i n ' J ( H ( R B ) , T ~ ) .
Let
DEFINITION 2 . 4
main nphead o u e h a 1 . c . s . amply-bounded)
v 06 x i n
cH(Cl)
=
id4 doh each x
H(C),
w h e t e C i d a do-
in n a i d t o b e e y u i - b o u n d e d E
<
a.
14 ( R , + , E ) i n a d o m a i n n p h e a d o u e h E
i d e y u i - b o u n d e d , t h e n t h e h e e x i n t n an a
t h a t jC C U J ( H ( Q a ) 1 a n d i d e q u i - b o u n d e d in H ( R a ) PROOF
Suppose
(oh
c t h e h e e x i ~ t oa n e i g h b o u a h o o d
nuch t h a t supfEz(( f l (
PROPOSITION 2 . 3
06
be a n u b n e t
E
and
An n u c h
.
C H ( R ) i s equi-bounded, t h e n t h e r e e x i s t s a
c h a r t V i n R s u c h t h a t supfE3(; I ( f
I/
<
a.
By Theorem 2 . 1
(3)
84
P.
t h e r e e x i s t s an a Hence f o r e a c h f
X c C
-t
E
BERNER
R and a c h a r t W C Ra s u c h t h a t V = J i l ( W ) .
A
,
-1 (0), a n d x
,
a
E
TI
f o (@Iv)-’($(x)
+
Xa) i s a bounded e n t i r e f u n c t i o n and
E
c1
E
t h e r e f o r e c o n s t a n t . I t f o l l o w s t h a t Da f l v
s o by Lemma 1 . 2 and Theorem 2 . 1
x a=
Let
(‘J)-l(Z)and
i s u n i f o r m l y bounded on V l
. 8
5 0,
all a
6
= {V
c RIV
i s open a n d 2
c o v e r s R so J c 1 ( 8 ) :{Jcl(V) ( V F
i s an open c o v e r o f Ra. C l e a r l y , X a i s u n i f o r m l y each U
E
Jcl(
8),
ZC
If
c
T
-bounded i n H ( R a ) ,
aJ(13(Ra) )
81 on
. -rI-bounded.
i 4
E
An,
and i s equi-bounded t h e r e . Hence (see [6] ) i t i s
s o by d e f i n i t i o n o f
DOMAINS SPREAD OVER A
3.
bounded
H ( R ) i s equi-bounded, t h e n f o r s o m e a
SF. 0
i s equi-bounded i n H ( R c l )
E v e k y e q u i - b o u n d e d n u b 4 e . t o d H(R)
COROLLARY 2 . 1
PROOF
so
vl1(o),
E
C L(ncl,R) C a J ( H ( R a ) ) .
(2)
let
Jal(W)
~
,
i s TI-bounded.
@r+ SPACE
L e t F be a l.c.s.,
PROPOSITION 3 . 1
T
theri t h e d o l l o w i n g
two
h t a t e m en24 ake e q u i v a l e n t : (a)
F in c o u n t a b l e a t i n d i n i t y aiid h e 4 e d i t a n i C y L i n d e l B d .
(b)
Each o p e n s u b n e t 0 6 F .In c o u n t a b l e a.t i n d i n i t y . 74 F
i 4
albo
c r e p a k n b l e , t h e n l a ! a n d I b ) ake
equivalent
to: (c)
E v e k q d o m a i n crpkead o u c h F i b s e p a k a b l e and c o u n t a b l e
at
indinity. PROOF
(b)
(a)
i s t r i v i a l . Suppose (a) i s
satisfied
and
U C F i s open. S i n c e U i s regular and L i n d e l o f , t h e r e is. a coug t a b l e open c o v e r of U,
8
= {Wi}i
such t h a t G i C U ,
a l l Wi
€8.
85
TOPOLOGIES ON SPACES OF HOLOPIORPHIC F U N C T I O N S
be a fundamental sequence of compact s e t s o f F , t h e n
L e t {Kn},
n wi
t h e s e t o f a l l f i n i t e unions o f s e t s of t h e form Kn n,i
E
N
,
,
i s a c o u n t a b l e fundamental s e q u e n c e o f compact
sets
o f U, so (b) i s s a t i s f i e d . ( c ) ==+ ( b ) i s o b v i o u s . Suppose F i s s e p a r a b l e , ( b ) i s
f i e d and ( R , $ , F ) i s a domain s p r e a d o v e r F. L e t xo
8
and l e t
=
IW
E
R he fixed,
E
be t h e s e t of a l l c h a r t s WC R s u c h t h a t @(W)
is
i s c o n t a i n e d i n a c h a r t . Now d e f i n e i n d u c t i v e l y :
convex and
x1
satis-
8
Ixo
E
~ =+ cw~ E
WI , x
8 / W xn
8 is
# g}. Since
an
o p e n c o v e r and R i s p a t h w i s e c o n n e c t e d , i t f o l l o w s t h a t
R
Un,
=
$Ix1 W2
xn.
i s i n j e c t i v e , s i n c e i f x,y E
8
s u c h t h a t x , xo
E
E
xl,
W l and y l x o
then t h e r e e x i s t s E
W2 thus
W1a
W2 # g
and $(\I1) 0 @(W2) is c o n n e c t e d ( s i n c e i t i s convex) IW1"
$(y)
w2
i s i n j e c t i v e (see e . g . :
x = y
. Therefore
X1
[lo:]
W1,
so
lemma 1 . 6 ) and so
i s homeomorphic t o a open
@(XI=
subset
o f F s o i t i s s e p a r a b l e and Lindel6E. A s s u i i e i n d u c t i v e l y Xn i s s e p a r a b l e . L e t yi
= C ~ SE
8
lei
E
arguing as f o r XI, Xn+l
{eiIi b e a dense s e q u e n c e i n
WI , i
E
IN. C l e a r l y
each Yi
and
Xn
x ~ =+ LJ ~i
I N 'i
E
that
i s s e p a r a b l e and L i n d e l o f .
let and,
Hence
i s s e p a r a b l e and L i n d e l o f .
8
T h e r e f o r e i7 i s s e p a r a b l e and
h a s a c o u n t a b l e subaver
{WiIi.
Each compact subset o f R i s c o n t a i n e d i n a f i n i t e u n i o n of c o g -1
($IE
,
where
i s a fundamental sequence of compact s e t s f o r F .
Hence 0.
i {Kn}
-0
IN
p a c t s e t s of t h e form
Xn)
n,i
E
is a l s o countable a t i n f i n i t y . COROLLARY 3 . 1
16 F i b a
@%@ ( k e d p :
a@%@)-npace,
and
I R , @ , F ) i d a d o m a i n d p h e a d o u e h F, t h e n ( H [ . Q ) , Tin ~ ) a Ffizchet
86
P . BERNER
MonteL ( h e s p : N u c L e a h ) s p a c e , ~ ~ - b o u n d tseL4 d a t e equi-bounde4 and
=
T~
T
~
.
S i n c e F s a t i s f i e s (a) o f p r o p o s i t i o n 3 . 1 , ( H ( n )
PROOF
m e t r i z a b l e and s i n c e F i s a k - s p a c e ,
8
Let
, T ~ )is
complete.
be t h e s e t of a l l c h a r t s i n R . I t i s e a s i l y
t h a t ( H (0),T
~
Now e a c h W
E
, T ~ is )
verified
h) a s t h e p r o j e c t i v e l i m i t t o p o l o g y i n d u c e d by the
r e s t r i c t i o n mappings {pW : f
(H(W)
(H(n)
is
,To)
@
E
H(Q)
flw
+
E
(H(W)
,
T ~ ) ) ,
€8
.
i s homeomorphic t o a n open subset o f F so each
,
5 M o n t e l ( r e s p . N u c l e a r ) s p a c e (see 1:8]
resp.
[5])
and a p r o j e c t i v e l i m i t o f l l o n t e l ( r e s p . N u c l e a r ) s p a c e s i s semi-Monte1 (resp. N u c l e a r ) . A semi-Monte1 F r g c h e t s p a c e i s Mon-
t e l . S i n c e equi-boundedness
i s a l o c a l p r o p e r t y , w e may
proposition G
p l e t e t h e p r o o f as i n [8]
(see also El]).
is caLRed Uehy
A s e q u e n c e {yn), i n a 1 . c . s . E
DEFINITION 3 . 1
s t k a n g t y cohzvehgent t o
i d d o h a L t carz-tieluoub
0
com-
semi-noKms a # n
E , a ( y n ) = o d o h n suddicienRLy L a h g e . LEMMA 3.1
Let (Z,+,E)
p a i n 2 i n C,
{nili
{yili p : f
a E
vehy
H(C)
b e a damciin s p h e a d u v e h a 1.c.s E ,
a s e q u e n c e i n I N , {xi)i
s t h o n g L y caiiv$tgent +
s u pi
E m
I
sequence i n E .
"i! ( D dni
Yi
f
( 5 ) (xi)
is a r U - c o n t i n u a u s s e m i - n o h m p a t t e d b y PROOF
a sequence i n E
5 a and
Then:
I
{
L e t V C C b e any n e i g h b o u r h o o d o f
5 . By s h r i n k i n g
w e may i d e n t i f y it w i t h a n open subset of E . L e t
~1
V,
be a contill
uous semi-norm on E whose u n i t b a l l c e n t e r e d a t 5, Ba(5, 1) , i s c o n t a i n e d i n V. C a u c h y ' s i n e q u a l i t i e s imply t h a t f o r i
E
IN,
each
81
TOPOLOGIES ON SPACES OF HOLOPIORPHIC FUNCTIONS
a ( y i ) . Since f o r
C = maxO <
n { 2 " i + ' ( a ( y i ) ) ( a ( x i ) ) '1,
<
-
-
Let E = surj l i m nEIN
THEOREM 3 . 1
so p i s E c ) - p o r t e d .
2
'mn'm
(En'
c o m p a c t c o u n t a b l e Auhjectiue limit 0 6
some
a
be
n
@f$ ( t e h p . @ F @ )
bpacu,
und l e t ( R , $ , E ) b e a d o m a i n A p e a d o w e & E . T h e n o n H ( R ) T~~
= T~
= T
i 6
I
(LF)-MunteC
a 6thic.t
(hebp.
NucLeah) 6pace and
t h e ~ ~ - b o u n d6 ~ex6 d ahe p h e c i n e l y t h e e q u i - b o u n d e d PROOF
6 etb
An i n d u c t i v e l i m i t o f a s e q u e n c e o f N o n t e l
c l e a r ) s p a c e s i s Monte1 ( r e s p . N u c l e a r ) , h e n c e i t
.
( r e s p . Nuimmediately
f o l l o w s from theorem 2 . 2 and c o r o l l a r y 3 . 1 t h a t r I i s a s t r i c t (LF)-Monte1 ( r e s p . N u c l e a r ) s p a c e . S i n c e by c o r o l l a r y 3 . 1 T~
= T~
~
N , nand by lemma
E
m a ,a r e
conti2
I t i s a l s o known
(see
on t h c d c f i n i n q s u b s p a c e s {H(Rn)} n n
1.1, t h e maps
J : (H(Rn)
u o u s , i t f o l l o w s t h a t T& [6]) that
T
5~
(
T~~
,T&)
5
T~
+
and , s i n c e every (LF)-space is
l o g i c a l and a s ( c o r o l l a r y 2 . 1 ) H ( R ) i s .rI-bounded,
( M ( R ) ,T&), n
on H ( R ) .
,
e v e r y equi-bounded
ultrabornosubset
t o complete t h e proof it s u f f i c e s t o
of show
t h a t e v e r y -ru-bounded s u b s e t of H(R) i s equi-bounded. Suppose
C
H(R) i s Tu-bounded, t h e n w e c l a i m t h a t
CnJ(H(Nnn) )
f o r some n
sequence { f i l i i n i n t e g e r s {mili
E
INR
.
I f not, then w e can
5 and an i n c r e a s i n g sequence o f
such t h a t
find
a
positive
88
fi
E
mi+l J(H(R
1) mi+l
6
Let
fi 4 ai
BERNER
P.
E
c
\
m. 3
i
R be f i x e d . Since
i
,.... .
,Rl
=J(H(Rm 1) 3 L(n
m 1. TI
= 1, 2,
11, i
'J(H(R,
and
r
mi
J ( H (Qm ) , i t f o l l o w s from lemma 1 . 2 t h a t f o r some -1 i (0) D f . 3 o i n any n e i g h b o u r h o o d o f 6. By t a k i n g m a 1 i i
,
a T a y l o r s e r i e s e x p a n s i o n a t 6 ( i n some c h a r t a b o u t 5 ) , this -n. f i ) ( F ; ) ( x i ) # o f o r some ni E IN and xi& E . i m p l i e s t h a t d '(Da i
d n ( D a f ) ( 5 ) ( x ) i s l i n e a r i n t h e v a r i a b l e a and n - l i n e a r i n x , n, s o w e may assume t h a t Id I ( D a f i ) ( 5 ) ( x i ) I > i . W e now have t h a t :
-
1
-n. (supid '(Da
f)
(5) ( x . ) 1 )
5
impossible i f Therefore
3C
> i for all i
7
j
IN
,
b u t t h i s is
i s ru-bounded b e c a u s e o f Lemma 3 . 1 .
nJ(H(Qn)
f o r some n
IN R.
E
i s a l s o -io-bounded h e n c e by c o r o l l a r y 1.1,
-i0
5
5
C n J ( H ( R n ) , ~ O ) i s bounded. Every bounded s e t i n ( H ( R J , T ~ )
-iu
so
i s equi-bounded
(corollary 3.1)
,
s o ("J1-l
on Q n . I t i s immediate now t h a t
i s equi-bounded
i s equi-bounded so the proof
i s complete.
4.
OTHER TOPOLOGIES ON H
(a)
I n t h i s s e c t i o n we study t h e
T~
and
T~
topologies i n
relation
t o t h e r I t o p o l o g y on H ( R ) under t h e h y p o t h e s i s o f t h e o r e m 3.1. I n [3] w e showed t h a t i f E was a n o n - t r i v i a l c o i i n t a b l e s u r j e c t i v e l i m i t of
@F#
spaces then
T ob
#
Tu
#
Tub
on H ( E )
when r e s t r i c t e d t o t h e s u b s p a c e o f 2-homogeneous lynomials
@(2E)
and
,
continuous pg
n e i t h e r T~ n o r -rob a r e b a r r e l l e d . A smallm-
d i f i c a t i o n of t h e proof
(see 12-1) shows t h a t t h e same i s
true
TOPOLOGIES ON SPACES O F HOLOMORPHIC FUNCTIONS
89
on H ( R ) when R i s a domain s p r e a d o v e r s u c h a 1 . c . s . E . W e w i l l use t h i s information i n proving:
L e t ( R , @ , E ) b e a d o m a i n 4 p t r e a d a v e h a non-LbLuiaP
THEORE?% 4.1
compact c o u n t a b l e 4 u h j e c t i u e L i m i t 0 6 E = surj l i m TWb
(I)
= T*
n
E
IN n '
E
= T
[hebp.
@3@ 4p,ace~, 7,
Then on H ( Q ) : ( L F ) -h(on.teL
i 4 u 4tfiict
I
@%#
( h e n p . 8 ~ u c ~ e aopace. h)
1 2 ) T h e b o u n d e d ~ ~ - b ~ u n d4 e td4 ufie p h e c i s e l y t h e e q u i - b o u n d e d
and
T
~
T ~ % ,f T~
#
T~~~
net4
T,
~
and ,
T ~ I*)
c i U c o i n c i d e on h e equi-bounded
T~~
4Qt4.
(3)
.te ( a n d id R (4)
T~~
(5)
T~
i 4
and
i b 4emi-montelI hence quani-cumpee-
T~
a b,ulanced n u b s e t
i n n o t bufifieVled, hence,
in
06
El
T~
ib c o m p l e t e ) .
n o t 4e.mi-complete.
v ~ o tbahtreLLed n o & b e m i - c o m p l e t e .
( 6 ) J h e f i e a h e no u l t h a b u f i n o l v g i c u l t o p a l o g i e n w e a k e h t h a n
T
~
S t a t e m e n t (1) i s j u s t t h e o r e m 3 . 1 where w e a l s o showed
PROOF
t h a t a subset i s
T~
bounded i f f i t i s e q u i - b o u n d e d .
S i n c e each
equi-bounded s e t i s c o n t a i n e d i n a d e f i n i n g s u b s p a c e n J ( H ( R n )1, and
=
T~~
is s t r i c t ,
T
InJH(12
= nJ(H(nn)
, T ~ )= -ro
n
wb
I nJ H ( R n )
(see C o r o l l a r y 1.1) s o s t a t e m e n t ( 2 ) i s v e r i f i e d . The 'rw t o p o l o g y on r w - c l o s e d and bounded subsets c o i n c i d e s by ( 2 ) w i t h which i s compact on s u c h s u b s e t s s i n c e i s semi-montel.
If
T~~
i s Montel. Hence
R i s a b a l a n c e d subset o f E l t h e n
p l e t e by c o r o l l a r y 2 . 2 o f
T~~
T~
-rw i s COIJ
161 which u s e s a T a y l o r s e r i e s a r g u -
ment. The r e m a i n d e r o f s t a t e m e n t ( 3 ) f o l l o w s from t h e
remarks
beginning t h i s s e c t i o n . Let
is
5 T~
E
R be fixed, then for a
E
E l t h e map f & H ( Q )
+
-2
d f ( 5 ) (a)E C
continuous because f o r s u f f i c i e n t l y s m a l l s > 0 ,
~
.
90
BERNER
P.
l d 2 f (6)( a ) I
5
-2
2!
11 f 11 ,
where B i s t h e a p p r o p r i a t e
morphic image o f t h e compact s e t { $ ( E l T
0
( r e s p . -rob) w e r e b a r r e l l e d t h e n f supx
E
+
+
A a1
11
-2 d f (E))o(bII
Ihl
homeoNm i f
= s}. 5
-2 K Id f ( E ) (@(x)) I would b e c o n t i n u o u s f o r e a c h K C R
2
compact. Hence { f o @ I f
E
@ ( E)
H ( Q ) v i a t h e mapping f
E
H(R)
+
1 would b e a d i r e c t s u b s p a c e o f ( d 2 f ( S ) ) o @ , so {fo$lf E @ ( ~ E ) }
would a l s o be b a r r e l l e d f o r t h e . r o ( r e s p . -rob)
t o p o l o g y . But a s
p r e v i o u s l y remarked, t h i s i s n o t t h e c a s e , so
T
0
and
are
T~~
n o t b a r r e l l e d . A s a semi-complete b o r n o l o g i c a l s p a c e i s b a r r e l l e d , t h e p r o o f of ( 4 ) i s c o m p l e t e . The b o r n o l o g i c a l t o p o l o g y a s s o c i a t e d t o a semi-complete
space
so t h e p r o o f o f ( 5 ) i s c o m p l e t e .
i s semi-complete
S i n c e a c o n t i n u o u s map from a n ( L F ) - s p a c e o n t o an u l t r a b o r n o l g
g i c a l s p a c e i s n e c e s s a r i l y open ( 6 ) i s t r i v i a l . REMARKS
Examples 2 . 1 and 2 . 2 b o t h s a t i s f y t h e h y p o t h e s i s
Theorem 4 . 1 .
I n p a r t i c u l a r , w e have an example o f a s p a c e w i t h
two d i s t i n c t n o n - t r i v i a l N u c l e a r t o p o l o g i e s : I f R i s a spread over a space of d i s t r i b u t i o n s
& I ,
then ( H ( R ) ,
domain T
~
a s t r i c t ( L F ) - N u c l e a r s p a c e , and, by t h e r e s u l t o f Boland Waelbroeck
of
[5], (H(R),
-io)
i s a l s o Nuclear, b u t not
i~ s
and
barrelled
o r semi-complete. FURTHER RESULTS sidered H(R,G)
,
I f G i s a normed 1 . c . s .
,
we c o u l d h a v e
con-
t h e space o f h o l o m o r p h i c mappings from a domain
s p r e a d R w i t h v a l u e s i n G I i n p l a c e o f H ( R ) . I n t h a t case
all
o u r r e s u l t s would remain v a l i d ( w i t h v i r t u a l l y t h e same proofs), except f o r corollary 3.1,
and t h e o r e m s 3 . 1 and 4 . 1 ,
where
we
would h a v e t o r e q u i r e t h e c o m p l e t e n e s s o f G and d r o p t h e words Monte1 and N u c l e a r from t h e c o n c l u s i o n s ( u n l e s s G were
finite
)
91
TOPOLOGIES ON SPACES OF HOLOMORPHIC FUNC'I'IONS
dimensional)
.
I n t h e o r e m 4.1 w e r e q u i r e d . E t o be a c o u n t a b l e s u r j e c t i v e l i m i t . I f w e allow t h e more g e n e r a l case of a compact i - s u r j e c t i v e l i m i t of t h e n -rl
@$&
s p a c e s (see [7]
symmetric
i s n o longer an ( L F ) - s p a c e , b u t w e s t i l l h a v e
that
and s t a t e m e n t s (2), (31, (4) a n d (5) are I s a t i s f i e d (see a l s o [ 8 ] ) . 'ub
=
8 '
,
for definitions)
still
=
REFERENCES
1.
J. Barroso, M. Platos, and L . N a c h b i n , o n b o u n d e d
hetb
h o l o m o t p h i c m a p p i n g b , L e c t u r e n o t e s i n Math., V o l . 364
,
(1974), 123-133.
Springer-Verlag
2.
oh
P , B e r n e r , UoLomokphy o n b u f i j e c t i v e L i m i t b
eocaLly c o ~
v e x b p a c e n , 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). 3.
P . Berner, S U X
l a t o p o l o g i e d e Nuchbi-n d e c e t t a i n n e b p a c e
de d o n c t i o n h h o l o r n o t p h e s , C.R. Acad. S c . P a r i s , t . 280 (1975), 431-433. 4.
P. Berner, A gLobal d a c t o t i z u t i o n ptopehty
aunctionb
06
a domuin
bptead oveA
h a t hoeomohphic
a b u h . j e c t i . v e L i m i t , S6-
m i n a i r e P.Lelong,1974/75.Notes i n Math. 524, Springer-Verlag(1976)
5.
P . B o i a n d and L. W a e l b r o e c k , '3!t t h e n u c L e a 4 i t y
.
0 6 H(U) ,
C o l l o q u e D ' a n a l y s e F o n c t i o n e l l , 9 o r d e n u x , Elai 1975. 6.
S.
Dineen, Uolomofiphic aunc-tionb ud l o c a l l y e u n v e x hpaceh :
I. P o c u l L y convex t o p o l o g i e d a d H(U), Ann. I n s t . F o u r i e r , G r e n o b l e , t . 23, f a s c 3 (19731, 155-185. 7.
S. D i n e e n , Subjective l i m i t s ad Luca.LLy convex npucels and
t h e i 4 application t o i n d i n i t e d i m e n n i o n a L h o l o m o t p h y ,
92
P.
BERNER
B u l l . S O C . math. F r a n c e T . 1 0 3
8.
(1975).
oh
S . D i n e e n , H o ~ o m v 4 p h i c ~ u n c ~ ~ oo nn b h t t o n g duaLh
Fhzchet-ManReL b p u c e b , T h e s e proceedings. 9.
L. Nachbin, S u t t e n e n p u c e b w e c t o t i e t n f o p o . t o g i y u e b d ' u p -
p l i c u t i o n h c o n - t i n u e b , C.R. Acad. S c i . P a r i s , t . 2 7 1 ( 1 9 7 0 1 , 596-598.
10.
M. S c h o t t e n l o h e r , R i e m a n n d o m u i n o ; R u n i c p t o b t e m n , L e c t u r e n o t e s i n Plath., V o l . -Verlag
(1974)
,
hebU&b
and o p e n
364, S p r i n g e r
-
196-212.
S c h o o l of Mathematics
,
T r i n i t y College, Dublin 2 ,
IRELAND.
C u r r e n t address: D e p a r t m e n k of M a t h e m a t i c , L e lloyne C o l l e g e L e lloyne H e i g h t s
S y r a c u s e , New Y o r k 1 3 2 1 4
Infinite Dimensional Holomorphy and Applications, Matos (ed.) @ North-Holland Publishing Company, 1977
NUCLEARITY AND THE SCHWARTZ PROPERTY I N THE THEORY OF HOLOMORPHIC FUNCTIONS ON METRIZABLE LOCALLY CONVEX SPACES
By K L A U S - D I E T E R B l E R S T E V T ANU RElNHOLV M E I S E
PREFACE:
I n w r i t i n g t h i s p a p e r , t h e a u t h o r s were
by t h e t w o r e c e n t a r t i c l e s
[lo]
Boland and Waelbroeck
[lo]
and
[22].
influenced
F i r s t l t h e work of
on n u c l e a r i t y of
(H(U)
for
open s u b s e t s U of q u a s i - c o m p l e t e d u a l - n u c l e a r l o c a l l y convex spaces E
i n d i c a t e d t h a t c e r t a i n s t r o n g t o p o l o g i c a l vector space
properties of
-
(E o r I r a t h e r ) E '
f a r from t h e Banach space casc,
h o w e v e r - m i g h t l e a d t o i n t e r e s t i n g s t r o n g r e s u l t s € o r t h c spacx:x
of h o l o m o r p h i c f u n c t i o n s on s u b s e t s o f
E
.
( T h i s i d e a was a l s o
[6] t h a t
,TJ
is
i s t h e s t r o n g d u a l of a n s - n u c l e a r
(F)-
s u p p o r t e d by t h e t h e o r e m c o n t a i n e d i n even s - nuclear i f
E
(H(U)
s p a c e . ) The main r e s u l t s of t h e p r e s e n t paper confirm the strength of t h e g e n e r a l p r i n c i p l e .
The s e c o n d s o u r c e of i n f o r m a t i o n f o r o u r paper w a s Mujica's
thesis [22J on H ( K )
and
(H(U)
f o r compact s u b s e t s K a n d
open s u b s e t s U of m e t r i z a b l e l o c a l l y convex s p a c e s E .
Al-
t h o u g h w e make u s e o f many d e f i n i t i o n s , i d e a s , c o n s t r u c t i o n s and r e s u l t s from
[22]
here, our general impression
is
,
that
M u j i c a w a s c h i e f l y i n t e r e s t e d i n t h e Banach s p a c e case and hence
93
BIERSTEDT
94
& DIEISE
h i s main c o n d i t i o n s are s o r e s t r i c t i v e as
exclude
even
( H o w t h i s c a n b e r e m e d i e d i s sham
Frschet-Schwartz spaces E . i n P r o p . 4 below)
to
.
A c o m b i n a t i o n o f t h e i d e a s w e g a t h e r e d from a s t u d y
[lo]
t h e two p a p e r s
and
of
l e a d u s f i r s t to i n v e s t i g a t i o n s
[2 2 ]
on t h e s p a c e H ( K ) o f h o l o m o r p h i c germs on a compact of a m e t r i z a b l e S c h w a r t z s p a c e E
. It
subset
K
t u r n e d o u t i n t h i s case-
by a r a t h e r e l e m e n t a r y a p p l i c a t i o n of t h e Cauchy i n e q u a l i t i e s , and of t h e A r z e l ~ - A s c o l i theorem by t h e w a y - t h a t H ( K ) with i t s u s u a l topology i s i n f a c t a S i l v a space.
f see
(a)
Theorem 7
]
8. (a) h e r e
show t h e (DFN)
3,75,-327 ( 1 9 7 6 ) . )
- property
of H ( K )
Acad. S c i . P a r i s
A s w e then proceeded
t o t h e p r o o f of p a r t ( b ) of Theorem 7
-
analogous
and b a s e d on t h e "exten
t h a t i s a l s o needed a t some o t h e r p l a c e s i n t h i s
d e v e l o p e d a t t h e same t i m e . I t s h o u l d be
paper-was
to
f o r n u c l e a r m e t r i z a b l e E (W-
orem 7 (b)), a ( s l i g h t l y ) d i f f e r e n t proof o f 7 ( a )
s i o n " lemma 6
of
was s k e t c h e d i n a announce-
ment of some r e s u l t s of t h i s p a p p r i n C . R . S h r i c A, t . 2 8 3 ,
(This f i r s t proof
t h a t an e s s e n t i a l p a r t of t h e
remarked
Boland-Waelbroeck theorem is made
u s e o f i n t h e p r o o f of 7 ( b ) , too. Theorem 7
i s o u r f u n d a m e n t a l r e s u l t , and s e v e r a l c o n s e -
q u e n c e s a r i s e from it: F o r i n s t a n c e , t h e s p a c e
(H(U),T~)
of
a l l h o l o m o r p h i c f u n c t i o n s on an open s u b s e t U of a m e t r i z a b l e Schwartz pology
fresp. nuclear
T~
i s a complete Schwartz
(Prop. 1 6 . ) . I f open
U c E
space E w i t h Nachbin's p o r t e d to-
-
H(U)
fresp. s-nuclear
i s t o p o l o g i z e d by
t h i s t o p o l o g y i s d e n o t e d by
r e m a r k e d , T~ and
' I ~c o i n c i d e
]
space
p 5 o j KC=" H ( K )
for
as ; Mujica
had
T
~
f o r open s u b s e t s " w i t h t h e Runge
95
NUCLEARITY AND THE SCHWARTZ PROPERTY
-
property"
then t h e sheaf
m e t r i z a b l e Schwartz
of holomorphic f u n c t i o n s
[ r e s p . nuclear]
space E
is
convex ( t o p o l o g i c a l ) s h e a f o f c o m p l e t e S c h w a r t z ar
]
on
a
locally
[ resp.
s-nuclg
s p a c e s (Theorem 1 9 ( a ) ) . F o r some o t h e r c o r o l l a r i e s , r e s u l t s of t h e a u t h o r s
9
]
the
a
and
[ 71
from
a r e used. W e o b t a i n e . g . t h e r e p r e s e n t a t i o n s of
€-products
H(Kl) c H ( K 2 ) = H(K1 x K7) rcsp. (H(U1) , T ~ ) (H(U2) E
, T ~ ) =
( H ( U 1 x U 2 ) , T ~ )f o r compact s e t s
K . C E . r e s p . open subsets 3 3 U . C E . of m e t r i z a b l e S c h w a r t z s p a c e s E j , j = 1 , 2 ( C o r . 2 2 ( c ) 3 3 and P r o p . 2 3 . ) . Moreover, w e would a l s o l i k e t o m e n t i o n t h e m v e r s e of Theorem 7
(Prop. 9 )
,
t i o n p r o p e r t y of H ( K ) and ( H ( U ) 2 0 ) and on v e c t o r
- valued
some r e m a r k s on t h e approxima, T ~ )
( C o r . 11
,
f u n c t i o n s (Prop. 2 1
,
Prop. 25 1 .
The l a s t p a r t of t h e a r t i c l e d e a l s w i t h p o s s i b l e a r i t y t y p e s of H ( U ) : Among t h e power series s p a c e s i n f i n i t e t y p e ) , P r o p o s i t i o n 26
, Prop.
Prop. 1 2
nucle-
Am(a)
g i v e s a r a t h e r r e s t r i c t i v e (neg
e s s a r y ) c o n d i t i o n f o r t h e sequences a with t h e p r o p e r t y (H(U)
, T ~ )
[resp.
(H(U) , T ~ > ] i s
anced open s u b s e t s U
open
U C Cm
,
AJa)
]
space E
( H ( U ) , T ~ )i s i n d e e d
if
- nuclear
for all
that bal-
o f an i n f i n i t e d i m e n s i o n a l F r S c h e t or S i l v a
[ r e s p . m e t r i z a b l e Schwartz 28. p r o v e s t h a t
(of
.
On t h e o t h e r h a n d , Prop.
AJa)
- nuclear
for
a s a t i s f i e s t h e c o n d i t i o n of 26.
each
(Examples
o f s u c h s e q u e n c e s a are e x h i b i t e d i n Remark 27 ( b ) ) .
ACKNOWLEDGEMENT:
We t h a n k R i c h a r d Aron and M a r t i n Schottenloher
f o r some h e l p f u l c o r r e s p o n d e n c e (and s e v e r a l r e m a r k s ) d u r i n g the p r e p a r a t i o n of t h i s p a p e r and Dietmar Vogt f o r a remark l e d t o t h e p r e s e n t form o f P r o p . 9 .
which
96
BIERSTEDT & MEISE
For o u r n o t a t i o n from t h e t h e o r y of locally coz
PRELIMINARIES:
vex (1.c. ) t o p o l o g i c a l v e c t o r s p a c e s (which i s q u i t e s t a n d a r d ) ,
,
w e r e f e r t o Horvzth [lS]
1191
Kothe
,
and
Floret
-
Wloka
[ 151 . The r e a d e r may a l s o c o n s u l t G r o t h e n d i e c k [16] , P i e t s c h [ 271 , and Martineau
[16]
a r i t y . W e mention
c o n c e r n i n g n u c l e a r i t y and s
[20]
,
[31]
Schwartz
g i c a l t e n s o r p r o d u c t s and t h e
,
and
- nucle-
1 9 1 f o r topolo-
€ - p r o d u c t . Concerning
holomor-
p h i c f u n c t i o n s and mappings on i n f i n i t e d i m e n s i o n a l s p a c e , t h e books of Nachbin
[23]
and Noverraz
and t h e i r
notarion
a r e u s e d . W e remind t h a t , f o r complex 1 . c . s p a c e s E
and F and
U C E
open, a mapping
f : U
i s c a l l e d holomorphic i f and
F
-f
[24]
o n l y i f i t i s G - a n a l y t i c ( i . e . G z t e a u x - a n a l y t i c ) and c o n t i n y ous. We d e n o t e by f : U
3
H(U,F) t h e s p a c e of a l l holomorphic mappings
and d e f i n e
F
H(U) : = H(U,C)
g i e s can be i n t r o d u c e d on
H(U)
,
. Several
e. g . t h e topology
uniform convergence on compact s u b s e t s of ed t o p o l o g y
U
T ~ ( = C O )of
o r Nachbin’s p o r t
which i s d e f i n e d by a l l semi-norms p on
T~
t h a t a r e p o r t e d by a compact s u h s e t K of K i f f o r e a c h V open w i t h
such t h a t
n a t u r a l topolo-
5
p(f)
U .
K C V C U, there
C ( v ) s u p If (x) I
for a l l
H(U)
(p is ported exists
f E H(U) .)
by
C(V) > 0
For m e t r i
XEV
zable spaces E and
‘cW
,
f o r i n s t a n c e , i t i s known ( c f .
have t h e same bounded s e t s , b u t
nological. I f t e r m i n o l o g y of
E
then
that
and
T~
T~
need n o t be bor-
( i . e . a S i l v a space i n
i s a (DFS) - s p a c e
p ]) ,
T~
[12])
the
c o i n c i d e on H ( U ) , For
more i n f o r m a t i o n on t h e s e t o p o l o g i e s see Barroso-Matos-Nachbin [4].
F o r any l o c a l l y convex s p a c e E
, G(mE)d e n o t e s
t h e space
of a l l c o n t i n u o u s m - homogeneous c o m p l e x - v a l u e d p o l y n o m i a l s on E. If
E
i s a ( s e m i - ) n o M space,
Q
(mE) w i l l be endowed
with
NUCLEARITY AND THE SCHF7ARTY PROPERTY
97
i t s n a t u r a l ( c o m p l e t e ) norm t o p o l o g y . O t h e r d e f i n i t i o n s w i l l be given l a t e r on.
From now o n , l e t a l w a y s E b e a m e t r i z a b l e
1 . c . s p a c e o v e r E . The t o p o l o g y o f E system
(P,)
of s e m i - norms.
Em
. Let
1
(Hausdorff)
i s g i v e n by an increasing denotes t h e set
Bi
b e t h e s p a c e E , endowed with (n) t h e semi-norm pn o n l y , and l e t E n b e the quotient space E/ -1 Pn (0) w i t h t h e norm 11 * I I n i n d u c e d by p, : i t s c o m p l e t i o n i s denoted {x E E; pn(x) < 6
by
(in, /I * ) I n
rnm :
- EnEm
+
Gn
En,
m > n
for
-+
-
:E
E n , r n m* ,Em
-+
-+
and
En
a r e t h e c a n o n i c a l mappings. Obviously d
n
En;
fine
I/
= {y E En;
rn(B6)
{y E
rn :E
1.
E
/In
Ily =
U
lIn
y
< 6
1
= : B"
holds t r u e .
+
Bg
and
UnI6
c
-
rn(UnI6)
Bg
i n E we
< 6 1 . For a f i x e d compact s e t K
K
/'n
Let
+
= Tn(K)
r, B6 .
=
de-
"1
1.
K w i l l alwayn d e n o t e a non
DEFINITIONS:
bet od
E
(a)
,
Y
Foh
-
empzy compac-t nub-
a complex Banach npuce.
i n t h e l o u p - n o t m e d l Baruch
U C E open, Hm(U,Y)
Apace o h a l l b o u n d e d h o e o m o h p h i c m a p p i n g n dham i n t o Y and (b)
Let
Hw(U)
= Hm(U,E).
( r n I n E be a o t h i c t l y d e c t e a b i n g
06 ponitive H(U,Y)
u
-
flumbetd rn. We dedine U n = U
= ind Hw(UnrY)
and
H(K)
AtqUtnCe
r i a
C
V
,U =Un
nfrn n
= H(K,E).
,
,rn (An
n-+
one c a n e a n i l y d e e , t h i n d e a i n i t i o n d o e n n o t depend on t h e neyuence p1 < p 2 <
(rnIn
that
[ n o t on t h e nemi-notmn
...j.)
Def. 1 i s t a k e n from M u j i c a ( i n Prop. 2.5)
~
[22]
, where
it
i s proved
H ( K ) i s a l w a y s H a u s d o r f f . I n t h e case
we
a r e m a i n l y i n t e r e s t e d i n a p r o o f o f t h i s w i l l a l s o b e contained
98
BIERSTEDT
i n Theorem 7
2.
&
MEISE
below.
REMARKS: (a)
For each
n E 3N and
: H
An
6 > 0 , t h e mapping
m
m -
( U n I 6 ) -+ H ( U n I 6 )
,
g i v e n by
An(f) = f
i s a ( s u r j e c t i v e ) i s o m e t r i c isomorphism, see
Lemma 2 . 2 .
nn,
0
[zzJ,
T h i s i s o f c o u r s e s t i l l t r u e f o r Banach
s p a c e v a l u e d mappings. (b)
2.4
In
,
Def.
Thm. 3 . 1 , M u j i c a p r o v e s r e g u l a r i t y o f t h e
in-
H ( K ) = i n d ( H ( U n ) , T ~ )by n+ and P r o p . 2 . 3 .
W e a l s o have
[22]
,
[22J
m
H ( K ) = i n d H ( U n ) . I n h i s Def. 1 . 5 ( b ) he defines n+ t h e t e r m "Cauchy r e g u l a r i t y " f o r i n d u c t i v e l i m i t s and shows i n
ductive l i m i t
Theorem 3.2 t h a t
H ( K ) i s even Cauchy r e g u l a r , i f
t h e c o n d i t i o n (B) of
[22]
,
Def. 3.1
satisfies
( S e e Remark 5 ( a ) below.)
On t h e o t h e r h a n d , w e i n t r o d u c e d t h e n o t i o n of edlyretractive inductive l i m i t " i n
E
[8],
" s t r o n g l y bow-
§ 1,l. The two notions
c o i n c i d e i n many i m p o r t a n t c a s e s :
3. a&
LEMMMA:
L e t { F a , iaB 1 b t an i n j e c t i v e i n d u c t i v e
Ranach n p a c e n
ayntem
F a . T h e n t h e i n d u c - t i v e Limit i n n-OLtrrangLg bound
edeg h e t h a c t i v e .id and o n l y i& it in Cauchy h e g u l a h .
PROOF:
From t h e d e f i n i t i o n i t i s immediate t h a t e a c h s t r o n g l y
boundedly r e t r a c t i v e i n d u c t i v e l i m i t o f Banach s p a c e s i s quasi c o m p l e t e * ) and h e n c e Cauchy r e g u l a r . The c o n v e r s e c a n
be seen
.............................................................. *)
Note t h a t two 1.c . t o p o l o g i e s which c o i n c i d e on a convex b a l anced s u b s e t even induce the s m uniform structure on this set.
NUCLEARITY AND THE SCHWARTZ PROPERTY
99
as follows: Let B be the closed unit ball of Fa. Then is bounded in F bounded in F
a
hence there exists
I
2
such that
c1
F -Cauchy net. Therefore the completeness of F a R that the topology on B, induced by F coincides with Fa
Ba
is
and such that each F - Cauchy net in Ba is already
an
induced by
i,(Ba)
implies the
one
.0
Mujica's condition ( B ) is satisfied (trivially) for normed spaces. Among distinguished Frgchet spaces E with a continuous norm, however, only normable spaces satisfy (B).
Therefore
it
that for
a
is important to take into consideration once more
set M bounded in some ed in @CmE (n))
for
Hm(Un) , {img ( 5 ) ; g E M,5
m = 0,1,.
..
(Here we follow the terminology of
by the Cauchy
Corollaries in
[22])
inequalities.
1221 . ) Hence (after analyz-
ing the proofs of Lemmas 3.1, 3.4, and 3.5) Mujica's main results
K} is bound-
E
(like Theorems 3.2
it turns and
remain true under the
3.3
out that and
their
weaker
condition
bjmce
bntib,(i/inq
(BM) mentioned in our first proDosition.
4.
PROPOSITION:
L e t B be u methizuble R . c .
th e A o l L u i u i n il c o n d i t i a n :
*)
We thank J. Mujica and P. Aviles for pointing out that uniformness in m is necessary here.
We would also like tonote
that Prop. 4 was proved independently in Thesis de Magister.
Patricio
Aviles
100
BIERSTEDT
&
MEISE
H(K) = i”,d Hm(Un) i n u 6 . t t o n g L q bouvt dud /rc?Rkactiwe n i n d u c - t i u e L i n i i t 0 6 Eunuch n p c t c e n and h e n c e u c vm pk e t e u L t t u b u R. -
Then
nokogicak (DF)- s p a c e . 5.
REMARKS: (a) Mujica’s condition ( B ) requires that the inductive limits ind (?(mE (n ) are strict. For m = 1, for iE n+ stance, this means that ind (En) = ind (En); is n+ n+ strict). Hence (BE{) is weaker than ( B ) . And for (FS)spaces E , ( B ) is satisfied if and only if all the spaces En if
are finite dimensional, i.e. if and n E = C” or E = c for some n E IN.
(b) Condition (BPI)
E For then the induc-
is always satisfied,
is a metrizable Schwartz space:
only
however, if
in+d @fmEn) are even compact (i. e. n Silva), as an argument similar to the one used in tive limits
the proof of the theorem of Schauder Wloka
[15
(see Floret-
] , 19, 2.1) immediately shows.Yet we will
prove much more for such spaces E in 7
below.
The ideas used in the proof of the next lemma are already well - known (see e. CJ. Schottenloher 1291 ) . We shall need this lemma several times in the proofs of our main results, however,
PROOF:
For
f E Mm(U,Y), x
converqent Taylor expansion
E
K f (x
and
+
h E FrCF, m
h)
=
C
n=O
we have
the
r
li(x , h) ,
where
101
NUCLEARLTY K13 THE SCHPJARTZ PROPERTY
l nf ( x ) = l nf ( x ,
n-homogeneous con
is a (uniquely determined)
)
t i n u o u s p o l y n o m i a l on F w i t h v a l u e s i n Y
and
( n = O,l,. . . I .
by t h e Cauchy i n e q u a l i t i e s
Now e x t e n d 1; ( x ) t o some c o n t i n u o u s n-homogeneous p o l y w f n o m i a l ln on j? ; t h e e s t i m a t e s above a r e p r e s e r v e d . Hence t h e 00 f c (--)t n c o n v e r g e s €or series c l l l n ( x p h ) l l 2 I1 f lLm(u,y) n=O n=O
.
- -
all
, + 6)
w
=
L
fx(x
If
x,y E K satisfy
(by d e n s i t y o f
f Y)
n (x,G) i s holomorphic w i t h
I
n=O (x
d e f i n e d on
+ ir
x
-
by
and
fx
t < r , and t h e f u n c t i o n
h E Bt
(x
+
Br)
+ 6,)
+ Gr)
(7 ( y
f) ( y
+
(x+Br).
i t i s e a s y t o see
-
and by c o n t i n u i t y o f
Br)
fx
t h a t t h e r e s t r i c t i o n s of
+ g,
fxl (xtBr) = f l
and
-f Y
to the
fx
inter-
ez
s e c t i o n c o i n c i d e . A s holomorphy i s a l o c a l p r o p e r t y , t h e r e
i s t s some
2
E H(V) with
f p
nu
=
qv
n U,
and
h o l d s by d e n s i t y a g a i n . Then i t i s o b v i o u s t h a L F : f
-$
f
i s well-defined,
THEOREM:
mapping
c o n t i n u o u s , l i n e a r , and i n j e c t i v e a
The n e x t i s o u r main t h e o r e m on
7.
the
H(K).
L4.t
E
b e a mettrizabPe 1 . c . d p a c e and F
E
i d
u S c h w a f i t z bpace., t h e bpe&uni
cE
cum
pact. W
(a)
16
h conipact
UMd
heWt H(K) a (DFS)-JpUCe.
H (Un); )p,
102
BIERSTEDT
&
MEISE
I n t n e n o t a t i o n i n t r o d u c e d i n 1. ( b ) wemy assume r
PROOF:
1.
C:
1 -
...
I t i s enough t o show t h a t t h e s y s t e m of semi-norms p , < _ p 2 1
c a n be c h o s e n i n s u c h a way a s t o g e t a l l t h e c a n o n i c a l
(un)
: H"
pnln+l
pings
l u t e l y summing
]
(n
E
m
H
-L
.
IY )
see F l o r e t - W l o k a
space
s
~a b +s o - ~
,
F o r t h e n H(K) i s a ( D F S ) - [ r e s p .
(DFN)d
[resg.
Un+l
Un+;,]
.
W e fix
n E IN
rn+l < s < rn and d e f i n e V
= en(K) +BE
En.
C
I n t h i s case w e may assume w i t h o u t l o s s of g e n e r a l -
a)
nnln+l
ity that
~
fi
IR w i t h
E
P
-
[ 1 5 ] ) , b e c a u s e n o component o f Un h a s
a void intersection with and
( u ~ + compact ~ ) [ resp.
map
. *
En+l
,.
-* En i s p r e c o m p a c t a n d h e n c e
'il
n,n+l
compact, Then
i s a compact s u b s e t o f leLmma 6 .
,
because of
V
we n o t i c e , t h a t
Here A n ,
An+l
pn
5
Using 2 . ( a ) a n d
can be represented
'n,n+l
--
Pn,n+l
are l i k e i n 2 . ( a ) , F i s d e f i n e d as i n
6.
,
A
and B i s g i v e n by
B(f) = f
0
u,u
B can be w r i t -
= n n , n + l , UI n - +l. m
t e n i n a n a u r a l way a s Bo
: C(Q)
-+
CB(U,+l)
B = B0
0
R , where
R : H (V)
[ C B = c o n t i n u o u s a n d bounded]
by R ( f ) = f 0 a n d B o ( f )
=
f
e
+
C(Q)
are defined
u . A s a l l mappings a r e l i n e a r
c o n t i n u o u s , i t i s enough t o show c o m p a c t n e s s of
R
and
only.
and This,
i n t u r n , f o l l o w s form a s i m p l e a p p l i c a t i o n o f t h e Cauchy i n t e g r a l f o r m u l a a n d of t h e t h e o r e m of A r z e l s - A s c o l i : in f
En
b e 3 6 , and f i x x , y E Q
E Hm(V)
with
Ilx-y
t h e following estimate h o l d s :
/In
L e t dist(Q,
6. Then f o r
-
1 03
NUCLEARITY AND THE SCHWARTZ PROPERTY
Hence
R maps t h e u n i t b a l l o f
Hm(V)
o n t o a u n i f o r m l y bounded
a n d e q u i c o n t i n u o u s f a m i l y of c o n t i n u o u s f u n c t i o n s on t h e p a c t s e t Q , a n d so R (b)
i s compact by A r z e l s - A s c o l i .
The ? r o o f o f
( b ) proceeds s i m i l a r l y . W e assume
o u t loss of g e n e r a l i t y t h a t ( p k ) k E~
1 2 1
,
Gk,k+l
:
8 . 6 . 1 Thm.)
Satz 8.2.6) Hence f o r r
,
6k+l
.
-f
canonical
b e o f c l a s s 1" w i t h p < 1 (cf.Pietsch
A s mappings o f c l a s s 3."
are precompact
([27]
-
t h e set Q1=
E 1R
Ek
with
i s c h o s e n i n s u c h a wayas
t o make a l l t h e Ek H i l b e r t s p a c e s and t o l e t a l l t h e mappings
com-
T ~ + ~ , ~ + ~ ( U E ~n + +l ~ i) s p r e c o m p a c t .
n+ 1 t h e r e e x i s t m E Il -in+ 1 B~ I and so we have:
w i t h r n + 2< r < r
and
~~
with
Q1
c mu
j =1
(kj
+
A f t e r t h e s e p r e p a r a t i o n s w e are g o i n g t o show now, how a s u i t -
a b l e factorization allows us t o use a r e s u l t Waelbroeck t o p r o v e t h a t t o r i z a t i o n of
Pn ,n+2
PnIn+2
H(V)
,
Boland
and
i s a b s o l u t e l y summing. The f a g
w e need i s given as follows:
(Here, f o r a compact s u b s e t
t h e space
of
S
of
V
,
endowed w i t h t h e s e m i
w e d e n o t e by
- norm
(H(V),pS)
p S ( f ) = s u p l f (x)l X E S
104
BIERSTEDT & I E I S E
J d e n o t e s t h e c o n t i n u o u s i n c l u s i o n , R t h e i d e n t i c a l mapping,
and B a r e as i n p a r t ( a ) of t h e p r o o f w i t h e . g . B ( f ) = f
0
F
u ,
N
(5
IUn+2
= I T" n , n + 2
L3
As f o r
and L i s t h e compact s e t
,
L a l l mappings i n t h e above f a c t o r i z a t i o n a r e
co"
i n s u c h a way t h a t R be
t i n u o u s , i t w i l l s u f f i c e t o choose comes a b s o l u t e l y summing.
TO d o t h i s , w e remark f i r s t t h a t by well-known
theorems
on t h e r e p r e s e n t a t i o n o f compact o p e r a t o r s i n H i l b e r t (cf.
[ 271 , 8.3) t h e condition
n,n+l
normal s y s t e m s
i n En w i t h t h e a d d i t i o z
4
l i m (1 + j ) x
j +m ( e j )j E IN
-
in En.
= 0
j
and
o f c l a s s IP" implies that
-
f o r an a p p r o p r i a t e s e q u e n c e ( x j )
a1 Property t h a t
spaces
E N in
(fj)
( F o r some o r t h o
En+l
1
resp.
m
and a s e q u e n c e
('j)j E N with
and
0
hj\
1 A T j=1
m
get
'n , n + l ( y ) =
~ ( +1 j l 2 h j
fj
1 A . ( y [ e j ) f j . Now t a k e j=1 3
and n o t i c e t h a t
C =
m
C
<
,
we
1 2IXj
j = 1 (l+j)
(1 + j ) ' l P * h j
sup
m
En
<
m.)
-Be
j EW
c a u s e of
r < rn+l < 1, l i m (1 + j ) 4 x j +m
j
= 0,
and t h e above
COG
s i d e r a t i o n s , t h e p r o o f of t h e theorem o f Boland and Waelbroeck
-
[lo]
implies t h a t f o r
p a c t s e t Q3 w i t h
Q 2 = ;n,n+l(Br
n Q 2 CQ 3 C Bi
Xi.1)En
t h e r e e x i s t a corn
and a p o s i t i v e Radon A
1.1 on
Qj
such t h a t f o r any SUP
x EQ2
I f(x)I
f
E H(B:)
./, I f l 3
the inequality dv
measure
1.05
N U C L E A R I T Y AND THE SCHWARTZ P R O P E R T Y
h o l d s t r u e . T r a n s l a t i n g t h e s e t Q3 and the measwe p by Gn,n+l(kj),
we f i r s t obtain
( j = 1,.
'1
2
s u r e v o n t h e compact s e t that for all
.. , m ) =
and f i n a l l y a p o s i t i v e m e a n,n+l(kj)
j =1
+
-
f E H(V) :
PL(f) = sup
I
f (XI
XEL
-
,
m
E
'yu
x s u p I l f (XI I ; x E ;in,n+l (k.+Brn+l1 n j =1
I 5
m
f L
T h i s p r o v e s t h a t R and h e n c e
8.
'n,n+2
i s a b s o l u t e l y summing. c]
REMARKS:
(a)
The a p p l i c a t i o n o f lemma 6
i n t h e proof of
part
( a ) of t h e Theorem c a n be replaced by t h e o b s e r v a t i o n t h a t a u n i f o r m l y e q u i c o n t i n u o u s f a m i l y o f funs t i o n s on a p r e c o m p a c t s e t P e x t e n d s t o a n equico; t i n u o u s f a m i l y o f f u n c t i o n s on t h e c o m p l e t i o n (b)
i .
The p r o o f of t h e t h e o r e m i n d i c a t e s t h a t f o r a comp a c t s e t K i n E ( m e t r i z a b l e ) , by l e m m a 6 , H E ( K ) = r H ( K ) c o n s i d e r e d as a s p a c e of f u n c t i o n s i n
E]
is
t o p o l o g i c a l l y i s o m o r p h i c ( b y r e s t r i c t i o n ) t o %(K). We t u r n t o t h e c o n v e r s e of t h e t h e o r e m .
9.
PROPOSITION:
Let E
b e a mettizabee L . c .
n o n - v o i d compact n e t K i n (D~N)-npace,
PROOF:
E, H(K)
in a ( a )
n p a c e . 16 doh borne (DFS)-hedp.
(b)
t h e n E hn a ( a ) S c h w a t t z h e n p . ( b ) nuceeah Apace.
H ( K ) and H ( K
+
e ) are topologically isanorphic by t r a n s l a t i o n
BIERSTEDT
106
MEISE
&
a r b i t r a r y ) , h e n c e w e may asctume
(e E E
we o b t a i n
H~
(un) n E '
= E'
r
+
Un C ( A
s o for
rn)Vn;
1E
i m p l i e s H~(U,)
El0
:
.;, .
n E'C
v:
for s u i t a b l e
K cAVn
On t h e o t h e r h a n d , from deduce
c un
: B:
v::
0 E K.Then w i t h Vn
1 > 0 we
the inequality
n ' sup
1
l ( x ) I 5 ( A + rn) s u p
I
l ( x ) I h o l d s and
XEU,
XEV,
I t i s a l s o clear t h a t
Hm(Un)
n
1 E Hm(Un)
.
E'
i n d u c e s on its ( c l o s e d ) s u b s p a c e
t h e c a n o n i c a l ( c o m p l e t e ) norm t o p o l o g y of t h i s s p a c e .
E'
v:: H ( K ) i s a (DFS)
If
c
m
H (Un)
- space,
t h e i n d u c t i v e system
'Prim 1
N
'
i s compact, and hence t h e s y s t e m p a c t . The remark
6;
i s com-
IE'
";o,.Pnm
v: of
t o g e t h e r w i t h t h e theorem
= Ell = E '
v:: Schauder
and
pnm
I
E'
v:: n E N hence
,
t h e r e e x i s t s an
=
'nm
then implies t h a t
m 2 n
rnm p r e c o m p a c t . S o E
such t h a t
Grim
for
every
i s compact and
is
i s a Schwartz space. I f H ( K )
a (DFN) - s p a c e , a s i m i l a r a r g u m e n t shows t h a t E i s a
nuclear
space. o The m a i n i d e a i n t h e p r o o f of t h e f o l l o w i n g
i s d u e t o Aron and S c h o t t e n l o h e r
10. PROPOSITION:
Let E
be a L . c .
[2],
-
proposition
T h . 2 . 2 , p r m f of (c)
(a):
m e t t l i z a b l e Schwatltz Apace
.
Then dok any n u n - v o i d compact h e 2 K i n E, H ( K ) c u n t a i n n a c o n tLnuounL?y p h o j e c . t e d t o p o l o g i c a l dubbpace c a n o n i c a l l y
to
( E ' ti3
PROOF:
(El
,!.Avmvtrpkic
rE) 1*
A s i n t h e p r o o f of 9.,
w e assume
0 E K.
The
mapping
NUCLEARITY AND THE SCHk7ARTZ PROPERTY
P : H(K)
-+
viously
P2 = P
H(K)
107
-1 i s d e f i n e d by P ( f ) = d f ( 0 ) ( = f ' ( 0 ) ) . Then oh n E IN
P(H(K)) = E l . For every
and
,
i s c o n t i n n o u s , b e c a u s e , by t h e Cauchy 1 I and by t h e i n e q u a l i t i e s , w e have s u p s u p I P ( f ) ( h ) 1 5 nllfil9 h q r p r o o f of 9 . t h i s i m p l i e s s u p /I P(f)I(Hm(un,< m Hence P : H(K) -+ E '
PIHOD(U,) : H m ( U n )
Hm(Un)
-+
.
llfll51 i s continuous, i f
E ' C H(K)
i s g i v e n t h e i n d u c e d t o p o l o g y . By
Theorem 7 ( a ) , t h e p r o o f o f P r o p . 9
E'
w i t h E'
t h e p r o o f of 9
,
1141, 1 4 , w e
and F l o r e t
H(K) i n d u c e s t h e t o p o l o g y of
know t h a t identify
,
on E '
ind E ' n - + V:
.
I f we
i t f o l l o w s a g a i n from t h e a r g u m e n t s
and t h e g e n e r a l t h e o r y t h a t
( g ' , B ( i ' 1 6 ) ) = ind
in Elo.
vm (k',@(ef,6)) =(El ,p(E',E)). 11-t
I,
T h e r e f o r e t h e p r o o f i s f i n i s h e d , if w e shcw
But t h i s i s a c o n s e q u e n c e of t h e f o l l o w i n g f a c t s :
Themetrizable
S c h w a r t z s p a c e E i s s e p a r a b l e (see e . g .
[26])
h e n c e by Kothe
1193
t?j
Pfister
2 9 , 6 (1) e v e r y bounded s u b s e t i n
c o n t a i n e d i n t h e c o m p l e t i o n of a bounded s u b s e t of
REMARK:
a
# F
The f i r s t p a r t of t h e p r o o f of 1 0
C. E
phic t o El
and
m e t r i z a b l e , t h e s u b s p a c e of
i s a l w a y s complemented.
w a s o n l y needed t o a s s u r e t h a t
E
is
.0
shows t h a t f o r any
H(K) c a n o n i c a l l y ism02
(The assumption of "E Schwartz"
H ( K ) i n d u c e s t h e t o p l o g y B(E',E)
on t h i s s u b s p a c e . ) T h e r e i s a n i m m e d i a t e c o n s e q u e n c e of 1 0 . f o r t h e approxA mation p r o p e r t y ( a . p . )
11.
COROLLARY:
In
A
of
H(K).
Apace E
AA
i n 10
, t h e 4.p.
home n o n - v o i d compact K Z E i m p e i e b t h e a . p .
06
06
H(K)
60h
(E',B(E',E)).
108
BIERSTEDT
& lllEISE
REMARKS : (a)
Under t h e a s s u m p t i o n s o f ll.,
(E",O(E",E'))
and
(see t h e p r o o f of 1 0 ) .
(k',R(k',G)) =
(E',B(E',E).)
Hence t h e remark a f t e r
1, Satz G i n
[8]
t h e a , p . of a.p.
i.
of
t h e a.p. (b)
E=
of
(El
,R (El ,El) i s
(And t h e a . p .
shows t h a t
then equivalent to t h e
of
i m p l i e s of
course
E .)
In particular, if
E
i s a n ( F S ) - s p a c e w i t h o u t a.p.,
t h e n b y 11. find ( a ) , H ( K ) d o e s n o t h a v e t h e
a.p.
f o r any
with
K C E .
(The e x i s t e n c e of
(FS)-spaces
o u t a . p . w a s deduced from E n f l o ' s counterexample by Hogbe (c)
- Nlend. )
By 7 ( b ) , H ( K ) h a s a l w a y s t h e a . p . ,
c l e a r . And from 4
and
if E 3
K i s ng
[ E l , 1, S a t z 2 i t is obvious
t h a t u n d e r c o n d i t i o n (BM) on E 3 I<, H ( K ) h a s a.p.,
i f a l l t h e spac<:s
Hm(Un)
the
( o r , by 2 ( b ) , a l l
s p a c e s ( H ( U ) , T ~ ) ,U 3 K) c a n b e c h o s e n t o h a v e t h e a.p.-Another
12.
PROPOSITION:
p o s i t i v e r e s u l t w i l l b e proved n e x t .
Let E be a L . c . rnettiizable S c h w a h t z
npace
whone t o p o l o g y can b e g i v e n b y a n i n c h e a b i n g n y ~ t e t t i ( p n I n E N a6 bem~Vmtund
H(K)
w i t h t h e p n o p e h t y t h a t aLL
ApUCt/s
Gn
h a v e Z f , ? a.p. Then
han t h e a . p . d o h e a c h baLanced c o m p a c t n e t
PROOF:
K C E.
W e u s e t h e e q u i v a l e n c e f o r t h e a . p . mentioned i n
I n o u r case i t i m p l i e s t h a t
H(K) has t h e a.p.,
if
for
[el. each
Banach s p a c e Y , H ( K ) 8 Y i s d e n s e i n H ( K ) E Y . W e have ( H ( K ) E Y ) ~=~
109
NUCLEARITY AND THE SCHk7ARTZ PROPERTY
H ~ ( U ~ ) E=Y H ~ ~ P ( U , , Y = )E f
[5]
f o l l o w s from
,
E H ~ ( u ~ , Y ) f; ( u n )
i s precanpact i n Y I
31.
r N o w Remark 2 (a) r e m a i n s t r u e f o r Banach s p a c e
valued
bounded holomorphic mappings. Hence t h e c o n s t r u c t i o n u s e d
in
t h e proof of 7 . ( a ) i s e a s i l y s e e n t o show = i n d H m f P ( UNn , Y )
ind HmfP(Un,Y)
= i n d Hm ( U n , Y ) , n-+ n-+ rv /v b e c a u s e h e r e nn,n+l(Un+l) i s precornpact i n U and lemma 6. n a p p l i e s . So ( H ( K ) & Y I b o r = i n d H m ( U n , Y ) . ] n-.
n+
= i n d H C c (l UJ n I Y )
n+
A f t e r t h e s e p r e p a r a t i o n s , it obviously s u f f i c e s t o W
d e n s i t y of
H (Un+l)
8 Y
in
of H(K). We f i x a f u n c t i o n
P,,,+~(H
f
E
W
Hm(U ,Y)
?
(Un,Y))
show
t o g e t t h e a.p.
and t a k e V
as defined
d
i n t h e proof of 7 .
Q = ?n,n+l(Un+l)En
i s t h e n a compact sub-
s e t of t h e b a l a n c e d s e t V . By a s s u m p t i o n , L so by Aron - S c h o t t e n l o h e r
[2],
Theorem 2 . 2 ,
n
has t h e
a.p.
f o r any given
E
, >0
one can f i n d
REMARK:
each
remains t r u e f o r e v e r y compact s e t K
12
Gn(K) C
En
h a s a neighbourhood b a s i s of open,
such
that
finitely
Runge s e t s . For t h e n one can c o n c l u d e as above by taking V smaller and r e p l a c i n g , i f n e c e s s a r y ,
r e m 2 . 2 of
[2]
A s Chae
n + 1 by a s u i t a b l e m > n. (Theo-
i s s t i l l v a l i d i n t h i s case.) [ll] and Mujica
[22],
Ch. 5 and 6 have done before
BIERSTEDT
110
MEISE
&
u s , w e w i l l now a p p l y t h e i n f o r m a t i o n on H(K) c o n t a i n e d i n t h e r e s u l t s above t o p r o v e some c o n s e q u e n c e s f o r N a c h b i n ' s topology
Tu
on H(U)
. We
ported
a r e m a i n l y i n t e r e s t e d i n open subsets
U of a m e t r i z a b l e S c h w a r t z s p a c e , however,
p e r t i e s of H(K) d e r i v e d i n 7
and t h e s t r o n g pro-
allow b e t t e r r e s u l t s than i n t h e
Banach s p a c e c a s e . I t i s o b v i o u s t h a t an i n j e c t i v e i n d u c t i v e system of
s p a c e s i s boundedly r e t r a c t i v e ( [ 8 ]
,
,
the f i r s t
of t h e f o l l o w i n g lemma i s n o t h i n g b u t a r e w o r d i n g o f [22]
d
1,l.) i f and only i f it is
even s t r o n g l y boundedly r e t r a c t i v e . So, by 3
lemmas 5 . 1 . and 5 . 2 . o f
n
part
Mujica's
i n (much) more g e n e r a l terms and
can b e proved by ( a l m o s t l i t e r a l l y ) r e p e a t i n g h i s arguments.For t h e s e c o n d p a r t of t h e lemma, t h e n u c l e a r c a s e i s
e . g. by combining P i e t s c h 3.3.5
05
3.2.5,
3.2.4,
3.2.13,
Let
{Xn; j n m } b e a c o u n t a b l e i n j e c t i v e inductive AYA
Banach 4paceb w h i c h
eah nubbpace. y
06
i 4
boundedly h e t h a c t i v e .
X = i n d X n ue d e d i n e
Yn =
n o n i c a l i n d u c t i v e AyAtem
xn
id also a
; jnm 1
?
i d
compact
(DFSI- [ t e n p .
Even i n t h e case o f
open p r o b l e m , w h e t h e r
a lin-
Y n Xn w i t h t h e
Then Ahe ca
-X { Y n n ; jnm I y n X n 1 -& again (o,f%onglyJ
boundedey h e t t a c t i v e and t h e c o m p l e t i o n ? ( t o p o l o g i c a l l y ) . 16 I
Foh
n+
i n d u c e d nohm t o p o l o g y and t o p o l o g i z e y a4 i n d Yn. n+
REMARK:
and
(in t h i s order).
1 3 . LEMMA:
Rem
[27],
established
06
Y equaln indx'n
(DFNI-J
(DFS)-spaces
n-t
[ hedp.
nucleat
3,
Apace.
X = i n d Xn i t i s an
n+
= i n d x X n as i n 13. must be a topological n+
N U C L E A R I T Y AND THE SCHWARTZ P R O P E R T Y
s u b s p a c e of
X
. There
111
a r e several other equivalent
fomlations
of t h i s q u e s t i o n which w e d o n o t s t a t e e x p l i c i t l y h e r e . The f o l l o w i n g d e f i n i t i o n s needed i n t h e r e s t of t h i s pa-
[22].
p e r are t a k e n from
14.
DEFINITIONS:
L e t E be a m e t h i z a b l e L . c . Apace and U C E
T h e n y n t e m &u
= CK C U ;
campact 1 i n dineoted up-
K
wahd b y i n c & L h i a n .
Fat
K E x u
take
We
X = H ( K ) = ind Hm(Un)
n+
d e d i n e Y = H K ( U ) an t h e image
Y i n
topolagized by
i n called
K € F U
t i a L l y denne i n We h a y t h a t U
b e t 6 ahe
15.
M = 5
n+
-
H(K)
By 1 3
06
E
Hm(Un).
i n nequen-
. &,.
i d s a i d t a be c-b&nced
6 u t a balanced n e t
. If
nuclear] space, then
Mo
A K i K ( U ) = €I ( U )
w e have
f i e s c o n d i t i o n (BM)
(b)
p (H(U1)
n
.
han t h e Runge p h a p e h t y , id U - Runge
+ Mo
and 4
n
Runge, i d
REMARKS:
(a)
p :H ( U ) + H ( K )
Y = ind Y , w h a e Y n = Y
codinad i n
A nubhet M
id
U
ad H ( U ) un-
p (H(U))
d e t t h e c a n o n i c a l heht&iCtiVn mapping
and
,
(6ah[EEj,
. if
E
satis-
i s even a Schwartz [resp.
E
cK(U)
i s a (DFS)-[resp.(DFN)-,
hence s - n u c l e a r ]
s p a c e by 7
(H(U) , T ~ )= proj +KE&
HK(U) = proj
and 13. G K ( U ) h o l d s for any
+WU
112
BIERSTEDT & PIEISE
open s u b s e t U Mujica
of a m e t r i z a b l e 1 . c . s p a c e
[22] ,Lemmas 5 . 5 , 5 . 6
plete, i f
E
see
E,
(So ( H ( U ) , T ~ )i s corn
s a t i s f i e s c o n d i t i o n (BM)). From t h e s e
r e m a r k s and w e l l - known permanence p r o p e r t i e s
of
S c h w a r t z r e s p . s - n u c l e a r s p a c e s ( c f . Martineau
1201 )
we o b t a i n immediately:
16.
Let E b e a m e t l r i z a b l e L . c . Schwahtz
PROPOSITION:
and U open i n E I t i n even
.
Then
(H(U)
17.
[lo]
,
and
I d E io
LEMMA:
We have a L g e b h a i c a l L y than t h e topoLogy
T~
- nuclearity
161 ,
of
( H ( U ) , T ~ ) see &land-
1.12.
a methizable l . c . b p a c e and U H(U) = proj H ( K ) , +KE&" 06 p r o j M ( K ) . +KE;F(
(continuous) r e s t r i c t i o n s
and
T~
open,
c E
bthongeh
i b
"
This i s straightforward:
PROOF:
in a c o m p l e t e S c h w m z bpace.
nuclecth, id E in n u c t e a l r .
s-
F o r n u c l e a r i t y and s Waelbroeck
, T ~ )
bpaCe
xKL) w i t h t h e canonical
H(K);
rKL: H ( L )
p r o j e c t i v e s y s t e m . The l i n e a r mapping
+
H(K) for
A : H(U)
-P
is
L 3 K
proj
a
H(K) d e
+KWU
f i n e d by i n d u c e d by
A(f)
= (fKIKEFU
,
where
f K d e n o t e s t h e germ on
f E H ( U ) , i s w e l l - d e f i n e d , i n j e c t i v e , and
t i n u o u s by Remark 2 . ( b ) . S u r j e c t i v i t y of Remark 1 5 ( b ) )
18.
A
T
K
~ COG -
i s also clear (cf.
.
REMARKS:
(a)
The t o p o l o g y
T~
had b e e n c o n s i d e r e d i n t h e
o f Banach s p a c e s by H i r s c h o w i t z
[ 171 and
Chae
case
Ill].
NUCLEARITY A N 3 THE SCHWAXTZ PROPERTY
The problem of w h e t h e r t r a r y open s u b s e t s
T~
= T
113
holds
w
for
arbi-
o f , s a y , a Schwartz s p a c e
U
i s r e l a t e d by 1 5 ( b ) t o t h e q u e s t i o n m e n t i o n e d t h e remark a f t e r 1 3
(see [ 2 2 ]
,
E
in
l a s t l i n e s o f Ch.
5).
(b)
F o r open s e t s U
w i t h t h e Runqe p r o p s r t y i n a m e t -
r i z a b l e 1.c. space s a t i s f y i n 9 condition (BM), can f i n d a system
HK(U) = H ( K ) f o r
we
3 c o f i n a l i n gJ s u c h t h a t each K E & ( a l g e b r a i c a l l y ,
and
so e e n t o p o l o g i c a l l y by a g e n e r a l open mpping thee_
rem, cf. ( H (U
[22]
,
Lenna 6.1). Hence by 15.(b) w e
proj H(K) also topologically i n c K "u c a s e ( I I 2 2 1 , Thn. 6 . 1 ) .
(c)
, T ~ )=
get this
Each 6 - b a l a n c e d open s e t h a s t h e Runge property (cf. [22],
Ch. 6 , where other examples of open s e t s with
t h e Runge p r o p e r t y and some e q u i v a l e n t
assertions
are g i v e n , t o o ) . So t h e r e a l w a y s e x i s t s a b a s i s
a
o f open s e t s w i t h t h c p r o p e r t y t h - t
on
H(U)
f o r each
U E
?.!. -
Pt.
T(* =
T ?I
S c ! l o t t e n l o h e r 7 3 0 1 re-
marked r e c e n t l y t h a t e a c h pseudoconvex open s e t i n an a r b i t r a r y p r o d u c t @'
h a s the Rungc p r o p e r t y .
I t t u r n s o u t below t h a t t h e t o p o l o g y
T~
on H(U) hasmany
p l e a s a n t p r o p e r t i e s , a t l e a s t f o r ( g e n e r a l ) open s u b s e t s U
of
Schwnrtz s p a c e s . So some o f o u r n e x t r e s u l t s a r e f o r m u l a t e d i n
t e r n s of t h i s t o p o l o g y r a t h e r t h a n
1 9 . THEORE'I:
(a)
Let
E
T h e Ahead
be a L.c.
T~
.
methizubi?e S c h a h t z Apace.
o h h o l o t : i o t L p h i c 6uncAion.J o n E , endowed
BIERSTEDT
114
(b)
T~
i h
&
MEISE
t h e u n i y u e L y deRehmined 1 . c . bheufj
Ropvdogy
on % Rhat coincideb i u i t h t h e potrted t o p o l o g y
on t h e bpuceh H(U)
doh
E-balanced open
betb
UCE.
(a) For open subsets U , V C E with U 3 V , the canonical
PROOF:
-
restriction p uv rKopUV : H ( U )
-f
*
(H(U),
(H(V),
T ~ )
-f
T ~ )
is continuous, because
H(K) is continuous for each K E&
tion 05 the topology
-rTI
on
by defini-
H(U) .
Let (Uili I be a system of open subsets Ui of U : =
-riA
iyI Ui . We notice already that
than the projective topology
T
T~
E
with
is stronger on H(U)
with respect to the mappings
: H(U) + (H(Ui), T ~ , ) and we have to show that the conPU,Ui verse in also true. Let p be a continuous seminorm on (H(U),T~).
By definition of
T
IT
there exists a compact X c U and a contin
uous seminorm q on H(K) with p 5 q o nK. As K is compact and
E metrizable, we can write K =
m K. with compact subsets j=1 J
(1 = I ,...,m; use the existence of a Lebesgue number j for the covering u Ui). The natural (injective) linear mapi €1 m m ping A = ’ 1 : H(K) + ;7 H(K.) is continuous. BY regular j:~1 j=1 1
K .c Ui 3
ity of the inductive limits in tho definition of H(K.1 , j=l,. 3
n1
..,m,
(B) is bounded in H(K) for each it is easy to see that m m bounded set B H(Kj). Furthermore g E ( K j ) is a (DFI-space
c 1I’
NUCLEARITY AND THE SCHWARTZ PROPERTY
115
.
and 7 (a) imp]-ies the Monte1 property for H (K) Hence apply Baernstein's lerma from 1 3 1
we
A is open, and so there ex
:
ist continuous seminorms qi on H(K+), j = I , ...,m r such J J m q(g) 2 max q . (A.(9)) for a l l g E H ( K ) . By Aj 0 v K = v K j=l
J
j=1
m max q
But
hence
T~
we
'tJ,U.
3
( f ) ) for all f E H(U). 13
is a continuous seminorm on
(H(Ui),TT),md iE I is equal to the projective topology T defined above. o n
j
j=l
that
j
J
m p(f) 5 q(nK(f)) 5 max q . (rK
obtain
can
Kj
863
It follows that
is a 1.c. sheaf with respect to
.As in
T.,,
Prop. 16 the locally convex properties (Schwartz, s-nuc1ear)can be derived from the definition of
T
~
from , 7 , and from well-
known permanence properties. (b) It has already been remarked in 18 on H(U) for all 5 - balanced open sets
(c) that
'TT = T~
U C E and that these sets
are a basis for the open subsets of E . Hence (b) follows from 1 . 2 . b ) . 13
[63
(H(U), U
T ~ )or
(H(U)#
T ~ ) has
the a. p. for each open subset
of a nuclear metrizablc 1.c. space E by 19
and 16. We give
another result corresponding to what we proved in 12. 20.
PROPOSITION:
LeX E b e cin -in 1 2 . and annume t h a t
6-buLunced optiz n u b n e t
04
U
in a
E , T h e n (H(U),-rn)= (H(U),-rIw) has
t h e a . p.
PROOF: then
If U
is an open subset of E with the Runge property,
(by the very definition) the
{H(K) i TKLIK E$U where
&,=
{K
E
projective
system
is equivalent to the system {H(K); vKLIKE&,
xu;
K is U-Rungc }
.
And the last
system
is
BIERSTEDT 6 b1EISE
116
r e d u c e d by d e f i n i t i o n of
U-Runge.
IE U i s even 6 - b a l a n c e d for sane 6 by K E
8 of
t h e system
El o n e c a n r e p l a c e
c - b a l a n c e d compact s u b s e t s of U .
it i m m e d i a t e l y f o l l o w s from 1 2
Hence 2 0
E
But
t h a t H ( K ) has t h e
f o l l o w s from t h e remark (see e . g .
r8i,
&
for a . p.
Introduction)
t h a t t h e p r o j e c t i v e l i m i t o f a r e d u c e d s y s t e m of q u a s i - c a n p l e t e spaces with a.p.
a g a i n has t h e a.p.
U
By u s e o f methods i n v o l v i n g t o p o l o g i c a l t e n s o r
and t h e € - p r o d u c t
(see i:21:
and
[!I]), t h e topological vector m
space p r o p e r t i e s of H ( K ) = i n d H ( U n ) n+ of t h i s a r t i c l e ( 4
d e r i v e d i n t h e f i r s t part
and 7 ) a l l o w t o t r e a t v e c t o r - v a l u e d h o l o -
morphic germs a s w e l l ,
a t l e a s t i n c e r t a i n cases.
F o r t h e f o l l o w i n g p r o p o s i t i o n w e remark regular
products
that
compactly
( i n j e c t i v e ) i n d u c t i v e s y s t e m were d e f i n e d i n [i3]
And F o r open s u b s e t s complete 1 . c .
U
i n a m e t r i z a b l e 1 . c . space E and a,
s p a c e F w e d e n o t e hy Ii (U,F) a g a i n ( c f .
t h e s p a c e of all bounded holomorphic mappings from U w i t h t h e t o p o l o g y o f u n i f o r m c o n v e r g e n c e on U . Prop. 2 1
Satz 6 ,
f o l l o w s from 4
, 1,l.
and 7
-
any
1. ( a ) )
into
F
The p r o o f o f
above t o g e t h e r w i t h
[9],
4,
4 , S a t z 3, and 3 , Remark b e f o r e K o r o l l a r 11 a s w e l l as,
s a y , [ 5 ] , 31.
( N o t i c e t h a t one c a n show t h e e q u i v a l e n c e of m
i n d u c t i v e systems {H (Un,Fn) ; p,
the
1 and {Hmrp(Un,Fn);prim} belm
i n t h e c a s e o f Schwartz s p a c e s E by a i d o f 7 ( a ) j u s t as i n t h e p r o o f of 1 2 . )
21.
PROPOSITION:
cur;ipac,t.
Ahb[ime
Le,t E be. a m e t h i z a b l e L . c .
Eunuch a p u c e d
I < cE
XhuX t h e compLe,te ( f i u u b d o h d ~ )L.c. h p u c e F
Xtke i n d u c - t i u e . L i m i t
06
h p u c e and
0 6 a c o u n t a b L e i i z j e c t i u e i.nduc.tiwe
Fn.
i h
sy.5tem
NUCLEARITY AND THE SCHWARTZ PROPERTY I I ; E n u t i 4 I ; i e h c o n d i t i o n ( B M ) u n d .;I;
lUhe4.e
(
.. .)
=
If
compact 1 w i R h t h e induced E
F
i 4
i 4
i6
W
H (Un,Fn); f(Un)h phg
F
ALlp-MOhm
S c h w r z ~ t zn p u c e , n o
CL
F = ind Fn n+
d e n o t e 4 .the u A A o c i u f e d botnolagicuL
4 p U c e and Hm”(Un,Fn)
76
117
needed t o
Ub4Uhe
topology.
6 1 1 ~ t h e . h undunip.tion
on
Rhc e q u u l i t y
m
(H(K) E Flbor = ind H (Un,Fn). n-t 16 E i n a S c h k W t t Z 4 p U C e Ctnd i d F LA even
U
(DFS)-
Apcice, we g e t ;the d o e l o w i n g i m p t o v e n i e n t :
H(K)
m
W
F = ind H (Un,Fn) = ind H (Un,F). n+ n+ Fvh n u c l c u t E o n e o b t u i n n :
H(K)
REMARK: [2]
,
E
GT
F=H(K)
v
F = H(K)€F=ind Hm(Un,Fn)= ind Hw(Un,F). n+ n+
For Banach spaces E and F
,
Aron
and
Schottenloher
Sect. 6 denote ind Hw’’\Un,F) by %(K,F) and toplgize this space n-t .-
by taking the induced topology
of ind Hw (UnlF). If K is even n+ (even) balanczd, they are able to show H(K) E F = (HK(K,F),T T
topologically. (In this case the topology of H(K) can be charas terized explicitly by representation of its semi-norms,cf. &on [l],
Sect. 4
)
-
It is still an open question whether
T
is bog
nological. 2 2 . COROLLARY:
compact
Let E
j
with K . C E
b e methizuble l.c. b p a c e d
1
1
4 u t L 4 6 y c o n d i - t i o n (BM),
uJe
( j = 1,2).
(a)
16 b o t h El
ge-t (II(K1)
E
und
E2
H(K2) )bar
= isd
Hwrp(Uk
x
U:),
IOhuLe
,the
118
BIERSTEDT & MEISE
notation U 1
and
n
c f ( x l , ; x1 c f ( , x 2 ) : x2
i b o b v i o u s and whehe
U:
E
un1 1
E
un2 I A ptecompact
LA p ~ e c o m p a c tin
in
H"
2 (un)
H"
(u:) I
and
w i t h X k e i n d u c e d bup-nokm. (b)
16
Lo a S c k w a ~ t zo p a c e , we g e t
El
(BM):
clu&hdieA
(H(K1)
(c)
E
H(K2)
[q,
Ibor
= H(K1
El and E2
16 bu2h
an e q u U L i t y
PROOF:
a n y E2 w h i c h
dotl
H(Kl)
x
K2).
ahe S c h s u h t z b p a c e o , t h e f i e E
= H(Kl
H(K2)
X
K2).
( t h e proof o f ) 2 1
( a ) i s an easy consequence of
i d
(a)and
43. ( b ) w e have
( b ) By 2 1 .
(R(K1)
EH(K ))
br
.
= ind Hw(U~,Hm(rJ~))
n+
But t h e r e i s a c a n o n i c a l norm isomorphism H m ( U1 n , H m ( U n2 ) )
?
H m ( U n1
U n2) :
X
I n f a c t , t h i s isomorphism i s g i v e n by J d e f i n e d a s J(f) (XliX2)
S e p a r a t e G - a n a l y t i c i t y of
= f (Xl) (X2)
-
J ( f ) i s o b v i o u s , so J ( f ) i s G-analy
t i c b y t h e c l a s s i c a l H a r t o g s theorem and h e n c e (by boundedlness) 1 2 1 2 J ( f ) E H"(Un x U n ) For t h e converse l e t g E Hm(Un x Un) and
.
put
I ( g ) (x,)
:
x2
by a n a l y z i n g e . g .
+
3,
g(x 1 ,x2 1 . Then one can shaw I ( g ) Lemma 2 of
[21].
Obviously
2
Hm(Ul-,Hm(Un))
F
J
I o J = i d , h e n c e t h e isomorphism above i s p r o v e d and
0
I = id,
(b) f o l -
lows i m m e d i a t e l y . ( c ) T h i s i s a c o n s e q u e n c e o f ( b ) and o f t h e a u t h o r s ' res u l t from
[g]
( a l r e a d y used i n t h e proof of 2 1 ( c ) ) t h a t
the
119
NUCLEARITY AND THE SCHWARTZ PROPERTY
E-product of two (DFSI-spaces is again (DFS) and hence bornolo gical. 23.
CI 1e-t E
PROPOSITION:
be R . c . m e t f i i r a b R e Schwafitz
npacen
j U . C E . open (j = 1,2). Then t h e equality 1 3
with
PROOF:
The system
is cofinal in
P pu 1
2
=
{K1
x
K
K2;
U . compact,l=1,2)
1
1
3
,hence (H (Ul x U2 1 , T ~ )= pro] u1 u2 holds already. But by a simple reasoning using the fundamental properties of projective limits and
171
4.4 as well as 2 2 (c),
I
we get (canonically):
2 4 . COROLLARY: U . C E j ( j = 1,2),
1
Fox
E1?E2
C L ~i
n 2 3 . and 5 - b a l a n c e d open b e t n j
6 o U o t o ~t h a t ( H ( ~ J ~ ) , T ~ ) F ( H ( U ~ =) , (' HT( U~I )X
This is obvious from 23
,
because the product of
sets is ( E l ,c2)-balanced, and hence
T
IT
=
Tw
on
U2)r~W).
5 . - balanced 1
H(Ul
U2),
x
too, There are more representation theorems forvcctor-valued holomorphic germs and functions of which w e only mention
the
following one (as an example): 25.
PROPOSITION:
and
U C E
Let E
be n m e t f i i z a b L e n u c l e a f i R . c .
o p e n . Anbume t h a t t h e c o m p l e z e R . c .
bpace
npace F
ib
120
BIERSTEDT & MEISE
proj (H(U), + a
T ~ E )
Fa
=
proj proj f
H(K) E F
pZoJK E&u
=
pro] +
a
H(K)
E
F, =
KEfrCU
t
proj H(K) E Fa =
KE&
t ,
pro] pro] H(K,Fa). f
f
( A g a i . n , vne c a n h e p l a c e t h e Runge pxopexty)
PROOF:
'cT
by
'cu h e x e ,
id .the o p e n s e t U has
.
Obvious from a repeated application of [ 7 ] ,
4.4
and
from 21 (d).0 The
rest of the article is dedicated to the question
"how good" the nuclearity of H ( U ) under the topologies TIL
can be. A s s-nuclearity of ( H ( U ) , - r o )
)
I
and
1
Irresp. of (H(U),T~))
was already proved in some cases (see [ 6 ] of this paper]
T~
of
I
1.12 [resp. Prop.16
it is natural to ask more qcnerally,what types
of X-nuclearity (cf. Dubinsky-Ramanujan [13]
)
can
occur
for
H(U), if U is an open subset of an infinite dimensional 1. c.
space E . T o start with, the followinq Proposition 2 6
confirms
the obvious conjecture that thz type of nuclearity cannot better than for open subsets U of
be
C N l where N may of course
be any natural number. The main tool in the proof of 2 6
is the
result I , 4. 12 of Petzsche [ 2 5 ] .
So let
(a ) n n E N be an "exponent sequence" of positive numbers, i.e. "n 5 'n+l for all n E 7N and lim an = m n+w a
=
.
Let
A o o ( a ) be the associated power series spacc of infinite type
of which wc assume nuclearity. We do not define Am(w)-(md Am (a)-) nuclearity here, but refer to the Memoir
1131
of Dubinsky and
NUCLEARITY AND THE SCHVJARTZ PROPERTY Ramanujan and to the article [28]
PROOF:
121
of Ramanujanand Terzioglu.
A well-known corollary to the Hahn-Banach theorem im-
plies that E can be rcprcscntcd (topologically) IN, where EN with
f o r arbitrary
n
subspaces of E
. EN
E
and Ei belong to
dim EN
,$
=
open subset of
Ei, U = UN
by (31, ( H ( U )
=
,T)
nm(a)-(or at least
(H(UN), T )
m
x E
UN
UIq =
ii
EN
00
and
EN are N -f
-’
j
(DN)I
Ut # @ is some balanced
is balanced and open and hence,
(H(Ui)
,T) .
Plorcover, (H(U), T I
A m ( a ) - ) nuclear by assumption, (H(UN), T )
(H(UN),~o)by (21, and this space is also Am(a)-[resp. nuclear as
x
by (1). Let j : EN
be a topological linear isomorphism and define where D is the open unit disk of C.. If
N
m
E = EN
topological linear subspace of (H(U),T).
Am(d-]
is =
122
BIERSTEDT & MEISE
On the other hand, (H(UN),~o) is topologically isomorphic to ( H ( DN , T ~ )= (H(D),-ro)671... Gr (H(D),ro) = N - fold projective tensor product of
. Now
ill (n)
Petzsche [ 2 5 ] , I,4.12
proves that this N - fold tensor product is topologically isanog &-I) . phic to A , ( R ) , where 6, = [ N fi] ( = largest integer Finally, it follows now from an application of E&mnujan-Terzioglu [20], where
‘n lim n+m Pn N E IPJ was arbitrary. D
Prop. 2.12, that
0 and hence
=
lim n-tm
‘n -
= 0,
N~Il
2 1 . REMARKS:
(a)
As thc proof shows, wc actually need much lessthan (1) and ( 3 ) of 26. Yet assumptions (1)- (3)in Prop.
26 arc satisfied for
or
(b)
=
T
= -i0
,
if
Silva spaces (see e.g.
,
if
mctrizablc Schwartz spaces.
it is a l s o allowed to take
Of course, condition
restricts
(*)
Frkhet spaces 43 ) . By (23
[5],
and) 2 4 =
=
T = T ~(or T ~ ) ,
the
possible
Am(a)-nuclearity types for H ( U ) considerably. (*)
is still satisfied by the following
sequences
a(p)
,
1 < p <
,
But
exponent
which lead to differ-
ent nuclearity types stronger than s-nuclearity:
1 < p < q <
m
,
and so A_(B(~)),but not Am(a(P)),
is d_(c~(~))-nuclcar by 2.13.
[28]
[ In particular , A m ( a
,
Prop. 2.12 and
)
and
Cor.
A m ( a (q)) cannot
bc topologically isomorphic 2 . 1 thc
Aftcr the examples in 2’7 (b) , the next question is whether Am(a)-nuclcarity types not excluded by condition (*) of 26
NUCLEARITY AND THE SCHWARTZ PROPERTY
123
are i n f a c t r e a l i z e d , i . e . : D o t h e r e e x i s t i n f i n i t e d i m e n s i o f l s u c h t h a t f o r e a c h ( s a y b a l a n c e d ) o p e n subset
a1 1.c. spaces E U of
t h e space (H(U)
E
gy T
,T)
i s Am ( a ) - n u c l e a r , where t h e t o p o l o -
( 2 ) and ( 3 ) of P r o p . 2 6 a n d ,
agrees with conditions
,
u =
instance,
1< p <
,
m
question i n t h e affirmative f o r
for
l i k e i n 2 7 ( b ) ? W e answer t k i s
em
by t a k i n g E =
T = T~
in
our last proposition. An e x p o n e n t s e q u e n c e c1
<
SUP
nEN
.
m
i s c a l l e d s t a b l e , i f it s a t i s f i e s
c1
a r c all s t a b l e
(The s e q u e n c e s
n '
28.
PROPOSITION:
L e t t h e n t a b l e exponent neyuence
nutiodg
CY.
c o n d i t i o n ( * ) a h 2 6 . T h e n t h e o h e a a @ 0 6 fiolarnahphic &ncfiam on
a''
,
e n c i o r ~ ~td~ ~ i t,the h topology
ale H(U)
Tr
, u c CI:N open),
io A,, (a)- n u c l e a x . PROOF:
As
i t i s enough ( c f . I , 5.4
-
,
[6]
1.2.c)
h e r e w e need t h a t
a r i t y of
,
( 8 $ l ~ 7 Ti)s a l o c a l l y c o n v e x s h e a f b y Theorem 1 9
(II(U) ,
bourhood b a s i s
T ~ )only
&
a
, 1131 ,
Thm. 2 . 1 0 ,
and/or
i s s t a b l e ) t o show A,, ( a ) - n u c l e
f o r open sets U
-
i n a s u i t a b l e neigh-
z e r o . I3y t h e p e r m a n e n c e p r o p e r t i e s
of
11251 ,
of
A m ( a ) - n u c l c a r s p a c e s t h a t wc h a v e a l r e a d y u s c d , it then s u f f i c e s
t o prove
A,,(a)-nuclcarity
f o r each H ( K )
a n a p p r o p r i a t e c o f i n a l subsystem choosing
21
and t h e systems
&,,
where K
guof aUlU U E
2
A m ( a ) - n u c l c a r i t y of H ( K )
n e e d o n l y show
,
E
@.
i s taken fran
.
Hence, by
I
i n a n o b v i o u s way,we
,
if K =
Kj
, where
j E N
Kj = { z E
ind Hm(Un n +
a;
1
z
with
1 5
s
j
Un =
} with
jem
s . > 0.
3 -
un , U n J
1
=
In t h i s case, for
j > n
H(K) =
and
124
BIERSTEDT
Un = { z E C ; l z l
c s . +in}
&
MEISE
for j 5 n, and hence
7
3
We represent the canonical impping n
E
M
Here
,
Gn = n ?Cen n
j=1 +
H (Un+l)l
as follows to get its Am (a)-nuclearity:
on satisfies
tion, and B(f) = f
IT^,^+^
defined by
on < r n' R is the canonical restric 1 ~ ~ , ~ + ~ where l 6 ~ +IT n,n+l ~ , . an+' + an is
rn+l 0
.
,..., z
(zl
~ + = ~ )(z,,
... ,zn).
(Any othernotg
tion is obvious-) Similarly as in the proof of 2 6
,
fro3 [ 2 5 ]
where Bk
=
I, 4.
12
that:
,
it follows
(H(Gn),-c0)= ( H ( U ~ ) , . I G71...G71(H(~),~g)= ~)
[ "A] , and hence this space is certainly
clear for any
-
maps (H(Un) ,
a satisfying ( * )
T )~
P,,,+~
Am(a )-nu
(by [ 2 8 1 , , Prop. 2.12).
into a Banach space, R is
factorization of ariQ
m
m
n,rttl : H (U,)
p
.
As
A m (a)-nuclear.
given above implies the
R The
Am(a)-nucle
of this napping, as claimed. tpn,n+l is again A m (a)-nuclear As the transposcd mappin9
for each
n E lN , we obtain from the representation
proj Hm(Un)i c n
(by the very definition) that
H(K)L
=
H(K)A is Am(a)-ng
clear by Ramanujan - Terzioglu 1 2 8 1 , Cor. 3.7. 0
NUCLEARITY AND THE SCHWARTZ PROPERTY
condition ulz b y 604
(*)
i n 26.
EA i n uL4eady A c u ( a ) - i i i i c l e-
[R@ma4k - t h a t
1 2 t i ] , Co4. 3.71.
each open n u b h e t U
I n t h e n (H(u),T~)u
06 E
125
n m ( a )-nucLam npace
?
FINAL REMARKS: In a private communication, Richard Aron mentioned that, some time ago, he had already given a proof (unpublished) for regularity and the Monte1 proper ty of H(K) for each compact K in a 1.c. metrizable Schwartz space E . Generalizing a result of Carroso (1970)cfor U Martin Schottenloher (in a private has recently proved
T~
= T
0
remains true with
am.
replaced by
T~
T~
T~ = T
71
So
T
0
= T
balanced and open ? T
#
T~
w
always hold on H ( U )
0
E
,
(We know of no examples
28
.
like
ask: For which (FS)-[ rcsp. (FPII-] spaces T
3,
- Tw)
Prop.
or
In connection with b), the authors would
the equality
IN
a:
communication)
(and hence
on H(U) for each open subset U of
=
to
does U C E
with
in this case).
REFERENCES
[ 11
R. ARON: Tensor products of holomorphic functions,Indag. Math. 35, 192- 2 0 2 (1973).
[ 21
R. ARON, 14. SCHOTTENLOHER: Compact holomorphic mappings on Banach spaces and the approximation property,J. Functional Analysis 21, 7-30 (1976).
126
[ 31
BIERSTEDT
&
MEISE
A. BAERNSTEIN 11: Representation of holomorphic functions by boundary integrals, Transact. Amer. Math.
SOC.
160, 27 - 37 (1971).
[ 41
J. A. BARROSO, M. C. MATOS, L. PJACHBIN: On bounded of holomorphic mappings, Proceedings on
sets
infinite
dimensional holomorphy , University of Kentucky 1973, Springer Lecture Notes Math. 364, p. 123-134 (1974).
[
51
K. - D . BIERSTEDT: Tensor products of weighted spaces,Funs
tion spaces and dense approximation, Proc. Conference 3onn 1974, Bonner Math. Schriften 81, p.26-58 (1975).
[ 61
K.
- D.
BIERSTEDT, B. GRAMSCH,
R.
IIEISE: Approximations
-
eigcnschaft, Lifting und Kohanolcqie bei lokalkonvexen
.
Produktgarben,manuscript math. 19 (1976)
[ 71
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Ann. 209, 99-107 (1974).
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- D. BIERSTEDT,R.MEISE: Induktive Limites gewechtiter stctigcr und holomorphcr Fun1:tioncn
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J. MUJICA: Spaces of germs of holomorphic functions, to appear in Advances in Math. (cf. also Bull.
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L. NACHBIN: Topology on spaces of holomorphic
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Springer Ergebnisse der Math. 47, 1969. 124 ] P. NOVERRAZ: Pseudo- convcxitg, convexit6 polynomiale et domains d'holomorphie en dimension infinie, North Holland 1973. [25 ] IT. J. PETZSCHE: Darstellung der Ultradistributionen
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1 3 1 1 L. SCHWARTZ: T h g o r i c des d i s t r i b u t i o n s les I , Ann. Inst. F o u r i e s 7 , 1 - 1 4 2
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(1957).
( R . Meise)
F a c h b e r e i c h 1 7 , Mathematik,
Mathematisches I n s t i t u t
Gesamthochschule D 2
der Universitat
Warburger S t r . 1 0 0 P o s t f a c h 1621
Universit'atsstr
3 - 4790 P a d e r b o r n
D
Bundesrepublik Deutschland
Bundesrepublik Deutschland
- 4000
.
1
D'ksseldorf
This Page Intentionally Left Blank
Infinite Dimensional Holomorphy and Applications, Matos (ed.) @ North-Holland Publishing Company, 1'377
DUALITY AND SPACES OF HOLOMORPHIC FUNCTIONS
By
I.
PHILIP J . B U L A N Q
INTRODUCTION
A c l a s s i c a l problem of c o n t i n u a l i n t e r e s t i n holomorphic
f u n c t i o n t h e o r y i s t h e f o l l o w i n g : when may o n e r e p r e s e n t
the
d u a l of a s p a c e o f h o l o m o r p h i c f u n c t i o n s a s a s p a c e of holomoz p h i c f u n c t i o n s ? T h i s p r o b l e m was e l e g a n t l y s o l v e d i n t h e complex v a r i a b l e c a s e by G r o t h e n d i e c k , S i l v a - D i a s , i n 1 9 5 2 - 53. T h e i r r e s u l t s t a t e s t h a t i f o f t h e complex p l a n e and
U
one
and K o e t h e
i s an open
subset
H(U) i s t h e s p a c e of h o l o m o r p h i c fung
t i o n s on U w i t h t h e compact open t o p o l o g y , t h e n t h e d u a l
of
H ( U ) may b e c h a r a c t e r i z e d a s t h e s p a c e of h o l o m o r p h i c germs on
t h e complement of
U
i n t h e Riemann s p h e r e which v a n i s h a t i n -
f i n i t y . The s e v e r a l complex v a r i a b l e case i s much h a r d e r ,
but
M a r t i n e a u ( 1 9 6 6 ) h a s p r o v e n some r e s u l t s i n t h i s d i r e c t i o n f o r
U an open convex s u b s e t o f C n . I n t h i s p a p e r w e w i l l c o n s i d e r some s p a c e s of
holmrphic
f u n c t i o n s on open a b s o l u t e l y convex s u b s e t s of d u a l of F r e c h e t n u c l e a r s p a c e s (dDFN s p a c e s )
,
and c l a s s i f y t h e i r d u a l s as spaces
of holomorphic f u n c t i o n s .
131
P. J . BOLAND
132 11.
PRELIMINARIES
[
For n o t a t i o n and t e r m i n o l o g y w e r e f e r p r i m a r i l y t o
B21
, [Mu] , [N]
and [ P I .
The p r o o f s o f t h e p r o p o s i t i o n s
t h i s s e c t i o n may be f o u n d i n
are
H'(C) ,
C C , Cn,
N
/$IA
5 1 1 and
and
a', and 1' . I f /$/A<
o u s m - homogeneous p o l y n o m i a l s on E
of c o n t i n u o u s p o l y n o m i a l s on E ist (@n)C E '
z
Kl:$l
<
A C E l t h e n Ao={$:$EE',
'1
*
and P ( m E ' ) a r e r e s p e c t i v e l y t h e s p a c e s o f c o n t i n u -
P(%)
m
[B2].
i t s s t r o n g d u a l . Examples o f @ k "spaces
= {$: Q E E',
A'
,
in
Q F N s p a c e (the strong dual of a Frechet-
E w i l l denote any
n u c l e a r s p a c e ) and E '
[Bl]
BlIl,
+
.- I f
and E '
.
P ( E ) i s t h e space
p E P ( m E ) , t h e n t h e r e ex-
x
such t h a t p ( x ) =
C $:(x) for a l l n=l f o r a l l compact K C E .
m
and
E E
n= 1
W e may d e f i n e two e q u i v a l e n t t o p o l o g i e s E
& TI
on P ( m E ) .
i s t h e compact o p e n t o p o l o g y d e f i n e d by t h e f a m i l y o f
norms T
E
E~
where K
i s a compact s u b s e t of E and
i s g e n e r a t e d by a l l semi-norms of t h e t y p e
F
semi-
(p) = s u p ~ p ( xI ). xEK
n K where .,(p)
=
m
f o r a l l L compact i n E } .
PROPOSITION
1.
I n duct g i v e n K
ir
and
E
U J L ~c q u i v a t e n t
topo-togie~hon
compact i n E , t h e k t e x i s t
compact L H E s u c h t h a t
K G Kl
and
C > 0
and
P ( ~ E ) .
K1
D U A L I T Y A N D S P A C E S OF H O L O M O R P H I C F U N C T I O N S
PROPOSITION 2.
P ' (mE)
P(?EI)
v i a t h e mapping
T($")
doh
,
t h e duaL 0 6
B : T
P(%)
E P'(mE)
+
,ib
133
to
ibomohphic
BT E P ( m E ' ) w h e t r e f3T($)
$ & E'.
S P A C E S O F HOLOMORPHIC FUNCTIONS
111.
L e t U d e n o t e an absolutely convex open set i n t h e Q F N space H ( U ) t h e s p a c e o f h o l o m o r p h i c f u n c t i o n s on U .
E , and
D E F I N I T I O N 1.
The
t o p o e o g y on
E
by a l e
H ( U ) i c l t h a t dedined m
bemn - nohmd
f E H(U).
dot
p o l o g y on
(K compact i n U ) whetre ~ ~ ( = f C)
E~
7 n duct t h e E
H(U),
DEFINITION 2 . C
H(U)
HN(U) =
W
nK(f) =
and
vK (
Ef
topology
i n f (0) 7 ) < +
w
1
L A a F k e c h e t bpace., and
TI
P( f ( 0 ) ) 7
f E H ( U ) , and d o h a&? compact K C U ,
n=O t h a t g e n e h a t e d b y aLL b e m i - n o t m b %(U),
~
i n a Fhechet nucLeah Apace ([Bd).
,E
:
i d
E
n=O t h e compact open t o -
. The
v
t o p o L o g y on H N ( U )
n K f whe.te HN(U) , n
-+
K
i b
H ( U ) ,E
i d
compaci in U.
i n a con -
tinuoud i n j e c t i o n . REMARK 1.
HN(U)
Ly c h a t a c t e a i z e duct on
i t b
HN(U) = H(U) H(U)
i n a n a t t h a c t i u e Apace becaude One can L c g
,'TT
in
duaL. 1 wou.td L i k e t o c o n j e c t u h e t h a t [and i n t h i b Cube it wouLd d o L & o w t h a t
71
= E
1.
REMARK 2 .
(a)
T h e doLlowing commentn axe in o t d e t concetrning % ( U ) : T h e apace
H(E)
npace 0 6 H N ( U )
0 6 entihe 6unctionb o n
E
i b
a
bUb
land h e n c e id U = E , Men %(U) = H ( U ) ) .
134
P . J . BOLAND
let
DEFINITION 3 .
Let
be an abdoLuteLg c o n v e x open b u b b e t o 6 E .
U
be an i n c k e a b i n g bequence 0 6 a b n a l u t e l y c o n v e x cum -
(K,)
p a c t bubbe.t.5 0 6 U b u c h t h a t U
then
K C- K n
hence we L e t Foh
each
that
Uo
n
H(Uo)
I
5 K,'
K',
i b
and
.
n
doh d o m e
h) Kn
U =
Now Uo
and id
logy
Y L U ~ ~We.
l;rn
H~(K:).
DEFINITION 4 .
tonb
06
and
I
.
Uo
an abboLuteQy c o n v e x open
6UCh
=
n
K.:
H(uO)
in a Q
and
(K,)
the
Banach
endowed w i t h t h e
F
Apace ( n e e [ n u j ] 1 .
be
ab
an > 1 b u c h t h a t
geMehaLity ( b y trepLacivzg (L,)
KE
E'
d,th the inductive L i m i t t a p o -
H(Uo)
Let U
in
bet
w i L L deno,te
H~(K:)
endow
c e n b a h y ) , w e may abbume
in
(K,)
g i v e n in U e a i r z i t i o n
5U
Ln = a n K n
.
U
w i t h a bubnequence
nuch t h a t
Uo
=
2
3.
Without
a n iMCkeUbiflg b e q u e n c e a 6
s o l u t e l y convex compact b u b b e t b o Q
4
ne
ab-
L , '
and
and
Xn
= lim H ~ ( L ~ ) .
H(u')
+
Now suppose
such that Tm = T
let
Tm
El
.in
be ,the bpctce o & holomu4phic ge4mb on
Uo
e a c h n , chaobe
Fah
i n compact
compact i n
i d
Apace 0 6 bounded holomohphic 6UnCtionb on bupkemum
K
E
P' (mE)
T(f)
T
I 5
P (mE) '
E Hi(U).
CT
Kn
Then
Now let
Fm
(f)
Then t h e r e exist
for a l l
Trn(p)
5
CIT
Kn
f E HN(U).
C > 0
For each m we
( p ) for a l l p
E
P(mE), and
= B T E~ P ( ~ E ~ )(see Proposition
2).
DUALITY AND SPACES OF HOLOMORPHIC FUNCTIONS
1 Tm(@m ) 1 5
IFm(@)1 =
Then
CTI (@m) for all Kn
135
Finally
$ E El.
L-r
define F on L,'
by
F(@)=
m
m
c
JF(+)J5
m= 0 that F E Hm(L:) where
and
TFEOREM 1.
(IF I]
w i t h t h e bpace
06
PROOF.
8
(a)
@ is well defined, and
getrmb
8
need only show
06
E
for all
(%) p
(bttrong
Uo.
P'(mE) and
1 - 1. To show that
P(mE') for each m
2
,
f3 is
it fol
is an isomorphism,
H(Uo). There exist n and
M
=
F
) I F 1)
E
Hm(K,')
such that m
I
and
F
=
X Fm be m= 0
For each m let
BTm = Fm It follows that Tm(p) 5 Mn (p) Kn P(mE). We now define T on HN(U) by
be such that
E
scu'.
/T(f)I 5 MrK (f) for all f E HN(U) ,and hence T E n E T = F, we have shown that 2 is an isomorphism between
Clearly Since
we
is onto.
i s the germ of F on Uo. Let
E P'
the
HN(U) may be idevttidied
the Taylor series representation of F in K,'.
Tm
06
is an isomorphism. Since by Proposition 2
is
Let
on
is
H(u').
Hence ,the driat
an isomorphism between lows that
.
d e d i n e b a homeomotphibm b e t w e e n Hi(U) H(Uo).
Now
(a l m
L e t U be an abboLutei?y conuex opehi n u b b e t
Then
E.
.
0
Ln
@ E
= (Can)/(an-l). It is clear n 5 (Can)/(an- 1). We define 8 T = F,
HE;(U) to
t o p u t o g y ) and
F
m=O
Fm(@) for all
is the germ of F on Uo
a linear map from
&FN
c c
I F ~ ( $ )5I
C
m= 0
Hi(U) and
H(Uo).
(b) tinuous. Let
i s a homeomorphism. First we show that
Ta
+
0
in
8
is con
-
HA(U) , and we will show that @T,=Fa
-+
0
136
P . J. BOLAND
i n H ( U O ) . Suppose now t h a t W i s a n e i g h b o r h o o d o f 0 i n and l e t W n = 1;'
(W)
f o r e a c h n where
In
Hm ( L z )
t h e r e e x i s t s a s e q u e n c e ( bn ) o f p o s i t i v e c o n s t a n t s s u c h Vn = {F : F E H m ( L z )
,
11 2
IlF
f o r each n
Wn
bn)
t o show t h e r e e x i s t s a' s u c h t h a t when
Fa
has a r e p r e s e n t a t i o n
there exist
a 2 a'
C > 0
f E HN(U).
,
T~ E BO. s u p p o s e now
and K n
] T a ( f )[
such t h a t
Ta E Bo,
But a s
ITa(f)
. Since
= ,8Ta
$ is
I L
5
CrK (f)
Then
a'.
for
n
(f)
for
'n it fol-
Fa E Hm(LZ) I ) V n ,
t h i s axpletes the proof
continuous.
i s c o n t i n u o u s . As
in
2
(l/Cn)r
3-l
W e c o m p l e t e t h e p r o o f by showing t h a t
that i f {
a
I] Fa!/-< (an/Cn)/(an - 1) =bn
lows t h a t there exists Fa E Hm(LZ) such t h a t
that
.
From t h e c o n s t r u c t i o n i n D e f i n i t i o n 4
f E HN(U).
Fa
3T
=
.
s u c h t h a t when
and
Fa
-
f o r each n }
all
suffices
B = i f : f E H ~ ( u ) , rK ( f ) < cn = a n / ( b n ( a n 1)) n Then B i s bounded i n H N ( U ) and w e may f i n d a '
Define
all
f o r some n
that
. It
a > a', t h e n
E H m ( L Z ) f'l Vn
,
. Therefore
H (Uo)
-+
H(Uo)
H(Uo)
: H(Uo)
s(U)
-+
i s bornological, it s u f f i c e s t o
Fa
}
i s bounded i n
{
Fa
} be bounded i n
-
-1 Fa 1 i s H ( U o ) , t h e n { f3 A
show
bounded
Hi(U). Let
{ F a } bounded i n
H(Uo)
. Then
Fa
Hm(KZ) such t h a t
t h e r e e x i s t Kn
i s t h e germ o f Fa on Uo
/ / F aI [ . From t h e a p a r t of t h i s proof, it follows t h a t t h e r e e x i s t Ta E f o r each a
f o r each a
(see I M u j ] ) .
such t h a t
M = sup
Let
,8Ta =
Fa
f E H N ( U ) . Hence w e see t h a t i f t h e n V i s a neighborhood o f each a
. Hence
0
and
IT,(f)
I 5
MrK
n
V = { f : f E H N ( U ) ,r
in
{Ta} i s bounded i n
and
Hi(U)
(f) f o r a l l (f)
Kn
H N ( U ) s u c h t h a t TaE
Hi(U).
first
2
lm, for
DUALITY AND SPACES OF HOLOMORPHIC FUNCTIONS
I37
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[BMN]
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BARROSO, M.
nets
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On
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06
Conf. on I n f i n i t e D i m e n s i o n a l Holomorphy,
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[
Bl]
P . J . BOLAND
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M a l g h a n g e Theohem dax e n t i h e 6 u n c t i v n s a n
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B e r l i n and
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[ B2] P. J.
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npace~,
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06
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Ban 47, Springer-Verlag New York, 1969.
[ P ] A. PIETSCH
-
NucLeah L o c a L l y c o n v e x
ApUCeA,
Ergebnisse der
Mathematik und ihrer Grenzgebiete, Band 66, SpringerVerlag, New York, 1972.
[S
C. L. DA SILVA DIAS
-
Enpacon VectohiaiA T o p a l o g i c o n
e
nua a p p L i c a C a e n nun eApaCon d u n c i o n a i b a n a l i t i c o n ,
Boletim d a Sociedade de Matematica d e Sao
Paulo
,
vol. 5, 1952.
Department of Mathematics University College, Dublin 4, Ireland.
A HOLOMORPHIC CHARACTERIZATION OFBANACH SPACES WITH BASES
BY s v o 8 a n g Chae
ABSTRACT:
Let E be a Banach space with a monotone normalized
basis i bn 1 .Every holomorphic automorphism on the open m
ball El of E is of the form
C xnbn n=l
W
+
C n=l
---
n
unit bn
W
C a b E E l ; lhnl = 1 (n E: N); a permutation of n=l n n if and only if €3 is isometrically isomorphic to co.
where
AMS (1970) Subject Classifications. Primary
32A30,
N
461145,
46B1.5, 46699. Key Words and Phrases. Holomorphic maps on Banach Spaces,Basis, Mobius transformations, automorphism, isometry.
139
140
S.
1. INTRODUCTION:
B.
CMAE
On the open unit ball El of a. complex Banach
space E with a normalized basis { bn}, we define
the
MZbiuh
t f i a n nd o h m a t i o n $a : El
+
E
by xn-a n 1 1 - B x bn n=l n n m
m
$a( 1 xnb,) n=1
=
m
where a =
C anbn E El. Then $a is an irjective holomorphic n=l (Frgchet differentiable) map on El. St.and.artresults about hg
lomorphic functions on Ranach spaces may he found. in [ 2 ] .
If
E = C, it. i s well known that the Mobius transformat.ion charac-
terizes the injective analytic maps (i.e., the conformal maps) of the open unit disk onto itself. In this paper w2 show that for the Mobistransformations to characterize the holomorphic automorphisms of El onto itself it is necessary and sufficient that the Ba.nach space E is :is0 metrically isomorphic to the Banach space co of sequences cog verging to 0. 2. AUTOMORPIIISMS:
Let E be a complex Banach space
U C E be a nonempty open set. A mapping f : U be
+
and
U is said
h o l v r n o h p h i c a u t o m o t p h i n m if f is a bijective
let to
holomorphic
map with the holomorphic inverse. A u t ( U ) will denote the space of the automorphisms on U and I s o ( E ) 1irill dcnote thet set of the linear isometries of E onto itself. Unlike the finite dimensiofl a 1 case, J. bijective map may not have the holomorphic inverse.
LEMMA 1 :
1c.t
El b e Rhe a p c n u n i R b a L L
04
a Ranach b p u c e
E
HOLOMORPHIC CHARACTERIZATION
1. 4 1
o u c h t h a Z & a h ewehy N E El t h e h e e x i b t h fa E Rut(E1)
f,(o)
= a
.
q
Then 604 evehy
E
Aut(El) Z l z e h e
ex&&
With
sE
ISO(E)
ouch th&
g
PROOF:
We have
lemma [l]
=
fN
s,
0
g(0)
=
a
.
-1 -1 fa (a) = 0 and fa o g ( 0 )
there exists
=
By
0.
S E I s o ( E ) such that
S
=
Schwarz's fil o g
on
*
Let I3 be a Banach space with an unconditional basis {bn}. The norm
1) x 1)
is called Aymrne,t'Lic
N and for any sequence
if for any permutation 4 on
{An 1 in C with lhnl
=
equality holds: m
m
We state the following lemma from [3],
m
m
$(I x b ) = C A x n=l n n n=l n n(n)bn
fn (x)
=
xn
X
(n)-"n n n (n)
1-8 x
p . 265.
1, the following
142
S. B.
PROOF:
Let
=
ct
( an ) =
@,(XI
E B.
c n=l
i s a n automorphism and
CHAE
Then t h e Mgbius t r a n s f o r m a t i o n
x -a
n n e 1-Bnxn n (0) =
-
en 1 d e n o t e s t h e
ct (
standard
b a s i s f o r c o ) . I t i s a n e a s y matter t o c h e c k t h a t
f E Aut(B)
Let
f = @
(y.
. Then
such
that
o S by Lemma 1, a n d h e n c e w e o b t a i n t h e d e s i r e d r e p r e -
f
s e n t a t i o n of
a s a c o n s e q u e n c e of Lemma 2 .
3 . A CHARACTERIZATION OF co: basis
S E Isc ( c o )
there exists,
{bnl
E
be a Banach s p a c e w i t h
a
i s s a i d t o be m o n o t o n e i f
bnl
*
Let
k
k+ 1
for a l l k
[4].
THEOREM 2 :
L e t E b e a Banach ~ p c ~ cwei t h n m o n o t o n e n o l r m a d i z e d
baAih
{
bnl
06
E
i d
El
.
7 6 evetry a u t v m o l r p h i ~ m f
m
f ( 1 xnbn) = C A n n= 1 n=l T
,
bade
0 6 t h e dokm m
(iuhehe
o n t h e open u n i t
an
,
An
caddy i d o m o k p h i c t o
ahe
ad
X
-a
l-d 1,
n n n bn n n(n)
i n Theotreni I ) , t h e n E in i n o r n e t h i -
co.
W e need t h e f o l l o w i n g lemmas.
1.4 3
HOLOMORPHIC CHARACTERIZATION
Let
LEMMA 3 :
n
2
2
b e u 6 i x ~ di r z t e g e h . 2
(a)
1
E R hUCh t h a t
X
(b)
a =
F v / r O ( h z n + 2 ,
Let
m~
Zhe/ze
N.
QXihth
= ) -x - q
1-ax
X
5
n
, thehe
2
n
E R hr
-a 1- x1-ax 1
and h =
m m+n+2 ,a=mtn m + l n+
z h ,that
1 x( ( lil+
'
and
.
W e u s e t h e f a c t t h e tl6bius t r a n s f o r m a t i o n on t h e
PROOF:
'
m+ 2
F O R each A , O z h 5 x
eXihth
n +2n-2
open
u n i t d i s k of t h e complex p l a n e n a p s c i r c l e s t o c i r c l e s and l i n e segments t o l i n e segments. I n p a r t i c u l a r , t h i s
transformation
maps a r e a l l i n e segment t o a n o t h e r r e a l l i n e s e g m e n t . W e prove o n l y ( b ) s i n c e ( a ) c a n b e shown i n e x a c t l y t h e same way a s (b). For
a =
m , let m+n
W e d e n o t e by S ( r ) t h e c i r c l e
lz
E C
: IzI
1
= r
m+ 1 m+ 2 Then c a r d 4 a ( S ( m + n + 1)) 0 S ( m + n +2 ) = 2
.
.
In fact,
m+ 1 m + 2 m+l m+2 -1 < a =$Ic1 (- m + n + l1 < - m + n + 2 < 0 < b = @ a (-m + n + l ) < m + n + 2 and t h e i n t e r v a l m+ 1 @a (' (m + n + 1) )
[ a , b]
. Therefore,
1x1 5
LEMMA 4 :
i s t h e d i a y o n a l of t h e c i r c l e
m + l
m+n+l
we can f i n d
and
-A
x E R such t h a t
X - a
=- 1-ax
U n d e f i t h e h y p O t h Q h i h v d Thevheni 2 ,
a
we lzave
144
/I
bl
S . B . CHAE
+
..,+bn
11
= 1
d o h each
n E N.
I t i s s u f f i c i e n t t o show t h a t f o r e a c h
PROOF: A libl
+
b2
...
+ b2 +
11
+ bn
A
< 1. L e t 0 < A < 1. Then
,
0 <
<
1,
m+2 m+n+2
-
0
for some m.
We u s e i n d u c t i o n o n m . 2
A L -n + 2 , t h e n t a k e
If
u
n
=
n2
1 XI
such t h a t
1x1 < n
,
f;
<
@,(x)
and
11
w e have
...
+
a(bl
-
=
+
A.
bn)
X I 1 bl Let m = l ,
+
... +
i.e.,
/I
bn
@,(XI
3 n + 3 .
0 < A <
-
Lemma 3 , t h e r e e x i s t s x E R s u c h t h a t
X
=
1
x - U
.
Since
IIn(bl
.. .
+
I/
11
< 1;
(bl
I
x
and
n
x(bl+ ...+ bn)/l < L
. .. +
+
Take
x E R
2 n+2
-
11
=
Since 0 < u <
0 < A <
By t h e h y p o t h e s i s o f Theorem 2 , i f
and
+ 2n - 2
c1
=-
then bn)ll < 1.
1 n + l
*
2
1
5 n+2
By and
+ b n ) 11<1 and Ilx(bl+ ...+ b2)11 < 1
(by t h e p r e c e d i n g c a s c ) , w e must h a v e
X /Ibl +
...
+
bn
11
< 1.
I n d u c t i v e l y we o b t a i n f o r e a c h 9 < X
Now t a k e
X
-f
1 a n d we h a v e
11
bl
+
5
z2-. m+n+2 IIbl+...+b 11 <1. n
.. . +
is monotone a n d n o r m a l i z e d , we conclude t h a t
PROOF OF THEOREM:
I
bn 1 1
11
bl
5 1. S i n c e I bn 3
+
S i n c e every automorphism f i x i n g
netry by S c h w a r z ' s lemma
. .. + b n 1 1 0
= 1.
i s a n is0
[ill it i s i m c d i a t c l y a p p a r e n t t h a t
HOLO?IORPHI C CHARACTER1 Z A T I ON
f o r any p e r m u t a t i o n
TI
on N
lXnj
and
145
= 1. From t h i s f a c t
we
have m
m
Therefore, t h e b a s i s
11
{ bn)
...
,
i s u n c a n d i t i o n a l (see [ 4 ]
p . 500).
is n 11 = 1. f o r a l l n E N , { b n ) e q u i v a l e n t t o t h e u n i t v e c t o r basis I e n 1 o f c i.e., there Of
Since
bl
M I N > 0
exist
+
ib
such t h a t m
I _<
N sup
1 xn
m
(see
[4
1,
C
p. 5 0 4 ) . T h e r e f o r e ,
n=l
xnbn E E
i f and
only
if
1l=i
m
Xnen
F
co
b y t h e d e f i n i t i o n of e q u i v a l e n t b a s e s (see [4],
p. 6 8 ) . m
03
T : E
Let
-t
co b e d e f i n e d hy T ( C
xnbn) =
n=l T i s an isomorphism. W e want t o show t h a t
I @,I
Let
5 1. W e c a n f i n d numbers
u n i t c i r c l c of C s a t i s f y i - n a m
Bn
=
1/2 ( A n
xnen.Then
C
n=l T
An
i s an i s o m e t r y . pn
and
+ vn).
on t h e
Then
m
T h i s i n e q u a l i t y shows t h a t i f
11
k C
n=l
ynbn
11 5 I!
Iynl
5 lxnl ,
k C xnbnll ( f o r a l l n=l
From t h i s and Lemma 4 w e o b t a i n
k
E
N).
146
S . B.
CHAE
Therefore,
From ( * ) w e a l s o o b t a i n
or
m
I1 c
n=l
Xnbn
T h i s proves t h a t T
I1 2
SUP
n
I
xn
I
i s an i s o m e t r y .
REFERENCES
[ 11
L . A. HARRIS,
S c h w a r z ' s lemma i n normed l i n e a r
P r o c . N a t . Acad.
[ 21
L. NACHDIN,
r o p o h g y on
Springer-Verlag,
[ 31
Sc. 6 2 ( 1 9 6 9 ) , pp. 1 0 1 4 ApaCf?b
Berlin,
-
ad h o l o m o t p h i c
spaces, 1017.
mappingb,
1969.
S. ROLEWICZ , Matrric L i n e a t b p a c e b , P W N - P o l i s h
Scientific
P u b l i s h e r s , Warszawa, 1 9 7 2 .
[ 41
I. SINGER,
Basen i n Ranuch
Berlin,
-
Verlag,
U n i v e r s i t y of South
Florida
bpUCeb
I , Springer
1970.
New C o l l e g e
Sarasota, F l a . 33580
Infinite Dimensional Holomorphy and Applications, Matos (ed.) 0 North-Holland Publishing Company, 1977
HOLOMORPHIC FUNCTIONS ON STRONG DUALS OF FRECHET-MONTEL SPACES
By S E A N D T N E E N
1. INTRODUCTION
Many interestinq results have been obtained
about
morphic functions on strong duals of Frgchet-Schwartz spaces and in [4]
holoC0s$3 )
and [6] a number of these results are extend
ed to holomorphic functions on strong duals of
Frgchet-Monte1
(OF#)
spaces. However the methods of proof differ qreatly,
Qrfi
spaces frequent use is made of the fact that such spaces
on
are countable inductive limits by compact linear mappings
in
the category of topological spaces and continuous mappings while on
@8& spaces great
reliance is placed on the fact
that
BF+
spaces are hereditary Lindelof k-spaces. It is easily seen that
in the
+
@3d spaces, auoted above, collection of be& spaces (which is
the property of
characterises @g& strictly larger) and
consequently the non-linear properties of @.%+ maces are essential tool in the study of holomorphic functions on
an such
spaces. In this paper we investigate various questions concerning 147
148
S.
DINEEN
holomorphic functions defined on open subsets of @$c#
spaces
and show that all the usual topologies coincide on such
spaces
and the pseudo-convex open subsets of a
@%% space
are
domains
of existance of a plurisubharmonic function. E will, otherwise stated, denote a
unless
F
Q"d space over the complex
field
and our notation, will, generaJ.ly, be the standard notation
2.
of
@g# SPACES A
locally convex infrabarrelled space in which everybot.@
ed set is relatively comnact is called a Montel space. zable Montel space is a Frgchet snace and its stronq also a Montel space and is called a
A
metri-
dual
is
@$$space. Thus E, a @%#
space, has a fundamental sequence of compact sets,
m
(Bn)n,l
,
and
in-
which we may, and will, sunnose are convex, balanced
creasing. For each n let EB denote the vector space spanned by n Bn and endowed with the norm generated by the Minkowski funcis a Banach space. EBn E is isomorphic to the inductive limit
tional of Bn.
EB in the can n tegory of locally convex spaces and continuous linear mappings.
Since Frgchet-Monte1 spaces are separable ([9]
p370) and reflex
ive it follows that the compact subsets of E are complete separable metrizable spaces. A topological space is a Souslin space ([12])
if it is the continuous imaqe of a complete separable me
trizable space. Countable inductive limits of Souslin spaces are also Souslin spaces. Since E is the continuous imaqe of lim Bn, the inductive limit of the spaces Bn endowed with the
3
topology
HOLONORPH IC FUNCTIONS
149
induced by E in the category of topoloqical spaces and continuous mappings, it follows that E is a Souslin space and quently a hereditary
conse-
Lindel6f space. Using once more the above
inductive limit we see that if
U is
an open subset of
then
E
there exists a countable dense subset of the boundary of U, bU, of which each point is the limit of a sequence in A
U ( ‘41).
topological space is a k-space if continuity on compact
sets implies continuity.
E
PROPOSITION 1 PROOF
i6
d
k-hpace.
Let X denote a topological space and let
f :
U 7
denote a mapping which is continuous on compact sets.Let xo be arbitrary and let V denote an open neiqhbourhood of It suffices to show that f-’(V)
E
X U
f(xo).
is a neighbourhood of xo. Since
1 > 0 such that f(xo + AIB1) c V. Suppose X2,...,Xnr positive scalars, have been chosen so that n f(xo + C AiBi) C V. i=l
Bl is compact we may choose
X
such that xitmc Bi If for each integer rn there exists n 1 ) 4. V, then, since Bi is a comand f ( x , + C Aixilm + m xn+l,m i=1 pact metrizable space, we may use a diagonal process to find a n m 1 I rm=ll n consubsequence of ( C Xixi,, + ;xm,n+l (Yn)n=l~which ,i=1 verges to y E 1 AiBi. Hence f (y,) f (y) as n m but f ( y,) 4 V i=l for any n and V is a neighbourhood of f(y). This is impossible I,
+
-+
> 0 such that n+ 1 f(xo + C AiBi) C V. i=l rn m By induction we may now find (Xi)i=l such that f ( C XiBi) (*k V. i=l
and hence we can find
(*)
This means all finite sums.
150
S . DINEEN
W
Since
C AiBi is convex, balanced and absorbs all the bounded i=l at subsets of E and E is a bornological space f is continuous
x o . Hence E is a k-space
E = lim Bn i n t h e c a t e g o l l y 0 4 t o p o l o g i c a l n
COROLLARY 2
Apace4
and COntinUOUA m a p p i n g d .
COROLLARY 3 betd
06
PROOF
S e q u e n t i a l l y c o n t i n u o u d mappingd {torn
open
dub-
E dlle c o n t i n u o u d .
Follows from proposition 1 and the fact that the
com-
pact subsets of E are metrizable. m
COROLLARY 4
Seth
o { t h e {ollm ( C AnBn, An > 0, a l l n) {oam
n=l
n e i g h b o u ~ t h o o d b a d e a t 0 i n E.
PROOF
a
The proceeding proof applied to the identity mapping on
E shows that every neighbourhood of 0 in E contains
a
neigh-
m
bourhood of the form scalars.
Z
n=1
3 . HOLOMORPHIC FUNCTIONS
nBn for some sequences of
positive
@v # SPACES
(H(U;F) (resp. H(U)) will denote the set of all F (resp. C)
val-
ued holomorphic functions U.
( p ] ) (H(U) , To)
PROPOSITION 5 PROOF have U =
S i n e E is a k-space
un
(H(U)
I
i 6
a F ~ t E c h e td p a c e .
To) is ample&. Since U is Lindelof we
Un where each Un i s a t r a n s l a t e of a convex balanced apen sub
set of E. Let Un
=
xn
+ Vn.
The s e q u e n e xn
+
{Bj
0
1 . form a f u n n I ~ K denotes an increag
Xic V a,
d a m n t a l s e q e n c e of ampact subsets of Un when (hn)n=l and l i m An = 1. nfundamental family o f
ing sequene of p i t i w n&rs Hence
U
contains a
compact s e t s
and
HOLOMORPHIC FUNCTIONS
151
(H(U), To) is metrizable. This completes the proof. Ls ("El is the vector space of continuous symnetric n-linear functionals on E endowed with the topology of uniform vergence on the compact subsets of E. Ls ("E) ,
and
P("E)
conare
isomorphic a s locally convex topological vector space. PROPOSITION 6
04
T h e bounded AubAetA
(H(U), To) a t e
locally
bounded. PROOF
We may assume, without l o s s of generality, that U is a
convex balanced open subset of E. Let B denote a bounded subset
of (H(U), To). We complete the proof by finding a neighbourhood of 0, V, such that sup fEB XIBIC U and
Let
E
1)
f
!Iv
<
m.
Choose X1 > 0
> 0 be arbitrary and suppose X 2 ,
so that
K
C XiBiC U and i=1
Let L1 =
... X K
such
have been
that
chosen
Y
K
C XiBi and let E ' > 0 be arbitrary. Choose 61 > 0 so i=1K that L2 = C A B + 61BK+1C U. Since L2 is a compact subsetd i=1 i i we can find 6 2 > 1 such that S2L2c U. Hence
is a To-continuous semi-norm on H(U) (to check this use Cauchy's
inequalities and the fact that U is balanced). Since B is bound ed we can find a posltive
For f E H(U) let
integer N such that
&%?-?denote the n!
continuous symnetric n-linear
form which is canonically associated with the
n-homogeneous
152
S.
DINEEN
polynomial d"fn!( 0 ) ' For X > 0, we have, by expanding each polynomial,
-
Since L1CL2 we have n-1-n El sup I I I: -~ d f (0) -< Y + E Y - . fcB n=O n! Since B is bounded
so that A,+1
Hence we can choose
< 61 and
It now follows that
Since
E
sup IIL1 + A B < M + E + E' fEB K+1 K+1 and E' were arbitrary we may follow an inductive
pro-
m
cess to find a sequence of positive integers, (Xn)n=l,such that
I/
00
C XnBn -<- 2M fEB n=l Hence B is locally bounded at 0 and this completes the proof. SUP
11
f
We now obtain a result which was proved for@A
[l].
spaces in
We refer to L5-J for the definitions of the different topo-
logies on H(U) PROPOSITION 7
. O n H(U) , To = Tu = T6.
HOLOMORPHIC FUNCTIONS PROOF
153
every
We always have To 5 Tu _<. Tg. By Droposition 6
To-bouncled subset of H(U) is ecruihounded and hence
Tg-bounded.
.
Since T0 is a bornological topology it follows that T o = T6 A locally convex is said to be semi-Monte1 if its bounded subsets are relatively compact. PROPOSITION 8 PROOF
(H(U), To) i . 4 a Fhechet-AfoflAe! Apace.
Semi-Montel spaces are closed under arbitrary products
and closed subspaces, hence, we may assume that U is a
convex
balanced subset of E. Since (H(U), To) is metrizable it
suf-
m
fices to show that any hounded sequence in H(U), (fn)n=l, has a convergent subsequence. By taking subsequences and using a diaqonal process if necessary, we may assume, since U is separable, m
that (fn)n=l converges pointwise on a dense subset of U. 00
Since (fn)n=l is esuibounded we can, siven xo
E
U,
find a
con-
vex balanced open set, V, such that m ;imf (XO) sun I: 2m 1 l-J-l I V-< M < m m! n>l - m=O Hence, given E > 0, we can find 6 > 0 such that
sup Ifn(xo) - fn(XO + x’l: E . n>l XF6V m This shows that the seouence (fn)n=l is equicontinuous.
By
a
simple argument it follows that {fn)iZ1 converqes at all points of U to a function which we call fo. By the classical theorem Eo is G-holomorphic and since the seauence (fn);=l locally bounded the function fo is a s o locally bounded
Monte1 is and
hence continuous. m
By Ascoli’s theorem the sequence (fn n=0 is a compact subset of m (H(U), To). Hence (fn)n=l contains a converqent subseouence and
154
S.
DINEEN
t h i s completes t h e proof. The above method shows t h a t equibounded s e t s o f holo-
REMARK
morphic f u n c t i o n on a r b i t r a r y l o c a l l y convex s p a c e s
are
equi-
continuous. We now show t h a t weak and s t r o n g holomorphic f u n c t i o n s c o i n c i d e
o n open s u b s e t s o f
@5
P
spaces.
l e t E and F d e n o t e ahrbithah!t l!acaUrt c o n v e x
LEMMA 9
14 ( o h each o p e n n u b d e t U a,( E t h e bounded b u b h e t h
ahe e q u i b o u n d e d t h e n H ( U : F )
= H(U: (F, u ( F ' ,
F))
04
dpaced. (H(U),TO)
.
W e may suppose t h a t F i s a normed l i n e a r s p a c e . L e t
PROOF
d e n o t e t h e u n i t b a l l o f F ' . Suppose f i s a compact s u b s e t of U and $
E
II(U;
E
B
(F, a ( F ' , F ) ) . I f K
F ' t h e n ( $ o f ) ( K ) i s a bound-
ed s u b s e t on C. Hence f ( K ) i s a weakly bounded s u b s e t o f F
and
by Mackey's theorem t h i s i m n l i e s f ( K ) i s .a s t r o n 7 l Y b o d e d
sub-
set of F . Thus ( $ o f
$ EB
( H ( U ) , To). By o u r hypo-
i s a bounded s u b s e t o f
t h e s i s t h i s i m p l i e s t h a t ( $ o f ) $ € * i s a n equibounded s u b s e t of H ( U ) and hence w e can f i n d , f o r e a c h
xo
E
U , a neighbourhood o f
x o r V, such t h a t sup I I f ( x ) l I = sup x EV x EV @ EB
o f(x)l
5
M,
i . e . f i s l o c a l l y bOu4
ed and hence c o n t i n u o u s . S i n c e H(U;F) C H(U;(F, u ( F ' , F ) )
for
any p a i r of l o c a l l y convex s p a c e s E and F w e have comnleted t h e proof. COROLLARY 1 0
Let F d e n o t e a n aRbithahr{ Locatl!i{ c o n v e x
t h e n Y ( L J ; F ) = H(U;(F, a ( F " , F)) L i
11 i h n n open a u b h e t
hpace od
a
HOLOMORPHIC FUNCTIONS COROLLARY 11
155
Let F d e n o t e am a h b i t h a h y t o c a t L y c o n v e x
and L e i U d e n o t e an o p e n b u b d e t 0 6 a
m+
Apace
F
s p a c e . Then f: U-
i b hoLomohphic i d and o n L q id f i b bounded o n t h e compact
sub-
s e t s 0 4 u and i b G - h o t o m o h p h i c . PROOF
We may assume that U is convex and balanced.If f : U - + F
is G-holomorphic and bounded on the comnact
U
subsets of
it
suffices by proposition 1 and corollary 10 to show @ o f is con tinuous on each comnact subset of U for each @ in F'. Let @ denote a fixed element of F ' and let B denote a compact subset of U. By Cauchy's inequalities there exists a X > 1 such that
Hence it suffices to show n. Let Pn =
-2
h n! q
( 0 ) and let
(0) is continuous for
6n denote the
each
associated sym_
metric n-linear form. As in proposition 6 it suffices to prove the following; if K and L are convex balanced compact of E and
1 (PnlI K 5
M then for each
E
subsets
> 0 we can find h > 0 such
n
(xli-lsup IPn(x)"-i(y)il x EL YEK follows and sup IPn(x)n-i(y)il < = for all i, 0 5 i 5 n, this xeL YEK immediately. J:
i=l
COROLLARY 12
A locatLrr c o n v e x v a l u e d p o l i { n n a i a l d e 4 i n e d
e i b c o n t i n u o u b id and on!!{ i4 s p a c e s EB i b c o n t i n u o u s n PROOF
A
40h
itb
hebthiction to t h e
on
Ranach
e a c h n.
polynomial on a Banach space is continuous
only if it is bounded on bounded sets and each bounded
if
and
subset
156
DINEEN
S.
of E is contained and norm bounded in some Bn. Restating corollary 11 we have the following result. COROLLARY 13
E = 1 3 EB
COROLLARY 14
S e p a 4 a t e l y continuoub polynomialb dedined
i n t h e CategOhy 0 6 l o c a l e y n n Apace4 and c o n t i n u o u s p o l y n o m i a l m a p p i n g & .
phoduct oil
@F$&s p a c e d
convex
on
a
a t e eontinuoub.
w$!
then and F = l=r FCn are spaces n E x F is also a&~+~space and E x F = 9E inFCn n Bn ductiue limits being taken in the category of locally convex PROOF
If E = 1 2 EB
spaces and continuous linear mappings). If P is a separatelycan x F n n' separately continuous for each n and hence is continuous
is
tinuous polynomial on E x F then P restricted to EB
(by
Hartogs' theorem on separate analyticity for Banach spaces (see [lo])). Hence P is continuous by corollary 11. COROLLARY 15
76 F
b p a c e and ( U , V )
a bequentially complete l o c a l l y
i d
convex
a 6 - e x t e n b i o n p a i t ( * ) 0 6 domainb bphead o v e t
i b
E t h e n (U,V) i h a n f - e x t e n h i o n p a i t .
PROOF
Apply corollary 10. m
m
1 6 ($n)n=l i
COROLLARY 16
ib and o n L q id $n
+
o a6
n
b
a sequence i n E' then
-+
m
If
Hence $n
-+
m
C 4: n=l 0 as n
E
H(E) then
+ m
Conversely suppose $n 370 (*)
L$~
+
0 as
n
+
In
)X(,$I
C n=1
<
ists a unique f
+
m
o
as n
+ m
for all all x
uniformly
pointwise on E.
t E.
BY 191
p.
on the bounded subsets of E
-
f
00
in ( E l , o ( E , E ' ) ) .
(U,V) is a F-extension pair if each f
-
c H(E)
i n (El, a(E,E')).
m
PROOF
C 6; n=l
H(V;F) such that flu = f.
E
H(U;F) there
ex-
HOLOMORPHIC FUNCTIONS W
157
m
n C $n I 5 C ( I I $n\IBIn is bounded if B is a bounded n=l n=1 m the subset of E. By corollary 11 C $: E H(E) .This completes n=1 proof. Hence
IB
I1
We now show that the bornoloqical topoloqy associated with Tw,b, is equal to the T6 topoloqy on certain surjective of
w#
are @'
spaces. Examples of spaces which satisfy our
( n ) , R an open subset of Rn, and
IT * a (a)
I
E
a
limits
criteria
a
space f o r each u and ( a ) may have any cardinality. We [4)
Twr
854 refer to
for background material to this result (especially sections
7 and 8). E will denote an arbitrary locally convex space.H(Eo) is the vector space of qerms of complex valued holomorphic funs
tions at 0 in E. We endow H(EO) with the inductive limit topolo
3 Hb(V)
where V ranges over all open subsets of E which V contain 0 and Hb(V) = if, f E H(V), 1 ( fI J v < m ) is endowed with
gy
its norm topology. Since Hb(V) is a Banach space H(EO) is
bar-
relled and bornological (in fact ultrabornoloqical) and the canonical injection (H(V), T I--+ w
H ( E O ) is continuous for
each
open neighbourhood of 0, V. PROPOSITION 16
L e t 0 = (Ear~a)aEA denote a compact, o p e n , s i f z
m e t h i C , i ( h e b p . j ) A u h j e c - t i v e h e p h e s e n t a t i o n 0 4 E and d e n o t e a c o n v e x b a l a n c e d open n u b d e t AuncXion o n
u
06
E. 14 each
let
u
holomohphic
has minimal 8 - s u p p o h t and t h e ~ ~ - b a u n d e dAubAetd
( t e s p . s e q u e n c e s ) i n H(T t h e n t h e bounded s u b d e t d
c1
(u))
ahe equibounded d o h e a c h a i n
IheAp. b e q u e n c e s )
04
A
(H(u), T ~ ) ahe
equibaunded and (H(U), Ts) = (H(U), Twrb). PROOF
Let B denote a bounded subset (resp. seffuence)
(H (U), Tw). For each f in B let A ( f ) denote a minimal
in 0-sup-
s . DINEEN
158
u
A ( f ) is a n E-bounded s u b s e t o f A f€B t h e r e q u i r e d r e s u l t f o l l o w s immediately. O t h e r w i s e , u s i n g
p o r t f o r f . I f A1 =
then the
f a c t t h a t 8 is an i ( r e s p . j ) s u r j e c t i v e r e p r e s e n t a t i o n when
is a s e t ( r e s p . a s e q u e n c e ) , we c a n f i n d a secruence o f m
s u b s e t s of A,
B
E-open
with t h e following properties;
(Wn)n=l,
(1) A1 Cl Wn # $ f o r each n , if a E A t h e n t h e r e e x i s t s a p o s i t i v e i n t e g e r
(2)
s u c h t h a t a E W:
for a l l n
Hence w e may choose o f e l e m e n t s i n B ,
+
(3)
fn(xn
(4)
S(Y,)C
n (a)
2 n(a), m
( f n ) n =l, such t h a t
y n ) # f n ( x n ) f o r a l l n and
wn m
By ( 2 ) t h e sequence (yn)n=l converqes v e r y s t r o n g l y t o 0 . Hence, by L i o u v i l l e ' s theorem, w e can suppose
+
fn(xn
(5)
-
yn)
f (x n
> n f o r a l l n. -~ (x)I <~m f o r a. l l n and ~ t h u s~
S i n c e B is bounded s u p I fEB (5) w e c a n f i n d , u s i n g i n d u c t i o n , t w o i n c r e a s i n g s e q u e n c e s m
positive integers, K
n+ 1 d K f a
'
(6)
I
n
and
( K ~ ) ~ =s u ~ c, h
(xen
+
of
that
ZKf
(0)
K!
K = K ~ + ~
CJ
by
( X q
'n
1'" K! --
Now c o n s i d e r t h e f o l l o w i n g seminorm on H ( E O ) Y
I f f E H ( E O ) t h e n , s i n c e ( y j ) j = l is a v e r y s t r o n g l y sequence, t h e r e e x i s t s j i K f (0) ( x . K!
3
+
y.) 3
-
aKf(o)
~
K!
0
convergent
such t h a t
(x.) = 0 for a l l 3
K
and a l l j
L
jo.
159
HOLOMORPH I C FUNCTIONS
n+l C
JO
C n=l
Hence p ( f ) =
+
sup
+
1 nE:j
K = K ~
yi)
-
K:
j5-j is finite f o r all € in Ii(EO). Since
1 ‘ n7 f ( ~ ) (x)I
p(f) =
is a continuous semi-norm on
for any x in E and H ( E O ) is barrelled it follows that
H(EO) is
p
continuous semi-norm on H(EO). By ( 6 ) the imaqe of B in
H(EO)
is not bounded and this contradiction implies that R1 E-bounded suhset of
A.
Hence there exists an a in
a set of G-holomorphic functions on I T , ( U )
for all f
B. Since
E
8
a
is
an
and R=(f)fEIy
A
f
such that
I
=
f o ncl
is a compact surjective limit it follows
that B is a To-bounded and hence esuicontinuous
subset
of
Hence B is an eauibounded suhset of H ( U ) and this corn_
H(na(U)).
pletes the proof. L e t 0 = (E
COROLLARY 17
m e t h i c hephebentation
04
I
F: bu
n a ) a e A d e n o t e a c o m p a c t , o p e n bUm-
4
pace^.
T i each
{ u n c t i o n d e j i i n e d on a c o n v e x b a l a n c e d o p e n n u b b e t
holomohphic
04
E hab m i n -
i m a l e - ~ ~ p p t~h ehnt t h e h o l l o w i n g h e n u l t b hoCd { o h o p e n b u b h e t b 04
E.
(a) 2 4
8 i d a
j-buhjectivc hephebentation then
(H(U), Tw,b) = (H(U)I Ts). (b) 1 4 e
04
ed d u b d e t b bemi-h(on.tel and T
U I b
i d
a n i - d u t j e c t i v e t e p t e b e n t a t i o n t h e n t h e bound
(H(U), Tw) a h e e o u i b o u n d e d , ( H ( U ) , Tw) d p a c e , (H(U), Twlb) i
6
i b
a Montel Apace and Tol
i n d u c e t h e dame t o p o l o g r r o n t h e T w - b o u n d e d b u b b e t b
a
Tu
04
H(U).
PROOF
Since the restriction mapDing (H(U), Tw)+ (H(V) I Tw)
(U3V) is continuous we apply pronosition 16 to complete
the
S. DINEEN
160
.
proof of (a) and the first part of (b) Let H a ( U ) denote
the
set of all f in H ( U ) which factor locally throuqh Ea. The bound ed subsets of (Ha ( U )
r
To) are equihounded and hence ( H a ( U ) ,
To)
is a complete barrelled bornological locally convex space the methods of propositions 6 and 7). Hence, if 0 is an
(use
i-sur-
jective representation, the identity manpinq; (H,(U)
r
To)d ( H ( U )
r
Tu,b)
is continuous. If B is a bounded subset of ( H ( U ) , Turb) then is contained and bounded in ( H , ( U ) ,
To) for some
in
Q
B
Since
A.
fact
(H ( U ) , To) is a Montel space (use nroposition 7 and the
a
that ( H a ( U ) , T 0 is isomorphic to a closed subspace of n(H(Vi),TO) i a is where Vi is an open subset of Ea for all i) ( H ( U ) , Tw) have semi-Monte1 space and ( H ( U ) , T u r b) is a Montel space. We also shown, since To 5 Tw 5 Turb,that T o r Tu and Turb the same topology on the Tu-bounded subsets of H(U). REMARK
If 8 satisfies the "countable stability"
induce
condition
of [ll] then 8 is trivially a j-surjective representation
and
it is possible to prove a result similar to Corollary 17 without the e-minimal support property on H ( U ) . EXAMPLE
18
(H(@(fl),
Tu,b) is a Montel space.
It is possible (see
[5]) that
To = T w even when all the
conditions of proposition 16 are satisfied. An open subset of a locally convex space is said
pseudo-convex if its finite dimensional sections are
to
be
pseudo-
convex. PROPOSITION 19
P d e u d o - c o n v e x open d u b d e t b 0 4
@s@ dpaced
domain4 0 3 e w i d t a n c e 0 4 pLuhibubhaamonic . ( u n c t i o n 4
.
ahe
HOLOMORPHIC FUNCTIONS
PROOF
161
L e t U d e n o t e a pseudo-convex open s u b s e t o f t h e m
d e n o t e a n i n c r e a s i n g s e u u e n c e o f compact s u b s e t s
E. L e t ( K n ) n = l
of U such t h a t
u
Kn = U .
n
By theorem 2.3.6
of
[lo]
pose t h a t e a c h Kn i s e q u a l t o i t s p l u r i s u b h a r m o n i c
w e may suphull.
Let
such
that
and
the
U.
Let
m d e n o t e a d o u b l e sequence o f p o i n t s i n U ('n,m I nlm=l x ~ , ~ xn +
CFU (boundary o f
E
U) a s m+
m
f o r each n
m
sequence ( x ~ ) i~s = a d~e n s e s u b s e t o f t h e boundary o f
Q
: N--I
N d e n o t e a f u n c t i o n from t h e i n t e q e r s i n t o i t s e l f s u c h
t h a t +-'(n) X
space
Q (1)'ml k
i s i n f i n i t e f o r a l l n. Now c h o o s e ml K1.
By theorem 2.3.6
[lo]
of
such
that
there exists a plurisub
harmonic f u n c t i o n on U, f l r s u c h t h a t
S i n c e a e x p ( b f l ) i s p l u r i s u b h a r m o n i c f o r a l l p o s i t i v e a and
b
w e may sunpose
n
Suppose ( Ei ) i=l
and ( m i 1 i=l
g e r s , and ( f i ) i = l l U,
r
two i n c r e a s i n g s e q u e n c e s of
a s e q u e n c e of p l u r i s u b h a r m o n i c f u n c t i o n s
have been chosen s u c h t h a t R1
...) n
inteon
= 1,
3 K~ i CXa ( i ),mi 1 f o r i = l r. .. , n - 1 Choose fn+l s u c h t h a t 9.n+1 > f n and K p 3 KQ lx Q ( n ) 'mn 1 . n+ 1 n Next choose mn+l such t h a t mn+l > m and x n Q ( n + l ) rmn+l f o r i = 1,
and K Q
i+ 1
u
u
and fn+l a p l u r i s u b h a r m o n i c f u n c t i o n on U s u c h t h a t ) > 2"+1 > 1 > sup fn+,(X) -- 2n+1 -- X E K . fn+l ( x (~ n + l ) ,mn+l
_> 0 .
S. DINEEN
162
m
By induction we then define the senuence (fn):=l.
Let f
=
By our construction this sum converqes at all points of U
1 frI n=l and
isunbounded on each neiqhbourhood of xn, n arbitrary. Each fn is a positive function and a finite sum of plurisubharmonic tions is plurisubharmonic, hence it suffices to show upper semi-continuous to complete the proof. Since
U
func-
f
is is
a
k-mace it suffices to show that f is unper semi-continuous on each compact subset of U. Let I< denote an arbitrary comDact sub set of U and let C denote some real number. B y our construction < -1 K - 2" V. Choose M
we can find a Dositive integer N such that I Ifn/I
for
all n > N. Let V = Ix E K, f(x) < C). Let xo E N M+ 2 1 such that 3 < C - f(xo). Since C fn is plurisubharmonic 2 n=l there exists a neighbourhood of xo in K, W, such that M+ 2 1 sup T fn(x) < c X E W n=l 2M+ 1 !I+2 flence sup f (x) 5 sup I: fn(x) X EIJ XEW n=l
+
w 1
-- 1
n=r1+3 2"
Thus flK is plurisuhharmonic and U is the natural domain
of
existance of f.
j e c t i v e hephesentation bv open suhneto
Xunctions PROOF
+
I.( E , a PocatPfr c o n v e x s p a c e , has
PROPOSITION 19
04
@s h p a c e s
then the
E a h e domains 0 4 e x i s t a n c e 0 4
an o p e n huh -
pneudo-convex ptutihuhhahrnonic
I
Let 9
tation of E by
= ( E a , ~ a ) a Edenote A
s , snaces and
open subset of E. By [4]
the open surjective represeg
sunpose U is a
there exists an a in
nseudo-convex A
such
that
HOLOMORPHIC FUNCTIONS
U =
na-1 (n,(U))
163
and na.(U) is a pseudo-convex open subset of Ea..
By proposition 18 there exists a plurisubharmonic
function
on
f, which is unbounded on each neiqhbourhood of each b u g
s,(U),
ary point of na. (u) If 5
. Let f
U.
rats)
E
6 ( n a . ( U ) ) and a,(V)
Hence
1 I f ! ! na.(")
=
D
n a. .Iis a plurisubharmonic func
and V is a neiThbourhood of 6 in E
tion on
E 6U
= f
and
then
is a neighbourhood of na.(E) in na(EL
I I?' lv
=
I If
9
na.l
I" -- I I f 1 In#)
-
m.
Thus U is the natural domain of existance of f. This completes the proof.
For the sake of oonpleteness we include the folladng p u l t s . (a) is proved in [4] and (b) is proved for @gc#spaces in [ 6 ] . PROPOSITION 19
&&+s p a c e
(a) A holomohphicall"
c o n v e x o p e n dubbe-t 0 4
i n t h e d o m a i n 0 4 e x i b t a n c e 0 4 a holomohphic
a
(unc-
ti0n.
(b) 14 t h e l o c a t l r r convex s p a c e E ha6 a n 0pe.n b u 4 j e c t i V e t e p t e -
nentation by
=#b p a c e b
each
0 4 which
hub a S c h a u d e t
habin
t h e n t h e pneudo-convex open b u b b e t b 0 4 E a 4 e damaina 0 5 tance
04
holomohphic { u n c t i o n b . (b) Use the result in [S] for
PROOF
eXh-
W+ spaces and
exactly
the same method as used in pronosition 18. We have been unableto prove or disprove the following conjecture. CONJECTURE
tions on
+
Do8
SiPva
(04
hlackerr) COntinUOUb G - h o l o m o t p h i c
bpaceb a t e c o n t i n u o u b .
If this conjecture were true then it would follow
w#
(unc-
that
spaces are Zorn spaces (i.e. the set of points of continui
ty of G-holomorphic functions on open subsets of@# open and closed).
spaces is
S. DINEEN
164
This conjecture requires a deep study of
convergent se-
quences which are not Mackey convergent.Indeed it is equivalent
to showing that convergent sequences are bounding subsets
for
Silva holomorphic functions and a counterexample may not
be
found by usinq the usual techniques (this follows by corollary 15). Grothendieck's example of a
@$ space does not provide
a
=#space E which
is
not
counterexamnle.
This results from the following facts about E (which
do
not appear to be common to all @ 1) E =
9En n
a
$ spaces which are not @@
):
and each En is isometrically isomorphic to
9 00. m
n=l
2) If B denotes the unit ball in Pmthen Tn(Bn C,)
is
a fundamental seauence of bounded subsets of E. 3 ) Every element of H ( X B )
, X
> 1, is bounded on B n C o ([B]).
The results on surjective limits parallel some those in section 7 of [4]
of
and loosely speakinq we have
shown that results for the T o topoloqy can be extended to the To)topoloqy without the extension
requirement
on the surjective limit. The method of proposition 16 m a y also he combined techniaues in [2]
+
to study holomorphic functions
snread over surjective limits of
on
spaces and this
gation has subseauently been carried out in [ 2 ] .
with domains investi-
HOLOMORPHIC FUNCTIONS
165
B I B L IOC?.APHY
[l]
J. BARROSO, M. PqATOS and L. NACHDIN; On bounded sets of ho-
lomorphic maminrjs, Lecture Notes in Vaths, Vol. 365, Sprinqer-Verlan, (1973), 216-224. [2] P. RERNER;
A
nlobal factorization pronerty for
holomornhic
functions of a damain spread over a surjective limit,Seminaire P.Lelong,1974/75.Lecture Notes inMaths,524 Springer-Verlag(1976) [3] P. BERNER; Topoloqies on spaces of holomorphic functions of certain surjective limits (this proceedings). [4]
S.
DINEEN; Surjective limits of locally convex spaces
and
their application to infinite dimensional holomorphy. Bull. SOC. Yath. Fr. t103, 1975 (to appear). [5]
S.
DINEEN; Holomornhic Functions on locally convex snaces I, Locally convex topoloqies on €I(U), Ann Inst. Fourier, Grenoble, t23, 3, (19731, 155-185.
[6]
S.
DINEEN; PH. NOVERRAZ and M. SCHOTTENLOHER; Le nrohlemede Levi dens certains espace vectoriels toaoloqicyes localement convexes,Bull SOC. Math. Fr. t. 104(1976).
[7] A. GROTIIENDIECK; Sur les espaces (F) et (DF). Summa
Bras.
Math. 3, 57-123, (1954). [8]
B. JOSEFSON; Boundinq Subsets of Rm(A) , Thesis,
Uapsala,
1975. [g]
G , ROETIIE; Tonoloqical vector spaces I ,
Bd 159, 1969.
Springer-Verlaq,
S. D I "
166
[lo]
PM. NOVERRAZ ; Pseudo-Convexite , convexite polynomiale
et
domains d'holomorphie en dimension infinil, North-HoL land, 1973. limit
[11J PH. NOVERRAZ; On a particular case of surjective (this nroceedinqs) [12]
.
L. SCHWARTZ ; Radon measures on arhitrarv tonoloTica1 spaces and cylindrical measures, Oxford Universit~r
Press,
1973.
DEPARTMENT OF MATHEMATICS UNIVERSITY COLLEGE DUBLIN nUBLIN 4 , IRELAND.
Infinite Dimensional Holomorphy and Applications, Matos (ed.) @ North-Holland Publishing Company, 1977
D I F F E R E N T I A L EQUATIONS OF I N F I N I T E ORDER I N VECTOR-VALUED HOLOMORPHIC FOCK S P A C E S
BY T H O M A S A .
w.
P W Y E R ,7 r r
CONTENTS INTRODUCTION
1. Vector-valued holomorphic Fock spaces and their duals
2. Vector-valued convolution operators and their adjoints 3. Vector--valueddivision theorems
4. Vector-valued existence and approximation theorems 5 . Application to entire functions with entire function values
6. Application to vector-valued variational equations REFERENCES
INTRODUCTION
Various situations where power series in infinite
dimen-
sional domains naturally arise a l s o involve infinite - dimension a1 ranges: e.g., the Volterra series representation of the out-
puts of non-linear systems as analytic functions of input signal LlU, 2,3], [Bol. 1,2,3,4,5], [Br. 1,2,3], [w],and the variational equations related
to the representation of solutions of well posed boundary value
167
T. A. W. DWYER
168
problems as functional power series, where the variable is the boundarv value function IDL]
.
This last reference especiallv shaws
the desirability of extending the existence and
approximation
theorems on convolution equations and partial differential equa tions in infinite dimension of [ G 1 , 2,3]
IN
2 , 3 , 4 , < ~ , rDil,2]
[Mat 1,2], [ Dw 1,2,3,5,6,7,8], [ Bol 5 . 2 , 3 , 4 , 5 ] and [Bd
I
to vec -
tor-valued functions. Existence theorems do not hold for general convolution q u a tions
? * 3
-+
= g,
6
where $ and
are mappings from a (dual)vec
tor space E ' to a vector space F and
%
is an F-valued linear
m -
ator on functions from E' into F, even in finite dimension. The case when
+
T = T B A , where T is a scalar-valued form acting on
scalar-valued functions on E ' and A is a linear operator on was shown in [ Dw 9,101
to be more manageable: in the first ref%
ence the Malgrange-Gupta existence and approximation were shown to hold for
T
8
A
*
in the space
theorems
HNb(E';F)of F-Val
ued entire functions on E' of nuclear bounded tvpe, when in the dual of
F,
HNb(E';F) and A is the identity operator
is
T
on
F
(where E and F are Banach spaces). In the second reference those results were extended to surjective bounded linear operators A , and a basis was constructed for a dense subspace of the spaceof solutions of the homogeneous equation, associated with the zeros of A and those of the Fourier-Bore1 transform of T: the problem was approached by the representation of
T
8 A*
in
the
form
g'(d) C3 A , where g ' is the Fourier-Bore1 transform of T and the "differential operator of infinite order" g'(d) is defined
as
the sum of the homogeneous operators gA(d) given by gA(d)f ( X' =
2
dng' ( 0 ) > n, where <
, >n is the bilinear form on
DIFFERENTIAL EQUATIONS OF INFINITE ORDER
the pair
P N ( n E ' ) x ?'("El
n pN( E l )
t h e d u a l of and
P ("E)
anf ( x ' )
,
169
d e t e r m i n e d by t h e i s o m e t r y
between
( n u c l e a r n-homogeneous p o l y n o m i a l s on
E')
( a l l c o n t i n u o u s n-homogeneous p o l y n o m i a l s o n E ) .Here
i s t h e n-th Frgchet d e r i v a t i v e polynomial of f : E '
(complex f i e l d ) a t x ' The a d j o i n t o f
,
and h n g ' ( 0 ) i s s i m i l a r l y d e f i n e d
0:
-f
on
E.
g ' ( d ) d A r e l a t i v e t o t h e F o u r i e r - B o r e 1 isomor-
phism between t h e d u a l o f
f f N b ( E ' ; F ) and E x p [ E ; F ' )
(F' - v a l u e d
e n t i r e f u n c t i o n s o f e x p o n e n t i a l t y p e on E ) was shown t o b e t h e + operator f ' B A' o g' * f ' (where A ' i s t h e t r a n s p o s e o f A ) , and t h e Malgrange-Gupta d i v i s i o n theorem f o r g ' on extended t o A ' o g '
a
E x ~ ( E ; F ' ) . The e x i s t e n c e
and
Exp(E)
was
approxima-
t i o n t h e o r e m s t h e n f o l l o w e d by s t a n d a r d d u a l i t y a r g u m e n t s . I n t h e present a r t i c l e w e consider t h e operators g ' ( d ) d A
Fp
i n t h e spaces
N,?
w i t h t h e "p-summable" g r o w t h c o n d i t i o n s C n P n f o r weights
p > 0
defined spaces
3
( E ' ; F ) of F-valued e n t i r e f u n c t i o n s
11
inl(0)
on E '
11
<
m
(N d e n o t i n g t h e n u c l e a r n o r m ) . The s i m i l a r l y
FfIP
( E l ) of s c a l a r - v a l u e d
p r o p r i a t e holomorphy t y p e f3 i n t r o d u c e d i n
functions f o r an ap [Dw
5,7]
serve
to
c l a s s i f y e n t i r e f u n c t i o n s o f f i n i t e o r d e r i n i n f i n i t e dimension, r e p l a c i n g t h e c l a s s i f i c a t i o n i n terms o f e x p o n e n t i a l g r o w t h es/ f ( x ' )1
timates
5
no a n a l o g u e f o r holomorphy t.:"pes [ T r 11
u s e d i n [Mar 1 , 2 ] ,
C exp p j l x ' l /
o t h e r than t h e c u r r e n t type: cf
Ch. 11 f o r t h e i r e q u i v a l e n c e when p = 2 .
[Bol 1,211
whichhave
f o r a similar re-casting,
We
to
refer
i n t h e form of weighted p e r
series e s t i m a t e s , of t h e e x p o n e n t i a l estimates employed i n IT.]. When p = 2 and f3 i s t h e H i l b e r t - Schmidt "Fischer-Fock" spaces introduced i n
1Dw
type
one
gets
the
.
1,2 ,31 : t h e e x i s t e n c e
t h e o r e m s i n t h e s e r e f e r e n c e s h o l d o n l y f o r c o n v o l u t i o n operators
T. A. W. DWYER
170
T* for which the Fourier- Bore1 transform g' of T is a HilbertSchmidt polynomial, and follow the method of The results in the
Hilbert-Schmidt case
tended to vector-valued solutions, but here: cf [K 1,2,3,4,5], IIDw
41
[Ar],
can
also be ex-
will not be considered
and [Bon]
for related topics.The
"unbounded" case of HN(E') in [ G 1,2,3] results in
, Ch. 11.
[Tr 1 1
and [N 2,3,4]
requires
and will also be omitted. fol-
We now outline the results in the present article: lowing the scalar-valued case of dual of
,
[Dw 5,7]
FE,_(E';F) (projective limit of the
weights p ) is now shown to be F:'(E;F')
the
Fourier-Bore1
Fp
(E';F) for all
Nip
(inductive limit of the
Fp: (E;F), defined for the current holomorphy type by p'-sumMble P
estimates analogous to those for the nuclear type with pp' pup =l), 1 + - = 1 (corollary of proposition 1.1). The adjoint of where P P' g' (d) 8 A on (E';F) , where g ' E f:' (E) and A is a continu-
Fg l m
ous linear operator on F , is shown to be A' o g'
on ':F
(E ; F')
with respect to the Fourier-Bore1 duality (Proposition 2.3). By use of the similar duality between the operators g(d) on Fp' (E) and g
- on
0
F,"
(El )
(where g
:F
I
cia1 cases g = exp o < u, space
E
(El)
)
, applied to the spe -
I=
1,
u
E
E and g
=
>nl
v
E
FF' (E;F') is shown to be translation-invariant
lary to Proposition 2.2 1
El
(corol-
and the functions 3= exp o < u, > < v, >"
are sham to be solutions of g ' (d) 8 A
3
=
0
.y
if and only if either
u is a zero of g' with order higher than n in the direction v # 0
the
of
or y is in the kernel of A in p (Proposition 2.4). The
division theorem f o r g' in Fi: (E) given in [Dw 6,8] is then elf tended to the operator A' o g '
in
F F i (E;F') when
A
is surjec
tive (Theorem 3.1, in which all propositions mentioned above are used, together with Boland's extension to Banach spaces[Boll, 23
DIFFERENTIAL EQUATIONS OF INFINITE ORDER
171
of Taylor's estimate [Ta] on the maximum modulus of the quotient of two entire functions of finite order). The existence and approximation theorems are then extended to CJ'(d) @ A o n
FpN i m (E';F)
when A is surjective (Theorems 4.2 and 4.1: as in the case p'l, i.e. ,
ffNb( E l ;F) , treated in [Dw 101
, the approximation theorem
provides a basis for a dense subspace of the kernel ofg'(d) @A). The preceding results are then extended to the
czse
F
when
is a Frzchet space, leading to existence and approximation theo P1 p2 rems for the operator gi(d) 8 g;(d) on F N I m ( E ~ ; F ~ , ~ ( E ; ) ) for 1 pi Banach spaces El and E2, where g; E Fo (Ei) and - + - = 1 , Pi Pi i = 1,2 (Theorems 5 . 1 and 5 . 2 ) . Relations with operators G'(d,d) on
FNP,,(Ei
x
E;)
, where
GI
E ':F
(El
x
E2) : are then indicated,
as well as applications to equations of the from -f
(a/at) z(t;x') = g'(d) f(t;x')
+
G(t;x'),
with t E CC, x' E E' and g' E F F ' ( E ) (remarks following 5.2).
The space
an appropriate
F,"
m
rm
( L (M): F) and
measure
1
L (M) - (resp. L m ( M ) rivatives 6"; (XI of functions
space,
':F
(L1(M): F')
are then
,
Theorem
where M is
characterized
by
growth conditions on the variational de
)
...
(tl)
.
6x1 (tn) (resp. 6" f'(x)/6x(tl). .6x(tn1 ) +
-+
1
f : L ~ ( M ) F (resp. fl: L (MI -f
+
F')
(corollaries
to Propositions 6.1 and 6.2, where the casting of the domain of
3 as a dual Banach
space E' permits the use of the Dunford-Pettis
theorem). Finally, the existence and approximation theorems are
1.. .I
then applied to variational equations of infinite order of the forni m 1 -f A6"Z(x1)/6x'(t,) ..6x'(tn)xA(tll.. ,tn)dm(tl). .dm(t,) =g(x') Cn4 -f F and (where m is the measure on M) , for functions f: Lm(M)
.
.
.
-+
+
g: Lm(M) -+ F with variational derivatives 6n$/6x'n and 6";/6xFn in L1 (Mn) , and kernels x: in Lm(Mn) (Proposition 6.3).
172
T. A. W .
DWYER
1, VECTOR-VALUED HOLOMORPHIC FOCK SPACES AND T H E I R DUALS.
14 :
We u s e t h e n o t a t i o n of [Dw 9 ,
i n particular, El F are n complex Banach s p a c e s , E ' , F ' t h e i r d u a l s , PN ( E ';F)(resp. p (?E;F')) + t h e n-homogeneous n u c l e a r p o l y n o m i a l s Pn: E ' F (resp.continu+
-+
ous p o l y n o m i a l s P I : E
F')
+
n
d e r i v e d from t h e n u c l e a r E 8
... 8
(I
E 8 F (resp.
-+
I
I/ pnllNIn
-
*
11
PnllN t h e n u c l e a r n o r m
completion of t h e tensor product -+ -+ P ; l l l n = ]I PA11 t h e c u r r e n t norm = sup. on
t h e u n i t b a l l of E ) , f f ( E ' ; F ) i s t h e s p a c e
3: E '
of
* F w i t h d e r i v a t i v e p o l y n o m i a l s ;"$(x') +
similarly for
f ' :E
+
F')
. We omit
e n t i r e mappings E P("E';F)
arrow s u p e r s c r i p t s
e x p l i c i t i n d i c a t i o n o f t h e r a n g e s p a c e s F when
F =
(11
(and
arid t h e = complex
f i e l d . The c a n o n i c a l i s o m e t r y between P N ( n E ' ; F ) I and
P("E ; F ' )
(not
<
P ( n E " ; F ' ) ) i s r e p r e s e n t e d bv t h e b i l i n e a r form
PN(nEiF)
x
P ( " E ; F ' ) c h a r a c t e r i z e d bv < xn
xn: = < x , >
with cf [DW
91,
-+
,
so t h a t
+
I
< Pn
,
, >n,F on -f
y f p A > n I F= < V , P ; l ( X ) >
*
11
11
I
I
:
Prop. I . 1.
The h o l o m o r p h i c Fock s p a c e w i t h d e g r e e p > 1, w e i g h t p > 0 and holomorphy t y p e N ( n u c 1 e a r ) from E ' i n t o F i s t h e Banach space ( E ' ; F ) of f u n c t i o n s
F N , ~
e q u i p p e d w i t h t h e norm
3: E '
+
F
such t h a t
/ I ) 111 NiPiP
thus defined.
The
corre-
- + 1=
1 and pl/P p ' l / p ' = L P P' f o r t h e c u r r e n t holomorphy typeis s i m i l a r l y d e f i n e d and t h e c o r
FF: (E;F')
sponding s p a c e
with
r e s p o n d i n g norm i s d e n o t e d by
Fp
N l m
(E';F) : =
P>O
111
111 P ' I P '
Fp
(E';F)
NIP
. We
write
,
which i s a F r g c h e t s p a c e w i t h r e s p e c t to the norm We also w r i t e
F':
(E;F')
u
: = p,>o
F p '' ( E ; F ' )
,
(11:
111 NiPiP1 P>O'
e q u i p p e d w i t h the lo-
c a l l y convex i n d u c t i v e l i m i t t o p o l o g v i n d u c e d bv t h e n a t u r a l i"
173
DIFFERENTIAL EQUATIONS OF INFINITE ORDER
j e c t i o n s FP'(E;F ) P'
+
F;'(E;F').
The s p a c e F E : o ( E )
as
(defined
Fp' ( E ) b u t f o r t h e n u c l e a r holomorphy t y p e o n E ) i s n o t r e p r e 0
s e n t a b l e as t h e d u a l of F E ( E ' )
-
( d e f i n e d as F g m ( E ' ) b u t f o r t h e r
c u r r e n t t y p e ) , and t h e q u e s t i o n o f t h e r e g u l a r i t y o f i t s
open
s e t s i s a s y e t u n s e t t l e d , e x c e p t when E i s a H i l b e r t s p a c e o r a F r s c h e t - S c h w a r t z s p a c e . However, a n a l o g u e s o f a l l t h e r e s u l t s on c o n v o l u t i o n o p e r a t o r s g i v e n i n t h i s a r t i c l e are a l s o v a l i d
F::N
on
( E ; F ' ) a l t h o u g h t h i s case w i l l b e o n l y b r i e f l y o u t l i n e d ( c f .
[Dw 8 1
,
sec.l.6
and 2 . 6 ) . A d e t a i l e d
through 1.9,2.2,2.5
o f t h e s p a c e s F i ( E ' ) and
studv E and
( E ) f o r l o c a l l y convex domains
IEw 6,8] .
v e r y general h o l m r p h y t p s 0 and dual types 9 ' is t r e a t e d i n
W e b e g i n b y e x t e n d i n g t h e F o u r i e r - B o r e 1 d u a l i t y t o t h e Firs FErp
(E';F) , F F : (E;F') and
FEtw(E';F),
F E ' (E;F') , where as i n
[Dw 9 , 1 0 1 t h e F o u r i e r - B o r e 1 t r a n s f o r m KT: E
a1 o f F) o f a n a n a l y t i c f u n c t i o n a l T: by
< y,
8T(x) > : = T ( e X
*
F* ( a l g e b r a i c du-
FEIp(E';F)
y E F
y) for
-f
+
x
and
a!
is defined where
E E,
eX : = e x p o < x , > : B i n un i n o -
PROPOSITION 1.1. T h e Fautlietl-BatleL t t l u n b d o t l m a t i u n
m e t t l y dtlom F; PROOF:
r P
(E';F)
FF: ( E ; F ' ) .
onto
I
One f i r s t s h o w s , p a r a l l e l t o t h e case p = l ([mg], Prop.
11. 2) t h a t
< y , BT(x) >
i 5 11
T
/I
exp($llx
I/y 11
11')
x E E a n d y E F , ~ . ~ . , I I B T ( x ) I I5- IlTIl e x p ( $ l l x l , P ) BT(E) C F ' whenever
T E Fp
NrP
(E' ;F)
I .
p o l y n o m i a l t r a n s f o r m o f t h e r e s t r i c t i o n of T t o checks t h a t
m
1
PA(x) >
t i n g w e a k * - c o n v e r g e n c e t o 8T ( x )
series i s s t r o n g b e c a u s e
m
/I Cn=o
+
1'
Pnl (x)
+
Pnl
that
be the
PN(nE';F)
< y , BT(x) >
. Moreover ,
so
<m,
Letting then
for all
as
m
+
a
one r
get
t h e convergence o f t h e
11 5 1 1
T
exp
??
IIx
lip
174
f o r a l l m , hence
BT
W.
T. A.
&
BT(x)
DWYER
7$A(x)
m
= Cn=o
in
F.
To show
71,
F p ' ( E ; F ' ) one emplovs t h e F-valued a n a l o g u e o f [Dw
Lem
51, Lemma on p . A 1 4 4 1 , which has an i d e n t i c a l proof,
= [Dw
ma 2 . 1 . 1
that
and p r o c e e d s a s i n t h e p r o o f o f [Dw 7 3 , Prop. 2.1.3 =[Dd 511, P r o p .
2 . 1 . F i n a l l y , t o show t h a t B
i s s u r j e c t i v e and lliBTlllp',pl
=I1
11
T
( h e n c e 8 i n j e c t i v e ) o n e f o l l o w s v e r b a t i m t h e s c a l a r - v a l u e d proof o f [Dw
5,7],
loc. c i t .
73
, P r o p s . 2.1.3', + P r o p . 2 . 2 , l e t t i n g << 3, BT >>F: = T ( f ) o n e h a s : A s i n t h e s c a l a r case o f I D w
Ff: , ( E ' ; F ) (lredp. F i , m ( E ' ; F ) ) a n d F F : ( E ; F ; )
COROLLARY :
F':
(E:F')
6okm < < ,
3' E
a t e i n nepatating duaeity w i t h
>>F
1 <<
and
2.4.1' = [Ddg,
d e d i n e d b y <<
" >>F/5 111
'
?,
$I
IlIN,p,pIIl"
to the bX.h?m
lrebpeCt
1 n l < aT(o) , arb ( o ) > ~ , ~ ,
>>F = C n = o
wheM
111p',p'
(lrebp.
f
E
Fg
I P
(E';F)
and
FF: (E;F').
2 . VECTOR-VALUED CONVOLUTION OPERATORS AND T H E I R ADJOINTS.
F i , m ( E l ;F)
The c o n t i n u o u s l i n e a r o p e r a t o r s on
mute w i t h t r a n s l a t i o n s
'cut
,u'
a r e t h e convolution operators f
ear mappings
-f
T:
Fz,m(E';F)
Ch. I11 and [ D w lo],
-+
Sec. 2.
o p e r a t o r s a r e s u r j e c t i v e on
-+
* ?
+ F (where T
FE,_(E')
and
*
If F = Q ff ( E l ) = Nb
[Dw 6,8].
F,P,,(E')
g i v e n by continuous lin+ f ( x ' ) : = ' ? ( L x 1 6 ) : C f IIMS],
a l l non-zero
FN1
and on v a r i o u s w e i g h t e d s u b s p a c e s o f a s on
-
(where - r u t ? ( x u ) : = ? ( x ' - u ' ) )
E E'
+ T
t h a t com
I
convolution
m ( E ' ) , [G1,2,3], [N 2 , 3 , 4 ] ,
f f N b ( E ' ) , [B1,2],
well
F > 1
then not 2 a l l c o n v o l u t i o n o p e r a t o r s a r e s u r j e c t i v e : indeed, i f E = Q , F = C ,
let ?$ b e d e f i n e d on by of
$($):
?
= (fl
= ( f l ( 0 ) ,O): then
, f 2) $
x
I f dim
as
with f i scalar-valued,
$(XI)
i=1,2
= ( f l ( x ' ) , O ) , and the range
$ * excludes a l l 3 with f 2 # 0 . Following t h e
case
p = 1
175
DIFFERENTIAL EQUATIONS OF INFINITE ORDER
treated in [Dw 9,ld -+
with T
=
-+
, we shall consider here the operators T *
T 8 A, where T
linear operators on
F i m(E')'
E
I
and
A
E
L(F;F) (continuous
F). Similar considerations hold for convo-
lution operators on Fp' (E;F'), as indicated where appropriate. NiO We first represent convolution operators by differential operators of infinite order, starting from the homogeneous differential operators as in [ Dw 9 ,101 : given P' E P (nE), by PA (d) n is meant the linear operator on P (m+nE') given by N PA (d)Qm+n(~'1 : = < 2nQm+n (x') , PA > in terms of the canonical bilinear for < cf [Dw 92. Ch. 111. Given
A
canonical linear operator on
E
I
'n on pN(nE') x P("E) :
L(F;F) we denote by PA(d)8 A the
P ~ (El) ~ 8+ F: ~ cf [Dw 101 .Def. 2.1.
We have :
PROOF:
IIDw 9 1 , Prop. 111.1' and We now fix g'
E FE'(E)
[Dw lo], Prop. 2.1.
and write : : 4
=
1 n!
an,' ( 0 )
to de-
fine the differential operator of infinite order g'(d) 8 A g'(d) B A
wherever
n=O gA(d) 8 A the defining series converges.
PROPOSITION 2.2. g'
E
3:
F F : (E) we h u u e
Given
= Cm
p > 0
and
by
3
choobing o
1
p
buch
thux
T. A. W. DWYER
116 m und
do& e v e k y O ~ ALineah
E
Fp
n,m
opetrutotr v n
(E';F) h e n c e g ' (d) Q A LA a c o n t i n u -
FN~
(E';F
I
PROOF:
.
By use of the estimate (i) in Proposition 2.1
to PA = g1',
one can check as in the case p = l of [Dw
applied
lo],
Prop.
2.1 (cf. also [Dw 83, Prop. 1.2.21, that
~lllNlolp
llIs1',(d) 8 A
< (n!on)l'pI\
IIA !I
gAll
I/!'/~\N,2ulp *
It is enough then to apply this inequality term by term to the ex pansion of g ' ( d ) 8 A 1.3.1 =
[Dw 6
1
(cf.[Dw 101
,
Prop. 2.2 and [Dw 81 , Prop.
Prop. 1.1, setting
r = log2 P I
II
in
the latter
IIP
IIp=Pll
II;=2rll
on Er = E l with dual norm 1 1 I)'rp' = p ' E') , and similarly for s = l o g 2 u ) .
/I
Illp'
(dual norm of
in terms of the F in the corollary of Proposition 1.1 that
It follows from the expansion of <<, >> bilinear forms <
I
>
nlF -f the operators g'(d) 8 A are the convolution operators T
T'
with
T 8 A, where g ' is the Fourier-Bore1 transform of T. Given v' E E' and setting g ' = e- v' as well as A = lF(iden-
tity operator on F) we get g'(d) Q A = COROLLARY:
F;
(E'; F )
thunALatian-invatriuMt.
i 6
Given
E
:F
(E';F) und 1m
06
3, 6 ' >> F
R'
E F:'(E;F')
we
+
h' >>Fli . e . , t h e <<,>. Fg ' ( d ) 8 A i b muLtipLicution by g ' 6vLLvwed by cvmpv-
<
udjaint
8 lFI hence we have:
T~~
I,=
PROPOSITION 2.3.
have
*
A'
o
g'
h i t i a n with A ' .
PROOF:
Follows from the term bv term application of the iden-
tity (ii) in Proposition 2 . 1 to the expansion of <<, >>F in
the
corollary of Proposition 1.1: cf [Dw 101 I Prop. 2.4 and [Dw
81
Prop. 1.5.1.
I
DIFFERENTIAL EQUATIONS OF INFINITE ORDER
177
We s h a l l need t h e t r a n s l a t i o n - i n v a r i a n c e o f FfS(E:F') (which d o e s n o t f o l l o w from P r o p o s i t i o n 2.2) and t h e €act t h a t the << ,>>-adj o i n t of
(a/av)"
( d i r e c t i o n a l d e r i v a t i v e along v E E ) i s multi-
p l i c a t i o n by v" =
v
,>n
(which d o e s n o t f o l l o w from P r o p s i t i o n
2.3). W e d e r i v e t h e s e r e s u l t s from t h e a n a l o g u e s o f P r o p o s i t i o n s
2.1., 2 . 2 and 2 . 3 f o r t h e o p e r a t o r s P n ( d ) on P ( m + n E )
on
and g ( d )
d e f i n e d below.
F:'(E)
Given Pn E P N ( n E ' ) I by P n ( d ) w e mean t h e l i n e a r on
P (mtnE) g i v e n by
PROPOSITION 2 . 1 ' .
P,(d)Q;+,(x)
5
(ii)
PROOF:
operator
.
>
: = < P n l d Q;+,(x)
P n ( d ) E L ( P (m+nE) : P (%))
Qk+n E P ( m + n E ) , Qm E P , ( % ' ) (i)
^n
and
doh
each
we. h a v e
Gn
IIPn(d)GnlI m _<_ llpnl I N,n %IPn(d)C&n>m = (m n 1 < Pn
$- <
sIGn>mn
The argument i s d i f f e r e n t from t h a t f o r P r o p o s i t i o n 2 . 1
i n u s i n g t h e Hahn-Banach t h e o r e m o n t h e b i d u a l of P N ( % ' )
fol-
lowed by Alaoglu's theorem ( d e n s i t y o f PN ( m E ' ) i n i t s b i d u a l ) ,t o f i n d polynomials
1 /lPn(d) E
QA+Jm
E
QmrE
<
such t h a t
(m2) I <
+
E
PN(%')
Pn
> 0 , then passing t o t h e l i m i t as E
QmlE +
IIQmlEIIN , Q;+m
5
>ml
and
E
for
each
0 t o g e t t h e estimate (i)
from ( i i ) ,f i r s t f o r P n o f f i n i t e t y p e ( f o r which t h e ( i i ) c a n b e proved d i r e c t l y ) and t h e n f o r a l l
identity
P n i n P N ( n E ' ) by
t h e d e n s i t y t h e r e i n of t h e p o l y n o m i a l s o f f i n i t e t y p e : cf
ID+?81,
Prop. 1.6.1. Given now g E F:
m ( E 1 ) and l e t t i n g gn: = I
f i n e g ( d ) a c t i n g on f
'
E
F i ' (E)
e v e r t h e series c o n v e r g e s .
by g ( d ) f ' : = C=:o
1
i n g ( 0 ) w e deg n ( d ) f ' wher-
T. A. W. DWYER
178 PROPOSITION 2.2': f'
Fp": (E)
E
m
PROOF:
and e h o a d i n g
5
0
gn(d) f' I l l u '
Il19111N,2u,pI I I f '
5 21'p'
that
Illp'
I
FOP' (E;F').
a c a n t i n u a u d Lineatr a p ~ k a t a ko n
i b
P nuch
m w e have
6 0 ~ .e a c h
IIIcn=o h ence g(d)
Given P > 0
Follows from estimating II/gn(d)f'I I I u l
by the term-
IP'
wise application of the estimate (i) in Proposition
2.1'
and
then the termwise application of the resulting estimate to
Ill gn (d)f ' I l l u r
:
cf [Dw 81, Prop. 1.6.2, 1.7.1 and
1.7.2,
as well as the proof of Proposition 2.2 above. Given y
-+
F and f'
E
E
-+ + Fp' (E;F'), letting f' (x): = < y,f'(x) >
v
E E
we have
-+
-+
-rV f' = (-rv f')y, hence
Y
for y # 0, so that
0
(- <
1113' I l l p ' II Y II. Given 1 ~ ~ ~ ~ =-v ?~ ~ '~ -~r v f~~ ~ ~~ ~ p l , p l =
p'fp IlYll
/ I ) T ~ 3'1 1 1 p ,
ting now g = e-V = exp
Y
P'
Ill?'Y 111 P'rP'
it follows immediately that
<
v,>
)
Q)
< a .Set whenever I1l-r ?'I11 v Y P'rP'
we get g(d)
=
-rV,hence
from
Proposition 2.2' we conclude: COROLLARY:
(E;F') i d tkannLaAion-invakiant.
':F
G i v e n f'
PROPOSITION 2.3':
FF'(E) and
E
h
E
F; -(El)
we have
I
h, f' >>
<> = < < g PROOF:
.
Follows from the termwise application of the
identity
(ii) in Proposition 2.1' to the expansion of <<, >> in terms
, > n in Proposition 1.1:
the bilinear forms <
of
cf [Dw 81 , Prop.
1.9.1. In particular, given u , v f 0 in E and y
g
=
E:
F , by
setting
zr
an and recalling that v" = < v , >n, g ' = + a vn -+ =<<eU y, g ' >>F ,
as well as << we conclude:
-+
3
-
y, gl >>F = << ;i , -f
G;
>>
by
[DW
9j , Prop. 11. 4,
DIFFERENTIAL EQUATIONS OF INFINITE ORDER
Given f '
COROLLARY:
i n E, doh each n
,
$ 1
3' E vn
( u ) = << eU
(a/av)"
FE' ( E ; F ' ) , y
, +f ' > >
( u ) > = << e u
$ 1
u and v#O
E F,
we have:
= 0,1,2,...
(i) (a/av)" (ii)< y
F E ' (E),
E
179
v
n
y
, +f '
>>
F '
1. The a n a l o g u e of the corollary above holds on F p
REMARKS:
NJrn
(E';F) ,
a s f o l l o w s from P r o p o s i t i o n 2 . 3 , b u t w i l l n o t b e u s e d . A d i r e c t proof f o r p = l
i s given i n
~ o J ,Lemma
3.1.
~ D W
2 . The a n a l o g u e s o f t h e e s t i m a t e i n P r o p o s i t i o n 2 . 2 '
)I A I/
the factor
Proposition 2.3' g E FZ(E')
,
as i n P r o p o s i t i o n 2 . 2 ) , and 2 . 4 ' ,
as w e l l
h o l d f o r g ( d ) on
as
those of with
FE:o(E;F')
and are l i k e w i s e d e r i v e d from P r o p o s i t i o n 2 . 1 .
[Dw 81, S e c . 1 . 6 t h r o u g h 1 . 9 when
(with
I :
cf
F = Q.
A f a m i l y o f s o l u t i o n s o f homogeneous equations f o r g ' ( d ) % A
i s g i v e n by t h e f o l l o w i n g p r o p o s i t i o n . PROPOSITION 2 . 4 : G i v e n u and v # 0 i n E ad weLL a d y i n F , t h e + y i d a n o L u t i o n 0 6 g ' ( d ) 8 A? = 0 d u n c t i o n f = eU vn id
-
and o n l y i6 e i t h e h Ay = 0 o h u in a z e h o than n i n t h e dihection
06
v . Moheoveh, d u c h dunc,tionb
e a h l y i n d e p e n d e n t d o h d i d t i n c t exponenth PROOF:
g ' w i t h oadeh k i g h m
06
and atib&ahy
u
ahe L i n n , v ,y.
The argument i s t h e same a s f o r t h e c a s e p = 1 i n [DWlO],
P r o p s . 3 . 1 and 3.2:
-
e'
t h e conditions f o r
vn
y
to be a sg
l u t i o n f o l l o w from c o n s i d e r i n g t h e i d e n t i t y g' (d) 8 A (eu
vn
y) = {ZL=o
n -k
t h e l i n e a r independence of
(vn-k),
The l i n e a r i n d e p e n d e n c e o f
{eUj
continuous polynomials
6i '-
E'
+
g ' (u) (v)eu
v"'~>
Ay
,
and t h e n o n - v a n i s h i n g of eU P .} j
j
f
i n fact for arbitrary
F and d i s t i n c t u ' s ,
from d e r i v i n g by i n d u c t i o n t h e i d e n t i t y
.
j
follows
180
T . A. W .
DWYER
. gj
from t h e h y p o t h e s i s C jk+l =l euj
= 0 through
differentiation
a l o n g u 1 E E ' c h o s e n s o t h a t ( a / a u ' ) n+ Pk+l = 0 and for j
5
#
< U ~ - ~ + ~ , U ' >0
k.
1. I n t h e p r e c e d i n g p r o p o s i t i o n ,
REMARKS:
l i n e a r independence + and h o l d s f o r f u n c t i o n s eU P w i t h d i s t i n c t e x p o n e n t s u E E -+ a r b i t r a r y c o n t i n u o u s p o l y n o m i a l s P : E ' + F a s shown i n the proof.
-
The a n a l o g u e s o f a l l p r o p o s i t i o n s up t o t h i s p i n t are
2.
t r u e f o r v e r y g e n e r a l holornorphy t y p e s and t h e i r d u a l t y p e s (ill c l u d i n g t h e compact, c u r r e n t and H i l b e r t - S c h m i d t t y p e s i n l o c a l l y convex s p a c e s ) a t l e a s t i f F = Q: cf [Dw 5,6,7
I
81.
The r e s u l t s
i n t h e n e x t t w o s e c t i o n s are c o m p l e t e l y known o n l y f o r t h e c u r r e n t type-nuclear t y p e p a i r i n g (and p a r t i a l l y f o r t h e Schmidt t y p e :
3 . VECTOR
cf [Dw 1 , 2 , 3 , 4 ] ,
- VALUED
[Bon],
[K 1 , 2 , 3 , 4 , 5 ] .
DIVISION THEOREMS.
The d i v i s i o n theorem f o r t h e o p e r a t o r g i v e n i n [Dw 101, Th. 4 . 1 on ponential type) r e s u l t on
,
Hilbert-
g'
+
A'
0
g'
-+
*
h',
E x p ( E ; F ' ) ( e n t i r e mappings o f ex-
F':
i s now e x t e n d e d t o
(E;F')
.
The
analogous
P ' ( E ; F ' ) w i l l a l s o be d e s c r i b e d . W e b e g i n Fm
with
a
s t r e n g t h e n e d v e r s i o n o f [Dw 1 0 3 , P r o p . 4 . 1 : PROPOSITION 3 . 1 . A E L(F;F)
Given
buch t h a t
f
l
E ff(E;F')
,
g ' E ff ( E ) buck ththat g ' f 0 ,
and a t o t a l b u b b e t Y
AF = F -+
l e t t h e " b c a l a h componentd" f 1 y Y'
E
F
06
06
A-l(O)
,
+ f ' have t h e 6 o L t o ~ n g
phopehtieb: ( i ) 16 y ji! Y t h e n
3' in Y
divinibLe by g'
a6
an e n t i h e duns
t i a n along a l e complex L i n e d i n E Whehe g ' d o e b n o t Vanid h . ( i i ) 16
y E Y
then
2;
= 0.
181
DIFFERENTIAL EQUATIONS OF INFINITE ORDER
From the case F = Q
PROOF:
Lemma 2.3.1 with Er
=
of [G 2 3
, 58, Prop. 2, or [Dw
81
,
E, it follows from the hypothesis (i) that
z' .
Y there is some h' E H (E) such that g' h' = (Y) (Y) Y By the hypothesis (ii), if y E Y then ?;l= 0 and we may set for each y
+ = 0. We now observe: h' (Y)
2' - 8' =I1
lows from the hypothesis (ii) that
=O y1 y2 y1-y2 (because y1 - y2 can be approximated by linear comb&
nations of elements of Y) , so that g' Since g' # 0, there is neighborhood
# 0, so that h'(yl)'U = hiy,)
g'[
by [ H I ,
I u'
*
h' = g t * h' (Y$ (Y,) U C-E such that hence h'(y,) ="'ry,)
111. 1.3, th. 3 ( b ) .
Th. 4.1, we may then define 6 ' : E + (algebraic dual) by < z , h' ( X I > : - hiy, (x) for every x E E Following [Dw lo],
z
=
AY
E
-+
F*
and
F. AS in [DW 101, loc. cit., we get:
(b) A'
0
&'
g'
=
3'
(from the definition of
C-x' (by considering
+
(c) h' (E)
6').
+
h' (x) = limr,g' (xnylg'(xn)EF'
on a sequence xn + x where g ' (x,) # 0, using [ H] ,lot. cit. and the uniform boundedness principle). (d)
2'
is Gzteaux-analytic -
(by the
analyticity
of
~
< z , > 0 6 ' = h' for each z = Ay in F). Y + (e) h' is bounded on compact - -sets (by showing -f
1 (z"
0
z"
F"
E
;I)
(K) I < 1
that
+
max {,hi (x)l: x E K I for each (Y) and each compact K C E, where z = Ay E F is
chosen in the unit ball with center
z"
in
F"
by
Alaoglu's theorem. It follows from (d), (e) and [H],III.
2.2,
Prop. 1
(ii)
T. A. W.
182
that
g'
E
DWYER
H(E;F').
The n e x t p r o p o s i t i o n d i f f e r s f r o m [Dw 103 , Lemma 4 . 1 t h a t t h e M a l g r a n g e - G u p t a estimate o n q u o t i e n t s
exponential
r e p l a c e d by t h e T a y l o r - B o l a n d
estl
o n maximum m o d u l i o f q u o t i e n t s o f e n t i r e
func
g r o w t h estimates i n
m a t e i n [ B o l 21
4 is
of
in
[G
t i o n s o f bounded t y p e . PROPOSITION 3 . 2 .
With A
g'
then t h e r e are c o n s t a n t s C
-f
,f
'
and
'1'
a s i n P r o p o s i t i o n 3.1,
> 0 (depending
PIPIV
only
on p
and
(depending only on A) such t h a t
Ill '1' Ill V I P The proof r e q u i r e s t h e e s t i m a t e s i n t h e lemma below,where M ( R , f ' ) : = Max{ f ' ( x ) 1
:
IIx
11 5
R}
H b ( E ; F ' ) i s t h e space of
and
e n t i r e f u n c t i o n s f r o m E t o F ' w h i c h a r e bounded o n bounded s e t s (same f o r
F =
( i )Id
LEMMA:
a): f ' E F p : ( E ) t h e n 6 0 l ~ e a c h R > 0 we h a v e P
M ( R -"pI ( i i )G i v e n
f '
f') 5 and
t h e n ,504 e a c h
~
g'
and
PROOF:
h' E
v >
Hb ( E ; F ' ) p > 0
~
i n Hb(E)
R > 0
M(R,f'/g') 5 lg' (0) (iii)16
~
~ (1F R P~ ) .
exp
, id
f'/g'
I
E H(E)
I and g'(O)#O
we h a v e
{1+M(2Rtf')
I3
C1+M(2RIg')l3
t h e n h a t each nequence
( i ) f o l l o w s from t h e d e f i n i t i o n of M (
06
,)
and of
Rn > 0
456 a n d [Bol 2 1 , Lemma 4 . 4 .
f r o m t h e Cauchy estimates o f
"11,
111
~ ~ ~ l l p l :
The e s t i m a t e ( i i ) i s
d u e t o T a y l o r when E = C a n d B o l a n d when E i s a Banach p.
.
we h a v e
cf [Dw 8 3 , Lemma 2 . 3 . 3 = [ D w 6 3 , Lemma 2 . 6 .
cf [ T a l l
I
Finally,
5 6 , P r o p . 3: c f
space :
(iii) f o l l o w s
LDW 81,
Lemma
~
DIFFERENTIAL EQUATIONS OF INFINITE ORDER
183
2 . 3 . 5 = [Dw 6-1, Lemma 2.5. The e s t i m a t e ( i i i ) i n t h e lemma
REMARK:
is the onlv point
t h i s e n t i r e a r t i c l e w h e r e t h e Cauchy e s t i m a t e s a r e u s e d , t h e o n l y r e a s o n f o r t h e r e s t r i c t i o n of t h e
in
hence
d i v i s i o n theorem t o
t h e c u r r e n t holomorphv t y p e ; c f t h e same d i f f i c u l t v i n [G 1 , 2 , 3 _ 1 ,
, [Bol
[N 2 , 3 ]
1,2,3],
f e r t o [Dw 1,2,3 ]
[Mat
(But w e
re-
functions
of
polynomialsfand to [ D i
1,A
and [Dw 5,6,7,8,9,10].
1,2]
f o r t h e d i v i s i o n of
H i l b e r t - S c h m i d t t y p e by H i l b e r t - S c h m i d t
entire
f o r t h e d i v i s i o n o f p o l v n o r n i a l s bv p o l v n o m i a l s f o r more g e n e r a l holomorphy t y p e s . 1 PROOF OF PROPOSITION 3 . 2 .
S i n c e AF = F , t h e image u n d e r
A
of
t h e u n i t b a l l i n F h a s non-empty i n t e r i o r by the open mapping t h e 2
rern, i . e . , t h e r e i s some Iy
IIAYII
E F:
6A > 0
5 1l3tzl
hence f o r each z E F w i t h
11 y I /
51
have
I
11
such t h a t
E F:
z
11
I/
z1
11
5 6Alf
5 1 t h e r e i s some
s u c h t h a t Ay = zl: = 8 A z f t h u s f o r e a c h + -1 -t < z , h ' (x) > 1 = 6A IhAy(x) I , so t h a t M(R~P-'/P
,
?it)
Z
= 6-1 M(R,P A
-Vp,
'Ip
R = 2R,p
as f ' : = g '
M(Rnp-l'p,?i;)
we
Setting
as w e l l
we g e t
(where w e used
1
1 1 + M(2Rnp1/p,g') 3
III?;IIlpl
-
,p, -
Ill$' Illpl
<
exp(2
1.2' P
RP,)
-f
I/ Y /I 5 I I I f ' Illpl,
when
A p p l y i n g t h e n t h e estimate ( i i ) i n t h e lemma, now with
-'4 ,
= Rnp
below).
and a p p l y i n g ( i ) i n t h e lemma t o f ' : = f $
t l + l l l l ' l l l p ' , p ' l {I+ Ills' I l l p ' rP'
.
E: E
-f
{I + M ( 2 R n p-'Ip, ?);
/ / y / I5 1)
x
with
+
f o r a n y s e q u e n c e o f Rn > 0 ( t o b e s p e c i f i e d
R:
y E F
we get
5 6*-1 lg' (0) \-3{l+lil?'
P ' rp
3 tl+lllg':lLl
3 ex~(~-6*Z~*PfJ
P
T. A . W. DWYER
184
From (iii) in the lemma we then obtain
that is ,
where
The series defining C verges when
P V
V P
converges when
PlPlV
$ (and di-
> 6
if we choose the radii of
6 )'2
modulus estimates to be
Rn: =
n
the
maximum
as follows from an ap
6 - 2p
plication of Stirling's estimate and the root test for cf [Dw
81 ,
Proof of Lemma 2.3.2 = [Dw 61
,
Lemma 2.7.
We now combine Propositions 2.4, 3.1 and 3.2
(by use
Proposition 2.3 and the corollaries of Propositions 2.3'1,
to obtain the division theorem for
series:
A'
+
A'
o
and
2.2' g1
-
of
6'
on
F F ' (E;F'): THEOREM 3.1 (Division Theorem). buch t h a t Y
g'
K1 ( 0 1 ,
# 0,
A E L(F;F)
E
b u c h t h a t AF
3 , 3' >> F
in
(c) << eU
F:,~
*
y,
g'
*
wheneum
= 0
(E' ;F)
vn
o
-f
h' =
2 A
?I
Fi'
(E;F'),g'E
Uhe
hub
FoP' (E)
and a t o t a l b u b e t
= F
t h e n -the d o e l o w i n g c a n d i t i o n b
(a) T h e e q u a t i o n A ' F P' o (E;F'). (b) <<
G i v e n f'
equivalent:
a balutian
a boldan
06
6'
in
q'(d) 8 A 2 = O
. ?I
>>F = 0 wlieneuex t i t h e x u i.s a z e t a
185
DIFFERENTIAL EQUATIONS OF INFINITE ORDER
otrdek h i g h e t t h a n n i n t h e d i h e c -
0 6 g' w i t h t i o n 06 v # 0 in E
f' i h d i w i n i b l e b y g '
(d)
y
oh
Y.
E
an an entihe ,junction o n
Y
complex L i n e d i n E w h e h e
'y
E Y,
(a) implies (b) by Proposition 2 . 3 and (b) implies
(c)
3' = Y
and
PROOF:
abR
whenewm
0
whenevek
f 0
g'
Y.
'y E
by Proposition 2 . 4 . Let now (c) be satisfied: if v is a zero of
g'
?j
with order m in the direction of
Y
and
u
v # 0 in E
then we conclude by the corollary to Proposition 2 . 3 ' that + (a/av)"ll (u) = << eU vn y , f' >> = o Y F + for every n m, hence u is also a zero of f' with order 2 m Y on u + d: v , i.e. , ? ' / g ' has an analytic continuation on all cog Y plex lines on which the zeros of g ' are of finite order. If
-
Y E Y
then for each
x
E
E
I Y' (x) =
arbitrary), to get
.
u = x, n = 0 (thus
we may set << e
X
. y, f ' >>F +
v
0, that is, (d) is
=
satisfied. Finally, let (d) hold true: it follows from Proposition 3 . 1 that
A'
0
+
+
h' = f'
g'
for some
first g ' ( 0 ) # 0: by hypothesis we f'
Let
g'
E
F F : (El
and
F p ' (E;F') for some p > 0, and we choose v > 6
E
F Z : (E;F')C F r ' (E;F') by Proposition 3 . 2 . If g' ( 0 )
P'
tion 2 . 2 '
3'
T- u
u
E
E (since g ' # 0 by
I
,
to get
2'0 =
0 then
hypothesis). Since
T-u9
we may apply the same argument as when g ' ( 0 ) # 0
'
and T-~$', getting
for a sufficiently large weight =
TUT-ug'
REMARK: g'
H(E;F').
(E;F') is translation-invariant by the corollary of Proposi-
Fo
g'
E
E
g'(u) # 0 for some P'
have
L'
6
E
T -U
v,
2'
E
so
F:: we
to
(E;F')c F r ' (E;F') still
have
FOP' (E;F'), that is ( a ) is satisfied.
The division theorem holds verbatim for F E ' (E;F'), when
FE' ( E ) , by use of the analogues of Propositions 2 . 3 and 2 . 4 ,
186
T. A. W .
DWYER
as w e l l a s t h o s e o f t h e c o r o l l a r i e s t o P r o p o s i t i o n s 2.2' and 2.3', f o r t h e d u a l p a i r c o n s i s t i n g of S e c . 2 . 2 a n d 2 . 5 , o r [Dw
61,
F E I O ( E ' ; F ) and F t ( E ; F ' ) : cf[Lw8],
Th. 2 . 1
( i i )when
F = b.
4 . VECTOR-VALUED EXISTENCE AND APPROXIMATION THEOREMS. We s h a l l now e x t e n d t o
F;,_(E';F)
p r o x i m a t i o n t h e o r e m s on t h e s o l u t i o n s t e d i n [Dw 9 , 1 0 1
.
t h e e x i s t e n c e and
3 of
g'(d) Q A
"f
ap-
trea -
=
analogues f o r
f o r H N b ( E ' ; F ) , and i n d i c a t e their
[Dw
lo] ,
t h e v e c t o r v a l u e s y of t h e e x p o n e n t i a l - p o l y n o m i a l
funs
Fp
(E' ;F)
Th.
5.1,
N t O
Improving on t h e a p p r o x i m a t i o n theorem i n
t i o n s i n t h e k e r n e l o f g ' ( d ) 8 A now n e e d o n l y be i n a total
SL@
s e t of t h e k e r n e l of A: THEOREM 4 . 1
( A p p r o x i m a t i o n Theorem)
A E L(F;F) w i t h AF = F and a Ui ' n
Zi
= e
dintinct
ui
tiona
doh
06
in a z e a o i
u
hection ad t h e dpace PROOF:
ckii
cj
v
ij
06 all
.
L e X t h m e be g i v e n g ' E Ff (X),
t o t d AubAet Y u d
vtj. yij
(XI.
J
denoting a d i n i t e
dintinct indicen i
,
5.1:
Proposition 2.4,
durn) 1j
w i A h d i n i x e u m f e 4 izigheti t h a n n i j
g'
# 0
i n
E,
nolutionn
with
whehe e i t h e h y . . E Y o h i n the
di-
dohm a b a n i n d o h a denne nubnpcice oh -+ f 0 6 g ' ( d ) B A 1 = 0 i n Fp (E';F). N tm
The a r g u m e n t i s t h e same as i n [Ow
[Dw 101 Th.
A - l ( 0 ) .The dune-
9-1, Th. I V .
2
and
t h e l i n e a r independence of t h e f i follows f r o m w h i l e t h e v a n i s h i n g o n t h e k e r n e l of g ' ( d )
Q A
of t h e a n a l y t i c f u n c t i o n a l s which v a n i s h on t h e f i f o l l o w s f r o m t h e i m p l i c a t i o n (c) t i o n s 2.3 and 2 . 4 ,
+
( a ) i n Theorem 3 . 1 t o g e t h e r w i t h P r o p o s i -
l e a d i n g t o t h e t o t a l i t y of t h e
Hahn-Banach
theorem.
THEOREM 4 . 2
{ E x i s t e n c e Theorem).
g' # 0
and A
E L(F;F)
nuclz
G i w e ~g '
that
AF = F
fi
E FF' (E)
evehy
by
the
nuch
that
eyuution
DIFFERENTIAL EQUATIONS OF INFINITE ORDER
PROOF: ever A'
A s w i t h [Dw A'
g'
0
-
101 , Th. 5 . 2 ,
187
the vanishing of A'(;'
( x ) ) wher-
h ' ( x ) = 0 and g ' ( x ) # 0 (by t h e i n j e c t i v i t y
of
as t h e t r a n s p o s e of a s u r j e c t i v e o p e r a t o r ) g u a r a n t e e s t h e i E
j e c t i v i t y o f t h e << 3 ( b )1 .
1.3, th. A'
0
The
, >>F - a d j o i n t
map h '
A'
+
*
g'
0
h ' ( I H ] ,111.
<<, >>'>F -weakly c l o s e d c h a r a c t e r o f
the range
FP'(E;F') f o l l o w s from i t s r e p r e s e n t a t i o n as t h e i n t e r 0
g'
s e c t i o n of t h e sets { f ' E F:'(E:F'):
2
lutions
of g ' ( d ) 0 A
2
= 0 in
<< f , f ' >>
Ff
(El ;F)
for all
F = 0)
(by t h e
so
implication
I m
(b) * ( a ) i n Theorem 3 . 1 t o g e t h e r w i t h P r o p o s i t i o n 2 . 4 ) .The s u g j e c t i v i t y o f g ' ( d ) 8 A f o l l o w s as u s u a l f r o m t h e Dieudom&Schwartz theorem on s u r j e c t i o n s i n F r g c h e t s p a c e s : c f ( i ) ,p .
Fz'
F
Prop.
5.1
25.
REMARKS: g'
[Tr 3] ,
(E))
1. The a p p r o x i m a t i o n t h e o r e m h o l d s o n
, as
F,PIo(E';F) (with
F z ' (E;F')
f o l l o w s f r o m t h e d i v i s i o n theorem on
a n d t h e a n a l o g u e o f P r o p o s i t i o n s 2.3 and 2.4 on F p
NtO
(E';F)
x
F?@;F')
(see t h e r e m a r k s f o l l o w i n g P r o p o s i t i o n 2 . 4 a n d Theorem 3 . 1 I . T h e
,:F
e x i s t e n c e theorem l i k e w i s e h o l d s on
( E l ; F ) by t h e
theorem
o n s u r j e c t i o n s i n s p a c e s w i t h F r g c h e t d u a l s ( [ T r 3 1 ,Prop. 5 . 1 ( i i ) , p.
25) , p r o v i d e d t h e bounded s e t s of
F N r r o (E'; F ) are regular (i.e.,
c o i n c i d e w i t h t h e bounded s e t s of t h e s p a c e s [FW],
9 2 3 , no. 5 , p . 1 2 3 a n d [ R R ] ,
Fftp (E';F))
Ch. 4 , 5 4 , P r o p .
l a r i t y c o n d i t i o n i s known t o h o l d f o r
FP'
NtO
I
15.This r e p
( E ) when E is a H i l b e r t
space, a p r o j e c t i v e l i m i t of a sequence of H i l b e r t spaces Frgchet-Schwartz
by
space (with an inductive l i m i t d e f i n i t i o n
or a for
Fp' ( E ) o n l o c a l l y c o n v e x d o m a i n s E w h i c h c o i n c i d e s w i t h t h e one NtO u s e d h e r e f o r t h e Banach space case): cf[LW5],Prop. 2 . 3 ( i ) ,(ii)=[Dw7],
188
T. A . W.
1 . 5 . l I ~ ~ m 6th. ~ , 3.1=[Dw8],Th.
c o r o l . toProp.1.4.1andProp.
2. with o
p(not
shown i n [ Dw 4
D
]
< p even i f g ' i s a polvnomial and
F E l p ( E ' ; F ) when
theorems on
0 < p <
m
t h e e x i s t e n c e theorem f o r t h e and
existence
instead
v i a p e r h a p s a f i n e r c h o i c e of t h e r a d i i R
2'p,
0 < p <
m
a, as
would r e q u i r e a s t r e n g t h -
ening of P r o p o s i t i o n 3.2 t o p e r m i t ch o o s in g v > 4 p v > 6
F =
f o r p = 2 , a p p r o x i m a t i o n and
a n d [Bon]
2.6.2.
P FE140(E';F) onlv i n t o F (E';F) NiP
S i n c e g ' ( d ) B A maps
2
DWYER
of
( b u t see
n
p = 2
Hilbert-Schmidt tvpe with
f o r polvnomials g ' i n I D w 1 1 2 , 3 ] ) .
5 , APPLICATION TO ENTIRE FUNCTIONS WITH E N T I R E FUNCTION VALUES.
W e b e g i n by e x t e n d i n g t h e p r e c e d i n g t h e o r v t o more general r a n g e s p a c e s : i f F i s a F r g c h e t s p a c e w i t h a c o n t i n u o u s mrm then
i t s t o p o l o g y i s d e t e r m i n e d by a f a m i l y o f norms
Ilr
iradexed by
a n o r d e r e d s e t w i t h a c o u n t a b l e c o f i n a l s u b s e t ( i t i s enough t o a d d t h e c o n t i n u o u s norm t o s u f f i c i e n t v manv c o n t i n u o u s semirmrms determining t h e topology of F ) . F i s t h e n a complete
countably
normed s p a c e i n t h e s e n s e o f G e l f a n d . L e t t i n g Fr d e n o t e t h e cog p l e t i o n of F w i t h r e s p e c t t o I / mappings F
+
I/r
it follows t h a t t h e n a t u r a l
Fr a r e i n j e c t i v e and F h a s t h e p r o j e c t i v e l i m i t
p o l o g y i n d u c e d by t h e s e m a p p i n g s . W e mav t h e n d e f i n e FE
as
n
F$,,(E';F,)
and
ff
(E;F') as
u
(E;FA)
o b v i o u s i d e n t i f i c a t i o n o f mappings i n t o F ( r e s p . F;) p i n g s i n t o Fr
(rerp. F')
.
Fp
Nim
( E l ;F) is then s t i l l
,
l m
to
(E';F)
with t h e w i t h map-
a
Fr6chet
s p a c e when r e g a r d e d a s a p r o j e c t i v e l i m i t o f t h e F f e c h e t s p a c e s F P w ( E ' ; F r ) , and t h e p a i r i n g of Nt
F:lm(E';F)
w i t h Ff(E;F')
given
by t h e c o r o l l a r y t o P r o p o s i t i o n 1.1 s t i l l h o l d s : now I<>l< + (E';Fr) a n d P ' E :F (E;Fi) , IIl'llC ,p , r I p I l l ? ' I l l p l l r l n l i f f E F; I P
DIFFERENTIAL EQUATIONS OF INFINITE ORDER
189
a r e t h e norms i n the c o r r e s p o n d i n g s p a c e s of maps E l en
+
Fr
and
E
F;
+
respectively. Giv-
A E L ( F , F ) , t h e estimates o f P r o p o s i t i o n 2 . 2 and 2 . 2 '
hold with
11 A I/
still
r e p l a c e d by t h e a p p r o p r i a t e g r o w t h e s t i m a t e
on
A r e g a r d e d a s a map from Fs i n t o Fr f o r e a c h r and a n a p p r o p r i -
a t e s ( g i v e n by A ) s a t i s f y i n g tween g ' ( d ) b A
and A '
g'
o
) I / l r 5 1) ) I s
.
The d u a l i t y
w i t h g ' E F F l ( E ) of
be
Proposition
2 . 4 c l e a r l y r e m a i n s v a l i d . S i n c e t h e open mapping theorem ramins
t r u e for t h e F r s c h e t s p a c e s , P r o p o s i t i o n 3 . 2 a l s o r e m a i n s t r u e . I t f o l l o w s t h a t t h e d i v i s i o n theorem 3 . 1 , h e n c e t h e a p p r o x i m a -
t i o n theorem 4 . 1 and t h e e x i s t e n c e theorem 4 . 2 ,
hold
unchanged
whenF i s a c o m p l e t e c o u n t a b l y normed s p a c e . L e t now E
i'
i = 1 , 2 b e complex Banach s p a c e s , and l e t there 1
pi > 1 and g;
be given
E F F ( E ~ ,) i = 1 , (~w i t h
[Dw 101
i f pi = 1). S i n c e , u n l i k e
,
P!
~ ~ 1 4 = ~~x p) ( :E ~ )
t h e v e c t o r values of
solu-
t i o n f u n c t i o n s now need b e o n l y i n a t o t a l s u b s e t of t h e k e r n e l o f A , w e o b t a i n t h e f o l l o w i n g a p p r o x i m a t i o n theorem f o r g i b g i ( d ) : THEOREM 5 . 1 .
gi
+
p2
T h e 6unctiann f i : E i
U
= e i 1'. x n i j vk j k=O i j
t h e h ui i d a zehv
06
. gi
-
"j
v" i j ij
-+
'
FNtm(E;)
the
ad
dohm
w i t h dintinct ui, whme ei-
w i t h ohdeh h i g h e t . t h a n
n
ij
a l o ng
o h pi; i n a zeho 0 6 g i w i t h ohdeh h i g h e h t h a n i j # 0 i n El m i j a l o n g v i j # 0 i n E 2 , 6ohm a b a d i n doh a d e n n e nubnpace o d t h e Aolution Apace 0 6 g i ( d ) b g i ( d ) z = 0 i n F N" I m ( E 1' ' - F pN2, m (E;)).
v
PROOF:
By t h e a p p r o x i m a t i o n theorem when F = C w e know t h a t t h e
functions i n F;:JE; f o r each
' i j vmi form a t o t a l s u b s e t o f t h e k e r n e l of g i ( d ) ij i s a t r u e norm on (E;) 1 . Moreover, 1 1 1 I~~N,p,p t m p > 0 . I t i s enough t o l e t F = Fp2 (E;) and l e t Y b e
e
t h e s e t of f u n c t i o n s
~Np2
m
ep i j
*
v 1j i j
Nrm
i n Theorem 4 . 1 .
190
T. A.
DWYER
W.
W e a l s o have an e x i s t e n c e theorem f o r g i ( d ) b g;(d) :
14
gi # 0
gi(d) B ga(d)l =
with
THEOREM 5 . 2 .
tion
"f i n
PROOF:
and
"4
g;
# 0
then
t h e dame A p a c e .
F = Q
By t h e e x i s t e n c e theorem when
and A = g ; ( d )
w e know t h a t g;(d)
9
p2I m ( E ; ) . FN
i s s u r j e c t i v e on
REMARKS:
equation
evehy
F9 N l m ( E i ; Fp2 N I m ( E ~ ) hub ) a AOLU-
LM
I t i s enough t h e n to set F = FN, & E i )
i n theorem 4 . 2 .
1. The r e l a t i o n s h i p w i t h t h e o p e r a t o r s ( g i b 9;)
o r t h e more g e n e r a l o p e r a t o r s G ' ( d , d ) w i t h G ' E F r i P l ( E l a c t i n g on t h e s p a c e s
Frlp(Ei
even f o r t h e n u c l e a r t y p e
Fp
Nim
(Ei
x
setting
E = E
2. I f
and
x E2
El
=
Q,
,
E2)
x
f = Q
,
hold
however, as f o l l o w s
p1 = p 2 = 1 w e a r e
I
on from
reduced
lf(a!, f f N b ( E ' ) ) ( o f f f N b ( E ' ) - v a l u e d e n t i r e
t o t h e spaces
,
i n s e c t i o n s 1 through 4.
and
E2 = E
E2)
x
i s s t i l l t o be c l a r i f i e d
8 = N . A l l r e s u l t s here do
w i t h G ' E F['(E1
E;)
E;)
x
(d,d),
t i o n s on Q). The r e l a t i o n s h i p w i t h
HNb(Q
x
func-
E l ) would b e a p a r -
t i c u l a r l y i n t e r e s t i n g t o p i c o f i n v e s t i g a t i o n , as i t s s t u d y leads t o i n i t i a l v a l u e problems i n i n f i n i t e dimension: indeed,
2
and
tion
$
in
f f ( C ; H N b ( E ' ) ) a s w e l l a s g'
a/at z(t,x')
equation
= g ' (d)i!(t,x')
G'(d,d)z =
f i n e d by G' ( t , x ) : = t
"g,
-
Ff m(C
,
x
+ G(t,x')
Exp(E) , t h e equa
g' (x) , provided x
$ and
-
i s e q u i v a l e n t to the
where t h e f u n c t i o n G ' : C
s c a l a r - v a l u e d f u n c t i o n s o n CC regarding
in
given
G
x
E
+
C
is d e
a r e r e g a r d e d as
El. S i m i l a r c o n s i d e r a t i o n s
hold
E l ) , a s w e l l as f o r o t h e r holomorphy t y p e s .
6 . APPLICATION TO VECTOR-VALUED
VARIATIONAL EQrJATIONS.
The r e s u l t s i n t h e p r e c e d i n g s e c t i o n s a p p l y t o h o l m r p h i c
191
DIFFERENTIAL EQUATIONS OF INFINITE ORDER
mappings defined on spaces of essentially bounded measurable fun2 tions, with integrable variational kernels. More precisely, let M be a measure space with a positive measure m and set E = LI (M) (all function spaces on M are relative to the measure m) . A function
8 on
Lm (M) with values in a (Banach or Frgchet) space F is
said to have an n-th variational derivative at x'
E
Lm(M) if its
1
(syrranetric n-th Frkhet derivative at x' has a kernel in L s ( M ~ ; F ) integrable functions M X . . X M F ) denoted by
.
(tl,... It*)
8"1/6*'": so that
-nf ( x ' ) (v') = d -+
I.../
for every v'
+
+
6"3(x')/6x' (tl). . .6x' (tn)
...6x'(tn)v' (t,) ...v' (tn)dm(tl)...dm(tn)
S~(x')/Gx'(tl)
Lm(M). This kernel is then the variational derivg
E
tive of f. We have: PROPOSITION 6.1.
A 6unction
Sn: Lm(M)
F in a n
n-homogeneou~ + nucLea4 p o L y n o m i a L .id and o n l y .id thekt LAa h m n c l xn E L : (M~:F)
nuch t h a t
1:n (x') = doh e v e k y
x'
1 1 hn [ I N
I... j 1.--I
q t lI . . . ,tn x'(t,)
E
=
Lm(M). 'In thin cane
11
WQ
+
... x' (tn)dm(tl)...dm(tn) have
.
.
zn(tl,.. ,tn) [ I dm(tl). .dm(tn).
PROOF:
PN(nLm(M);F) can be identified with the nuclear comple tion of L 1 (M) 0 ... 0 L 1 (M) 8 F (where 0 denotes the symmetric tensor product). The repeated application of the Dunford-Pettis and the nuclear complg theorem (on the isometry between L'(M;F) 1 1 n tion of L (M) 8 F) then leads to the identification of Ls(M ;F)
with PN(nLm(M);F): cf. [Tr21, Th. 46.2. Together with the translation-invariance of F;
I m
(E';F) coy
ollary to Proposition 2.2), by application of the root test
to
192 the
T. A . W.
norms lll'lllNl
2
(*I
( O h
I
a l l ohdekn a t nume x '
06
F N ~ , ( L ~ ( M ) ; F)
in
: L ~ ( M )+ F i n
id
; F ) when p = 1) id and o n l y
HNb(L
tiven
r e g a r d e d as power series i n p w e t h e h a v e :
p,p
A dunction
COROLLARY:
DWYER
han v a k i a t i v n a l detriva -
E L ~ ( M ) nuch
that
limn {n!
I n t h i n cane we have
doh each p > 0, and
The a n a l o g u e o f t h e c o n d i t i o n ( * I
REMARK.
norms
x ' = 0.
11
dng(0)
]Io
i n p l a c e of t h e i n t e g r a l s clearlycharacterizes
( E ' ;F) f o r g e n e r a l holomorphy t y p e s
F:
above a p p l i e d t o t h e
0.
1m
We c a n a l s o c h a r a c t e r i z e p o l y n o m i a l s and h o l o m o r p h i c map1
pings of c u r r e n t
t y p e on
PROPOSITION 6 . 2 .
A dunctian
L (M):
1
: L (MI
+
magenevw p o l y n o m i a l on I., 1 ( M ) id and o n l y
1
x n' E L ; ( M ~ ; F I ) nuch t h a t -+ Pn' (XI = ;;(tl,. . . r t n ) x ( t l )
...
doh eveky
x E L1(M).
/j PROOF:
-+
PA
j[
position
ketrnel
{ [ I -+x;(t
l l . . . , t n ) I :i ti
E
MI.
P, ( n E ' ; F ) '
and
1 n P N ( n L m ( M ) ; F )and Ls(M ;F) (Pro 1 n 6.1), and f i n a l l y t h a t b e t w e e n Ls(M ; F ) ' and Li(I?;F').
C D w 9 l 1 t h a t between
Proposition 2 . 2 ' )
Ill$'
COROLLARY:
a
We t h e n have
= ess. s u p
The t r a n s l a t i o n
norms
- i d t h e h e in
n-hc
...x(tn)dm(tl) ...dm(tn'
F o l l o w s from t h e i s o m e t r v between
P("E;F')
F ' LA a coni'inuaros
Illplp'
- invariance
of
F P0 ' ( E ; F ' ) ( c o r o l l a r v
and t h e a p p l i c a t i o n o f t h e r o o t
test
to
to the
r e g a r d e d a s power s e r i e s i n p ' t h e n g i v e us:
A dunction
$ 1
: L ~ ( M )-+ F'
in i n
F ~ ' ( L ~ ( P;IF) ' )
DIFFERENTIAL EQUATIONS OF INFINITE ORDER
The analogue of condition
REMARK:
11
an?' ( 0 )
types
characterizes
(+)
applied
to
193
the
norms
F i : o (E;F') for general holomorphy
0.
We may then treat variational equations on PROPOSITION 6.3:
The
eXihteMCt
doh v a & i a t i o n a L ? e y u a t i o n h
06
Lm(E4) :
and a p p t o x i m a t i o n t h e o t e m d hoLd
t h e dohm
$ j..,/APZ(XI)/GX'(t,) ...6x'(tn)x;l(tlf...,tn)dm(tl).. .iim(tn) =
(a) z~~
+
g(x') I 6n-.f/6xfnand
what t h e uatiationaL? dehiuatiweh i66y
(*)
hatindy
bat
-
i n t h e c o t o C L a t y t o P h o p o d i t i o n 6 . 1 and t h e he4neLh x;l (+)
i n t h e CuhoeLahy t o
PtOpUhitioM 6. 2. + g = 0 t h e n L i n e a t combinationn o d
I n pahticuL?a4, id
tiono
6"6/6x'"
Lm(M)
+
F
06
t h e dkom
~ U M C -
I
x' b Iexp {ju(t)x' (t)dm(t) I * { v(t)x' (t)dm(t)Inly,
w h e t e e i t h e h y L i e n i n a t o t a l d u b n e t a d A-l(O) vtr u ivith vhdet h i g h e t t h a n n i n t h e d i t e c t i o n g'
x
06
i d
a
zeko
v 0 6 t h e dunction
given by
bZ:mn=o \...Ix~(tlf...ltn)x(tl)...x(tn)dm(tl) ...dm(tn)inL1(MI,
me d e n s e i n t h e hpace
PROOF:
06
doLutiond
It is enough to set F
-+
f.
and $(6ng'/Gxn)x=o = : x' n in the corollary to Proposition 6.2, and observe that the varig = Cfx' =gt
194
T . A. W.
DWYER
g ' ( d ) 8 A? =
t i o n a l equation (a) i s t h e equation
Fm :,
in
( L m ( M ); F ) a n d g '
REMARK:
Fr'
E
t
with
-
f
,g
f
(L1(M) 1 .
P r o p o s i t i o n s 6 . 1 and 6 . 2 ,
hence t h e proof of P r o p o s i -
t i o n 6 . 3 , d o n o t h o l d o n LP(M) w i t h a r b i t r a r y p: e . g . ,
if p = 2
then t h e functions
1.. .I
...x' ( t n ) d m ( t l ) . . . d m ( t n )
z n ( tl,... , t n ) x ' ( t l )
x'
for
2 -+ 2 n x ' E L ( M I a n d x n E Ls ( M ; F ) a r e t h e n-homogeneous Hilbert-Schmidt
2 p o l y n o m i a l m a p p i n g s f r o m L ( M ) t o F , s o t h a t i t i s t h e treatment i n [Dw 1 , 2 , 3 ]
that applies.
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U.S.A.
Infinite Dimensional Holomorphy and Applications, Mat05 (ed.1 0 North-Holland Publishing Company, 1977
ON THE RANGE OF ANALYTIC FUNCTIONS
INTO A BANACH SPACE
By J . G f o b e w n i k
L e t A be t h e open u n i t d i s c i n C and l e t X b e a s e p a r a b l e
complex Banach s p a c e . Does t h e r e e x i s t a n a n a l y t i c
function
f : A-cX s u c h t h a t t h e convex h u l l of f ( A ) i s c o n t a i n e d and dense
i n t h e u n i t b a l l of X ?
T h i s problem, r a i s e d by D . P a t i l
at
t h e I n t e r n a t i o n a l Conference on I n f i n i t e Dimensional HolomorMy, U n i v e r s i t y of Kentucky, 1 9 7 3 , was t h e m o t i v a t i o n t o s t a r t s t u d y i n g c e r t a i n r a n g e p r o p e r t i e s of a n a l y t i c f u n c t i o n s i n t o a Barn& space. Throughout, A is t h e open u n i t d i s c i n C and
aA
is
its
boundary. L e t X b e a complex Banach mace. The closure of a set S
cX
i s d e n o t e d by
g.
I f r > 0 w e d e n o t e by Br ( X ) t h e open b a l l
i n X , of r a d i u s r , c e n t e r e d a t t h e o r i g i n . By H ( A , X ) w e
denote
t h e sDace o f a l l a n a l y t i c f u n c t i o n s from A t o X I by H o o ( A , X )
we
d e n o t e t h e Ranach s p a c e of a l l bounded a n a l y t i c f u n c t i o n s
from
A t o X w i t h s u p norm and by A ( A , X ) w e d e n o t e i t s c l o s e d
sub-
s p a c e of f u n c t i o n s having c o n t i n u o u s e x t e n s i o n t o a l l
w r i t e H"
2.
We
f o r H m ( A I C ) . I f K is a compact Hausdorff space w e d e n 0
t e by C ( K , X ) t h e Banach s p a c e o f a l l c o n t i n u o u s f u n c t i o n s 201
from
J . GLOBEVNIK
202
K t o X w i t h s u p norm. L e t X b e a s e n a r a b l e complex Banach s p a c e . Does t h e r e ex-
ist an f
€ H(A,X)
whose r a n q e i s d e n s e i n some b a l l ? The
posi
t i v e answer i s p r o v i d e d by THEOREM 1
(see
[lo],
p.157,
[15], p.298)
Let
X be a
Banach b p a c e . G i v e n a n y i n j e c t i v e s e q u e n c e { z n ) C A
compee,x
with
no
c k h A t e k p o i n t i n A and ant! bequence { w n ) C X t h e h e e x i n t b fEH(4X) A a t i A h u i n g f ( z n ) = wn { a h a l l n. Theorem 1 d o e s n o t D r o v i d e t h e a n s w e r t o whether t h e r e e x i s t s an f
€ Hm(A,X)
the
question
whose r a n q e i s d e n s e i n som
b a l l . A n a t u r a l way t o show t h a t s u c h a f u n c t i o n e x i s t s i s i n t e r n o l a t i o n o f bounded s e q u e n c e s of v e c t o r s i n Banach
the
spaces
by bounded a n a l y t i c f u n c t i o n s . C a l l a s e q u e n c e { z n ) C A i n t e h p o -
e a t i n g Aeque.nce (see
[111) i f
q i v e n a n y bounded s e q u e n c e {wn]CC
t h e r e e x i s t s a n f E Hm s u c h t h a t f ( z n ) = wn f o r a l l n. The w e l l known t h e o r e m of L. C a r l e s o n q i v e s a n e c e s s a r y and
sufficient
c o n d i t i o n f o r a s e q u e n c e t o b e i n t e r p o l a t i n q (see [ l l ] ) .C a l l a s e q u e n c e { z n ) C A a g e n e h a [ i n t e k p o l a t i n q Aequence if g i v e n
any
bounded s e q u e n c e {wn) i n a n y complex Banach space X
ex-
there
i s t s f E H m ( A , X ) s a t i s f v i n a f ( z n ) = wn f o r a l l n . U s i n q t h e t h e o r y o f HP-spaces o f v e c t o r - v a l u e d
f u n c t i o n s it w a s shown i n [2]
t h a t t h e p r o o f of t h e C a r l e s o n i n t e r n o l a t i o n t h e o r e m d u e
to
S h a p i r o and S h i e l d s (see [ll]) c a n be m o d i f i e d t o o b t a i n THEOREM
2
(see [ 2 ] )
A bequence { z n ) C
e a t i n g s e q u e n c e i{ and o n l y i{ it
i d
a i n a genekat i n t e k p o -
an intehpotating b e q u e n c e .
Note t h a t Theorem 2 c a n b e d e d u c e d a l s o f r o m a r e s u l t P . B e u r l i n g (see [ 2 ] ,
[3]).
of
ON THE RANGE OF ANALYTIC FUNCTIONS
203
I t was proved i n [ 2 ] t h a t q i v e n a n i n t e r p o l a t i n g s e q u e n c e
{zn)CA
t h e r e e x i s t s a u n i v e r s a l bound f o r t h e norms
i n t e r p o l a t i n q functions, i.e. t h e r e e x i s t s M <
of
the
such t h a t
m
any secluence {wn] i n any complex Banach s p a c e X t h e r e
for
exists ! l f ( z )I
f E H ~ ( A , x ) s a t i s f y i n g f ( z n ) = wn f o r a l l n and
I 2
.<_M.sun 1 ( w n l 1 ( z E A ) . Even i n t h e s c a l a r c a s e i t seems t o b e n a n open problem what i s t h e l e a s t lower bound €or s u c h M. T h e r e
for which t h i s bound i s close t o 1:
e x i s t sequences { z n ] c A
Given
(see [ 2 ] )
THEOREM 3
> 0 thehe e x i b t a a b e y u e n c e h 2 C A
E
auch t h a t { o h art!{ complex Banach apace X a n d d o t any
Aequence {wn)CX
thehe
(il f ( z J (ii)
1 If
(2)
exibtb
= wn
11 5
batib4Ning
f E H"(A,X) 40t a . U
(1 + E ) .
sup n
bounded
n
I \wn( 1
(z E A).
Observe t h a t q i v e n a s e p a r a b l e complex Banach s p a c e X Theorem 3 g i v e s an a p p r o x i m a t e s o l u t i o n t o P a t i l ' s problem s i n c e E
> 0 it g i v e s a n f E H"(A,X)
and d e n s e i n B1 ( X )
. However,
given
whose r a n g e i s c o n t a i n e d i n
as o b s e r v e d i n [ 2 J s u c h a
3+Eo$
method
d o e s n o t g i v e t h e e x a c t s o u t i o n o f P a t i l ' s problem. The s o l u t i o n of P a t i l ' s problem i n t h e f i n i t e d i m e n s i o n a l c a s e was found i n [5] a s a n a p p l i c a t i o n o f t h e f o l l o w i n g g e n e r a l i z a t i o n o f t h e w e l l known Rudin-Carleson theorem.
(see [5], [13] , [ 1 6 ] )
THEOREM 4
G i v e n a c f . ! o ~ e db e t F C a A thehe
exinta
aequenttq
f
it( x
buch t h a t f(A) REMARK 1
L e t X b e a cornp-tex R a n a c h d p a
0 4 Lebengue meabuhe
E A ( h , X ) A a t i a 4 u i n g f l F = f and id 4inite
0 and f E C ( F , X )
11zI 1
dimenbionat t h e m e x i b t a
LA c o n t a i n e d and d e n b e i n
=I I f ! Z
!.Con
E A(A,X)
B1(X).
To d e d u c e t h e second p a r t o f t h e above theorem
from
J. GLOBEVNIK
204
its first part, let F C a A be a Cantor set of Lebesgue 0 and let f be a continuous surjection from F onto
exists by the compactness of
-_B1(X) (see
[12!,
measure
-B1(X)
which
p.166).
In the infinite dimensional case Patil's problem wassolve3 independently in [l] and 161. In 111 this was done in a typical ly infinite dimensional way: by constructing first a
function
that solves the problem for co and then composing it by an anathe
lytic function mapping B1(cO) into B1(X) densely. In [ 6 ]
solution was found as an application of the following generalization of an interpolation result of [g]: THEOREM 5 F
c
aA-E
(see [6])
Let E C a A
be a c l o n e d
and
bet
let
be a h e l a t i v e l y c l o d e d 6 e t 0 4 L e b e b g u e m e a b u t e 0.GLvw
a c o m p l e x Banach 6 p C e X and a bounded COntinUOU6 , ( u n c t i o n f:F+X
t h e h e e x i b t d a bounded c o n t i n u o u b { u n c t i o n
g:
if-E
-+
X, analytic
on A and satisfying
(i) (ii)
?IF = f sup I I f f z ) l l = ZE;?;-E
SUD
llf(s)ll
sEF
C o n 6 e q u e n t L f { i4 X i n d e p a h a b l e t h e n t h e h e e x i d t d
2
E
H(A,X)Khbe
h a n g e i 6 c o n t a i n e d and d e n d e i n B1(X). REMARK 2
To deduce the second nart of the above theorem
its first part, let E = 111, let F = (zn)caA-{l) tive sequence converrJing to 1 and let f (2,)
from
be an injec-
= wn where {Wn)cB1(X)
is a sequence, dense in B1(X).
An interesting consequence of Theorem 5 is that the space Hm is isometrically universal f o r all complex Banach spaces pos
sessinq countable determining sets, i.e. that every such space is isometrically isomorphic to a subspace of IIm(recall
that
a
determining set for a Banach space X is any subset S of thecbal
205
ON THE RANGE OF ANALYTIC FUNCTIONS
The r e s u l t s p r e s e n t e d above l e a d n a t u r a l l y t o v a r i o u s Prcb
l e m s . F i r s t , l e t u s a s k when d o e s Theorem 5 h o l d i f B1(X) r e p l a c e d by some o t h e r set. L e t us s a y t h a t A i s
is
analytically
denbe i n a s u b s e t P of a s e p a r a b l e complex Banach s p a c e there exists f E H(A,X)
if
X
whose r a n q e i s c o n t a i n e d and d e n s e i n P .
PROBLEM 1
Ubtain a ( g e o m e t h i c a l , topoloqicablchahactehization
04 the
i n which A
betd
i d
anaPutical.tr/ d e n h e . was
T h i s problem seems t o b e h a r d . R e c e n t l y a p a r t i a l s o l u t i o n found f o r t h e c a s e when t h e sets i n a u e s t i o n are onen: THEOREM 6 (see [ 7 ] )
P
A i d anaPr/ticak'lrt dende i n a n open h u b A e t
0 4 a b e p a a a b j e complex Banach dpace i4 and o n l 1 4 id
con-
P id
nected. L e t X be a complex Banach Apace and b e t
be
a
b e p a t a b l e complex Banach Apace. G i v e n anri o p e n connected s e t
8
COROLLARY 1
in Y thete
eXidtA
an a n a l r l t i c m a p p i n q F
: B1(X)
+
Y
Y whode hanqe
i n c o n t a i n e d and denae i n 0 . The p r o o f of C o r o l l a r y 1 is t r i v i a l . One a p n l i e s u E
t o map B1(X)
u
X I ,
o n t o a n open d i s c i n C and t h e n by Theorem 6
# 0 one
mans t h i s d i s c a n a l y t i c a l l y and d e n s e l y i n t o 0 . S i n c e t h e r a n q e of f E H ( A , Y )
i s always s e p a r a b l e such a c o n s t r u c t i o n
is
p o s s i b l e i f t h e s p a c e Y i s n o n s e p a r a b l e . C o n s e n u e n t l y we PROBLEM 2
have
L e t Y b e a n o n d e p a a a b l e complex Fanach Apace. D e f eh
m i n e t h e CPadA
04
a l l complex Ranach bpacea X havina t h e doRRow
i n g phopehtri: q i v e n anu o p e n connected an a n a t r r t i c mappinq F : B1(X)
dende i n 0 .
not
+
Y
b e t
n i n Y thehe
tvhode h n n q e
i d
exiatd
contained
and
J. GLOBEVNIK
206
CONJECTURE
7 4 P i n a n open c o n n e c t e d
AP.t
i n a complex
Apace X t h e n t h e h e e x i n t d an a n a l y t i c napping F
: B1(X)+
Eanach X ruhone
h a n g e i d cantai.ned a n d denbe. i n P. Since it is easier to construct the continuous
functions
havinq some prescribed range nroperties than the analytic
ones
it is obvious that various qeneralizations of the Rudin-Carlesm theorem are an efficient tool when proving the existence of analytic function whose range has certain density
an
pronerties
(see Remarks 1, 2). In particular, when provinrJ the
existence
of an analytic function whose ranqe is dense in some ball
we
first constructed a continuous function on a suitable subset of ah whose ranqe was dense in this hall, and then we extendedthis
function to a function analytic on A whose ranqe was
contained
in the same ball. It is a natural rruestion whether such an
ex-
tension is possible for some other classes of sets in place
of
the balls. Let us say that a subset P of a complex Ranach space
X
has
the a n a l i f t i c e x t e n d i o n p h o p e h t y if given any closed set
E C a A , any relatively closed subset F C 2 A - E of Lebesgue sure 0 and any continuous function f : F there exists a continuous extension
-
f
+
X satisfyin9 f ( F ) c P
-
__
: A-E
mea-
-+
X, flF = f, ana-
lytic on A and satisfyinq f ( z - E ) C P. PROBLEM 3
04 t h e
O b t a i n a ( p ~ o m e t h i c a l ,t o p o l ' o q i . c a l ] chahncte.h.izntian
n e t n h a v i n g t h e a n a t r i t i c e x t e n n i o n paopeatit.
This problem seems to he very hard. Recently a partial solution was found for the case when the sets in question are onen: THEOREM 7 (see [Sl)
An ope,n n u b n e t o.( a complex panach
Apace
han the. annk'qtic e x t e n n i o n phopextit i.( a n d a n l u i4 i t i~
can-
207
ON THE RANGE OF ANALYTIC FUNCTIONS
nected. By Theorem 5, any closed ball in a complex Banach space X has the analytic extension property. To prove the same for more general closed sets seems much more difficult. For examnle,does the closed shell
{x
E
X : 1/2 5 11x1 I 5 1)
have the analytic
extension pronerty? ACKNOWLEDGEMENT The author acknowledqes qratefully the supDort from the Boris Kidric Fund, Ljubljana, Yuqoslavia.
REFERENCES
[l]
R.M.ARON: The range of vector-valued holomornhic maminqs. Proc.Conf.Anal.Funct., Krakow 1974, Ann.Polon.Math. 33 (1976).
12J
.
R. M ARON ,J.GX%EWIK,M.
SC3-R l:
Interpolationby vector-valued
Analytic functions.Rendiconti diMat42) V.9,serie VI (1976). [3]
L.CARLESON: Internolations by bounded analytic
functions
and the corona problem. International Congress
of
Mathematicians, Stockholm 1962. [4]
-.-...-__.. .
Representations of continuous functions. Math.
Zeit. 6 6 (1957) 447-451. [5]
J.GLOBEVNIK: The Rudin-Carleson theorem for vector- valued functions. Proc. Am. Math. SOC. 53(1975).
. Analytic
[6]
functions whose range is dense
in
a
ball. Journ. Funct. Anal. 22(1976). [7!
I_--I_.
.
The range of vector-valued analytic functions.
To appear in Ark. far Math.
The range of
vector-
J . GLOBEVNIK
208
v a l u e d a n a l y t i c f u n c t i o n s 11. Ark. f o r M a t . 14(1976)
r 81
: A n a l y t i c e x t e n s i o n s o f vector - v a l u e d f u n c t i o n s .
P a c . J o u r n . Math. 6 3 ( 1 9 7 6 ) .
[
9 1 E.A.HEARD,
J.H.WELLS:
b r a s o f Ha.
[lo]
Y.HERVIER:
An i n t e r p o l a t i o n t h e o r e m f o r s u b a l g e
Pac.Journ.Math.
2 8 ( 1 9 6 9 ) 543-553.
On t h e W e i e r s t r a s s p r o b l e m i n Banach
P r o c . on I n f i n i t e D i m e n s i o n a l
spaces.
Lect.
Holomorphy.
Notes i n Math. 364, S p r i n g e r 1 9 7 4 , p p . 157-167.
[11] K.HOFFMAN: Banach s p a c e s o f a n a l y t i c f u n c t i o n s . P r e n t i c e
-
H a l l 1962.
[13]
J.L.KELLGY:
[ 131
D.ii.OBERLIN:
G e n e r a l t o p o l o g y . Van N o s t r a n d 1 9 5 5 . I n t e r p o l a t i o n and vector-valued
functions.
J o u r n . F u n c t . A n a l . 1 5 ( 1 9 7 4 ) 428-439.
[14]
P:.RUDIN:
Boundary v a l u e s o f c o n t i n u o u s a n a l y t i c f u n c t i o n s . P r o c . Amer. Math. SOC. 7 ( 1 9 5 6 ) 808-811.
[15] k7.RUDIN:
Real and conplex a n a l y s i s . McGraw H i l l 1966.
r-1 6 1- E.L.STOUT:
On some r e s t r i c t i o n a l q e b r a s . F u n c t i o n s A l g e b r a s .
S c o t t , Foresman 1 9 6 6 , pp. 6 - 1 1 .
ADDED I N PROOF:
B.
J o s e f s o n (Some r e m a r k s o n Banach v a l u e d p l y
n o m i a l s on co(A), t.o a p p e a r i n t h e s e P r o c . ) p r o v e d t h a t t h e above c o n j e c t u r e i s f a l s e by c o n s t r u c t i n g a c o u n t e r e x a m p l e i n X=co(A) w i t h A u n c o u n t a b l e . On t h e o t h e r h a n d t h e a u t h o r
(on t h e
ranqes
o f a n a l y t i c s maps i n i n f i n i t e d i m e n s i o n s , t o a p p e a r ) s h o w e d t h a t t h e a s s e r t i o n i n t h e c o n j e c t u r e i s t r u e f o r X = l P ( A ) (1 5 p < F u r t h e r , t h e a u t h o r (The r a n g e o f a n a l y t i c e x t e n s i o n s ,
.
t o appear
209
ON THE RANGE OF ANALYTIC FUNCTIONS in Pac. Journ.Math.) obtained partial solutions of Problem
3;
in particular, he gave a complete topological description o f t h e subsets of C having the analytic extension property.
INSTITUTE OF MATHEMATICS, PHYSICS
AND
MECHANICS
UNlVERSITY OF LJUBLJANA LJUBLJANA,
YUGOSLAVIA
This Page Intentionally Left Blank
Infinite Dimensional Holomorphy and Applications, Matos (ed.) 0 North-Holland Publishing Company, 1977
10-SPACES AND a-CONVEX SPACES
By E R I K GRUSELL
I n t h i s p a p e r w e g i v e a n example of a l o c a l l y convex top o l o g i c a l v e c t o r s p a c e E , s u c h t h a t E i s a n w-space,
but i s not
a-convex w i t h r e s p e c t t o t h e onen s e t s . However, E i s
a-convex
w i t h r e s p e c t t o t h e open and c o n n e c t e d pseudo-convex s e t s . ( F o r d e f i n i t i o n s see b e l o w ) . To p r o v e t h i s w e show t h a t E i s
not
a-convex w i t h r e s p e c t t o t h e open s e t s , and t h a t e v e r y p l u r i s u _ b harmonic (and i n f a c t a l s o e v e r y c o n t i n u o u s ) f u n c t i o n
in
E
depends o n l y on c o u n t a b l y many c o o r d i n a t e s . DEFINITION 1
[Dineen, N o v e r r a z , S c h o t t e n l o h e r 33 L e t
PCVX(F)
d e n o t e t h e f a m i l y o f open and c o n n e c t e d pseudo-convex
subsets
m
of t h e l o c a l l y convex s p a c e F. L e t a = { p j )
be a c o u n t a b l e j=1 s e t of c o n t i n u o u s seminormson F , and l e t Fa d e n o t e t h e space F
endowed w i t h t h e t o p o l o g y g e n e r a t e d by t h e seminormsin a . F c a l l e d a-convex w i t h fienpect t o t h e pneudoconvex domainn
is ( or
a-convex) i f : PCVX(F) = U PCVX(Fa) a
where t h e u n i o n i s t a k e n o v e r a l l c o u n t a b l e sets a of continuous seminorms on F . 211
E . GRUSELL
212
DEFINITION 2
L e t F , a and Fa be as i n D e f i n i t i o n 1,
OPEN(F) d e n o t e t h e f a m i l y o f open s u b s e t s of F .
F
let
and
is
called
a-convex w i t h t e h p e c t t o t h e open d e t ~i f : OPEN(F) = U OPEN(F,)
a
,
where t h e u n i o n h a s t h e same meaning as i n D e f i n i t i o n 1. DEFINITION 3
[Dineen 1, 21
and l e t @ ( U )
be t h e set o f complex v a l u e d a n a l y t i c
L e t F b e a l o c a l l y convex
space, functions
d e f i n e d on U , where U i s a n open and c o n n e c t e d s u b s e t o f F . T h i s means t h a t f
E
@(U)
i f f is c o n t i n u o u s and A
+
a n a l y t i c i n a neighbourhood o f z e r o f o r e v e r y x F i s c a l l e d a n w-Apace i f f o r e v e r y f E @ ( U )
f(x E
U
+
and y
where
open and c o n n e c t e d , t h e r e e x i s t s a s e q u e n c e { p i )
m
i=l o u s seminorms s u c h t h a t f is a l s o c o n t i n u o u s i n t h e
is
Xy)
U
E
C F
F.
is
of continutopoloqy
g e n e r a t e d by t h e s e q u e n c e {pi S i n c e OPEN(F) 3 PCVX(F) it i s t r i v i a l t h a t i f F is a-co"
vex w i t h r e s p e c t t o OPEN(F), t h e n F i s a l s o 6-convex w i t h
re-
s p e c t t o PCVX(F). I t is a l s o clear from t h e d e f i n i t i o r s t h a t
if
F is a-convex w i t h r e s p e c t t o OPEN(F), t h e n F is a n w-space.
The converses o f t h e s e i m p l i c a t i o n s are n o t true.We show t h i s by g i v i n g a n example o f a n u-space E , which i s n o t
a-con-
vex w i t h r e s p e c t t o OPEN(E), b u t w i t h r e s p e c t t o P C V X ( E 1 . L e t E = { ( x ~ ) ~ Clxr/ <
-1
on
and l e t t h e t o p o l o g y
E be d e f i n e d by t h e set of seminorms of t h e form
where A C IRis a c o u n t a b l e set. W e f i r s t show t h a t E a-convex w i t h r e s p e c t t o OPEN(E) and t h e n g i v e a lemma i m p l i e s t h a t E is a-convex w i t h r e s p e c t t o P CV X ( E ) , and t h a t E is a n w-space.
is
not
which also
213
w-SPACES AND a-CONVEX SPACES
@PEN(E) and
uGth hedpeCt t o
E
i6
an U-bpaCe.
To p r o v e t h a t E i s n o t a-convex w i t h respect t o
PROOF
not
E i b a - c o n v e x w i t h R e s p e c t t o PCVX(E) b u t
PROPOSITION
OPEN(E)
it is s u f f i c i e n t t o f i n d o n e o p e n s e t U C E s u c h t h a t UdOPEN(Ea) f o r any c o u n t a b l e s e t a o f c o n t i n u o u s seminorms on E . L e t ar = fr
+ k, k
E Z ) , for r E [O,l[
n
= I.
a , e a c h ar i s d e n u m e r a b l e , and ar as = r . F u r t h e r , f o r r E I l e t xL = ( x ; ) ~E E~, ~where
IR=
U
-
r E I
xr = t D e f i n e Ur = x
r
+
that A
n
9
if
1
if r =
t.
11, and
U =
f o r some a . Then U
U
Ur.Then UEOPEN(E1.
r €1
r E:
OPEN(EP) w h e r e P={pA)
U B . S i n c e A i s c o u n t a b l e , t h e r e is an ro PBE a
ar
0
=
r
/3.
x
E
r # s.
if
r # t
0
( x ; pa ( x ) <
Suppose U E OPEN(E,) and A =
{
Then
such
I
E
‘0 U by c o n s t r u c t i o n , b u t s i n c e pA(x
e v e r y pA-open neighbourhood o f xro c o n t a i n s z e r o . S i n c e
0
r
t h i s shows t h a t x U,
h a s no nA-open n e i q h b o u r h o o d c o n t a i n e d
so t h a t U c a n n o t be pA-openr which c o n t r a d i c t s t h e
t i o n . W e have t h u s shown t h a t U
OPEN(E,)
FO, U
,t!
in
assump-
f o r any a .
To p r o v e t h a t E i s a n w-space and t h a t E is
a-convex
w i t h respect t o P C V X ( E ) w e g i v e a lemma (Lemma 1), which
says
t h a t a plurisubharmonic f u n c t i o n i n E o n l y depends on a
coun-
t a b l e number o f c o o r d i n a t e s . T h i s shows i m m e d i a t e l y t h a t
E
a n w-space,
and a s a c o r o l l a r y it f o l l o w s t h a t E i s
is
a-convex
w i t h r e s p e c t t o PCVX(E). Lemma 2 s a y s t h a t t h e c o n t i n u o u s f u n c t i o n s on E a l s o h a v e t h i s p r o p e r t y . LEMMA 1
16 i
E
P S H ( U ) , wheae U i d an o p e n and c o n n e c t e d d u b b e t
0 4 E, t h e n f dependb onLi4 o n c o u n t a b t l r rnanr! v a t i a b t e b
in
U,
E. GRUSELL
214
i . c , thche
exioto
a
supp y c
==> f(x
+
couMtabte
A
bet
y ) = f(x)
c
1R
x
E
doh euefirr
ouch
that
u,
lohehe
supp y = {r; yr # 01.
Choose xo
PROOF
U so that f(xo)>
E
-
m.
Because
semicontinuous in xo, there is a countable set constant
E
> 0 such that pA(x-x0) <
If sunn y c [ A and pA(x-x0) < for every X E @, and since X
E
+
E
==> x
x,,
+
SUPD
yc
1J such that n ( x l - x O ) >
h(xl-xO) E U for
1x1
and
A C R
a
U and f(x)
f(x+Xy) is a suhharmonic functbn
f (x+y) = f (x) if pA(x-xo) < E and F
upper
this means that f(x+Xy)
it cannot be bounded without reducinq to a
exists a point x1
E
f is
c
A.
E~
constant.
Thus
Assume that tAere and
such that
< a for some a > 1. Let K > 1
be
a
constant such that f (x,) < f (x,) + K. The set V=fx;f ( x )
is a nseudoconvex set, so that = inf
%(x,y) in V
X
-+
x
(1x1;
x+Xy ,k V}, is a nlurisubharmonic
If we choose y with
(E\{O}).
b,
y c
then
in a neiqhbourhood of zero. It must be enual to
-m
everywhere so that -109 dv(xl,y) = +
sunp
function
-107 dV(xO+ X(xl-xo),y) is a subharmonic function which
equal to
X
where
f(xl
+
Xy) is bounded by f(xo)
-m,
which
means
is -m
that
+ K and therefore constant.
Since U is connected, every point in U can be reached
in
this
manner in a finite number of stens, startinq from xo.It is thus proved that f(x
+-
-7)
= f ( x ) if x
E
U and s u m y c
c..
This lemma canalso benmved h a vav somewhat similar the way in which Lenxta 2 is proved, but the proof Tiven
to
above,
which was suqgested by C.O. Kiselman, is simnler. COROLLARY E PROOF
If U
i b E
6 - c o n u c x w i t h h e o p e c t t o PCVX(E).
PCVX(E) , then -log dU(x,y) is a
nlurisubharmonic
U-SPACES AND a-CONVEX SPACES function in U
X
215
where du(x,y) = in€ { 1 x 1 ; x + X y
(E \iO))
For the m a c e E defined above, it holds that E
x
U).
is topoloql
E
cally homomornhic to E . Thus we can apnlv Lemma 1 also to space E
the
E so that -107 dU(x,y) denends only on countably many
x
coordinates. But then there exists a continuous seminorm pA E
E, so that -1oa dU(x,y) is pA-continuous. Hence
x
m
PB-o?en, if R x B 3 A, since
U =
u
u
{X
k = l ~EE\{O}
on
must be
U
u;-log d u ~ , y ~ k ~
It is interesting to note that the lemma remains true if
"f
E
PSH(U)
is replaced by "f is continuous in U". We
"
state
this as a separate lemma, because the proof is different. LE?W 2
14 f
i 6
a c o n f i n u o u n , compLex uaPued ( u n c t i o n i n U , a n d
an open n u b n e t
U i b
04
E, t h e n f d e p e n d n onL!r o n c o u n - t a b t r l manil
v a h i a b e e d i n U. PROOF
Let xo E U . Let A
c 37 be
constant such that V = (x;
P
A
a countable set and E > 0
(x-x
0
)
< E ) C U. For every
there is a countable set Aa 2 A and a constant
fir
E
such that pA (XI-x0 ) < A ==> If(x')-f(xo) ' < &.Let Ax = a
Then supp y c [Ax
==>
f(xo+ y ) = f(xo). Let R be a
0
he a a > 0
> 6 > 0, m
U
k = l Al,k.
countable
0
set, I3 C I R . Then there exists a countable set R 1 3 B such that
x
E
U,
supp x C B and s u ~ py c [Bl
.
==> f (x+y) = f (x) To
con-
struct B1, let EB = {x E E; sunp x c I3). Then E B is tonoloaical ly homomornhic to k?,I and thus is separable. Let {x.ImCEB n U 3 1 be dense in the closure of E B f l U, and for every x let C . C IR j 7 be a countable set such that SUPP y c ==> f(x + y) = f(Xj). Then B1 = sun? y c
c
m
C has the desired j=1 j R1 then U
kj
nronerties,
j
because
if
E. GRUSELL
216
for some subsequence fxkj) of the sequence fx.Im
3 1'
Let B1 be the emptv set and define Bj+l so that x
E
U, supp x C R
m
a
and supp y c
j
some j and such that (f(x)
-
implies sunp y c
If(x + Y)
-
f(x) I
P
U with supp x a C B
j
B
Cj
'C [B, f (x+y) =
E
for
so that f (xa + y ) = f (xa). Thus
5 If(x + y)- f(x,+y) I+lf(xa)-f(x) I <
Since for every x XI
E
==> f (x+Y) = f (XI.
f (x,) I < a and If (x+y)-f (xa+V)\ < a .
Since a was arbitrary this shows f(x
SUP?
n
-
for every a > 0 there is an element xa
supp y c
pj+l
U, x = x'
+ +
2a
.
y ) = f(x).
x'
where sunn x' C B,
f (x) whenever x E U and supp
yc
c
B.
REFERENCES
[l]
Dineen,
S.,
Holomornhicallv comnlete locally convex vector
spaces, Szm. P. Lelonq, 1971/72, l e c t u h e
noted
i n
m a t h e m a t i c d 332 (1973), 77-111. [2]
Dineen,
S.,
Surjective limits of locally convex spaces and
their application to infinite dimensional holomomhy. To appear in J o u ) ~ . Math. [3]
PUhtb
e t Appl.
Dineen, S., Noverraz, Ph., and Schottenloher, M., Le
pro-
blgme de Levi dans certains espaces vectoriels topologiaues localement convexes. Bull. SOC. Math.France 104 (1976).
ERIC GRUSELL Dept. of MATHEMATICS SYSSLOMANSGATAN 8 S - 752 23 UPPSALA SWEDEN
Infinite Dimensional Holomorphy and Applications, Matos (ed.) @ North-Holland Publishing Company, 1977
SOME PROPERTIES OF THE IMAGES OF ANhLYTIC MAPS
T h r o u T h o u t t h e n a n e r : X d e n o t e s a complex Banach
letters such as
U
o r V , open s e t s i n X; & ( U , V )
( F r g c h e t - 1 a n a l y t i c maps o f U i n t o V;
o n e mans f o f U o n t o V s u c h t h a t f
space;
t h e class
of
8 ( ( U , V ) t h e c l a s s of o n e -
a n d f-'
E &(U,V)
E &(V,U);
f i n a l l y d d e n o t e s t h e d i s t a n c e i n X a s s o c i a t e d w i t h t h e norm.
1. A NEW PROOF OF EARLE-HAMILTON'S
[I] THEOREM, w h i c h
reads
as follows. THEOREM 1
Let
U
be open bounded i n X and f
E &(U,U)
t h a t t h e image f ( U ) P i E d a t a p O b i t i V e d i n t o n c e . ( t o m [U:
hub a unique 4 i x e . d p o i n t i n PROOF
(1.1)
L e t CP = { @ E & ( U , C )
Auch
then f
U.
< 1); f o r a E U, b E X,
:
a ( a ; b ) = s u p l @ ' ( a ) . b )1
set
.
@EQ
T h i s supremum i s f i n i t e : i n f a c t , f o r a n y C$ E a n a l y t i c f u n c t i o n of t E @,
be
It 1 < d ( a , [U)/l
lbl
+ I,
,
$(a+tb) is an
whose
deriva-
t i v e f o r t = 0 i s C $ ' ( a ) . b , hence (1.2)
ol(a;b)
5
On t h e o t h e r h a n d , i f b
l[b\l/d(a,~).
# 0 t h e same supremum i s p o s i t i v e : 217
in
218
M. HERVE
f a c t , i f u E X u ( t h e a d j o i n t space t o X) w i t h = I(u1
Ixl
= 1 / D i a m U r then @ = u
-
lu(b)
1/1
/bl
Ix
=
u ( a ) E 0, hence a ( a ; b ) ? - ( u ( b ) l
and
(1.3)
a(a;b)
r
.
we g e t
cllf ( a ) : f ' ( a ) .b]
(1.4)
U
t h e n 19 o f : @ E 6 ) 6 0: S i n c e ( O o f ) ' ( a ) . b =
If f E A ( U , U ) ,
= 4 ' [ f ( a ) ] . [ f ' ( a ) .b]
>- 1 ( b l l/Diam
5
a(a:b)
.
Now w e u s e t h e assumption made on f ( U ) i n t h e same way a s EARLE HAMILTON d i d , namely: s i n c e U i s bounded, t h e r e e x i s t s
~-r >
0
such t h a t
L e t f o = i d , f n = f o fn-l: f f ; ( a ).b = f
'
[fn-l ( a )
1. [fA-l
since
( a ).b]
,
t h e i t e r a t i o n of
(1.6)
yields
(1.7)
u [ f n ( a ) ; f;(a).b]
5
a(a;b) --( l + v ) ~ - Y a E U, b E X ,
n E N.
P u t t i n g ( 1 . 2 ) , ( 1 . 3 ) and ( 1 . 7 ) t o q e t h e r , w e q e t Diam
If
U 3 S = {a t t
V a
E
U, b E X , n E N.
(b-a): 0 < t ._ < 11, from ( 1 . 8 ) f o l l o w s
and f o r a r b i t r a r y a , b E U w e have
IMAGES OF ANALYTIC MAPS
219
With h = f(a), from (1.10) follows that (fn(affnEN is a Cauchy sequence; with arbitrary a and b, that 1 = lim fn(a)does not depend on a. By the assumntion made on f ( U ) , 1 E U and 1 is a unicrue fixed point of f.
2. THE DEPENDENCE OF THE FIXED P O I N T ON A'PARAMETER open bounded in X and, for each t
E
T, let ft
Let U
E &(U,U)
be such
that ft(U) lies at a nositive distance (denendin9 on t) [U:
be
from
then ft has a unicye fixed noint 1 (t) E U. L e t T b e n RopoPaqicat
PROPOSITION 2.1
point a
E
E
U has n b i e i q h b o t h o o d A s u c h that ft(x)+ f
unir(Okmk!!/ ,(Oh X E A , t h e n 1 (t)
PROOF
s p n c e , to
+
1 ( t o )a h t
T: i4 each ( x ) u s t+tO
to.
+
Take a = 1 (to) in the proof of Theorem 1: since
map X 3 b+j@'(a).b,
@ E @,
is linear, the map b * n(a;b)
each de-
fined by (1.1) is a seminorm; by (1.2) and (1.3) it is a
norm
equivalent to the given one, and we mav assume it is the
given
one. Then, if p o > 0 satisfies (1.5) for ft
, (1.6) means that
0
since ft
is continuously differentiable, A in the
assumption
0
of 2.1 may be chosen as an onen ball with center a, such that
1 I f;(XI I ! 2 0
(A)
+
~-A;'
A,
Ca
+
r+-j-=-Jz . V x E A ;
AA and, for t sufficiently near to, ft(A)Ca+
with the consequences:
a) ftln
has a fixed point, namely l(t);
220
M.
HERm
b) (1.9) can be written for ftin with b = ft(a)
a
and
number p depending only on A , therefore l(t) is the uniform limit of the sequence bn(t) defined by (2.2) P R O P O S I T I O N 2.3
id
t w ft(x)
i d
Let
T
b e an o p e n A e t i n n nohmed tine.alr Apace:
analytic
304
each x
E
U, t h e n t-
l(t) i . a~n a -
lytic. PROOF
Since the analytic maps t
they are equicontinuous ( [ 4 ] ,
.+
ft(x) are bounded toqether,
III.2.2), which entails
the assumption of 2.1 holds, and that (t,x)-
that
ft(x) is a contin
uous map of T x U into U. Since this continuous map is separately analytic, it is analytic, the composed maps bn defined by (2.2) are analytic, and so is the mar, 1 by the last statement in
the
proof of 2.1. 3. SOME E X T E N S I O N S O F THE THEOREM
Given a real Banach
let X = X 0 + i Xo be the complexified space, Banach space with the norm Xo,
a
space complex
which satisfies
THEOREM 3 . 2 f E &(U,U)
b: t h e n PROOF so
L e t V be. open bounded i n X o , U = V + iX 0 ' be buch t h a t f (U) !Led
a.t a p o d i . t i u e d i b t a n c e
f has a unique. X i x e d p o i n t i n
nnd (horn
u.
In the proof of Th. 1,(1.2) and (1.4) remain unaltered:
does (1.5) since, by (3.11, the nrojection of f ( U ) into
lies at a nositive distance from [V; thus we only have to
V find
IMAGES OF ANALYTIC MAPS
221
a substitute for (1.3).
1 = Iv(xo)- v(ao) I (1.3) , (1.8) , (1.9) , Diam
6 =($/2-$)€
jRe $(x)
< 1, therefore
in
U is replaced by 8a.Diam V.
REMARK 3.3
@,$'(a)=::
4
In the statement of Th. 1, the boundedness of f ( U )
may be assumed instead of the boundedness o f U : in fact,
if
ed subset of U , and f ( U ' ) lies at a positive distance from [U'. Similarly, in the statement of Th. 3.2, the boundedness of projection of f ( U ) into V may be assumed instead of the
the
bound-
edness of V. REMARK 3.4
In the classical case when X has a finite
dimen-
sion, a proof of Th. 1, under the corresponding assumption that f(U)
is relatively comnact in U , can be found in [3], chap. IV,
1; Th. 3.2 does not seem to have been stated before. PROPOSITION 3.5 UW'
had
Let
net
{c
f
6 1 ( u 1 , u ) be
E
W:
a+ gb
containing 0
&.,(t~,u),
PROOF
U\rrt)
E
nuch
R i v e d i n t n n c e (&om E
be open n e t n i n X n u c h t h a t : U a u ' ;
an e m p t r r i n t e h i o h ; doh a n y a
connected component E
U and U '
E U,
04 fr;
E
b
E
C: a
X, i d
+
II)
yb E U )
i o t i t h e 4 p o t a h oh w i t n e t { .
t h a t f ( u ' ) i n bounded and l i e n a t a
II
U:
t h e n f c a n be c o n t i n u e d
w h i c h had n u n i q u e { i x e d
into a
i n the
,
the
Lc t poni-
map
p o i n t i n U.
By the argument used in Remark 3.3, U may be
assumed
222
HERVE
M.
bounded: then Th. 6.9 in 151 proves the existence of a continua tion
E
&(U,U),
for which Th. 1 above holds.
4. ONE-ONE RESTRICTIONS TO OPEN BALLS WITH A GIVEN CENTER B = {x
E
F=If
X: 11x1 1 < 1) and
E
&(B,X):f(O)=O,f'(O)
The classical SCHCVARZ lemma for the
PROBLEM 4.1 f
i n t o a map map
function
B
0 (Ufl pB). Note t h a t SCH(QAR2'n Pemma h o t t h e N i e t d d P I 1x1 151 If (x)I I v x E uf and P 5 1.
in-
bf1hde.t
Uf 3 0
E
Find
a hadiud r d e p e n d i - n q o n f i r
o(rB, Vf) 404 e a c h f
The b e d t w a h e 1
1
-
p
Let qy(x) = x
-
f (x)+y: if
Phoblem 4 . 1 i d
p =
a positive distance from
o n 1.1 duch t h a t qrBE
and d o m e open d e t Vf i n X.
E $(p)
THEOREM 4.3 (partly in [2]).
PROOF
scalar
id);
04
PROBLEM 4.2 E
=
d e p e n d i n o o n l i r o n p duch t h a t each
p
can b e h e d t h i c t e d t o d o m e o p e n
E %(p)
wehde
F i n d a hadiun
Let
i X p
53 ,
04
p
anbwehing 1
p = 1/4p
i 4
p
-
p,
'ly(B) lies at
I IyI 1
< 1
0,and by Prop. 2.3
2.
the uniaue
fixed
point of q
in B depends analytically on y, therefore we answer Y -1 Problem 4.1 with p = 1 - p , Uf = B flf ( p B ) ; if r < 1 and 11yI ! < r - vr 2 , q (rB) lies at a positive distance from [(rB) ,
Y
therefore we answer Problem 4.1 with and may take r = 1/2p if p 2
p =
r - pr21Uf=(rB)flf-1bB),
I. 1
In order to show that the result is starp, choose with (lal(X= 1, u
E
XI with u(a)
IIuIIx, = 1, and
=
the example (4.4)
f(x) = x
-
pa u 2 (x)
:
a
E
X
consider
ANALYTIC MAPS
or x 1Zl;i.f
223
-
a = pa u ( x - a ) u ( x + a )
u-:T
1
,
im-
implies
f ( x ) = a/4p
-
a u , f ' ( x ) . a = 0 , f ' ( x ) non i n v e r t i b l e .
3 1,a
r a d i u s r a n s w e r i n q Problem 4.2 c a n n o t
u(x) = 1/2p,
fl(x) = id
REMARK 4.5
If p
exceed 1 / 2 p .
I n f a c t , with (4.4)
-
w e have f 1 ( a / 2 p ) = i d
au
non
invertible. THEOREM 4 . 6
1 L e t pzz
:
t h e n t h e b e n t vaRue
4 . 2 i n a.!L Banach b p a c e b X
Y f
e
,
r anbwelrinq PlrobLm
and
r = 1/4p,
f (B/4p) 3
~
3
1
B
Set g(x) = f ( x )
-
x , t a k e x and y
E
1 TB: y'(x). y
is
111.1.1 and 1 1 1 . 1 . 3 )
t h e mean value f o r 0_<_0_<_2n [ d e n o t e d
Now x E B/4p
I 19' ( x ) I
implies
t i b l e ; x and y E B / 4 P r
x #
yr
imply
1<1,f
I
(x) = I d +
g' (x)inver
1I~(X)-~(Y)(I
by t h e c l a s s i c a l i n v e r s i o n t h e o r e m f o r c o n t i n u o u s l y d i f f e r e n t i a b l e maps, f
IB/ 4 u
h a s a n open image Vf and a n a n a l y t i c
inverse
map. If (4.9)
-
&).
PROOF
([4]
in
04
y E -B,
3 161.1
xo -
t h e r e c u r s i o n formula Yr
xn = Y - 9 ( x n - l )
d e f i n e s a s e q u e n c e o f p o i n t s xn E B / 4 p ,
f o r it i s m a j o r i z e d
t h e r e c u r s i o n formula
t o = 3/16u,
tn = (3/16p)
2 + utn-l
.
by
M. HERVE
224
The more precise ineaualities
1 Ixn1 151 lyl I +
prove the existence of x = lim xn
2
IyI I+
utn-l(l
and f(x) = y.
8 B/4uf
As an example of a Banach space where the results
are
sharpf we choose X = C2 with the following notation. The 1 (Ixl-ix21+lxl+ix2\) of x = (xl,x2) E x is 11x1 I = 2 r f x
=
Y1+iY2
-
settinq
-
-isr/4
p
2
2
(x1+x2), y2 = x2
x1+iXZf y1 -iy2 = 11 =
c (XlfX2) E flu
we have
I' -
-
2
I
1
llJS(1-i) (Xl+iX2) (x1-lx2) i J
c 2 : Xl+iX2 # l/lJfi(l-i) I f
1x1 1<1, p is the upper bound of =
1
-
1 If (x) -
[(l-i)xl +
vanishes at some point x such that 11x1 = (l+i)x2 = 1/(2pfi); finally E
2
E d(uru).
J~ (x)
y
maps
einl4u (x1+x2)
2 2 Since 1 is the uDper bound of !xl+x2 I = lxl-ix2
for
norm
(xlfx2)onto y = (ylfy2)such that y1 = x1
or
1/16u
1
1 If-l(y) 1 1
XI 1
1
(xl+ix2I
:
(l+i)x2]
= 1/4pfnamely (l-i)xl=
= 1/4~1 for some
U such that I lyll = 3/16p, namely yl-iy2 =
1/8Ur
Doint
y1+iy2 =
= eiTi4/4p.
5. THE CASE OF HILRERT SPACES THEOREM 5.1
L e t X br a H i P b e h t n p a c s a n d u _ > .1: t h e n t h e b e n t
v a f u e 0 4 r a n n w e h i n p PRobeem 4 . 2 i n r = 1/2p, and f ( B / 2 p ) S B / 4 p
v
f E
PROOF
%5u). Since X is a Hilhert space, we have the special formula
IMAGES OF ANALYTIC MAPS
225
ie 2 2 IIx+e Y I + Ilx-eieyl12 = ~ ( I I X+lly112)l I ie 2 which turns the end of ( 4 7) into MV Ix+e yI I = 1x1 12+1 IyI I 2 1 B. So the 2111 1x1 I V x E 7 and ( 4 . 8 ) into I / g ' ( x )11-1. inversion theorem can be used for
If y
E B/4p
,
f
IB/2p
instead of f
JB/~IJ
( 4 . 9 ) is majorized by the recursion formula
with the same consequences as in the proof of Th. 4 . 6 .
Finally
the results are sharp by Example ( 4 . 4 1 , as it was pointed
out
in Remark 4 . 5 . The d i m p l e example X = @,
REMARK 5.2 f'(l/fi)
f(x)
=
x
-
px 3 ,
whene
= 0, b h o w 6 t h a t a lradiun r a n A w e R i n g Plrobtem 4 . 2
, and
H i l b e l r t 4paCed c a n n o t e x c e e d 1/,&
1/2p id p
3/4.
in
t h i n i n nmallefi
than
T h e n a b u b n t i t u t e { O R 1/2p has t o b c {ound
i{
p < 1.
2 1 If 2 5 IIxII < 1, theirqudit;l 119'(x).Yl~(v(IlxI12+11ul~ 1
proceeding from ( 4 . 7 ) may be written for
I IyI
I<
1 - 11x1
I
only,
and therefore only yieli,:
-
p
(
m
-
211x11)
1 so the radius r = (2v
4v
-
1
+
V
x
E
B :
/1+4v-4p2),
1 as 1/2p and decreases from 1 to 7
p
which is smallertha
increases from 0 to 1, can
be substituted for 1/2p in Th. 5.1, but the author does know if this value of r is sharp.
not
M. HERVE
226
6. THE CASE OF HAUSFORFF, LOCALLY CONVEX, SEQUENTIALLY COMPLETE SPACES
-
As an analogue of closed balls in such a space E l let
be bounded, closed, balanced and convex, let X be the subspace of E spanned by E l q the gauge of inf {A >
o
Since
: x E
AEI, x
E
linear
in X:
q(x)
X.
-
is closed, B = ix
E
B
X : q(x) 5 1); since
B
bounded, any continuous seminorm p on E satisfies p 5 cq
=
is on
for some constant c; since E is Hausdorff, q is a norm on
X XI
which from now on defines the topology of X. This is finer than the topology induced on X by the given one, and the
following
example shows that it may be actually finer. EXAMPLE 6.1
Let w be the open unit disc in C, E convergence,E = (x
the usual topology of -act
then X = Ix E E : x bounded}, with q(x)
E
=&(W,C)
E: Ix(c]< -1W 5
sup (x(<)1 if x
=
with E w}: E
X.
SEw
PROPOSITION 6.2
X
i d
a l3anach s p a c e .
Let (xn) be a Cauchy sequence in X for the norm
PROOF
x = lim xn in E l B = sup q(xn) : since
x
-
BB; now let n and n'
E
xn
-
$fi
THEOREM 6.3
Let B =
{x
is closed in
2 no imply q(xn-xn,) 5
EE is closed in El n 2 no implies x E
q l
E
E
:
Er since
xn- & or q(xn-x)2E.
X : q(x) < 11 and g be a
GZteaux-
a n a l y t i c m a p 0 6 B i n t o E, w h i c h o n t y mean6 t h e e x i d t e n c e i n 06
lim 1 [g:ga+tb)-g(a)] t-+O
v a
numbet p , t h e n g E &(B;X)
E
B, b E
x
:
if g(B)
with the topology
c VB
0 6 X.
E dome
IMAGES OF ANALYTIC MAPS
227
-
Let q(a) + q(b) < 1; since the closed convex set PB -in8 contains e g(a+eieb) W 8 E lR , it also contains, with the no-
PROOF
tation used in [4], 111.1.3,
and in the proof of Th. 4.6 above:
.
gnla:b: = MV [e-in8g(a+eieb)] Now the expansion g(a) + c tngn(a;b) converges in E to g(a+tb)
for It1 < 1; since the 2 n t g2(a;b)+ t gn(a;b) f o r each n 1. 2, it
...+
-
g(a+tb)
g(a)
-
also
'*.
contains
t (rl(a;b), which yields the inequalities
q[g(a+tb)t
-
(a)
showing that gl(a;b) = lim t*O q[g(a+b)
-
-
1
gl(a;b)] < 1- t [g(a+tb)-g(a)]
r
in X, and
vq (b) g(a)] < l-q(a)-q(b)
proving that q is continuous, for the topology of X, at
the
point a .
L e t f b e a G a t e a u x - a v i a l y f i c m a p 0 6 B into
COROLLARY 6.4
E,
w i . t h t h e p m p e h t i e s : f(0) = 0; lim f(tx)/t = x in E d ' x E X; t-r0 I f(x) - x E pB W x E B; p 2 7. T h e n , W i t h t h e t ~ p ~ l o~f i g y X r f/B,4P
E
@(B/4p,Vf)
EXAMPLE 6.5
(OZ
some o p e n s u b s e t
Vf
04
X,
containing
Let w be a bounded simply connected open set inC;
assume that, f o r some real number 6, any two points in w can be linked by a path in w whose length does not exceed 6. Then, for 2 any 4 E (w,C) with < 3/(16 6 1 , the unique solution van-
4
ishing at a given point a E tion u'
+ u2
=
I$
w
of the RICCATI differential equa-
is analytic on
w,
with [uI < 1/46, and the map
M.
225
4-u
HERVE
is analytic, with the topology of uniform
on X = {x PROOF
E cf?..(w,C)
:
convergence
x bounded).
Let E = &(u,Q:) with the usual topology of compact con-
-=
vergence, B
{x E E
:
)x(c)/ 5 1 / 2 S 2 W 5
fine g(x) as the function w
E LO); for
x E E, de-
5
3
5-
[Jax(~)drI2 (where the inte
gral is taken along any path in w with a length 5 6 ) , which has 1 and on the other hand a modulus 1/4S2 V x E B. Thus g (B) ZB,
c
g
E
d,(E,E).
3 1 : for any 6 E gBI By Corollary 6 . 4 with 1.1 = 7 the equation 1 x + g(x) = 4 has a unique solution x E TB, depending analytical
ly on $ for the topology of X; u(C)
=
J:
X(T)
dT
is the
re-
quired solution of the differential equation, and 1x(<)I <1/462 W 5 E w
implies lu(~,)l < 1/46 V 5 E w .
BIBLIOGRAPHY
[ l] C.EARLE and R, HAMILTON-A fixed pint theorem for holmrphic mappings (Proc. of Symposia in pure Math., XVI,A.M.S., 1970). [ 2 ] L.HARRIS
-
On the size of balls covered by analytic
trans-
formations (Preprint, University of Kentucky). 131 M.HERVE
-
Several complex variiables, local theory
(Oxford
University press, 1963). L4] M.HERVE - Analytic and plurisubharmonic functions in finite and infinite dimensional spaces (Lecture notes Math., 198,1971).
in
229
IMAGES O F ANALYTIC MAPS
[53 M.HERVE
-
Lindeltif's principle in infinite dimensions (Lec-
ture notes in Math.,
3 6 4 , 1974, p. 4 1 - 5 7 ) .
UNIVERSITE P I E R R E E T MARIE C U R I E
4 , PLACE J U S S I E N , P A R I S 5 et ECOLE NORMALE SUPERIEURE ,
4 5 r u e d'ULM, FRANCE
PARIS 5 ,
This Page Intentionally Left Blank
Infinite Dimensional Holomorphy and Applications, Mntos (ed.) @ North-Holland Publishing Company, 1977.
SOME REMARKS OM BAIJACII VALUED POLYNOMIALS ON
Co (A)
By BENGT J O S E F S O N
1. J. G l o b e v n i k 151
*
h a s proved that for every open ccolnected set
a s e p a r a b l e complex Banach s p a c e E t h e r e e x i s t s a f u n c t i o n f from t h e open u n i t b a l l B i n
u
holomorphic
a!, t h e complex l i n e ,
t o U such t h a t f (B) i s d e n s e i n U . I n [5]
in
even s h a r p e r
i"
results
are o b t a i n e d and i n t h e case U i s a b a l l t h i s i s a l s o p r o v e d i n
pi.
I n t h i s n o t e w e s h a l l show t h a t t h i s r e s u l t c a n n o t be gen-
eralized t o a r b i t r a r y , non-separable
Banach s p a c e s .More p r e c i s e
l y we s h a l l prove, P r o p o s i t i o n 2 , t h a t e v e r y
holomorphic func-
t i o n from a n open c o n n e c t e d set i n C o ( A )
EP(r),
s e p a r a b l e r a n g e f o r a l l i n d e x sets A and
into l'.
This
P > 0, has
proposition
follows more o r less o b v i o u s l y from r e s u l t s d u e t o A . P e l c z y n s k i and Z . Semadeni. We a l s o show, Theorem 3 , t h a t t h e r e open s e t U C C o ( A )
(*)
and a c o n s t a n t
E
e x i s t s an
> 0 s o t h a t , when A i s
un-
S u p p o r t e d by t h e S w e d i s h N a t u r a l S c i e n c e R e s e a r c h c o u n c i l c o n t r a c t N Q F 3435-004
and D u b l i n I n s t i t u t e f o r Advanced
Studies. 231
232
JOSEFSON
B.
c o u n t a b l e , e v e r y h o l o m o r p h i c f u n c t i o n f from t h e open u n i t b a l l
Bo i n C o ( A ) i n t o U
+
B
E
0
dense i n
h a s n o t a r a n g e which i s
U. c E b e a n open s e t o f a Banach s p a c e E and l e t
Let U
NOTATION:
F b e a Banach s p a c e . Then
z ( U , F ) d e n o t e s t h e s e t of a l l holo-
morphic f u n c t i o n s from U i n t o F , P("E,F)
i s t h e s e t o f a l l con-
t i n u o u s , n-homogeneous p o l y n o m i a l s on E i n t o F and P ( E ) d e n o t e s
all complex v a l u e d , c o n t i n u o u s p o l y n o m i a l s o n E .
We
note t h a t
i f U i s b a l a n c e d t h e n f E g ( U , F ) h a s a T a y l o r series e x p a n s i o n m
f ( z ) = C P n ( z ) where Pn E P ( " E , F ) . S e e [8_1 f o r d e t a i l s . C o ( A ) 0 i s t h e Banach s p a c e o f a l l complex v a l u e d f u n c t i o n s o n t h e i n d e x s e t A which a r e a r b i t r a r i l y s m a l l o u t s i d e f i n i t e A.
s u b s e t s of
P F i n a l l y 11 (A) d e n o t e s t h e Banach s p a c e
II P ( A ) = {Z = (Za } ~ E A ; Z / Z a I p < m].If V C A and Z = { Z 1 a ~ E CAo ( A ~) P o r 11 (A) s u p p Z = { u E A: Za # 0) and P r o j [vlZ={Zi}aEA where Zi = Za if u
E V and 2;
= 0 elsewhere. W e n o t e t h a t supp Z
al-
ways i s c o u n t a b l e .
2.
r e c a l l t h e f o l l o w i n g r e s u l t which
We
P e l c z y n s k i and Z . Semadeni. S e e [7] THEOREM 1:
due
is
to
[8].
16 F i b a B a n a c h bpace w h i c h d o e b n o t c o n t a i n
t h e n doh euehy n
E
.
A
N, P("Co(A) , F ) = Pk(nCo(A) , F ) iuhe4e Pk
n o t e d t h e p o t y n o m i a t b w h i c h map4 t h e u n i f b a t e
06
Co(A)
Co,
deinto a
h e e a t i u e e y c o m p a c t b e t ofi F.
W e n e e d also t h e f o l l o w i n g r e s u l t o f R . A r o n l see [I!. THEOREM 2 .
iuhe4e
r)
P("Co(A)) = t h e c t o b e d b p a n 06 t h e cottection
E II1(A)
t h e duaL bpace ofi
Co(A)
.
P
BANACH VALUED POLYNOMIALS REMARK:
233
Theorem 2 implies that every function
f
E %(Bold:),
where Bo is the open unit ball in Co(A), can be approximated on every strictly smaller ball by polynomials on Co(A) with finite spectrum. The spectrum Spf, of a function € on Co(A) is the intersection of all subsets
S
of A such that f factors through
and we will denote with PF(Co(A))
S
the continuous polynomials on
Co(A) which have finite spectrum. We note that a function is de termined by its spectrum. If
P E P("Co(A) ,C) then P can be written
. ..
r1 , ,r aall.. . , a n n
E
c.
In [3] and [7] the followina is proved P(z) PROPOSITION 2.
~ v e f q E
8 ( n o , t P ( r ) ) , whehe P
b i t h a h y , 6actohd t h h o u g h a c o u n t a b l e n u b n e t
PROOF:
06
>
o
and
r
I.
L A ah
A.
It is enouqh to prove the proposition for every P E P("C~(A),
aP(r))
because of the Taylor series expansion.
P = (P 1 where Y YEr P E P("Co(A) ,C) According to Theorem 1 P (Bo) is a relatively Y compact set in I? P ( r ) since it is wellknown that Co is not con-
.
tained in R
P
(r).
Rut it is also vellknown that every relatively P compact set of R ( r ) has its support on a countable subset of r.
Hence there is a countable set Y e hjljE*'
such that P Z 0 if 'Yj 1j EN Y But then the proposition follows from proposition
1, which gives that each P factors throuah a countable setland Y from the fact that a countable union of countable sets is acoun table set.
Q.E.D.
B . JOSEFSON
234
REMARK 1.
Theorems 1 , 2 and P r o p o s i t i o n 2 a r e a l s o t r u e i f
r e p l a c e Co(A)
we
by C ( K ) , t h e Banach s p a c e o f c o n t i n u o u s f u n c t i o n s
on a d i s p e r s e d , compact Hausdorff s p a c e . For e v e r y P E P("Co(A) ,II
REMARK 2 .
1
(r))
esti-
the coefficient
mate c o r r e s p o n d i n g t o P r o p o s i t i o n 1 h o l d s which c a n be seenfrom
Q(t,z) =
the f a c t t h a t
C
yEr
t
Y
P (2) can be r e g a r d e d a s a n+l-
Y
r)
homogeneous polynomial on C o ( A 0 where P = {P 1
i n Co(A)
If
P
t = it 1 E Co(l'). Y YEr P E ( U , I I ( r ) where U i s a n open c o n n e c t e d
1
and
Y YEr
REMARK 3 .
i n t o 6 such t h a t \ \ Q I I =
f E
( o r C ( K ) ) it f o l l o w s from t h e p r o o f of P r o p o s i t i o n
t h a t f h a s a s e p a r a b l e r a n g e b e c a u s e o n l y c o u n t a b l y many f
Y
set 2 !j 0
( f y = P r o j L y l f ) . On t h e o t h e r hand a m o d i f i c a t i o n o f Hirschowitz example [ 6 ] shows t h a t t h e r e e x i s t a n open, c o n n e c t e d and bound e d set U C C o ( A )
and a f u n c t i o n f E g ( U , 6 )
such t h a t f depends
on a l l v a r i a b l e s . L e t ea b e t h e a - t h u n i t vector i n C o ( A ) .
where Bg i s t h e u n i t b a l l i n
(II.
Put U =
Ua
Put
U
i s an
open
CI EA
bounded c o n n e c t e d s e t i n C o ( A ) . b a l l i n Co(A)
and
E
F i n a l l y l e t Bo b e t h e open u n i t
Bo be t h e open b a l l w i t h r a d i u s
E.
The s e t
U w a s s u g g e s t e d t o m e by R. Aron.
THEOREM 3 .
Let f
buch t h a t f ( B o ) U t 1/10
PROOF:
E
$G(BolCo(A) 1,
in denne i n
whehe A i b u n c o u n t a b L e ,
U. T h e n f ( B o )
i b
n o t contained i n
Bo.
Let f E S(Bo,Co(A))
have a r a n g e which i s d e n s e i n
There e x i s t a c o n s t a n t k > 0 and an u n c o u n t a b l e s e t H C A that
be
U. such
235
BANACH VALUED POLYNOMIALS
This follows becausc if we assume it is false there existsl for cvery sequence ( k . ) of positive numbers tendin? to zerol a se~j
quence
(ii )
z
of countable subsets of h such that
j j I f a (a)I 5 1/2 sup E (l-k.)Eo
+
a E P\H
1/10 if
j
'
3
Hence
m
which contradicts the assumption that f(I30) is dense in U since
x
*
ec1
x
if
E IJ
E
for all
Bq
a
E A.
From Theorem 2 it follows that we can take,for every a E P ,
a
such that
E P,,(C0(P))
hence I1
ff
c1
I'
( 1 - k ) Bo
> 1/2.
We need now the followin? lemma. LEIUEA:
(galaEB be an u n c o u n t a b l e c o l l e c t i o n 0 6 complex
Let
v a L u e d , c o n t i n u o u b p o L y n o m i a ~ &o n Co ( A ) w i t h d i n i t e and let
and 9,
5
Sp
E
> 0
CT?
. Then the.he e x i b t
n Sp(o;
s . Fuhthehrnohe,
ibomohphibm
hq
+
ull)
id Hl
= $
a #
3
y E
t h e h e e x i b t , doh e v e h y
: Co(Sp qy)
~ u c ht h a t gi(h?(z)) 1
an i r n c o u n t a b t e
= mq(z)
qi +
{oh
PROOF O F THE; TIIEOREP6 C O N T I N U E D .
€I1
a
E
bpecthum
d e t tflC
ioh a&?
H,
an
i,j and
i, a n i n t e g e h qi,arz 91
and a polynomial
all
-
qi
E P(C
)
El.
Take an uncountable set HI C H
236
B.
and
I
(l-IC)Eo (2)
a
a s i n t h e Lemma and t a k e
a;
z E
JOSEFSON
1
such t h a t
> 1 / 2 - SinCf!
~
9
crA(z)
I
> 1/2
-
. Take
y E 111
also
which i s p o s s i b l e i f
E
CL
and n . h a v e d i s j o i n t s p e c t r u m w e may a2
j
1
t h a t z i s c h o s e n so t h a t z = z o
SUme
and
+
S
0
z1 where s u p p z c i J Sp
q1
s 0 SI' qc/.
and s u p p z1 c
. Put
i
1
A
-
where
z1 I i
= ProjrSp qylz
Further
cr' ( z o fl
2
+ z1 +
s u p p z 2 n s p (q!
+
+
z ) = or (zo
Y
c o r d i n n t o t h e a s s u m p t i o n s . Cut because
1'
hy 1 1 = 1,
Now
Iq:(z')\
and
if Y ( 2 ' ) :
I'
-
> 1/2
1/10
-
i s small cnouqh b e c a u s e x ea
PROOF OF THE LEMVA.
z'
-
= zo
+ z1 +
z2 E
hence I f a ( z ' ) ) > 1 / 2
E
2 c . Thur f ( z ' ) P U
+y
+
ev E U if 1x1 and lyl
y E V when
€1' CL
-
.
1/10- 2~
1 / 1 0 Bo
aQ h a s i t s s p e c t r u m on a t
so t h a t e v e r y E 11.'
q
CI
"
c
if
1/10. Q.E.D.
most R
a i s less t h a n R Mo. L e t V C A be a s e t s u c h t h a t t h e r e e x i s t s a n
t a h l e set 1lVC
(1 - KIB,
0
a b l e s and s u c h t h a t t h e d e g x e of
if
+ z 2 ) = q'U (2' + z 1 ) s i n c e n s p ( o 1 + ail = @ ac-
T h e r e e x i s t a n u n c o u n t a b l e s e t II C H
an i n t e q e r R s u c h t h a t
E
.
zO+zl112 l + k and s u p p z 2 O s u p p ( z 0+ z 1 I = @
= la'(z') I
Y > 1/?
zl
= @ and s u p p z1
qi)
1
1-
c
and
vari-
for
all
uncount
depends on t h e v a r i a b l e
z
Y
R elements s i n c e each
V c o n t a i n s a t most
o n a t most R v a r i a b l e s . Let V' Le a maximal s e t w i t h 0 t h e p r o p e r t i e s above. V e x i s t s b u t c a n be t h e e m j t y s e t . Also
cI
" depends
s
t h e r e e x i s t s an inteqcr qC"l S = 1, g:"
s
C ai"*I 9; where 1 Co(A) w i t h spectrum i n Vo
such t h a t a
a r e p o l y n o m i a l s on
"
=
BANACH VALUED POLYNOMIALS
237
a
i f i < s and yi a r c p o l y n o m i a l s w i t h s p e c t r u m o u t s i d e V Now t h e r e a r e , f o r e v e r y f i x e d a E I!,
at most
0
.
countably
S
many y E ' :I because V
such t h a t
Va
r i Vv
# @, w h e r e
L'.
i s maximal and Vn i s f i n i t e .
V
=
11
a
live
,
supp 9i
i=l Hence t h e r e i s a n
1
un1
.
countable set I I I C s u c h t h a t V a f \ V = @ i f €I 3 a f y E H Y a,l 0 i s a n U n c o u n t a b l e s e t of p o l y n o m i a l s i n COW 1, {q, }nEII1 wherc Vo
i s f i n i t e , w i t h clearee l e s s t h a n R. Hence t h e r e e x i s t ,
f o r every
E~
2 0 > 0 , a n u n c o u n t a b l e s e t I1 C 111 and q l E P (Co (V ) )
- q l l 'Eo < c o i f a E € I 2 . Ve c a n r e p e a t t h i s 1 a r q u m e n t s - 1 times and n e t t h a t t h e r e a r e a n u n c o u n t a b l e set € I s C €is-'!= . tI1 1 and cri E P (Co (V 0 ) ) s u c h t h a t 1'
such t h a t
aarl
..
-
I l o : , ' P u t Vy
ui11
<
Eo
if
a E 1is.
OO
a = s u p p rrl.
There e x i s t s an uncountable set MS+&€IS
s u c h t h a t t h e number of e l e m e n t s q i i n Vy i s i n d e p e n d e n t a E HS+l. L e t ty : V g -* (1, ...,q i )
hq
: Co(Vq)
+
Q
O i
b e a one-to-one
)Je t h e mapping i n d u c e d Ly R i
.
of
map and
let
Put
C , ? ( z ) = n:((hi)a -1 ( z ) )
where z E Q
CJ i
1 c(
and
(11.
1
)
-1
a CL qi hi-GiW(C 1.
i s t h e n a t u r a l i n v e r s e of
Now by t h e same r e a s o n s as a h o v e t h e r e e x i s t a n u n c o u n t a b l e s e t Hs+2
0.
c €Is+1
and
Gi
s u c h t h a t I ' GY
E P (Q
-
G i i IBqi <
CT.
where Bgqi
a E
is the u n i t b a l l i n
P
1
Cut t h e n t h e 1 m a follows S
and
a
a
T: gi 1 xax (
=
s *
CZEA
ics
-
a
n i
if
E"
+ 0
KO( A ) )
E~
if
a
w i t h t h e Co-iiorm.
. a
f ere p u t I I = ~ I I ~ + ~ , G . = q~a ~r o, = 1 'i
i s c h o s e n so small t h a t
238
B.
The a u t h o r
is
JOSEFSON
grateful to Professors
R. Aron
and
J.
Globevnik f o r u s e f u l d i s c u s s i o n s on t h i s p ap er .
REFERENCES
R . ARON, Compact p o l y n o m i a l s a n d compact d i f f e r e n t i a b l e map
[l
p i n g s between Banach s p a c e s . T o a p p e a r .
12
1 -, The
r a n g e of a v e c t o r v a l u e d h o l o m o r p h i c mappings.
To appear.
[3
]
[4
] J.
S . DINEEN, U n p u b l i s h e d m a n u s c r i p t . CLOBEVNIK,
The r a n q e of v e c t o r v a l u e d a n a l y t i c
func-
t i o n s . T o a p p e a r i n A r k i v f o r Matematik.
"] -
The r a n q e o f v e c t o r v a l u e d a n a l y t i c f u n c t i o n s 11.
To a p p e a r .
[G ] A . IIIRSCI1OWIT2, Remarques s u r l e s o u v e r t s
d'holomorphic
d ' u n p r o d u i t d e n o m h r a b l e cle d r o i t e s . A n n a l e s
cle
1 ' I n s t i t u t Fourier 19 (1969). r7
B.
JOSEFSON, A c o u n t e r e x a m p l e i n t h e Levi p r o b l e m . L e c t u r e
Notes i n Plathematics 3 6 n ( 1 9 7 4 ) . [8
]
L . NACHBIN, Topoloqy on s p a c e s of h o l o m o r p h i c
mappings
S p r i n q e r - V e r l a q , C r n c h n i s s e i l e r Plath. 4 7 ( 1 9 6 9 ) F9
]
A.
.
PELCZYNSKI, A t h e o r e m o f D u n f o r d - P e t t i c t y p e f o r p o l y n g
m i a l o p e r a t o r s . B u l l . Acad. P o l . S c . X I , 6 ,
[lo]
.
A . PC1,CZYMSKI AND 2 .
t i o n s 111.
SCPIADCPJI,
S t . Math.
S p a c e s of c o n t i n u o u s
(1963). func-
XVIII ( 1 3 5 9 ) . Uppsala U n i v e r s i t y
D e p a r t m e n t of i l a t h c m a t i c s Syssloiwnsqatan 8 S-752 2 2 Ullpsala, Swxlcn
Infinite Dimensional Holomorphy and Applications, Matos (ed.) 0 North-Holland Publishing Company, 1977
DOMAINS OF EXISTENCE IN INFINITE DIMENSION
By G E T U L I O KATZ
INTRODUCTION Among the subjects studied in infinite dimensional
holo-
morphy the one of analytic continuation is nerhaps the most develoned. (See 161 for a survey and a fairly comDlete granhy)
bihlio-
.
The Levi problem asks whether it is true that a domain is pseudo-convex if and only if it is a domain of existence.In the first nart of this paper we rJive a positive answer to the
Levi
problem in the case of Riemann donains over Banach spaces with Banach apnroximation pronerty (R.A.P.).
This result
follows
from investiqations about nroperties of permanence for
domains
of existence under elementary set operations.
P
In the second part we m o v e that a convex set in (A
any index set) is a domain of existence. This result
ses that the behavior of the space of polynomials is
n
(A)
stres-
directly
related to the answer to the Levi problem. This naper is hased an the author's doctoral thesis written under the quidance of Leonoldo Nachhin at the University of 239
KATZ
G.
240
Rochester. 1.
Let E and F be locally convex
Hansdorff
spaces (1.c.s.);
U C E be an open set: @cmE;F) the set of all continuous m-ho-
of
moqeneous polynomials from E into F; C . S . ( F ) the set continuous seminorms in F. Denote by %(U;F)
all
the set of holomor
phic mappings from U into F as defined below. DEFINITION
5
E
all?
A {unction f
:
U
+
F
h o l o m o k p h i c i i $oh
i h
.
U a n d m = 0 1 1 1 2 1 . . t h e h e e ~ i Pm ~ Et @(mE;F) (F) t h e h e
f3EC.S.
lim P[f(x) m+m DEFINITION
-
m
c
n=0
:
P,(X-S)J
L e t E b e a l.c.s.,
R i e m a n n d o m a i n o v e h E id and p
X * E
i h
u
exiht.4 V(E)
X
#
o
nuch t h a t
uniformly on V.
t h e paih (X,p) i n c a l l e d
$,
a -!ocn-! h o m e o m o h p h i h m .
over the same basic space; da
(where A
a E
C.S.(E);
C $$(XI
a
X LA a C o n f l Q c t e d H a u h d o h 4 b bpace
Holomorphic functions; morphism between Riemann
tively to
(OR
,that
nuch
=
all
:
X
+
+ IR
domains
the distance in X rela-
The concept of A-domain of
holomorphy
1 : etc. I are all defined in the obvious way.
Let (Xllpl)and (Xz1p2)be two Riemann domains over and E2 respectively. Let X = X 1XX 2 and p = (pl1p2) : X then (X,p) is a Riemann domain over E1xE2.
+
El E1xE2'
Assume that Ell E2 are metrizable 1.c.s. whose topologies and 8, < p 2 respec are generated by seminorms al < a2
...
...
tively. Then E = E1XE2 is a metrizable 1.c.s. whose topology is
. . where
yi (x,,~,) = sup ( a i (x1 I Pi (x,) 1 . X2; Y = X x(x2); 9: Y El .Then ( Y l q ) is 1 (Xl1X2)+ P1(X1)
generated by y1
E
a Riemann domain over El such that Y
-+
=
X1.
DOMAINS OF EXISTENCE
Let (Xi,pi)
THEOREM 1
241
a n d (Y,q) ah a b o v e .
i = 1,2
ifi
Then
( x , p ) i 6 a domain o f i e . x i 6 t e n c e 6 0 i d ( Y , q ) . W e u s e some r e s u l t s o f 151 and a l l d e f i n i t i o n s r e a u i r e d
PROOF
i n t h e proof c a n be found i n t h i s p a p e r . Suppose (X,p) i s a domain of e x i s t e n c e . Then by
t h e r e i s a n a d m i s s i b l e c o v e r i n g v o f X I such t h a t X
4.3 o f [5!,
i s an
A--
domain o f holomorphy.
Let
Q. =
v
have t h a t X i s AV t h a t Y is A
1 v
(V f ) Y
E
V
'v may
and A U
So
I lfll l v n y
f (u)
51 I f l [
f ( w ) = f l (w)
#
Iv
<
m;
+
I If1 I v
Again by theorem 3.6
<
m
defined V
for a l l
v.
E
and f l ( u ) =
Y i s AZI - s e p a r a t e d .
.,
o f 15J
we
i n X s u c h t h a t dim ( x n ) + 0 f o r
s u c h t h a t s u p l f (xn) I =
Y s u c h t h a t dam (Y,)
Y
0 f o r a l l Y,.
%(Y)
t h e n by theorem 3.6
-convex,
a l l yml t h e r e e x i s t s f E Av
dXm(Yn) Y
v -separated,there
it follows t h a t f l E A
have t h a t f o r a l l s e q u e n c e s (x,)
L e t (Y,)E
As X i s A
f(w). Let fl E
then
. Therefore
Now as X i s A
prove
W e shall
-separated.
w.
such t h a t f ( u ) #
by f l ( y ) = f ( y ) . As f E A v
we
-separated.
Assume f u , w ) C Y C X , u # E AV
of Y.
a n open c o v e r i n g
B y p r o p o s i t i o n 4.2 o f [5]
-convex and A v
u -convex
be t a k e n c o u n t a b l e .
3 is
1.
a is admissible.
W e claim t h a t
exists f
theorem
+
0
for
W e have t h a t f l E A
of [5] w e g e t t h a t Y i s A
m.
all
am.
Then
and s u p ! f l ( Y n )
u -convex
I*. and
by p r o p o s i t i o n 4.2 and theorem 4.3 t h a t Y i s a domain o f e x i s t -
...
ence. q e d
DEFINITION
A Banach 6 p U C e E i 6 s a i d t o h a v e t h e Banach a p p h o x
i m a t i o n p R o p e h t q (8.A.P.) i.( E i b b e p a h a b e e and R h e R e . 6eque.nce 0 4 o p e h a t o m 0 4 f i i n i t e doh
ale x
E E.
Rank
( u n I n c-
exibt6
a
6 u c h t h a t un(x)+ x
242
G.
COROLLARY
KATZ
L e t (Xl,Pl) b e a Riemann domain o u e h a Ranach Apace
El w i t h B.A.P. T h w X1
pneudo-convex i4 and o n t ! j id
i d
X1 i n a
domain O X e x i n t e n c e . PROOF
It is known that a separable Banach space has B.A.P. if
and only if it is a direct subspace of a Banach space with
ba-
sis. (see C4-J). Let E = El
G where E is a separable Banach space
x
with
basis. Assume (X,p) is a R i m domain over E such that X=X1x G and p = (pl, id): X * E. As X1 and fX,P) * Now by
[l]
are pseudo-convex
G
so
is
(X,p) is a domain of existence, hence by the-
orem 1, X1 x { O ] is also a domain of existence.
Finally
as
X1x{O)=X1 it follows that (Xl,p$ is a domain of existence.q.e.d. In contrast with theorem 1 where separability was crucial, we shall state. THEOREM 2
~ e it = 1,2,
Ei. Then
l.c.6.
x
a2 i n
a domain o h e x i b t e n c e
Suppohe E i d a
LEMMA 1
i b t e n c e i( and E
k!.c.b.
BC'U), nuch t a h t ( o h neighboahood w (z-bak'anced
aU t h e h e doen n o t e x i n t a
LEMMA 2
Suppobe E and
G
PROOF OF THE THEOREM x
6E
%(Wl
2,
E 2 ) and f2 E
4 agheeb
and
iuith
0 4 w flu and z E aU fl a R t
ahe t w o t . c . n . ,
then
'u x
G ib
a
do-
E i n a domain 0 4 e x i n t e n c e .
main 0 4 e x i n t e n c e phouided
%(%1
i b a domain 06 ew
o n l i t i{ t h e h e e x i d t o 9 E
g i n a c o n n e c t e d component R
E
eXibtenCe.
and UCE.
and c o n v e x ) buch t h a t t h e h e e x i b t n
fl
06
a domain
a
in
The proof is based on two technical lemmas.
PROOF
att Z
q,
q i c E~
x
(x1
U 2= %(El
x
x
2,)
E ~n )
(
E~ xQ2).Let
such that:
'u1 x EZ
DOMAINS OF EXISTENCE is the domain of existence of fl and El
243
u 2the domain of
x
ex-
istence of f2.
a(%,
We have that (
3,
and x E Aopenc El fl
+
f2
I (2' 1,
E
A, then A n ( a 3 ,
x
a 2 )has
x
If not, there $(
such that
)
5
a ) Suppose Z E
a(%,)
21,.
x
+
f2 in
b) Suppose Z E
a,
z
x
c) If
(aU,)
E
claimthat
as natural domain.
u,)
I
w ( Z ) and
-
9
of
domain
i
x
wlC vlc-Q1
E2.
9 . Let 0 be a connected CXXJ
Q,).
x
-
8;
a%,.
x
E
B2CQ2'
f2
E
% ( wl) and
in 8. This contradicts the fact that fl has of existence.
We
3 2 ,
Then there exists
Wlc W, 0 (21, = fl
ff.
x
a(n).
E
Wl#
We have
#
a ?ll
E
fl + f2 in a connected component R
domain El' such that Z E B1 x B2 = B1 Besides as 2 E a ( R ) then D f l
n 0
a ,)
X
21, x 21, E ?(qlx
exists Z
Wn('?J, x U2)such that z
ponent of
u caql x Q.,N
aq,)
x
It i s easy to prove that for all x
3'21,).
x
(aul
21,) =
x
2,
x
9- f2=fl
E2 as
domain
Same proof as in (a).
(aQ2), let
W(Z) as in
lemma 1. sup
pose A1 and A 2 convex, Zi - equilibrated oDen neiihborhoods of Ai and Z E A1 X A 2 C . w ' , where Z = (ZlIZ2). As Z E a(Q),there exists (xlIx,)
E
R f) (A1
A,).
x
So (ZlIx2) E A1xA2; s=(Z1,x2)
E
ul) q,. Let A3 = connected component of %, 0 such that x2 Then A1 = v,;wl w ;vlC:EIX?$; wl '&, E2 and (xl,x2) R fl (A1 Thus reasoning as (2
X
A,
E A3.
C
X
s E
X
A3
E
in Dart (a) we would cJet that
X
a,
existence of fl. It follows that
X
21,
A2).
E2 is not the domain X
of
(1L2 is a domain of ex-
istence. q.e.d.
2.
In this section we shall follow the techniques of 121
to
244
G. KATZ
prove that in .f? ( A ) the open convex sets are domains of F tence. Let F be a Banach space,
F containinq the oriain and Let c > 0 such that
ac F be a convex open subset of
the closed unit ball in F.
Efi
C-U ; usinq the onen-mappinq
Hahn-Banach theorems, one can nrove that for all x exists $x
E
F' satisfyinq:
I lGxl
Re($,(Z))
<
1 whenever Z
E
U.
(Z) =
1
Let
f
X
have that fx
E
have t h a t
sup
Let Z
E U
$,(x)) 03)
and fix Y
,Y
E
= (1/2 (1
5 Re (4,(Z))
$,(x))
N is fixed.
E
duch t h a t Z
-
+
5 I 7
a ) . So
cpxfx) I 2 1/2 (1- I Let .f?,(A) , p
Z E
U.
+ ~vRCU,
+
2 y B c U . kfine
If
YEy
E
;
(l-Re($x(Z)).Therefore (1
-
-i 1 (Re($x(Z))-l). But Re(Gx(Z)) 5 1. , therefore
-
We
m.
such that Z
(Olm)
Re($x(Z+2Y)-@x(x)) 5 0, then Re $ ( Y ) 5
-
and
# 0 for a l l
(fx(Z+Y)I 5- M Z I y <
sup YEYB
a = sup (r>OlrZEU) and Vz
Re($x(Z+Y)
there
E aU
(z)-$x
Fofi ale Z E U , y E ( 0 , XEaU
PROOF*
, where i
and
= 1
1 < 2 / ~ , Re $,(x)
since Re($,(Z)-
%(U)
PROPOSITION 1 We
[a,
exis-
Re(bx(Z)) - 1 = IRe($x(Z+Y)
-
lfx(Z+Y) 1 5 M Z I y f o r all YEyR. q.e.d. be the set E(A)=(f:A+Cl
E[l,m),
endowed with the norm 1 If
I1
Banach space.Let
given by
= ( Z If ( a ) aEA
1)' 'ID
I a (PI =
c
E a€ A fp(A)
If
(a)lp<-l
is
a
1 if a = f?
0 otherwise
THEOREM 3
Let 0
E U
c t
P
(A)
b e an o p e n coyluex n e t . T h e n
U i n
__
*
This elegant proof is due to David Prill, nine was consider-
ably more complicated.
DOMAINS OF EXlSTENCE
0 4 existence.
u domain
If U =
PROOF
245
eP ( A )
the result is clear. So we may
consider
orderedl
U # l p ( A ) , and we may also assume that A is well
say
A = [O1$) where $ is the cardinal number of A. Finally it isp? sible to prove that there exists a dense subset (xnlaEA in aU. The proof of the theorem is based in three claims. CLAIM 1
T h e J w e x i b t d C:A+A ouch t h a t 1
injective
i d
1
= 0 if{; where CJ e ( P I is the (xY)l(P) - W P ) associated family of coefficients functionals.
and
corresponding
'1
? if p is integer,
Let
q
=
c
(p]
+ 1 otherwise, where
pI = integer part of p \
Z ) , c a n n o t be continued .f t(a) xu 0ue.4 xa. ( I n t h e d e ; ( i n i t i o n O X f t . . k e i = CJ + 1). xa C Ca Z:(a). f ( 2 ) where C a is defined as Set q ( Z ) = aEA a' f01lows
CLAIM 2
The .(unction Z
+
2'
.
We shall conclude the proof showing that q cannot be con-
tinued over any point xal
aEA;
and therefore g cannot be
tinued over any point of a U since {xn} deed,
J:
Y'P such that
.
a EA
is dense in aU.
C ( S X ~ + ~ L ~ ( ~ ) ) 'fxy(Sxy+ylt(B)) = 0
conIn-
for all 5 , y E C
Exg+yLZ(p) E U, because the way we defined 1 . Now if
G. KATZ
246
as r
+
0.
If as r
+
0
on a sequence of points with infimum 0. Indeed in this
case
= 1 and the behavior of the sum is qiven by the last term, P which goes to fininity by the claim 2. This finishes the proof.
C
BIBLIOGRAPHY
111 HERVIER, Y., Sur la problgme de Lev1 pour les e s p a c e s h G s Banachiques, C.R. W a d . SCI, V.275, ser A.p.
821,
1972. 121 JOSEFSON, B., Counterexample in the Levi problem.
Proce-
edinqs on infinite dimensional holomophy, Lecture no tes in Mathematics. Snringer Verlag, V. 364,
1973,
168-177. 1 3 ) NACHBIN, L., Topology on spaces
of holomornhic
maminqs.
Erqebnisse der Mathematik und ihere qrenzze
biete,
Sprinaer-Verlag, Hejt-47, 1969. 141 PELCZINSKI, A , , Any separable. Banach space with the bounded approximation pronerty is a comnlemented subspace of
DOMAINS OF EXISTENCE
247
a Banach space with basis. Studia Math: E 40,
1971,
p. 239-243. [5] SCHOTTENLOHER, M.,
Analytic continuation and regular clas-
ses in 1.c. Hausdorff snaces. Portuqaliae Matematica. (to appear). [6] SCHOTTENLOHER, M.,
Riemann domains: Basic results and open
problems. Proceedings on infinite dimensional holomornhy. Lecture notes in Math, Springer-Verlaq V.364, 1973.
DEPARTAMENTO DE MATEMhTICA UNIVERSIDADE FEDERAL DE PERNAMBUCO CIDADE UNIVERSITARIA RECIFE
-
-
TEL. 27-2388
BRASIL
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Infinite Dimensional Holomorphy and Applications, Matos (ed.) @ North-Holland Publishing Company, 1977
GEOMETRIC ASPECTS OF THE THEORY OF BOUNDS FOR ENTIRE FUNCTIONS IN NORMED SPACES
By C. 0. KISELAlAN
CONTENTS : 1. Introduction 2 . Geometric indicators of the growth of
a
plurisubharmonic
function. 3 . Zones where a qiven nlurisuhharmonic function is large. 4 . The inequality \ R ( x )
-
R(y)
I
< 1 Ix-yl 1
5 . Estimates for the Lipschitz constant of a radius
of
bound-
edness. 6. Prescribing the radius of convergence. 7 . References
1. INTRODUCTION
A
phenomenon in infinite-dimensional
complex
analysis
which has no counterpart in finite dimensions is that an entire function may be unbounded on a bounded set. For example the series
249
C. 0 . KISELMAN
250
f(x) = m c
X kk'
...1
x = (xl, x*,
E
LP,
1
converqes everywhere i n L
, 15
r)
p <
m,
and d e f i n e s a n
entire
f u n c t i o n which i s unbounded i n e v e r y b a l l of r a d i u s l a r q e r t h a n
t o t h e d e f i n i t i o n of a b o u n d i n q s e t , i . e . a
one. T h i s l e a d s e . g .
s e t which i s mapped o n t o a bounded s e t by e v e r y e n t i r e f u n c t i o n , and t h i s c o n c e p t h a s b e e n s t u d i e d b y many a u t h o r s . A
fundamen-
t a l r e s u l t is t h a t i n c e r t a i n spaces only t h e r e l a t i v e l y
com-
p a c t sets are boundinq, and, g e n e r a l l y s p eak in g , bounding
sets
h a v e no i n t e r i o r . I f o n t h e o t h e r hand we c o n s i d e r s e t s
where
a n i n d i v i d u a l e n t i r e f u n c t i o n i s bounded it is c le a r t h a t t h e s e s e t s may h a v e i n t e r i o r p o i n t s : t h e y a r e t h e s u b s e t s of t h e o p e n
sets = Cx E E; I f ( x )
wk , f
1
< kl
f o r some k . The g e o m e t r y of t h e s e s e t s is of i n t e r e s t . However, t o describe a l l p o s s i b l e families ( w k , f ) k , O
is a
formidable
t a s k , and o n e i s l e d t o m e a s u r i n g t h e g r o w t h of i i n some p l i f i e d way. The m o s t f u n d a m e n t a l n o t i o n i s t h a t of t h e
sim-
radius
of b o u n d e d n e s s : i f f i s e n t i r e on E, a normed space, t h e r a d i u s of b o u n d e d n e s s a t x
E
E i s t h e l e a s t u p p e r bound R f ( x ) o f
numbers r s u c h t h a t f i s bounded i n t h e b a l l {y:
I1y
- X I 1'
all
rl.
How d o e s R f ( x ) depend o n x ? T h i s i s a d i f f i c u l t and i n t e r e s t i n q
q u e s t i o n , o b v i o u s l y c o n n e c t e d t o b o t h complex a n a l y s i s and i n f i nite-dimensional geometry. I t t u r n s o u t t h a t even i f o n e i s i n terested o n l y i n R f ( x ) , s e v e r a l r e l a t e d c o n c e p t s come i n t o
the
p i c t u r e , and t h e main p u r p o s e of t h i s l e c t u r e i s t o p r o p o s e , i n 52,
a whole f a m i l y of r e l a t e d m e a s u r e s of t h e g r o w t h of a n
t i r e (or p l u r i s u b h a r m o n i c ) f u n c t i o n .
en-
251
THEORY OF BOUNDS FOR ENTIRG FUNCTIONS
First of a l l we must broaden the view to
the
Include
plurisuhharmonic functions. It then becomes natural to
04
the h a d i u d u: E
-+
[-m,
define function
b o u n d e d n e n d RU(x) of any numerical
+m[ as the supremum of all numbers r such that u is
bounded above in the ball of radius r and center at x . This, of course, is to allow u = log / f l , f entire, and we axe not
con-
cerned here with sets where f is small. We remark that the radl us of boundedness of u = log If( is then equal to the h a d i u n 0 4 c o n v e h g e n c e of f, i.e. the least upper bound o f a1.1 r such that
the Taylor series of f at x converges uniformly in the ball
of
center x and radius r (see Nachbin [7, p. 263 1, The radius of boundedness may be regarded as a kind
of
boundary distance. In fact, let E be a normed Bpace,
u
plurisubharmonic function in E, in symbols u
and put
Rk = {x
E
E PSH(E),
a
E; u(x) < k}.
Then Rk is a pseudoconvex open set and this implies that function
the
-log d k is plurisubharmonic in Rk, where d k ( x ) is the
distance from x
E
R k to aRk. Hence, =
lim dk = s u p dk, k + m k and by well-known properties of plurisubharmonic functions, RU
(1.1) -log RU
i d
p ~ u h i d u b h a h m o n i ci n E ,
for it is locally the limit of -1oq dk
E
PSH(Rk). A l s o ,
since
every dk is Lipschitz continuous with Lipschitz constant 1,
we
see that (1.2)
1RU(x)
-
RU(y)I 5 I!X-Yl], X I Y
E El
which h o l d s , of course, for any function u, not u
E
just
for
PSH(E). Property (1.1) was first proved by Lelonq [5,p.176].
Thus the radius of boundedness appears as the regularized bound
252
C. 0. KISELMAN
ary distance from (x,O)
E
E
x
E
x C
to a w
"direction"
in the
{Ol, where w = {(x,t) E E x
is an open subset of E
x C,
(II;
u(x) + log It1 < 0)
pseudoconvex if u
E
PSH(E).
Instead of Ru(x) one may of course define RuIA(x) = sup (r;u is bounded above in x + rA) for any set
A.
However, this is not so interesting unless
manage to impose a structure on the family of sets A .
we
In 9 2 we
shall do precisely that for certain sets A, viz. those
which
are linear images of the unit ball of some other normed
space.
We
may then study the dependence of R
(x) on A as well as on u,A x, and this leads to new problems as well as to some answers.In
53and 5 4 we give applications of this-it is hoped that they not the last ones.In § 3 we determine zones where a
are
plurisub-
harmonic function must be large if its radius of boundedness de cays slowly at infinity. This is proved using R
for A = AX a u,A linear image of the unit ball of the same space under a one-pa-
rameter family of mappings: AX =
In 5 4
(I + (X-l)a) (B) = { x
E
E;
I
1 - 1)nfx) I \ z l ) r x >o. + (7
we study when equality in (1.2) can occur: this
turns
out to be rather exceptional, and the result already shows that not every function R in co or L P , 1 < p <
w,
satisfying
(1.1)
and (1.2) is a radius of boundedness of an entire (or even plurisubharmonic) function. In 5 5
we state a precise estimate for
the Lipschitz constant of a radius of boundedness which
de-
creases slowly at infinity. Theorem 5.1 was (essential1y)proved in [4]
. The problem of constructing an entire function with pre-
THEORY OF BOUNDS FOR ENTIRE FUNCTIONS
253
scribed radius of convergence is discussed in the final Section 6 were we first give the main theorem from Section 4 of [4J ,and then discuss a special case (see Theorem 6.2) which is not covered by the methods of [4]. My basic conjecture concerning the radius of convergence has been this: any function R: L 1 + ] O r + m [ satisfying (1.1) and (1.2) is the radius of convergence of some entire function el. A
on
test for any new methods would be to prove this first for
functions R depending on, sayl C x . only. A more general conjec I ture is that Theorem 6.1 holds for e p l 1 5 p < and co without the finiteness condition. In fact in this geometric setting the natural concept is that of the r-local radius of
bounded-
ness RT T
and the conjecture concerning this is that for ,up = a(E,E') , E reflexive (say) and infinite-dimensional,-log RT,u
is just any plurisubharmonic function. It should however be remarked that if Theorem 6.2 is best possible or close to it,then the conjecture does not hold for b1 (but may still hold for the reflexive spaces P, 1 < p < -1. NOTATION
With the exception of Theorem 4.1 all spaces
normed linear spaces over the complex numbers. If E
are
and F are
normed spaces we let L(E,F), or just L, denote the
continuous
linear mappings from E into F, and A = A(E,F) the
continuous
affine mappings, identified in a natural way with F
x
L.In L we
use the topology given by the norm
and in A the product topology of F X L. The closed unit ball of E is denoted by BE or just B. We use vector operations also for
254
C. 0. KISELMAN
sets, thus e . g . A 1A 1 + A2A2 = {Xlal + h 2 a 2 ; a l ~ A land a2 We write PSH(R) and o ( Q ) f o r the set of, respectively,
E
A21.
all
plurisubharmonic functions and all analytic functions in Q. The star in f * ("&toile de Lelong" in the terminoloqy of
Martineau
[ 6 ] ) means upper regularization, 1.e. the operation of
taking
the Interior of the epigraph. By a uniform neighbourhood of set M in a normed space we mean a set containing M some
+
a
for
EB
> 0.
E
GEOMETRIC I N U L C A T O R S OF THE GRO\JTA OF A PLURISUBHAR~4ONIC
2.
FUNCTION
The definition of the radius of boundedness RU plurisubhai-monic function u on a normed space E means (x,t) x
+
E
E
x
a
of
that for
R U ( x ) if and only if u is bounded above
63,
on
tB. KnowinQ Ru is equivalent to knowing the set s1
*
{(X,t)
f
E
it\
C;
x
< RU(xfl,
and -log RU fs plurisubharmonic If and only I f 0 is
pseudo-
convex. More d e t a i l e d information is provided by the function ti(x,t)
a
-
IIYI
I
sup
U(X+Y) I
5 It1
and R is qotten from u in a simple way: it I s the Interior the set where "dunctiona!
is less than
+m.
We can thus say that
of
is
a
i n d L c d t a x " and that R (or Its boundary distance) is
a " g e o r n e t t i c i d L C a C Q k ' ' of the growth of u.
The pukndse of this section is to Introduce a whole famlly of "indicators" of the growth of a nlurisubharmonic function
u which are *elre general than
and R above, but still
enough structure t o allow calculations to be made.
have
THEORY OF BOUNDS FOR ENTIRE FUNCTIONS
255
The g e n e r a l v e r s i o n i s g i v e n i n t h e f o l l o w i n g t h e o r e m .
THEOREM 2.1
Let E be a notmed b p a c e , X
bpace 0 4 bounded mappingb
h : X t E
a
and H a
bet
lineah
normed by
and c o n t a i n i n g t h e c o n b t a n t mappingn. D e d i n e ,
(Oh
any n u m e t i c a l
(unction u on E, = sup u ( h ( x ) ) , h E H,
;(h)
and l e t fl be t h e
x EX o ( all h
bet
E
H buch t h a t u
i n bounded
04
h(X).
Then
PSH(E)
. 75
i n a UnifiOhm neighboxhood h ( X ) + €BE
-*
pbeudoconvex and u
x
E
PSH(S2) id u
E
I$ i
above Sl
i d
a (ilteh on
b
we p u t I
uo ( h ) = l i m ME$
s u p u ( h ( x )) XEM
and have t h e anatogoun c o n c l u n i o n d o t fl
= (h
cb
E
H; u
i b
E
If
E
functions),
-* u4,
On
bounded above .Ln h(M)
> 0 and bume
M
E
*;
bOme
E B ~{ o h
61.
i s r e g a r d e d as a s u b s p a c e o f and
+
become e x t e n s i o n s o f
H
(via the constant
u.
T h i s theorem i s too g e n e r a l t o b e of u s e a s it
stands:
f o r e a c h p a r t i c u l a r a p p l i c a t i o n o n e w i l l h a v e t o c h o o s e X and H
t o match t h e problem. It i s of special importance to find criteria for
*;
( h ) t o b e e q u a l t o u(h).The f o l l o w i n g i n s t a n c e of Theorem
2.1
t h e r e f o r e seems t o b e a t a r e a s o n a b l e l e v e l of a b s t r a c t i o n .
THEOREM 2 . 2
L e t G and E be notmed b p a c e b , and Let
A = A(G,
E)
b e t h e L i n e a t b p a c e 0 4 a t l a(4ine c o n t i n u o u b m a p p i n g b 0 4 G i n t o E.
Let
u
E
PSH(E) , l e t
256
C.
<(h) = and l e t R be t h e
i n h(BG)
-* u
E
+
bet
A ~ A Oh
Banach b p a c e b , h p a k t i c u l a t , ii open d u b b e t R
i b
1tl Aiso
KISELMAN
u(h(y) 1
SUP
1 lY1 lc, 5 04 a l l h
€BE doh dome
PSH(R).
0.
E
h
At
E
1
buch t h a t u
E A
i b
bounded
above
> 0. Then R i b pdeudoconvex i n A
R and ii*(h) = u(h) w h e n e v e h G and
E
b u h j e c t i v e and ii i b
i d lend than
pluhibubhahmonic i n t h e
and ahe
E
neah h . I n
+m
pbeudoconvex
RnAiso 04 A,
Aiso
denotinq t h e
ibomohphibmb O X G
o n t o E.
One can of course give a more general instance of Theorem 2.1 by letting H denote, say, a space of holomorphic
mappings
h : G - c E , bounded on the unit ball X = BG of G. However,
the
difficulty then is to decide, for a given h, whether ii*(h)=ii(h). Theorem 2.2 generalizes a result which has been and applied in [4]:
E
A ( E , E ) which are of the form
h(y) = x + ty, y E
to
we take G = E and restrict attention
those affine mappings h
for some x
proved
E and some t
E
E
E,
C. This leads to the function G(x,t)
considered at the beginning of this section. Letting
denote
9
the filter of neighborhoods of the origin for some topology
T
on E (e.9. T = a(E,E')) we get corresnondiny "local" objects of which the most important is the T - L O C U Lh a d i u b 0 4 R
TiU
boundedneab
( x ) which is the supremum of all numbers r such that
bounded above in x + r (BflW) for some r-neighborhood W
of
origin. We shall use these constructions in the proof of rem 4 . 4 .
On the other hand, in 5 3
we shall use Theorem
with affine mappings not of the form y - t x
+
From one plurisubharmonic function u
u
is
the
Theo2.2
ty. E
P S A ( E ) we thus get
a family of others, reflecting in various ways the growth of u. -t
THEORY OF BOUNDS FOR ENTIRE FUNCTIONS
257
for instance, in the notation of Theorem 2.2, d(h) denote
the
A(G,E) to the boundary of R measured in
some
distance from h
E
more or less arbitrary way (we may use any continuous norm A). Then -log d
in
P S H ( f l ) . More generally, if d(h,f) is
E
the
distance from h to a R in the direction f, i.e. d(h,f)=sup h+tf
E
(r;
R for all complex t with It\ < r), then -log d(h,f)
plurisubharmonic in 0
x
A (see Noverraz [8, Th6or&m2.2.1]
is
is no harm to allow f = 0). In particular, we may take h stant, h(y) = x
E
E for all y
E
it
;
cog
G, and f linear and then d(h,f)
= d(x,f) is the least upper bound of all numbers r such that
is bounded above in a uniform neighborhood of x
+ rf(BG);
u we
may justly call this number the h a d i u b 0 4 b o u n d e d n e b b a t x w i t h hebpect t o
f(BG). We state this result as a corollary of Theo-
rem 2.2, to be used in the proof of Theorem 3.1:
COROLLARY 2.3
L e t G and E be notmed bpaceb and u
m i n u b t h e logahithm
06
04 u
t h e h a d i u b 0 4 boundednebd
h e b p e c t t o f(BG) i b a p l u h i b u b h a h m o n i c ( u n c t i o n
PSH(E).Then
E
06
at
x wi2h
(x,f)
E
.
ExL (G,E)
Consider for a fixed x E X the continu-
PROOF O F THEOREM 2.1
ous linear map gX : H
Then u o 6, ization of
3
h
+
h(x)
E
E.
is plurisubharmonic in H so the upper =
regular-
supxEx u o 6, is plurisubharmonic in the
set 0 ' where the family (u o It is easily seen that
Q'
is locally bounded
is precisely
Q
open above.
as defined in thestate_
ment of the theorem. (The constant mappings are used in proving
0. KISELMAN
C.
258
Q ' CQ )
.
To see that Q is pseudoconvex we put
'U \ *
w =
ke N
is pseudoconvex in H if u E PSH(E) and the operakrx Finally, tions used to get wk and w preserve pseudoconvexity.
Clearly w
one sees easily that w = 0 . The statement about ii
@ part of the theorem to a set
now follows by applying the 14 E @
first
and then passing to the limit.
(The usual "lim-sup-star" theorem when one takes the limit a b r g
a directed set requires this set to possess a denumerable cofinal subset, but this is no longer necessary if, as is the
case
here, the family of functions decreases. For details in a similar situation, see [ 4 , proof of Proposition 2.21 1 .
PROOF OF THEOREM 2.2
Only the statement about surjections is
not a direct conseauence of Theorem 2.1. Let be surjective, and assume that borhood of hot say for
1 Ihl in A = A(G,E); x =
x + f(y), y
E
E
\ =
u
is less than
I
IE E
+
' I f 1 IL
+
f,(y),
tfo) = s u p u(xo YEBC
+
y
we
E GI
tfo(y)) 5
+m
for
I Ih -
ho! I A
2-
6
=
consider
+m.
Obviously $ depends on It\ only and is a convex
of log Itl. Since G(h) <
A
in some neigh-
+m
L = L(G,E) being defined by h(y)
G. If ho(y) = xo
+
E
Ih - h o \I A 5 6 where we now use thenonn
E and f
$(t) = ii(xo
> 0, let ho
E
we
function have
THEORY OF BOUNDS FOR ENTIRE FUNCTIONS
$(I
+
ti1) <
where
+m
convexity, $(1
+
S2) < $(1)
such that 2 6 3 B E C "fo(BG)
,
S/l Ifol I L
61 =
+
E
and by
for some S2 >
(we suppose now that
x
+
I If -
and
fO1 I 5 6 3
f (BG) C xo + fi3BE + fo(BG)
= xo
Hence 6(h) = 6(x
+
+
(1
logarithmic
0. Pick 6 3
G
Banach spaces and apply the homomorphism theorem).
IIx - xol I 5 fi3
259
and
>
0
E
are
Then
if
we obtain
+ 63BE C xo + fo(BG) +62fO(BG)=
+ 62)fo(BG).
f) < $(l
+
2
S2)
6(ho)
+
6*(ho) = 6(h) and at the same time that ho
E E
which proves that R.
We finally remark that Aiso is a pseudoconvex open set in A (possibly empty) (for the distance d(ho) from ho = xo + fo to -1 -1 the boundary of Aiso is at least I f o I IL(E,G) by the Neumann
I
series so -log
1 !fill I
~ ( ~ ,is~ a) plurisubharmonic function in
tending to +m at the boundary). Therefore R n Aiso is Ai so Aiso. This completes the proof pseudoconvex, and <* = 6 in S l /l of Theorem 2.2.
REMARK 2.4
We do not know whether in qeneral a point ho E
A
for all h near ho belongs to R.
This
is
spaces
in-
such that 6(h) <
+m
true, as we have seen, if ho is surjective and the volved are Banach.
REMARK 2.5
Let E and F be Banach spaces. It has been observed
above that the isomorphisms in L(E,F) form a pseudoconvex
open
set. Similarly, the epimorphisms, the monomorphisms with closed range, the direct epimorphisms, the direct monomorphisms, the direct homomorphisms form open sets in L(E,F). Are open sets pseudoconvex?
and these
C. 0. KISELMAN
260
3.
ZONES WHERE
A
GIVEN PLURISUBHARMONIC FUNCTION IS LARGE
By the very definition of the radius of boundedness,
a
function u assumes arbitrarily large v a l ~ sin every shell (y; Ru(x) where
E
-
E
< I1y
-
XI1 <
RU(x) +
E)
> 0. It is natural to ask if it is possible to describe
subsets of this shell where u is also unbounded above. We shall do so assuming that u is plurisubharmonic and that RU decreases slowly at infinity. The description will be in terms of a B n Bn-l(B) where
THEOREM 3.1 index beh
bet,
06
Let E = lP(J)
is a projection in E.
,
1 5 p < m,whehe J i b a n
and l e t F b e a b u b b p a c e
coohdinateb.
{unction u
E * E
IT:
E
slabs
Abbume t h a t t h e
06 E d p a n n e d b y hadiub
indinite
a { i n i t e num -
0 6 b o u n d e d n e b b 06
a
PSH(E) b a f i b d i e b an e d t i m a t e a { t h e dohm R ~ ( X )2
c 11x1 I-y
doh x
E
2 rl,
F, 11x1 I
whehe C > 0 a n d y > 0 d e n o t e C O n b t a n t b . Let
T:
E
-+
F b e t h e ca-
n o n i c a l p h o j e c t i o n o n t o F. T h e n u i n u n b o u n d e d i n aR
n
fin-’ (B)
{ O R evehy a and p b a t i n d y i n g
a > RU(0)
and
B > Ru(0) (&I
l/P.
Note that the number r1 may be very large, so no explicit assumption is made on the behavior of Ru(x) for small 11x1 \.For p = 2 , E is an arbitrary infinite-dimensional Hilbert spacelad
F any finite-dimensional subspace. The theorem then says roughly that if RU decreases slowly in a certain direction x, then u assumes large values close to the hyperplane (y;
PROOF
= 01
.
Changing the constant C if necessary we may, and shall,
THEORY OF BOUNDS FOR ENTIRE FUNCTIONS
261
assume that RU(x) >
C
IIX~I-~
R ~ ( x )>
c
r;Y
for
for x x
E
E
F, 1IxI( 2 rl, and
F, 11x11 5 rl.
Let K(a,b) denote an “ellipsoid” with half-axes a and b: K(a,b) = (x
E;
E
(1
la(x) 1 (/a)’ +
(1
1x
-
n(x) I I/bIp < 1).
We claim that u is bounded above in K(a,b) if a L r1 and b.c a-Y. In fact, u is bounded above in every ball x
+
blB, where x
E
F,
11x1 I 2 a, and bl < RU(x) for all these points x. We can choose such a number bl > b. Now let wx = x + blB where
E~
n
~
~
-1?
v (B)
is positive but so small that wx 3K(a,b) c\ (x + ~~?v-l(B)).
Since K(a,b)
F is compact, finitely many wx with x E K(a,b)n F
suffice to cover K(a,b)nF, and then they cover K(a,b) as well. Hence u is bounded above there. Letting now
R U ( h ; x) denote the radius of boundedness of
u with respect to fX(B) where f X is the linear mapping fh(X) = x + ( A
-
11 n(x),
we therefore have, since f X (B) = K(IX 1 ,I), > b = C a“ RU(X; 0) -
where
X = a/b, i.e.
Now the function
for
a
2 rll
a = (CA)l/(y+l). Equivalently,
= -log Ru(X;O) is subharmonic in h
E
Corollary 2.3.Furthermore, and this is most essential, $ ( X I pends only on
1x1,
since, as already noted, fX(B) =
When ( X I is large we have
C
by de-
K(IXIll)
.
262
C. 0.
+(A)
KISELMAN
-
5 $ 109 i 1x1
l o g c, Y+l
and t h e l o g a r i t h m i c c o n v e x i t y e n a b l e s us t o e x t e n d t h i s mate t o a l l X w i t h
1x1 2
1:
I n p a r t i c u l a r , g o i n g back t o R u ( A ; O ) number c > y / ( y Ru(l
+
+
esti-
,
w e obtain f o r every given
1) t h a t
t; 0) > Ru(l;O)
1
-
c t ) = R U ( 0 ) (1 - c t ) f o r t > 0.
T h i s means t h a t u is bounded above i n t h e " e l l i p s o i d " K ( a , b ) E x a = (1 + t ) b and b = R U ( 0 ) (1 - c t ) . L e t now f? b e f i x e d w i t h R > 2 > R U ( 0 ) ( Y / ( Y + 1 ) )'Ip. S i n c e u is unbounded above i n (Q(O)+ t )B, u must a l s o b e unbounded above i n ( R u ( 0 )
+
2
t )B\K(a,b)
for
t h e s e a , b and t . A s i m p l e c a l c u l a t i o n now shows t h a t t h i s set -1 is c o n t a i n e d i n fin ( B ) i f c i s close enough t o y / ( y + 1) and t
is s m a l l . ( T h i s is o n l y a matter o f f i n d i n g t h e i n t e r s e c t i o n
of
t h e t w o curves
6'
+
(ValP in
IF?).
+
qD = ( R U ( 0 )
+
t2)',
(n/bIp = 1
T h i s p r o v e s t h e theorem.
When p t e n d s t o
+m
t h e s l a b uBn p
TI
-1
(B) i n c r e a s e s
and
t e n d s , r o u g h l y s p e a k i n g , t o a b a l l . T h i s is t o b e e x p e c t e d s i n c e i n t h e l i m i t i n g c a s e w e have t h e f o l l o w i n g example. EXAMPLE 3.2
The e n t i r e f u n c t i o n m
f ( x ) = z (Xlxk)
2 has r a d i u s o f convergence
k I
x
E
col
Rf(x) = 2((1x1I2 + 4 ) 1/2
+
I n p a r t i c u l a r Rf ( 0 ) = 1 and Rf ( x) L C lxl
x1
-1
THEORY OF BOUNDS FOR ENTIRE FUNCTIONS
263
easily seen that if f is unbounded in the set aB fl {x;
XJ’
for every ci > 1, then P > - I I i.e. the slab contains the
unit
ball.
It can be shown that the numher
REMARK 3.3
best possible. For 1 < p <
m
(y/(y
+ 1)) lip is
one can qive an alternative
proof
of Theorem 3.1 using Theorem 5.1. For p = 1, however, this
is
not nossible.
4.
THE INEQUALITY
IR(x)
-
R(y) I < 1 x - YI!
We shall investigate when equality in the estimate IRU(x)
-
RU(y) I 5
I !x -
Y!I
can occur for x # y: in the classical coordinate spaces
15 p <
+ml
and co we can have equality only in l1 and co
.L’
r
for
continuous functions u and only in L1 for plurisubharmonic funs tions u . The first result of this section therefore belonas
to
real functional analysis.
THEOREM 4.1
L e t E be a l h e a l o h c o m p l e x ) t o c a l t y
uniAohml4
c o n v e x Apace, o h , m o h e q e n e h n l l r l , n notmed dpnce d a t i b ( r ! i n g t h e Weakeh c o n d i t i o n ( 4 . 2 ) b e l o w . T h e n ( o h evehr4 u p p e h b e m i c o n t i n u -
oub d u n c t i o n u: E+
[-m,
+m[
w i t h { i n i t e hadiun
04
boundednebb
RU we h a v e IRU(x) A
-
RU(y)l <
1 (X -
yl 1 ,
X, y
E
E,
x # Y.
normed space is called t o c n l ! r { u n i . ( o h m e r { c o n v e x if
C. 0 . KISELMAN
264
uni -
If 6 can be chosen indenendently of x, E is called @nmLy convex. We recall that the Banach spaces
e”(J),
1 < p <
m,
are uniformly convex. In any separable Banach space (Kadec [ 3 ] ) or, more generally, in any weakly compactly generated space (Troyanski [9]) there is an equivalent locally
Banach uniformly
convex norm. Now consider a weakened version of (4.1): (4.2)
Foh eweRq x
buch t h a t
E w i t h 11x1 I
E
doh evehr!
> 0
E
IIx+yl1 2 2
-
thehe
=
y
E
+
K
ib
a compact
K
bet
b u c h t h a t { o h IlylI=1,
i b a 6 > 0
implied
6
1 tCi‘cte
EB.
{XI.
On
the other hand, uniform c-convexity, a property introduced
by
Thus (4.1) is obtained from (4.2) by remiring K to be
is weaker than uniform convexity and does
Globevnik [l],
not
imply (4.2).
PROOF OF THEOREM 4.1
Assume that the conclusion were false.By
normalizing we may then assume that RU(0) = 1 and that%(hx)=l-X for some x with 11x1 1 = 1 and some X with 0 < A < 1. This means that u is bounded above in
IY; I Iy1 I say u
5
< 1
-
l/kI U Iy; I Y-XX
Mk there, and unbounded abov
Iy; IIy - XxlI < 1
-
A
+
1
< 1
-
X
-
l/kI,
in l/k).
Take zk in the last-mentioned ball such that u(zk) > Mkf hence
l\zkl\> 1
-
l/k and 1
-
X
-
l/k 5 llzk
-
Ax\[ < 1
-
X + l/k.
A1 so
IlzklI 5 IIzk so
that
I
\zk\1
-
X X ~ )+ 1Jhx)l< 1
1. Putting 1 Y k = n ‘k
-c
- - 1-X
-
X
+
l/k + X = l+l/k,
THEORY OF BOUNDS FOR'ENTIRE FUNCTIONS
(1
-
X)y,
I
=-
+
265
xx
\IZk
1-X Iyk(1 + 1 and
- Xx/I * I lzkl I
+
1. 1 which implies
that
zk is a convex linear combination of x and
yk
0,l. It now follows from ( 4 . 2 ) that there is a
compact set K such that the distance d(yk,K) from yk to K tends to zero; equivalently d(zk,K) K
+
0. Since u is bounded above
-c
in
EB for some E > 0 this contradicts the inequality u(zk)> Mk
for k large, and this contradiction proves the theorem. The spaces t1 and co are not uniformly convex and the c o ~ clusion of Theorem 4.1 does not hold for them. In fact, we have the following two examples. EXAMPLE 4 . 2
The entire function
,x
f(x) = C e-kxl x;
E
e 1,
2
has radius of convergence Rf(x) satisfying Rf(x) = 1
+
Re x1
Re x1 Rf(x) = e Thus IRf(X)
-
Rf ( y ) I =
ples of el = (1,0,0, EXAMPLE 4 . 3
when
Re x1 2 0,
when Re x1 5 0 . when x and y are positive multi
1 Ix-yl1
...1.
Put $(x) = (1
- 1 1x1 1 ) + , x
E
co. The
continuous
function m
g(x) =
2
k 4(k(x
-
ek)), x E
Cor
where ek as usual is the k:th unit vector, is bounded for lxll) > E > 0.
Its radius of boundedness satisfies
C. 0 . KISELMAN
266
R(tel)
SUP (1tIrl)r
=
in particular R(tel) = It1
when
It\
1.
It is not by chance that the function q in the above anple is only continuous. For plurisubharmonic functions
exthe
conclusion of Theorem 4 . 1 still holds:
Let u
THEOREY 4 . 4
E
PSH(C~(J)) Luhehe. J i~ a n i n 4 i n i t e
b e t , and b u p p o b e t h a t u
4 0 % a&? x,y
PROOF
E
co(J)
u n b o u n d e d o n clome h o u n d e d b e t . T h e n
icl
IRU(x)
index
-
RU(y) 1 < 11x
-
YII
x # y.
with
It suffices to prove that RU(x) > 1
-
11x1 I if O < I Ix''
and RU(0) = 1. For a fixed x, let Jx denote the set of J for which Ix.I = 11x1 I, and put
indices j
E
We let
denote the (non-separated) topology of convergence of
T
3
all coordinates y
boundedness j E Jx. The ?-local radius of j' was introduced in [ 3 ] ; the definition is qiven followinq the statement of Theorem 2.2 of the present paner. Ey [ 4 ,
Proposi-
tion 3 . 5 1 we must have (4.3)
R
for all y
E
T r u
(y)
2 Ru(0)
F with I1yI
1
=
1
< 1, where we have written F for
subspace of co(J) consisting of all points y with
y
j j d Jx. Since Jx is finite, the unit ball of co(J) is compact, so Proposition 3 . 7 of [4] (4.4)
inf
I IYI 1<1
= 0
the for
T-quasi-
yields
RTrU(x + Xy) -< h
YEF
for all X > RU(x). Now R
ru
is obviously Lipschitz
continuous
THEORY OF BOUNDS FOR ENTIRE F U N C T I O N S
267
with constant at most 1 in the variables yj, j L Jx, so we
get
from ( 4 . 4 1 , writing n for the canonical projection onto F,
and
notinq that
Now
I Irr(x) 1 1
= 11x1 1
suppose that RU(x) = 1
and
-
1 \rr(x)- x (I
= a < 11x1
1,
11x1 1 . We can then let A-1-1 1x1 1+6
where 6 is any positive number. Hence
or equivalently
z EF whereas, by (4.3),
This means that the function
which is subharmonic in t 4(1) < 0,
E
C, satisfies
$(1+6) ?-log
(l-IJxJ !+a) > 0
for all 6 > 0. But Q, is a convex function of l o g It! so it must, in particular, be continuous whenever it is finite. (It is thus essential to note that $(t) is finite for t # 0). This diction shows that we must have RU(x) > 1
-
11x1 1 ,
c0ntr.a-
i.e.
desired conclusion. 5.
E S l ' ~ W t E S FOR THE LIPSCHITZ 03EISTW OF A RADIUS OF BDUNDEDNESS
the
268
C.
x # yI for all u
0 . KISELMAN
PSH(E) (such that RU <
E
of the spaces t P ( J )1, < p < A c
+m)
where E is
or co ( J ) , there is no
m,
any
constant
1 such that lRU(x)- RU(y) I 5 A1 Ix-yl I holds for all pluri@
harmonic functions u. This follows from the existence
theorems
summarized in Theorem 6.1 of the present
paper.
which gives the P Lipschitz constant for plurisubharmonic functions in L (J)
best
in [4],
of 2 4
We recall here another result from [4]
, l
and co(J) whose radius of boundedness decays slowly at infinity. The following theorem is slightly more precise than Propositions 3.9 and 3.11 of [ 4 ] .
Let E = t P ( J )1, <
THEOREM 5.1
an a h b i t h a h q i n d e x 15
u
E
PSH(E) and +m
det,
m,
oh E = c ( J ) , w h e h e J i d
0
and L e t F be a c o o h d i n a t e dubdpace o b E . 0, y
Xoh d o m e C >
2 0,
cI Ix11-y d o t x c o n d t a n t 06 RU i n F
> R~(X) 2
t h e n t h e Lipdchitz
whehe E = co cohhedpondd t o p =
SKETCH OF PROOF
p <
m
E
F, IIxI1
2 rr
i d a t modt
and A(m,y) = y/(l+y).
Let G be a finite-dimensional coordinate sub-
space contained in F. It suffices to prove that IRu(x)- Ru(Y) for x,y
G.
E
T
A1
Ix-yl
I
In the proof of Proposition 3.9 of [4] we used the
weak topology topology
I 5
a ( E , E ' ) when 1 < p <
m.
We now use instead
the
of convergence of all coordinates spanning G and let
The result now follows as in [ 4 ] ,
using the logarithmic convex-
ity of v (cf. also the proof of Theorem 3.1) .We omit the details.
THEORY OF BOUNDS FOR ENTIRE FUNCTIONS
REMARK 5.2
269
The c o n c l u s i o n o f Theorem 5 . 1 i s t r i v i a l l y
true
f o r p = 1, and g i v e s t h e n no i n f o r m a t i o n on t h e problem s t u d i e d i n Theorem 3.1. Theorem 3.1,
6.
For 1 < p <
m f
but not f o r p =
Theorem 5.1 c a n b e deduced
w r
from
E = c0 '
PRESCRIBING THE R A D I U S OF CONVERGENCE
L e t R: E
+
]08
+m[
be a g i v e n f u n c t i o n on a normed
E s a t i s f y i n g (1.1) and ( 1 . 2 ) .
Is i t t h e n p o s s i b l e t o f i n d
e n t i r e f u n c t i o n on E s u c h t h a t i t s r a d i u s o f convergence e q u a l t o R? R e s u l t s i n t h a t d i r e c t i o n were g i v e n i n [ 4 ] .
space an
is The
e x a c t d e g r e e o f a r b i t r a r i n e s s o f a r a d i u s o f convergence depend i n g on f i n i t e l y many v a r i a b l e s i n L p ( J ) o r c o ( J ) is shown
by
t h e f o l l o w i n g theorem.
Foh any g i v e n p
THEOREM 6 . 1 011 E = c O f p =
set i n minq
+m
,
E
11,
brr
t e t E = t p , I 5 p < +=,
h c d p e c t i v e t t j . L e t w be a pdeudoconvew
,p+1c o n t a i n i n g a t 1 p o i n t h z
en+'
+a]
E
with z
open
~ =+ O.Noa~
270
C.
open
bet w i n
en+’
0. KISELMAN
duch t h a t
RU(x) = dP(xl,...,xn~0).
The proof is to be found in [4!, (For 1 < p <
m,
4.1.
Theorems 3 . 8 and
the last part of Theorem 6.1 holds without
the
finiteness condition). There are two obvious restrictions in the first part
of
Theorem 6.1. On the one hand Rf factors throurJh a finite-dimensional space: on the other hand the finitely many variables on which Rf may depend are coordinates. Let us consider a
case
where Rf depends on just one complex variable which is not necessarily a coordinate: E = a.1, c(x) = C s.x a linear form on I J e 1, and Rf a function of c(x) only. We note that the unit ball is compact for the weak star topolorjy o(t
L
,
co) and
quasi-com-
pact for the topology generated by 5 , but not for the
topoloqy
generated by co and €, toqether if 5 pI co. On the other the inner and outer moduli as defined in [4]
hand,
do not agree
for
the topoloqy defined by one linear form. Thus there seems to be
no topology in t1 which is suitable for the results of [ 4 J , for these require the inner and outer moduli to be the same and the unit ball to be Tuasi-compact. In the absence of mre general results it is therefore perhaps of interest to see what can be
proved
in this particular case which falls outside the scope of
l4J.
THEOREM 6 . 2
l e t ,€ be u t i n c u l t 4 o h m o n
e 1,
( s j lml
5 =
E
ernl a n d
i n t l t oduc e
a = sup
lcjI
=
1 1 ~ I1
tm
and P = lim sup
.
lcjl
=
11c)I
tm/co
271
THEORY OF BOUNDS FOR E N T I R E FUNCTIONS
z
w i t h z2 = 0, and L e t d
a, P
(2)
he t h e d i b t a n c e , an meaduhed br!
ll-llfl,y
(horn z
hadiud
c o n w e q e n c e Rf d a ? i d f ( i Q d
04
Then thefie e X i d t d f
w t o aw.
E
Rf(x) = d,,p(S(x),O),
x
d
E
1
E
8(L1)
whode
.
I d o n o t know w h e t h e r t h i s t h e o r e m c a n h e improved,
I lz/ la,B
i.e.
r e p l a c e d by a b e t t e r norm: t h e i d e a l would b e
c o r r e s n o n d i n q t o n = 1 i n Theorem 6 . 1 . On t h e o t h e r b a n d , Theo-
r e m 6 . 2 i s a l r e a d y much h e t t e r t h a n what could be e x p e c t e d from t h e b e h a v i o r o f t h e i n n e r a n d o u t e r moduli w i t h r e s n c c t t o
the
toDoloqy g e n e r a t e d by 5 . I n f a c t t h e s e s a t i s f y
m(x) = 1 MIX) =
1
-
11x1
1,-
f o r 11x1
+ 11x1 l 1
€or
Il
I IxIIl
< 1, and < 1,
IF;(x) I
< P,
so t h e r e s u 1 . t ~oi [ 4 ] only y i e l d a n e n t i r e f u n c t i o n w i t h r a d i u s of c o n v e r g e n c e s a t i s f y i n q a n i n e q u a l i t y d ' ( 5 (XI r 0 )
5
Rf (X) < d'
I
( E (XI1 0 )
d ' (zl,O) < d " ( z l , O ) .
where
PROOF OF THEOREM 6 . 2
Let h
@(w)
E
b e a f u n c t i o n which
b e c o n t i n u e d anywhere beyond t h e boundary o f w w i s t h e domain of e x i s t e n c e :
see e . q .
cannot
( i . e . f o r which
Ilormander 12,
Theorem
4.2.81)
and expand h i n a n a r t i a l T a y l o r series a r o u n d ( Z ~ , O ) E O W k h(z) = h k ( z 1 ) z 2 , z1 E @, 1z21 s m a l l .
Then hk
E
0
(6.1)
d(C)
and
( l i m s u p Ihkll'k)*(zl)
=
a-(zToT 1
I
Z 1
E @ r
where d ( z ) d e n o t e s t h e d i s t a n c e from z t o a w i n t h e
direction
(0,l). D e f i n e m
(6.2)
where
e
f ( x ) = C h k ( S ( x ) ) x k , x E 1, m 0 "k ( n k ) O i s a s t r i c t l y i n c r e a s i n q s e y u e n c e of i n t e q e r s such
272
C. 0. KISELMAN
that
(cnkl
,!3
+
k
as
+
t
m.
We s h a l l a l s o need a sequence (mk);
(tending t o i n f i n i t y
or
c o n s t a n t ) such t h a t
lcmkl
+
a
k
as
+
+
and w e may o b v i o u s l y p i c k (n,)
m,
so t h a t mk # n
and (m,)
j
for a l l
k and j. We s h a l l f i r s t prove t h a t Rf ( a ) < a
2
d a r B ( S( a ) ,O).
L e t r c Rc
( S ( a ) , O ) . I t is t h e n enough t o show t h a t f is bounded
+
r B . Now r c R < d
(S(a),O) implies t h a t
d(zlrO,) > R
(6.3 and , p r o v i d e d (6.4
U rB
f? < a
d(Zlr0) >
and
aR
-
-
when
r
-
\zl
c(a)\5 B r,
is close enough t o R ,
( z l - F(a) 1 a - P
when p r
5
lzl-
2(a)1 5 a r .
If B = a, i t is enough t o c o n s i d e r ( 6 . 3 ) o n l y . I n view o f
so c a l l e d H a r t o g s ' lemma (Hgrmander [2, p . 2 1 ] ) ,
+
imply t h a t f o r l a r g e k, w r i t i n g x = a l h k ( c ( x ) )1 = l h k ( c ( a )
-I-r
c(Y))
and s i m i l a r l y w e o b t a i n from ( 6 . 1 )
k
=
(a "k
+
r y , w e have
1 5k 1
R and ( 6 . 4 )
lk
ryn k
the
(6.1) and ( 6 . 3 )
when 1 c ( Y )
I 5
f o r l a r g e k:
To e s t i m a t e f w e m u l t i p l y t h e s e i n e n u a l i t i e s by Xnk
in
,!3r
THEORY OF BOUNDS FOR ENTIRE FUNCTIONS
provided (6.6)
1En1
+
< fi
(true for large n) and
E
1 [ y [1 5
273
Thus
1.
gives, if p < a and k is large,
Now the right-hand side in this inequality assumes its for 15 (y)1 = p (if r
+
E
maximum
< R, which we assume) , so we have
where q < 1 for a suitable choice of
E
> 0. Now ( 6 . 5 ) and ( 6 . 7 )
show that series defining f converges uniformly for x
E
a
+
rB,
hence that Rf (a) 2 r. Conversely, assume that f is bounded in a + RB. We then prove that d
U I B
( 5 (a),0) 2 R , which will yield daIB(5(a),O)?
+ t''R'Ie , where en denotes "k mk usual the unit vector of index n, and where 1 t ' 1 5 t' ' 1 5
> Rf(a). For x = a -
t'R'e
+
R"
(~(x)1( IS(a)l
+
0 < R' < R
Now
+
shall
< (c(a)I +
and R '
Ra
as 1I
= R we get
R'a
+
R l ' a = (S(a) 1
+
R a , and for lzll
5
we have an estimate of the form
tends to zero this shows thatthe j sum in ( 6 . 8 ) is bounded. It follows that the first term to the
for some constant
C.
Since an
right in ( 6 . 8 ) must be bounded for x large that
la "k
1 5
E,
E
a
+
R B I hence, for
k so
274
C.
where Rk = R'1c
I+
R"15
nk number smaller t h a n R'$
let r = RIB Rk
2 r
+
R"a
0.
mk
+
I.
KISELMAN
L e t r be an a r b i t r a r y
i f the l a t t e r is positive,
R"a
+
=O i f R ' P
positive
R"a
=O
( i . e . R"=
and
p =O).
Then
f o r l a r g e k so t h a t sup
Ihk(E(a)
+
It15 1
&k
tr)I1lk -< ---R' -
E
which i n view of ( 6 . 1 ) g i v e s inf
Itl< 1
d(c(a)
+ tr,
0)
> R' --
S i n c e r i s a r b i t r a r i l y c l o s e t o R'$
+
-
E.
R"a
and R '
-
t r a r i l y close t o R ' t h i s means t h a t d a r B ( E ( a ) , O ) ? -R ' I n d e e d , w i t h R"
arbi-
is
E
+
R"
=
= 0 w e have R ' = R and
where r is a r b i t r a r i l y close t o RP; t h i s i s what we need
f3 = a. I f f3 < a
if
w e have i n a d d i t i o n t o ( 6 . 9 ) :
- -
i n f d ( g ( a ) + t r , O ) ->. R ' = Ra r 6 Itl< 1 a - B where w e have d e f i n e d 6 by t h e e q u a t i o n r = R ' $ + R " a (6.10)
R.
-
6 ; now
( 6 . 9 ) and ( 6 . 1 0 ) t o q e t h e r mean t h a t d a r B ( [ ( a ) , O ) 2 R.
We c a n t h u s n o t e t h a t i n t h i s s p e c i a l s i t u a t i o n a t l e a s t , t h e t e c h n i q u e used i n p r o v i n g Theorem 4 . 1 i n [4]
survives,
in
s p i t e of t h e f a c t t h a t t h e r e i s no u s e f u l r e l a t i o n between
the
r a d i u s o f boundedness and t h e l o c a l r a d i u s o f
boundedness.
Indeed, h e r e t h e boundary d i s t a n c e d ( c ( a ) , O ) serves t h e pu,rpose as t h e l o c a l r a d i u s of boundedness i n [ 4 ] ,
but we
o n l y d e f i n e it f o r f u n c t i o n s o f t h e s p e c i a l form ( 6 . 2 ) .
same can
THEORY OF BOUNDS FOR ENTIRE FUNCTIONS
275
REFERENCES
Dl
GLOBEVNIK, J., On complex strict and uniform
convexity.
Ptroc. Amelr. Math. S O C . 47, 175-178 (1975).
PI
HORMANDER, L., An i n t t r o d u c t i o n t o complex a n a t l j n i d i n n e v g h a l v a f i i a b l e n . North-Holland, 1973.
131
KADEC, M.I., Spaces isomorphic to a locally uniformly con. no. 6 vex space. l z v . V y d ~ . U ~ e b n . Z a v e dMatematiha (13), 51-57 (1959), and correction no.6 (25),186187 (1961).
[4
1
KISELMAN, C.O., On the radius of convergence of an
entire
function in a normed space. Ann. Polon. Math. (1976), 39 153
33
- 55.
LELONG, P., Fonctions plurisousharmoniques dans les espaaes vectoriels topoloqiques. Lectutre N o t e n i n MaAhemc t i c 4 71 (1968), 167-190. MARTINEAU, A., ohal communication, 1968. NACHBIN, L.
, Topotogg
o n dpaced 0 4
hoeonotrphic
mappingb.
Springer-Verlaq, 1969. NOVERRAZ, P., P d e u d o - c o n v e x i t ; , c o n v e x i t ; domained d ' holomohphie
en
po!ynomia4?e
dimendion
et
indinie.
North-Holland, 1973. 193
TROYANSKI, S.L., On locally uniformly convex and different iable norms in certain
non-separable
Banach
spaces. S t u d i a Math. 37, 173-180 (1970-71). C. 0. Kiselman Dept. of Mathematics Sysslomansgatan 8 S- 752 23 Uppsala, Sweden
This Page Intentionally Left Blank
Infinite Dimensional Holomorphy and Applications, Matos (ed.) 0 North-Holland Publishing Company, 1977
HOLOMORPHIE ET T H E O R I E DES D I S T R I B U T I O N S EN DIMENSION I N F I N I E *
KREE
Par P A U L
Que peut Gtre la thgorie des distributions sur un e.t.cs. reel Y de dimension infinie? Peut-elle servir dans l'etude
de
l'holomorphie en dimension infinie? La transposition de la thsorie d6veloppge il y a une tren taine d'annies par L. Schwartz n'a pas et6 immgdiate, peut &re parce que cette transposition fait apparaftre simultanhent
de
nombreux ph6nomZnes spgcifiques 5 la dimension infinie.Par exem_ ple, si @ est une fonction l. fois Frechet derivable sur Y, d6riv6e ' D
@(y) ne prend Das ses valeurs dans 8 (Y"),
e
03
la Yvc
dgsigne le cornalexifig du dual de Y; mais dans un espace vectoriel plus grand. I1 existe A'ailleurs en dimension infinie d'ay tres calculs differentiels que celui de Frechet, par exempleles calculs differentiels de Gateaux et de L. Gross. Ainsi, la premi&e
question qui se pose est de choisir un calcul diffgrentiel
pour definir les fonctions d'6preuve. I1 est naturellement pr6ferable de n'en choisir aucun
s
~ J L ~ O ceR qui ~
277
motive la dgmar-
P. KF&E
2 78
che suivante [7].
Soit Y un espace de Banach et soit
N un cer-
.
Pour
L = 0, 1, 2... le compldt6 de @(YfC) pour la norme N est N EL. Au dgpart, toutes les fonctions d'6preuve considgr6es
not6
tain type de normes tensorielles, par exemple
II, E.....
sont
suppos6es cylindriques, ce qui permet d'utiliser seulement
le
calcul diffCrentie1 de Newton et de Leibnitz. Pour chaaue choix de la famille (E!),
une th6orie des distributions est ddvelopp&
en utilisant le formalisme des prodistributions et des seurs distributions [a].
proten-
Les op6rations sont d6finies par
des
arguments de transposition analoques 5 ceux de la thdorie de L. Schwartz. Fuis, en utilisant un argument de hitransposition, 5 N chaque choix d'une famille ( E L ) t , il est associd un calcul dig fgrentiel gdndralis6 sur Y.(Les fonctions d6rivables de
cette
th6orie sont suppos6es universellement L u s h mesurables, mais elles ne sont nas sunposdes continues). hinsi, le calcul diff6N = @- Y rentiel de Frechet est g6ndralis6 en prenant EL ', le caL
e
cul diffgrentiel de L. Gross est prolonq6 en prenant un triplet N = 6 - Xc.. de Wiener Y 'C XCY et EL ce qui donne d6j; une applL L cation de la thdorie des distributions.
.
Dans la confirence au colloquium, il a 6td tent6 de
don-
ner un aperqu d'ensemble de la th6orie. Ceci est assez long car le seul concept de distributions relatif 5 la dimension
finie
se diversifie en dimension infinie, en six concepts diffdrents:
-
prodistributions protenseurs distributions espaces de Sobolev scalaires espaces de Sobolev vectoriels
- distributions
HOLOMORPHIE ET THEORIE DES DISTRIBUTIONS
-
279
tenseurs distributions
Css deux dernizres th6ories ont &6
d6velopp6es jusau'ici seule
ment dans le cadre banachique [5] [ E l . normes tensorielles
flatUht?!~eA
sur
Et vue la diversit6
@Y",
a.
des
il apparait encore des
sous-classes. Ainsi dans le cas d'un triplet de Wiener, Y &ant suppos6 hilbertien, il apparalt trois types d'espaces de Sob&
scalaires et neuf types de distributions scalaires sur Y 1 Le texte aui suit est limit6 2 l'exposg de rgsultats rela tifs 5 deux cas particuliers qui sont reli6s 5 1'holomorphie;il s'agit de l'gtude des profonctionnelles analvtiques, des
fonc-
tionnelles analytiaues (avec une application 5 la physictue); et de l'gtude des distributions et du calcul diffgrentiel sur e.L.c.s.
Cette dernisre gtude est motivge nar le r6sultat
Grothendieck montrant la coincidence des topolosies
TI
et
des de
E
sur
B Y t c si Y'' est nuclGaire, d'oc un seul calcul diffgrentielra2 L sonnahle dans ce cas: alors qu'il y a beaucoup de calculs diffg rentiels si Y est un espace de Banach non nuclgaire. I1 est ra& sonnable de nenser que ceci est li6 au fait que
l'holomorphie
pour les espaces 5 dual nuclgaire semble se formuler plus facilement que l'holomorphie concernant les espaces de Banach
de
dimension infinie.
1. FONCTIONNELLES ANALYTIOUES DE TYPE EXPONENTIEL Ce paracjraphe prgsente une extension de la thgorie
tono-
loqique des mesures Ae Radon aux fonctions entigres de type exponentiel. Cette extension est motiv6e par la mgcanicrue quantique
.
(1.1) NOTATIONS Si Y est un espacc vcctoriel et si 0 est m e toplogie
P. KPGE
280
localement convexe sur Y, (Y, 0 ) dgsiqne l'espace convexe correspondant et (Y, 8 ) ' (Y, 8).
localement
d6siqne le dual topoloqiquede
s i x dgsigne un espace tovologique compGtement
lier, B0 (X) dgsiqne l'espace des fonctions continues 0
regubornges
.
@ : X-t C . Soit f3 la boule unit6 de B (X)
(1.2)
B = 19 E B o ( X )
;
I1911,
= SUP
I$ (XI) 5
1)
L'espace M(X) des mesures de Radon bornses sur X est l'es pace des mesures complexes born6es m sur la tribu borglienne de
X, telles que pour tout
E
> 0 , il existe une partie compacte
de X telle que Iml (X\K) 5
E.
K
0
Soit tk la topoloqie sur B (X) de
la convergence uniforme sur les parties connactes de X. La t o p logie stricte T sur B 0 (X) est la topoloqie localement 0
la plus fine sur B (X) qui coincide avec tk sur [3] que M(X) = ( R 0 (XI,
convexe
On sait
f3.
1_2]
TI'.
(1.3) INTRODUCTION DE POIDS
Des topologies strictes seront d g
finies sur des espaces de fonctions continues, de manigre
in-
duire des tonologies sur certains espaces de fonctions entikes. C o m e une fonction entisre born6e est constante, des poids sont introduits de mangxe 5 permettre une certaine croissance 5 l'ic fini. Soit 2 un espace de Banach r6el et soit m un entier positif. L'espace C Expm (2) est l'espace des fonctions continues sur
2
@
telles que
(1.4)
Il@llm
= SUP
I$
(211
exp(- n llzll)
Cet espace a une boule unit6 pm = I $ , gie tk, d'oG une topologie stricte Tm (C Exp"(2)
,
T ~ ) est
1 I @I l m 5
m
11, une
. Le dual M
topolo-
Exnam(Z)
l'espace des mesures de Radon
IJ
sur 2
de tel-
HOLOMORPHIE ET THEORIE DES DISTRIBUTIONS
281
les gue
L'espace C Exp(Z) =
u
(C Exp"(2)) est l'espace des fonctions m continues sur Z, 5 croissance exponentielle. 11 peut &re muni
6 = lim T ~ .Et le dual M ExT)' ( 2 ) de (C Exp&)B) + dgcroissance exponentielest l'espace des rnesures de Radon 1.1
de la topologie
le, c'est-5-dire des mesures 1.1
E M(X)
qui vgrifient (1.5)
pour
tout m > 0. (1.6)
FONCTIONNELLES ANALYTIQUES DE TYPE EXPONENTIEL S o i t Z un
ebpace d e Banach c o m p l e x e , e t
bait
Exp(2) l e b o u b - e b p a c e v e c t o -
hieL t o p o ~ o g i q u ed e (C Exp(Z) , 6) {ahrnz pah t e b XOnCtiOnb e n t i -
a" ChOibbanCe e x p a n e n t i e l e e . L e d u a l EXD' (Z) d e Exp(2)
heb
ebt
a p p e l z l ' e b p a c e d e b d o n c t i o n n e e l e b Una.tf/tiqUeb d e tr4pe exponen-
tiel
bUh
Z.
Soit 8' la trace de 6 sur Exp(2). Une application du
thgor6me
de Hahn Ranach donne immgdiatement la: (1.7) CARACTERISATION DE Exp'(2f
Soit T
une
40hme
k?inEaihe d z -
{ i n i e bu4 un bOUb-ebpaCk? denbe d e (Exp(Z), e').ALohb T E Ex@((Z) b i
e t beuLement
b i
T e b t h e p h z b e n t a b l e p a t une mebuhe de
Radon
a" d z c h o i b b a n c e e x p o n e n t i e e l e . (1.8) TRANSFORMATION DE FOURIER (T.F.) Pour tout 5 E Z', fonction e
5
: z
+
exp(-
J-7: z
5 ) appartient 5
d6signe la forme bilineaire de dualit6 entre
2
EX^ C, oG
la z 5
et Z'. Alors 4 T
est par definition la fonction suivante sur Z' :
JZ
03 l'intggrale d6slqne symboliquement le rgsultat de
l'action
P.
2 82
de T sur e
(1.9)
KREE
5'
HYPOTHESE (H) On d i t que L'edpace d e Banach
L ' h y p o t h z n e (H) b ' i k ? e x i n t e une d u i t e (u,)
Z
d ' o p i i h a t e u h n d e hang
S i n i d e Z, d e nohmen unibohmement m a j O h Z e d , t e n d a n t vehb
ve'hibie
dimplement
L'OpZhateUt identique d e Z.
(1.10) LEMME DE DENSITE p o t h z n e (H), ae0h.A Exp
CYl
-_
S i L'edpace d e Banach Z we'hidie L ' h y (Z) e t Pol (Z) CYl
den n o u d - e b p a c e d
AOnt
dended de Exp(2).
Par d6finition Exp (Z) et Pol (Z) sont les sous-espaCYl CYl ces de Exp(Z) constituk respectivement par 1es fonctions cylin driques et par les fonctions polynomiales cylindrlques. PREUVE
a) Pour f
E
Exp(Z), soit fn(z) = f(un
vient pour tout n et pour tout z
E
Z
2 ) .
Alors
il
:
De plus d'aprss [12] par exemple, la suite (f,)
converge vers f
uniformgment sur toute partie compacte de Z. Donc (f,)
converge
vers f pour 8'. b)
Vu a), il suffit de montrer l a densit6 de Pol(Z)dans Exp(Z)
si Z est un espace vectoriel de dimension finie. I1 suffit d'ap procher f par les sommes partielles de son dgveloppement
de
Taylor.
(1.11)
PROPOSITION
Ln thanAbohmiie d e Fouhieh d e t o u t e
(onc-
t i o n n e t t e a n a l y t i q u e d e t y p e exponentice e4.t e n t i h e . la t h a n d -
dotmation de Fouhieh
edt
injective d i L ' e ~ p a c e d e Banach Z
wt?&
d i e (HI.
DEMONSTRATION
a) D'aprss (1,7), 11 existe p
E
M Exp(2)
telle
HOLOMORPHIE ET THEORIE DES DISTRIBUTIONS
283
que
+(el =
(rTI
(5) =
La fonction $ est bornge sur les born&
/e-izC
du (z)
de Z' car
Or pour tout nombre cornplexe u
Donc Ceci prouve que
.?I
est C-d6rivable; et
est enticre et de
plus
D '?(a) = $(O). b) Par rgcurrence sur l'entier k, on peut montrer:
J
z) dp(z) k Si donc $ est nulle, T s'annule sur les fonctions polynomiales Dk ?(O)
= ( - i)k
(@
cylindriques. Vu le lemme de densit6, T est identiquement nulle. (1.12) IMAGE PAR UNE APPLICATION LINEAIRE
Soient Z et U
espaces de Banach complexes et soit X une application continue de
z
deux
lingaire
dans U. L'application lingaire:
est continue. Par transposition, une application a ' de dans Exp'(U) est dgfinie. Pour toute T
X est d6finie par
A T = a'(T), soit
E
Exp'(Z)
Exp'(Z), son image par
284
P.
KREE
Ceci entraine la relation suivante entre les transformges
de
Fourier de T et de 1 T
(1.13) PRODUIT PAR UN ELEMENT
Exp(Z)
(J E
C o m e en thgorie des
distributions, le produit 4 T est d6fini par < I$ T, J, > = < T, (J
8 >
pour toute J, de Exp(Z) (1.14) PRODUIT TENSORIEL
Soient Z1 et Z2 deux expacesde Ban&
complexes. Alors le produit tensoriel des deux formes lingaires associges 5 T1 E Exp'(Z') et 5 T2 E Exp'(Z 2 ) est une forme lingaire X sur le sous-espace E = Exp(Z 1 )@Exp(Z 2 ) de Exp
z = z1
x
z2 .
ri est supposg que
z
(2)
vgrifie (HI; il en est
avec par
exemple ainsi si Z1 et Z2 vgrifient (H). Alors E est un sous-es pace
dense de Exp(Z), car si T
E
Exp'(Z) est orthogonale 5
E,
alors F T = 0 , ee qui entraine l'annulation de T d'aprss (1.1). De plus, si v
j
reprgsente X .
reprgsente T alors v l @ p Z E M Ekp(Z) j' Donc d'aprgs (1.71, X E Exp'(Z). I1 est pos6 E
M Exp(Z')
X = T1 Q T2.
(1.15) FORMULE DE FUBINI L.SCHWART2
Si 2
1
et Z2 v6rifient (HI,
alors
pour toute 41
E
Exp(Z)
.
En effet: Soient respectivement (u,)
et (vL) des suites d 'op6ratEurs
de rang fini de Z1 et de Z2, de norme au plus un, tendant vers
HOLOMORPHIE ET THEORIE DES DISTRIBUTIONS
285
l'ldentltb unlformgment sur toute partle compacte. Alors T1(X) 4 2 cP(Uk X'"k I1 suffit alors de faire vendre k et 1 vers (1.16) ILLUSTRATION EC = E
+ fl E
+
Y)d T2(Y) m.
Solt E un espace de Banach r6el et
solt
le complexlflg de E. Soit m une mesure de Radon
sur E 5 dgcrolssance exponentlelle et soit m' = m (X) 8 6 o (y)1 ' ~ m 5 EC. Solt 8 un angle quelconque, et solt
tension de
la rotation d'angle
8
EC
dans EC EC
___.)
z ei e
z=x+Ciy-+ Alors cette transformation
QI
-
lln6alre de EC transforme m'
une fonctlonnelle analytlque R(B)m de transformge de G(ezie
2).
R(8)
en
Fourier
Cecl prolonge en dimension lnflnle un rkultat
blen
connu dans le cas particuller 06 E = IR; m =
1~
-1/2
exp(- 2)
X
L
dx ;
II
e = 3
En effet, dans le tome 1 de leur traitg, Gelfand et Silov ont not6 que la fonctlon ex?
XL 7 est
la transformge de Fowler
d'un Glgment de Z ' ( l R ) reprgsenth par l'extenslon au plan
com-
plexe d'une mesure gausslenne sur l'axe lmaglnalre. (1.17)
I1 faut noter, m6me dans ce cas partlculler, que R(e)m
aglt non seulement sur les glgments de Exp(EC), mals sur
des
fonctlons @ beaucoup plus ggngrales. En effet, 11 sufflt que la trace de
+
sur la var16t8 lingatre de EC dgdulte de E par
R(0)
puisse 6tre d6flnle; et gue cette trace solt lntdgrable par rap port R ( e ) m
.
2.
KREE
P.
286
PROFONCTIONNELLES ANALYTIQUES DE TYPE EXPONENTIEL Le concept de promesure qbnbralise le concept de
mesure
de Radon. De la m6me manisre, la notion de fonctionnelle analytique de type exponentiel est prolongbe par la notion de
pro-
fonctionnelle analytique. soit
(2.1) NOTATION
plexe et soit F,(Z)
Q
m6s de codimension
z
un espace localement convexe &par6
I la famille des sous-espaces feg
= (Ai)i
finie de Z, ordonnge par l'ordre inverse de
celui ddfini par la relation d'inclusion: i Soit s
ij
la surjection canonique de Zi = Z /
j signifie A i c T . Ai
sur J j = Z /Aj,
ddfinie si i 2 j. Soit si la surjection de Z sur Zi.Pour i on a une injection de Exp(Z
2
DEFINITION DE ExpIcyl(Z)
Une p h o d o n c t i o n n e t t e
fonc-
anatyti-
~ u de e .type e x p o n e n t i e t T b u h Z e b t u n e d o t m e t i n i i a i h e
Exp cyl(Z) d o n t
j,
dans Exp(Zi) , et la limite induc-
.)
3
( 2 ) de ces espaces s'identifie 5 un espace de tive Exp CYl tions cylindriques sur 2.
(2.2)
corn_
hebthiction
2
bU h
c h a q u e Exp(Zi) e b t h t p h & t n t g e
p a h u n e d o n c t i o n h i i e t t e Ti E Exp' (Zi). l ' e n b e m b l e d e ceb d o t m e 6
tinzairreb
ebt
.
n o t ; ExpAyl (Z)
Pour JI = Jli o si (2.3)
E
Exp (Z), on bcrit CYl
< T , J I > =
D'une manigre dquivalente, Exphyl(Z) est l'espace des T = (Ti)i avec Ti E Exp'(Zi), ces familles &ant
familles
cohsrentes
au
sens suivant: (2.4)
i > j
Tj = SijfTi)
D'une manisre bquivalente, on peut munir pour tout i,
l'espace
HOLOMORPHIE ET THEORIE DES DISTRIBUTIONS
287
de la topologie "stricte" sur C Exp(Zi),
Exp(Zi) de la trace 9;
Alor s (Z) peut &re muni de la topologie m:l 0;. puis Exp CYl (Z) apparait comme le dual de l'espace (Exp (Z),lim 0;). + Exp;yl CYl (2.5) INJECTION DE Exp' (Z) dans Exp' (Z) CYl relle (2.6)
(ExPcyl(z), lim e ; )
LJI
a) L'injection natg
(Exp(2),
e l )
est continue. En effet, vue la propri6t6 universelle des
limi-
tes inductives (P U L I), il suffit de montrer que pour i
E
tout
I, l'injection
(2.7)
(Exp(Zi1 t 0; 1
-
(EX~(Z),e l )
est continue. I1 suffit de montrer la continuitg de
( c Exp(zi),
ei)
(c EXP(Z), e l )
Vue P U L I, il suffit de montrer la continuit6 de (C Expm(Zi), Zy) e
JT
( c EXP(Z),
e l )
Vue P U L I, bl suffit de montrer la continuit6 de la tion de J T 5 la boule unit6 de C Expm(Zi). Vu
restric-
un lemme
de
Grothendieck il suffit de montrer la continuit6 5 l'origine; et ceci est clair. b) Si 2 est un Banach v6rifiant (H), alors J a une image
dense
d'aprgs (1.10) Par cons6quent, dans ce cas, la transpos6e de est une injection de Exp'(2) dans Exp' (Z). Les 616ments CYl ExpAyl(Z) appartenant 5 Im J sont caract6ris6s par (1.7). (2.8)
J de
OPfiRATIONS USUELLES SUR LES PROFONCTIONNELLES ANALYTIQUES
DE TYPE EXPONENTIEL
La transformation de Fourier, l'image par
une application lin6aire continue, la produit tensoriel la convolution se d6finissent naturellement. Par exemple la
T.F.
de
288
T
E
P.
KREE
Exp' (Z) est la fonction suivante d6finie sur le dual CYl
Z'
de Z
?
(2.9)
Soit
(5) =
'
/e--
dT(z)
une application linbaire continue de Z dans l'espace lo-
calement convexe complexe U.
L'application I$ -+
.
ExpCyl (U) dans Exp (Z) Et l'image X T de T CYl d6finie par V $
(2.10)
E
< X
Expcyl (U)
$ o X
applique (Z)
E
est
T,I$>=
Pour d6finir la produit tensoriel T = T1 d T2 des TjE Exp' (Zj), CYl on introduit d'abord les systgmes projectifs (Zk)k 1 et (Zl)l 2 associ6s respectivement 5 Z1
x
Z2. On veut dbfinir
< T I J, > = < T 1 0 T 2 1 J, >
(2.11)
pour toute $
Exp (Z) avec Z = Z1 x Z2.Comme $ admet une baCYl se du type Zk1 x Zt, J, admet la factorisation: E
Le produit de convolution T t U de T
Exp' cyl. (Z) et U E E q k 1 (Z) est difini c o m e 6tant l'image de T 0 U par l'application s m e
xI y+
x
+
v
de Z
*
E
Z dans 2.
(2.13) RADONIFICATION DES PROFONCTIONNELLES ANALYTIQUES L'injeg
tion canonique (ExpCyl(Z), lim 0;) -c
L--t (C Exp
CYl
(21,
nr:l
0,)
est continue. Donc toute promesure 3 dbcroissance exponentielle p
sur Z d6finit canoniquement une profonctionnelle analytigue
de type exponentiel. Soit X une application linbaire
continue
HOLOMORPHIE ET THEORIE DES DISTRIBUTIONS de Z dans un Banach U transformant dgcroissance
289
en une mesure de Radon
exponentielle. Alors X ( i )est une
5
fonctionnelle
analytique de type exponentiel d'aprss (1.7). Ainsi, les rgsultats de la thgorie des triplets de Wiener de L. Gross, ou de la thgorie des applications radonifiantes de L. Schwartz
permet-
tent de montrer aue certaines profonctionnelles analytiques scnt transformges par certaines applications lingaires en des
fonc-
tionnelles analytiques de type exponentiel. (2.14) APPLICATIONS t
Le fait que le changement du temps
t
en
permet de passer de l'gquation de diffusion de certains
processus de Markov
des 6auations du type Schrodinger, entrai
ne une analogie entre thgorie des diffusions et mgcanique quantique. Cette analogie conduit 5 rechercher l'analogue quantique de la mesure de Wiener. On rappelle que cette mesure P s'obtient par radofification de la promesure normale canonique v de l'espace de Hilbert r6el:
avec I = LO, 1! par exemple. Le nroduit scalaire dans X &ant 1 < @, > = act) $(t) dt. Plus prgcisgment, P est l'imaqe $J
de v par l'injection canonique X de X dans l'espace de rgel Y des fonctions continues sur I. Ceci a conduit C.B.
Banach De
Witt [I] 5 dgfinir la pseudo-mesure de Feynman W sur X c o m e h la collection des mesures Wi s u r les sous-espaces X i de dimension finie de X; 06 Wi admet pour transformge de Fourier la re5 triction S Xi de la fonction suivante d6finie sur X W (x) = exp 1- --
Siqnalons que la thgorie des prodistributions permet d'interprg
P. KREE
290
ter W comme une prodistribution 5 ddcroissance rapide sur X; et ainsi, W dbfinit une forme liniaire sur l'espace des cylindriques 2 croissance tr6s lente sur X
: [8].
fonctions
Le
de la "radonification" de W consiste 5 trouver une
problsme
application
lin6aire continue de X dans un certain espace de Banach E, trag formant W en une forme lin6aire sur un vaste espace de fonctions non cylindriques sur E. Une solution de ce problhe est
donn6e
par la thgorie qui prbcsde, en considGrant W c o m e une profonctionnelle analytique de type exponentiel sur le complexifi6
Xc
de X.
La c o m p l e x i d i c e Ac d e l'injection X de X dano
PROPOSITION .thanb(Ohme W bUJt
Y
en une donctionnelle analytique d e .type e x p o n e n t i d
Yc. La remarque (1.17) montre d'ailleurs que Xc(W) dgfiniture
forme linbaire sur une classe beaucoup plus vaste que Exp(YC). DEMONSTRATION
L'application Xc peut &re
consid6r6e comme
la
composde de trois applications:
-
La rotation d'angle
- ;r7l
dans Xc, transformant W en
la
promesure normale canonique v.
-
l'injection ,Ic transformant v(x) C3 6 0 ( y ) en la mesure
Radon P(x)
-
@
de
60(y) sur Yc.
la rotation d'angle
'II
dans Yc, transformant P(x)
@
60(y)
en une mesure de Radon sur Yc 5 d6croissance exnonentielle, COG centrde sur un plan faisant un angle de
avec Y.
Par consbquent, vu le th6orsm de reprgsentation (1.7),Xc(W) est une fonctionnelle analytique de type exponentiel sur Yc. Pour d'autres applications B la physique voir [lo].
HOLOMORPHIE ET THEORIE DES DISTRIBUTIONS 3.
291
DISTRIBUTIONS ET CALCULS DIFFERENTIELS GENERALISES
RELA-
TIFS A DES ESPACES LOCALEMENT CONVEXES Dans [7] ont gtb pr6sentgs la thgorie des
distributions
et les calculs diffgrentiels ggngralisgs pour les espaces
de
Banach. Le but de ce paragraphe est d'indiquer comment s'effectue le prolongement de ces r6sultats aux espaces localement con vexes. On se limite aux distributions borndes d'ordre
borng;
pour dgfinir des distributions non bornges d'ordre quelconque, il suffit d'utiliser la techniaue de localisation dgveloppge au paraqraphe 1 de [ll]. (3.1) MESURES DE RADON VECTORIELLES
La thgorie des
distrihu-
tions de L. Schwartz est un prolongement du thgorzme de
Riess
concernant la reprgsentation des mesures de Radon scalaires sur un compact K c o m e formes lingaires sur B0 (K). La thgorie
des
distributions en dimension infinie a pour point de dgpart la re_ prgsentation comme formes lingaires des mesures de Radon vectorielles sur des espaces comnl&ement
r6quliers: voir [5],[6]dans
le cas de mesures 5 valeurs dans des Banach et [ 4 ] dans le
cas
de mesures 5 valeurs dans des espaces localement convexes. Soit
x
un espace cornpliitement rdgulier et E un espace lo-
calement convexe. Soit BK(X, E) l'espace des fonctions dgfinies sur X, 5 valeurs dans E, dont l'image est une partie ment compacte de E.
A.
relative-
Katsaras introduit une famille filtrante
croissante ( p e ) de semi-normes dgfinissant la topologie Pour tout 1
E
L,.
d6siqne la tonoloqie localement
de E . convexe
sur BK(X,E) telle aue l'origine admette pour base de voisinages la famille des convexes disqugs absorbants W tels que pour tout r > 0, il existe un compact K de X et n > 0 avec
P.
292
{$;
sup {pe($(x)), x
E
K} <
KREE rl;
sup Ipe($(x)), x
E
XI < r ) C W
Puis la topologie localement convexe B est dgfinie
come
la limite projective des topologies Be. Si B est la tribu de Baire de X I Me(X,E') est dgfini corn_ me l'espace des fonctions additives d'ensemble m: B
de X et pour toute famille (siIide vecteurs de E
E' telles
-c
tels
que
pe(si) 5 1. Le dual de (BK(X,E), B ) est le sous-espaces M(X,E') de la rgunion des Me(X,E') formbe par les mesures vectorielles tendues. ( 3 . 4 ) LA SITUATION GEOMETRIQUE
Soit Y un e.C.c.s.
rgel
et
(Yi, sij) le systgme nrojectif d'espaces de dimension finie intervenant usuellement en thgorie des probabilitgs cylindriques: (YiIi I est la famille des quotients de Y par les sous-espaces fermgs Ai de codimension fine de Y; la surjection s :Yi+Y ij j est d6finie si i 2 j, c'est-5-dire si A i C A Soit k un nombre j' qui est ou entier 2 0, ou gqal 5 + m. Soit Bk (Yi) l'espacedes fonctions Qi
: Yi
*
Q! de
k classe C , dont les dgrivges
au plus k sont bornges sur Yi. Donc pour tout
e
5 k, la dgrivge
De 9, est une application continue bornge de Yi dans O Y ; ' . k
d'ordre
e
On
note Bk (Y) la limite inductive des espaces B (Yi), car cette CYl limite inductive s'identifie un espace de fonctions cylindrik ques sur Y. Plus prgcisgment pour $i E B (Yi),posons $=$, o si. La dgrivge Dc $ est une fonction vectorielle cylindrique admettant la factorisation Y (3.5)
-~
HOLOMORPHIE ET THEORIE D E S DISTRIBUTIONS
293
(3.6) DEFINITION DES T-DISTRIBUTIONS Supposons donnge pour T < k une complgtion El de 0 Y r C relativement 2 une certai tout e -
e
ne topologie localement convexe. On peut prendre par
sur les produits tensoriels T T T Q3 YIc. Soit T la faille des espaces Eo, El, E2 Noter que T :E = C, mais que El peut diff6rer de Y". On introduit le plonles traces des topologies
'IT
ou
exemple
E
.. . .
gement canonique
cb H($, D 4 ,
k D cb)
Chaque facteur est muni de la topologie induite par la topoloT Et BEy1 (Y) est muni de la topologie gie stricte de Bk (Y, El). k 8 induite par la topologie produit. L'espace B,+k(Y) des T-distributions borndes d'ordre au plus k est le dual de (Bk (Y),&. CYl (3.7) PLONGEMENT DE B,+~(Y)DANS LES PRODISTRIBUTIONS
Pour tout
i , on a un plongement canonique
k S i chaque facteur est muni de la topologie stricte et si B (Y,) est muni de la trace Bik de la topoloqie produit, alors le dual distribuBtk(Yi) de (Bk (Yi), 8,)k s'identifie 5 l'espace des tions intggrables au sens de L. Schwartz, d'ordre au plus
k.
Par transposition de la bijection continue
on obtient une injection canonique de B&k(Y) dans B t k (Y). En CYl l'espace particulier, pour k = 0, on obtient une injection de
P.
294
KREE
Br0(X) = M(X) des mesures de Radon bornees sur X, dans l'espace (Y) des promesures bornees. I1 faut noter que pour k # 0 et CYl pour Y de dimension lnflnle, on obtlent un phenom6ne analogue 5 B'O
celul obsev6e par L. Nachibln [13] en theorle de
l'holomorphle
en dimension lnflnle. Et pour divers cholx de T on obtlent
dl-
vers types de distributions. (3.8) STRUCTURE DES DISTRIBUTIONS
une
AOmme
de divetgenced de
mebU4eA
T o u t e T-dibtftibutionb
ebt
d e Radon v e c t o t i e l l e ~ .
La demonstration de ce thiorsme resulte du theor6me
de
Hahn Banach et de la thgorle des protenseurs distributions.
En
effet, pour toute U E B,i,k(Y), 11 exlste k' fin1 5 k et des mesu T ' 1 telles que res p E M(Y, (El) j
D ' O ~le th6orGme. (3.9)
THEORIE
DES DISTRIBUTIONS ET CALCUL DIFFERENTIEL
famllle T peut 6tre associi un calcul differentiel
A
chaque
g6n6ralIse
sur Y. La construction de ce calcul dlffgrentlel est tr&
diff5
rente de l'extenslon du calcul diffgrentiel dbdulte de la theorie des distributions en dimension flnle. Le principe consiste
5 partlr d'une opgratlon ttiviale de derivation
Mais c o m e cette application linealre a est continue pour
des
topologies du type strlcte, a est prolongbe par sa bltransposee
a". Cette bitranspos6e donne l'extenslon voulue pulsque a" peut
HOLOMORPHIE ET THEORIE DES DISTRIBUTIONS
&re
295
restreinte 1 cetaines fonctions bor6liennes. Voir [ 4 ] .
BIBLIOGRAPHIE
[I] CECILE B. DE WITT, Feynman's path integral.Definition with out limiting procedure. Comm.Math.Phys.
Berlin.
t 28. 1972 p. 47-67. [2]
D.H. FREMLIN, D.J.H.
GARLING et R.G. HAYDON, Bounded
mea-
sues on topological spaces. Proc.Lond. Math.Soc. 39 sbrie. t 25. 1972. p. 115-136. [3]
D.J.H.
GARLING, A generalized form of inductive limit top2 logy for vector space. Proc.London Math. SOC. 3Q sirie. t 14. 1964. p. 1-28.
[4]
A.K.
KATSARAS, Locally convex topologies on spaces
of
continuous vector functions. A paraltre.
Math.
Nachrichten. 151
P. KREE, Distributions sur les espaces de Banach.
Comptes
Rendus. 280. s6rie A . 1975. [6]
P. KREE, Mesures de Radon vectorielles d6finies sur des es paces compl6tement rbguliers. Comptes
Rendus.
20 octobre 1975. [7]
P. KREE, Thbories des distributions et calculs
diff6ren-
tiels sur des espaces de Banach. A paraftre
au
shinaire P.Lelong 1974-1975. Lectures Notes
in
mathematics. Springer. [8']
P. KRSE, Equations aux d k i v 6 e s partielles en dimension in finie. Shinaire P.K.-
le annbe. Publib le secr:
P. KREE
296
Poincarg.
tairat mathgmatique de 1'Institut H. 1975 [9]
-
(PARIS).
P. KREE, Courants et courants cylindriques sur les vari6-
t6s de dimension infinie. "Linear operators and approximation: Proceed of the conf. held at
the
Math. Institute at Obercvolfach 1971" p.159-174. Baseland Stuttqart. Birkhauser Verlag 1972.Inter national. Series of numerical mathematies, 20. [lo] P. KREE, et R. RACZKA, Kernel and symbols of operators in
quantum field theory. (en nrharation). [ll] P. KREE, Solutions prodistributions d '6quations aux d8riv6es fonctionnelles (5 parartre). 1121 J. LESMES , On the approximation of continuously
differen-
tiable functions on Hilbert spaces. Revista
Co-
lombiana de Mat.Vo1. VIII. (1974) pp. 217-223. 1131 L. NACHBIN, Topology on spaces of holomorphic
mappings.
Collection jaune. Springer Verlag. Berlin (1968).
P. Kr6e D6partement de mathgmatiques Universit6 de Paris VI Place Jussieu
-
Paris 5sme
Infinite Dimensional Holomorphy and Applications, Matos (ed.) 8 North-Holland Publishing Company, 1977
SUR L'APPLICATION EXPONENTIELLE DANS L'ESPACE DES FONCTIONS ENTIERES
Par P l E R R E L E L U N G
1. INTRODUCTION Soit E f(zl,
=
A(Cn) l'alqsbre des fonctions entigres
f (2) =
...,zn)
de n variables complexes. Muni de la topologie de la convergence uniforme sur les compacts de Cn I E est un espace
de Frkhet-Montel; on notera (1)
pK(f) = sup If ( 2 ) I
I
z E K, f E E
la semi-norme relative au compact K de Cn. On se propose ici de prgciser l'image de l'applicationex ponentielle; celle-ci notge (2)
exp
f _ j ef
:
sera considgrge c o m e une application de E dans E. rl =
L ' image
Exp.(E) est Gvidemment le c6ne de sommet 0 form6 des
tions entigres sur C" q u i ne s'annulent jamais. trg dans [3,a] que
rl
NOUS
fonc-
avons mon-
est un ensemble fermg et polaire (au sens
des fonctions plurisousharmoniques). Plus prgcisgment, si N est le sous-espace fermg dgfini dans E par f(0) = 0, i l existe
une
fonction plurisousharmonique Uo(f) dgfinie sur E et telle qu'on 297
298
P. LELONG
ait
= Exp.(E) = [f
11
(3)
E
E-N; Uo(f) =
-
m].
Dans la suite nous dirons qu'une fonction plurisousharmonique U est continue si eu l'est, ou encore si
u
est continue en
tout
point 03 sa valeur est finie. Avec cette d6finition Uo(f) continue sur E-N; on peut encore 6noncer en posant
est
Vo(f)
=
= exp. Uo(f) : l'imaqe de Exp. dans E = A(Cn) est un c6ne fern6
dans E-N et est l'ensemble des zeros d'une fonction plurisousharmonique Vo(f) positive et continue sur E-N. Ce r6sultat laissait ouvert un problgme essentiel: semble
q
l'en-
est-il un sous-ensemble analytique dans l'espace E* = E
-
10)
05 EO) dgsigne l'ensemble constitug par la fonction
identique-
ment nulle de E. On donnera une r6ponse n6qative 5 cette question. Pour cela on utilisera un r6sultat r6cent
[l]
de
Alexander: si e est dans le plan complexe un ensemble ferm6, il existe une fonction entisre q(x,y) pour tout y
E
E
A(C
2
)
H.
polaire
telle que
e, q (x) = c~(x,y) ne s'annule pas. H.
Y
Alexander
Qtablit ainsi une r6ciproque d'un r6sultat que nous avions donn6 en 1942 (cf. [3b] et [3c]) : les y tels que q ( x ) ne s'annule Y pas forment un ensemble fern6 de capacit6 nulle dans le planice r6sultat a 6t6 retrouv6 ind6pendamment par M. Tsuji [5]. La r6ciproque donnee dans 111 par H. Alexander caractgrise
l'ensem
ble des y pour lesquels une relation enti6re g(x,y) = 0 n'a aucune solution en x; elle r6pond ainsi 5 un problsme pos6 par G. Julia 121. On utilisera ici ce resultat de la dimension pour montrer l'dnonc6 suivant qui r6sout le probl6me pos6 haut
.
finie plus
3??3Id N 3 N O d X 3 NO IdV3I?ddV
GGZ
299
APP L I CAT1ON E XPONENT I E L L E
L'Qnonc6 obtenu est 5 rapprocher d'un r6sultat plus pr6ci.s obtenu pour les formes lingaires (c'est-5-dire les fonctlonelles analytiques lin6aires) : rl
, elle
SI
une telle forme
x (f)
s'annule sur
est la constante nulle. En effet soit lJ une mesure ' 2
support compact dans Cn reprgsentant
une telle
fonctionelle;
s l p(f) = 0 pour tout f E q r on a
dp(a) eg(a) = o n g = = C akzk, on 1 L n ce qul pour tout z E C
pour toute g
E
E; en particuller
pour
aura p =
=dp(a) = 0
entrahe p = 0 et la nullltg de la fone
P. LELONG
300
tionnelle analytique lin6aire. On notera que la m6thode suivie dans le cas non lingaire est t r k diff6rente et correspond
au
fait que
la
0
est (cf. le corollaire 1) le complzmentaire de
projection sur E, d'un ensemble analytique dgfini dans E,
PROPRIETES DE
2.
x
Cn.
(f)
Uo
Pr6cisons les rgsultats donn6s dans [ 3 , a ] en
rattachant
la plurisousharmonicitg de -log d(f) sur E-N 5 une
proprigtg
classique. PROPOSITION 1.
S o i t G un d o m a i n e d e Cn. Poux
f
E
A(G)
et
z € G, l'application
(5)
(f,z)+
@ :
f(z)
u n e d o n c t i o n a n a l y t i q u e dann G
eht
x
E.
B
En effet elle est continue de (f,z) en (fo,zo): soit une boule de centre zo, compacte dans G, et r > 0 son soit pB(f) = suplf(z)I pour z Pour
oii
-
B.
z o ) 1 5 r' < r , on a par la formule de Cauchy
Ifo(z) - fo(zo)I 5 c I I z - z o I I -1 = 2r pg(fo). D'autre part soit f E E; on a
C
I Iz
1 Iz
€
-
zoII
rayon:
aussi
pour
< r1
Jf(z)
-
fo(z)J 5 PB(f - fo).
D'o6 (6)
(f(z)
-
fO(ZO)
1
5 PB(f
-
fo) +
c I Iz
-
zol
qui. 6tablit la continuit6 de 9 sur E. De plus Cb est
que car on a pour u (7)
@(fl
+
E
I G-analyti-
C; h E C"
uf2, zo
+
uh) = fl(zO
= l$(U).
+ uh) + uf2(z0 + uh)
APPLICATION EXPONENTIELLE
301
I1 suffit de montrer que l'application C - i C donnee en (7) par
+ uh)
$(u) est une fonction analytique pour u = 0. Or u--,fl(zo
+
et u+f2(zo
dans E, pour h
uh) sont, pour fl,f2 fix&
dans Cn et pour zo
pris dans
G,
des fonctions analytiques de u
pour ( u ( I Ihl I < r, donc au voisinaue de u = S o i t M = Y m ( f ) , f E E,,
COROLLAIRE 1
fix6
0.
oii m(f) enk L ' e n n e m b l e
anaLyRique d e b z h o n d e f d a n n Cn : M e n t un e n d e m b e e A = E, x Cn , ek l ' i m a g e d e E x p . e n 2 Le camp.i?&nenkaihe E, d e La phojectian d e t ' e n n e m b k e ana.L!qtique M En effet E
x
c";
(a
dam dand
E, n e t o n C".
AUK
f , z ) = 0 dgfinit un ensemble analytique Mi dans
on a M' =
oc Ma = { O ]
MIU
M2
Cn est obtenu dans E
X
Cn c o m e ensemble des
x
26-
ros de la fonction identiquement nulle (oriaine de E ) . On gliml ne M2 en considgrant l'ensemble analytique M' n (E, PROPOSITION 2
n e d e Cn
a
Pauh f
A(Cn)
E
L ' e n n e m b l e m(f) d e b
continue AUp&ieUhQment
Cn)
=
M1
.
La dibtance. d(f) d e L ' v ~ g i -
= El Z&LO~
d e f nut E,,
x
d e f e n t une j i o n c t i o n
C?
Ua.i?eUhb davlb
[0,
bemA-
+-I.
On a seulement 5 Qtablir la proprigtg au voisinage de
fo pour lequel on suppose fo f 0 et d(fo) = do 06 do 2 0 est fini. I1 existe alors zo
E
Cn, I l z o l
I
= do
1.
0 tel qu'on ait fo(zo)=O.
D'autre part fo f 0 entrahe pour tout 6 > 0 donn6,l'exis tence de h
E
Cn, IlhlI < 1, tel que fo(zo
comme'seul z6ro pour que I$(pe
if3
)I
1.
+
uh) = $(u) ait u= 0
< 1. 11 existe alors p ,
o
< p <
E
tel
ait un minimum non nul, soit m > 0. Dans ces con-
ditions si K est un compact de Cn contenant le disque 6 d6fini par z = z o
+ uh, IuI
< p,
les € appartenant dans E au voisinage
P. LELONG
302
-
W de fo dgfini par pK(f boule
I IzI I
I lzol I +
<
E
d(f) < d(fo)
fo) <
s'annulent sur 6 donc dans la
ce qui 6tablit
+
pour
E
f
E
W
et la semi-continuitg supgrieure de d(f); le raisonnement
est
valable si d(fo) = 0. COROLLAIRE 2 me
band
L'image
q
d e 1 ' e w p o n e n t i e l L e e A t u n endembee deb-
p o i n t i d o L E d a n n E,. d(f) < c] est pour tout c > o
un
ouvert dans E, d'aprgs la proposition 2; il en est de m6me
du
En effet Ec = [f
complgmentaire de
r(
E
E,,
dans E, qui est la &union
des Ec pour
O
<m.
L'ensemble
est de plus parfait (c'est-2-dire sans point
En effet soit fo
i s o l g ) dans E,. f = uf0 pour u
r)
E C,
c
PROPOSITION 3
fo
j! 0;
u # 0, appartiennent 5
qe de fo dgterming par pK(f f = ufo pour u E
E r),
-
v6rifiant
fo) <
1.1
E;
> 0,
r);
les
fonctions
soit W un VOiSina-
W contient les fonctions -1 (fo). Iu - 11 < E pK
La d o n c . t i o n
Ul(f) = - 109
a(€)
e a t p 1 u ~ ~ i n o u n h a 4 m o n i q u au e voidinage de t o u t e Aonction
f, E E
p o u t laqueC1e o n a f0(o) # 0 .
Soit Ifo(@)
I
= b, b > 0; soit K la boule compacte I1z11'2,
et W(fo) le voisinacre de fo dgfini par b (8) W(fo) = [f E E, pK(f-fo) < 7 1
.
On a (9)
pK(f) < pK(fo)
+
= m
pour
f
E
W(f0).
Soit h un vecteur unitaire de Cnl et Dhf (z) la dGriv6e de f en
303
A P P L I CAT1ON EXPONENT I E L L E
[& f(~+uh)]~=~'
z dans la direction h, c'est-5-dire Dh f(z) = u E C. La formule de Cauchy et (9) donnent IDh f ( z )
I
m '
pour
Si zo est dans Cn le z&o
I I Z I )
< 1, et f
W(fo).
E
de f de plus petite distance 5 l'ori-
< 1: qine, on a alors si )]zoII -
I 2 m IlzolI If(zo)-f(o)I = If(0) I JzoII
Mais
= d(f): on a donc si d(f) < 1
d(f) > m-l If(0) 1 2 rn-l
(10)
.
pour f
E
W(fo)
et Uo(f) = -log d(f)
+
lou Jf(0)I(
loa m pour f E W(fo), d(f)c 1.
Soit d(f,k) la distance de l'oriaine de Cn 2
l'ensemble
m(f) des zeros de f parallslement au vecteur unitaire k de Cn: d(f) = infk d(f,k). Interpretons d(f,k) comme la distance dans E rallElement 5 k du point (f, z =
0)
(f
,z)]-':A
plurisousharmonique de f E
est
Cn dont on retranche l'ensemble M1 des zeros de a. D'aprgs
x
un resultat classique (cf. [5]), -1ou d(f,k) est une
f
pa-
5 la frontiGre du domaine A
qui est le domaine d'holomorphie de la fonction [$ E,
Cn et
x
E
E
*
. De plus prgs
fonction
(lo) et (11)'
pour
W(fo) on a -log d(f,k)
oc 109,
5 -lo? d(f) 5 log, [mIf-l(0) I ]
a = sup(10a a, 0). D'aprss (8) on a aussi
If(0)
1
b
2,
donc -10s d(f,k) 5 lou, 2m b-l pour f Les fonctions
E
W(fo).
-log d(f,k) sont ainsi majorges
uniformement
(par rapport 5 k) dans W(fo); leur enveloppe superieure requla-
. D'autre
part
d(f) est semi-continu inferieurement de f dans W(fo) car
d(f)
risge est donc plurisousharmonique, (cf. [3,d] 1
P. LELONG
304
est la distance de (fro) 5 l'ensemble ferm6 d6fini par @(f,z)=O dans
E,
x
Cn, distance prise parall6lement 5 Cn et d(f)
bornee infgrieurement dans W(fo) d'apr&i (10). Ainsi = -log
est
Ul(f)
=
d(f) est l'enveloppe supgrieure rggularis6e de -log d(f,k)
pour I l k [ / = 1, et est semi-continue sup6rieurement;U1 (f) est donc plurisousharmonique pour f E W(fo) , c'est-5-direr finalement au voisinage de tout point de E-N. On peut alors 6noncer d'aprss les proprigtgs 2 et 3 : La d o n c t i o n Ul(f) = -log d(f) e b t p e u h i b o u b h a h m g
COROLLAIRE 3
nique e t c o n t i n u e
blth
E-N; e l L e
t ' e b t en
pahticutieh
but
un v o i
b i n a g e d e i m a g e q d e t ' e x p o n e n t i e t t e d a m t ' e b p a c e E, = E - { O } .
D'autre part on a :
L'image
COROLLAIRE 4 ban
bommet
0
d e Exp.
Q
ebt
u n c8ne
b u h Q:
;point;
en
d ' i n t i i h i e u h v i d e e t dehm; danb E,.
Le corollaire 4 a d6jd 6t6 donn6 dans [3,a]; il est immgC, X # 0, donc
rl
est un c h e 6point8 5 son sommet qui est l'origine. De plus,
n
diat que f
E rl
entraine Xf
E
n , pour tout X
E
est exactement l'ensemble V (f) = 0 dans E - N ofi V (f)=d-'(f) 1 1 = exp [U1 (f)] ; V1 (f) est plurisousharmonique continue sur l'ouvert E - N et tend vers r(
+-
quand f tend vers fo
est un c6ne fern6 dans E,.
qu'il est polaire dans E
3 . L'IPAGE
rl
-
E N
-
{Ol;ainsi
Enfin son int6rieur est vide puis-
N.
DE Exp. N'EST PAS UN ENSEMBLE ANALYTIQUE DANS E *
La demonstration comprend trois parties: a) On Qtablira d'abord
APPLICATION EXPONENTIELLE
305
PROPOSITION 4 ~ a i tg(zlI...,zn,y) = q(zly) une donctian entib/re bUh Cn+l (Z,y): l'appeiCatiVfl G : y--jg(Z,y) = g (2) e b t Ufle Y application analytique.
En effet
est continue: posons M(R,R') = sup Ig(2,ylPour R' major6 1 x 1 IR, I~I'R'. Pour < R et < 2 , est R R : par la formule de Cauchy et l'on a, pour IYI 2 F, lYo[ 5 7 C
II~II
pK [G(yI
-
I?+\
I ~ I
G(yo)] 2 Iy-y0/ C2Rl-l M(R,R')]
valable en semi-norme pK, si K est un compact
I 1z1 I
contenu
< R. De mzme on majore le module de la deriv6e
lyo1 2
R'
2
I
IIzll < R.
Si h est un vecteur unit6 de Cn, u+G(y
aYq
dans pour
+uh) est analy-
0
tique de u au voisinage de u = 0 car on a
et l'on obtient une s6rie majorante en norme pK qui R' d'aprzs pour 1u1 < 2
Ainsi G(y) est une application holomorphe de
Q:
converge
dans E.
b) Rappelons alors l'&nonce suivant [l] de H.
Alexander:
?itant donn6 un ensemble ferm6 e de capacite nulle dans le plan 2
complexe C(y), il existe une fonction entigre q(z,y)
E
le que e soit l'ensemble des y pour lesquels g ( 2 ) Y s'annule pour aucune valeur z E C.
g(z,y)
=
A(C Itel-
Nous prendrons pour e un ensemble fermg sans point
ne
is016
de capacit6 nulle, construit S partir d'un segment ferm6 [a,bJde l'axe reel de C(y) en utilisant une suite d'entiers S={p1,pZ..J fortement croissante. Si
el
est la longueur de
(pl) consiste 5 retirer de [a&]
l'opgration
l'intervalle ouvert m6dian
de
P . LELONG
306
longueur el(l
-
-1
l'ensemble
p1 ) de manigre qu'il ne reste que
e(p,)
form6 de deux segments fermgs
[a, a
+
,
el(2p1)-l]
[b
-
Cl(2p1)-',
b] de longueur
C1(2p,)
-1
.
L'op6ration (p,) est appliqu6e 5 chacun de ces deux segments et donne e(p1,p2), etc. Ce proc6d6, imit6 de la construction l'ensemble de Cantor fournit une suite
d'ensembles
de
fermgs
.
E (pll.. ,pn); leur intersection e (S) est un ensemble ferm6 sans point isolg; si la sgrie de terme qdneral 2-'
diverge,
log pq
e(S) est de capacitg nulle (cf. [ 4 ] ) . Dans ces conditions on utilisera 1'8noncG suivant
Soit W
PROPOSITION 6
0
un v o i s i n a g e d o n n e dun6 E, de foE
n ' e x i d t e pas d ' a p p l i c a t i o n hotomohphe H : Wo+F
0;
ehpacc v e c t o t i e C LocaCement c o n u e x e sEpahE teLCe que
F
ebt
T)
e o t un ensembte detrme' maid n'est
t i q u e au v o i b i n a g e d'aucun de h e 6
E
(11)
-
wo
n
=
: Wo+F
10)
points. E et
n , il n'existe pas d'espace F localement convexe &par6
d'application holomorphe H
un
un ennembtennaLy-
Montrons en effet que si fo et Wo sont donngs dans fo
ie
onwo n o i t
d c d i n i duns Wo par H(f) = 0. E n d'authed tehmed danb E,=E
C'ensembCe
T):
et
telle que
won H-'(o).
Dans ce but nous utiliserons l'ensemble e(S)construit sur les reels de C et une fonction entigre g ( z , y )
E
2
A(C 1 telle que
APPLICATION EXPONENTIELLE
ensemble des y
E 6
307
oii l'on a q ( z ) = q(z,y) E q soit e(S).
Y Prenons un point yo dans e(S); g(zl,yo) = e s'annule pas
et on posera C J ' ( Z ~ ~ Z ~ , . . . ~=Z ~g(Zlry) ~Y) [g(zlry0)] L'application G : y+g;
= g ' ( z l,...,zn,y)
E
-1
E
fo(Z1r.-.rZn1.
est holomorphe
(proposition 4 ) et l'on a G(yo) = fo. De plus on a G(y)
E 11
si
et seulement si y appartient 2 e(6). L'application G' 6tantcontinue, il existe a > 0 de mani6re qu'on ait G(Y)
wo
E
IY -
pour
y0I < a .
On a alors c(y) E
won n
IY -
pour
Supposons que, dans Wo,
TI
y0I < a , Y
soit d6fini par H(f) = 0 oc
holomorphe, F &ant
est une application Wo+F,
e(s).
E
H
un espaceloca-
lement convexe sbpar6. Alors l'application compos6e H o G(y)
yest holomorphe de
C
dans F. Soit 4
E
F' une forme lin6aire
co"
= 4 o H o G(y) est une fonction ho1omorpheC-W. 4 Elle s'annule pour Iy - yo/ < a, y E e(S), c'est-5-dire sur un
tinue sur F: h
ensemble fern6 non vide n'ayant aucun point isol6;elle est donc la constante nulle; on a alors h
= 0 pour tout 4 E F', ce
4 -
entraine h(y) = II o G(Y) =
o
pour
1y
-
yOl < a.
On a alors ~ ( y )E H-'(o) On a d'autre part G(y) ~(y) E
pour E
~y - yOl < a.
pour Iy
Wo
wo n H-'(o)
-
pour
yo/ < a, donc
I Y - yo[
Si donc (11) 6tait rSalis6, on aurait
C(Y)
E
won 11
pour
IY -
y0I < a
< a.
quj
30 a
P . LELONG
...
-
zn)E rl pour tout y vgrifiant Iy Y 1 contrairement 5 l'hypothsse faite, ce qui 6tablit la c'est-%dire
q'(z
y o I < a,
proposi-
tion 6, et la seconde partie du th6orsme(l).
4.
PROLONGEMENT DE Uo(f)
On a montr6 dans [3a] que la fonction Uo(f)
=
d(f)
-109
demeurait born6e au voisinaqe du sous-espace ferm6 N dgfini par f ( 0 ) = 0. Elle se prolonqe donc 2 travers N . Montrons
que
prolonqement est continu sur N , c'est-z-dire que Uo(f)
ce
ainsi
prolonq6e est une fonction plurisousharmonique continue sur E. PROPOSITION 7 (12)
a
ofl
Uo(f) = -109 d(f) + 109 If(0)I 5 log pK(f)
IIZII 5
oii K e ~ ~a t b o u ~ ecompacte
2 dc
c".
En effet si d(f) < 1, on a en op6rant comme en (10) If(0) I pil(f)
d(f)
d'oc r6sulte (12); dans le cas d(f) 2 1, on a Uo(f)c loqlf(0) ' 1 log pK(f) , c'est-;-dire Soit alors f dans E par pK(f'
-
E N,
encore (12). f(0) = 0 et un voisinaqe W de f
f) < b; on a pK(f') 5 pK(f)
et Uo(f) demeure borne sup6rieurement dans D'aprgs un r6sultat connu (cf. [ S ] ) ,
+
b pour
d'aprss
W
Uo(f) se prolonqe
dgfini f
E
W
(12). alors
d'une manisre unique en une fonction plurisousharmonique 5 travers N , d6finie par Uo(f) = lim sup U(fm), f m j f
E N,
fm
E E
-
N.
Montrons que ce prolonqement est encore continu sur N. Posons z
= apt p > 0,
f ( z ) = f(ap) = f ( 0 )
+
l i ~ l l= pP1(a)
+
1, et 2 p P2(a)
+
...
APPLICAT1ON EXPONENTIELLE
309
06 Pk (a) est un polynome homoghe de degr6 k des
. Supposons P1(a)
de
coordonn6es
# 0. Alors on a
fm(ap) = f m ( 0 ) + P P l , m + P2P2,m(a)
...
+
la convergence fm----,f &ant
oii f m ( O ) d 0 , P ~ , ~ ( U-pl(a), )
uniforme sur tout compact selon la topologie de E.
IP
( a ) I pour I la1 I = 1. On a max 1 ,m -1 f,(O) I (1 + E ~ ) ,E m j O quand fmd f . d(fm) = Mm
soit M~
=
D'oc
lim Uo(fm) = lim log Mm.
m On a donc, pour valeur du prolonqement m=w
(15)
Uo(f)
=
log
qui montre que Uo(f) de f
s
max
,
Dfa(0)
I est continu en llal =1
f
E
f
E
N
N si la diffgrentielle
l'oriqine n'est pas nulle. Si l'on a f(0) = 0 et Daf(0) = 0 pour tout
c'est-5-dire f
E
N
-
(0)
mais f 3 0,
a,
la resultat est encore valable: le pro
--. En
lonqement de Uo(f) demeure continu avec valeur
effet, a3
besoin a p r k un changement de coordonnge unitaire, f equivant un pseudo-polynome
'TI
en z et on a
f(z)
= p(z)
valable dans un domaine D = [ z '
a(z) E
d,
I zn I
f ( z ) ne s'annule pas sur le compact z' E
un pseudo-polynome de degr6 q
c p]
p(z) # 0 dans D; les racines 5 1. ( 2 ' ) de
3, lznl
choisi tel
=
p;
a(z)
que est
TI
n-1) E d ; s'annulent 5 l'origine ( Z lr...,~
ainsi que les coefficients Ak(z') de
Alors soit fm+f
,
2
06 zn est la variable distinquge et z ' =
z' = 0
s
51.
est une suite de fonctions
entisres
qui converge uniformgment vers f dans un voisinaqe de l'oriaine
310
P . LELONG
contenant
5.
I1 e x i s t e mo t e l que p o u r m > m 0 , fm a i t d a n s
D
une d6composition holomorphe fm
Pm*
P I
. rm
- P,
p m ( z ) # 0 pour z E D
I
P m ( 0') # m0 , T
T
uniformgment s u r
5,
raci-
et a q
On
nes c i I m ( z ' ) d e module i n f 6 r i e u r 5 p p o u r z ' E d l c c d .
a
alors d(fm) L inf IciIm(z') i
I
CI f m ( 0 )= 1Pm(O)l - T - - & m ( z ' )
1
.
I
oii l e p r o d u i t a u second membre c o n c e r n e les r a c i n e s de rrn d o n t on a e n l e v 6 c e l l e de p l u s p e t i t module: comme on a d u i t t e n d v e r s 0 quand fm-+
#
f
,
q'2,
leprs
cependant qu'on a pm(0)+ p(O)#
0. L e premier membre d e ( 1 6 ) t e n d vers 0 quand f m + f
v a l e u r d e prolongement d e U o ( f ) e s t
-mI
et la
6cJale 5 c e l l e d o n n g e p r
(151, ce q u i s t a b l i t l a c o n t i n u i t s d e Uo a u v o i s i n a g e d ' u n e fonc tion f E N
-
{O}.
Enfin U ( f ) est encore continu 2 l ' o r i g i n e de 0
avec v a l e u r Uo(0) l a boule [ I z I
si f m j f
=
I -<
=
-m.
En e f f e t s i K e s t un compact
2 , on a d ' a p r s s ( 1 2 ) : U o ( f )
0 , pK (f,)
-+
0 e t Uo
5
E(f
01,
contenant
l o g p K ( f ) . Donc
(fm)+ --m.
F i n a l e m e n t U o ( f ) , compte-tenu du C o r o l l a i r e 3 , e s t f o n c t i o n plurisousharmonique continue s u r t o u t
@,
une
ce q u i achgve
l a d g m o n s t r a t i o n du thborzme. B I B L IOCRAP I1 I E
[11
ALEXANDER (11.)
,
On a problem of G. J u l i a . Duke Math.
t . 42, n? 2 , p . 326-332, 1975.
J.
311
APPLICATION EXPONENTIELLE 12
I
JULIA (G.), Sur le domaine d'existence d'une fonction implicite dgfinie par une relation entisre G(x,y) = O . Bull. SOC. Math. France, t.54, p. 26-37, 1926.
1 3
I
LELONG (P.), a) Fonctions plurisousharmonique dans lesespaces vectoriels topologiqucs.Colloque International du C.N.R.S., 1972, publig dans Agora Mathematica, vol. 1. 1974, p. 95-116 (Gauthier-Villars). b) Sur certaines fonctions multiformes,C.R.AG. Sci. Paris, t. 214, 1942, p. 53. c) Sur les valeurs lacunaires d'une relation 5 deux variables complexes. Bull.des Sciences Plath.,t. 56, p. 103-112, 1942. d) Sgminaire d'Analyse , Lecture-Notes Springer,NQ71, p. 167-190 et NQ 116, p. 1-20.
14
I
NEVANLINNA (R.) , Eindeutige analytische Funktionen,Springer, cf. p. 149, 1936.
15
I
NOVERRAZ (Ph.), Pseudo-convexitg et convexit6 polynomiale.
16
I
... North.
Holland, 1973.
TSUJI (M.), Theory of meromorphic functions in a
neigh-
boorhood of a closed set of capacity zero, Jap. J. Math., t. 19, p. 139-154, 1944. (1)
-
Rgcemment (Mai 1977), M. ZRAIBI, puis J.SICIAK ont
don&
une dgmonstration de la proposition 6 sans utiliser lergsultat de H.ALEXANDER:la nullit6 de Hrexp(f
+ tf)]
pour
f E A(Cn) au voisinage de t = 0 entraine celle du dgveloppement de Taylor de H en fo. DEPARTEMENT DE MATHEMATIQUES, UNIVERSITE DE PARIS VI, 4 PLACE JUSSIEU, PARIS 5, FRANCE.
This Page Intentionally Left Blank
Infinite Dimensional Holomorphy and Applications, Matos (ed.) 0 North-Holland Publishing Company, 1977
HOLOMORPHIC GERMS ON INFINITE DIMENSIONAL SPACES.
By J U R G E M U J I C A .
1. INTRODUCTION
d e n o t e s t h e v e c t o r s p a c e o f a l l complex-valued h o l g
$(U)
morphic f u n c t i o n s o n a n o p e n s u b s e t U o f a complex Banach s p a c e E.
T~
d e n o t e s t h e Nachbin t o p o l o g y on %(U).
W e recall t h e def&
.A
is s a i d t o
n i t i o n of
T ~ ;see
[ 111
seminorm p o n z ( U )
p o r t e d by a compact s u b s e t K o f U i f f o r e a c h open s e t V K
cVc
U,
for all f E
t h e r e e x i s t s c (v) > 0 s u c h t h a t p ( f 1
z(U).The
l o c a l l y convex t o p o l o g y
be
, with
5 c ( V ) supxEV1f(x)l
‘ I ~ is
defined f o r
all s u c h seminorms.
#(K)
d e n o t e s t h e l o c a l l y convex s p a c e o f a l l c o m p l e x - v a l
ued holomorphic germs o n t h e compact s e t K , endowed w i t h t h e
in
d u c t i v e t o p o l o g y coming f r o m %(K)
= ind l i m s m ( K E ) E > O
where
KE
= {x E E :d i s t
(x,K)<
E
}
and
a m ( K E ) d e n o t e s the
Banach s p a c e o f a l l bounded h o l o m o r p h i c f u n c t i o n s o n t h e s u p norm. I n t h i s s u r v e y w e show how t h e s p a c e s an important r o l e i n t h e s t u d y of t h e 313
locally
KE,
with
E ( K ) play
convex
space
314
J. MUJICA
(6&(u),
[s].
T ~ ) .Most Of
t h e r e s u l t s presented here a r e
t a k e n from
Some o f them were announced w i t h o u t p r o o f i n [lo].
2 . LOCAL MULTIPLICATIVE CONVEXITY O F B ( K ) AND ( B ( U )
TU).
W e begin with
Z(K) i b
THEOREM 1.
i,e.
i t b
a kVCaLey m u L t i p & c a t i w e l y
t v p o l v g y i n d e d i n e d b y ,the
6 p(f)
-that p(fg)
p ( g ) 6ffh aLe
For t h e proof w e w r i t e V
f2VMtiMUVUb
c o n v e x algebtra, p buch
bfmiVIVtrmb
frg E
n
where
- KE
( E ~ )i s
a sequence
n'
of p o s i t i v e numbers d e c r e a s i n g t o z e r o . F o r e a c h n and o < 6
< 1
n
we define
Since
E(K)
= i n d l i m &?(Vn)
t h e sets
U 6 = convex h u l l o f
d
n=l
U
nr 6n
form a b a s e o f convex, b a l a n c e d n e i g h b o r h o o d s o f 0 i n t h e sequence
6 = ( 6 ) v a r i e s . Since n
U 6 = { f i n i t e sums CAnfn: A n i t i s e a s y t o see t h a t
(@(u)
THEOREM 2 .
B ( K ) , as
L A n = 1, f n E U n I 6 n
u6 u 6 c u 6 ' i b
I
> 0,
1,
ending t h e proof.
a L o c a L L y m u L t i p L i c a t i w e L y c v n w e x a&
gebtra.
Theorem 2 a n s w e r s a q u e s t i o n r a i s e d by Matos [8] . W e sketch t h e p r o o f o f Theorem 2 . F o r e a c h compact K C U I w e l e t M t h e image o f t h e c a n o n i c a l mapping
we d e f i n e
MK = MK K
and endow M
n srn(KE)
I
= ind
E> 0
denote
%(U) * Z ( K ) . F o r e a c h
E
> o
w i t h t h e norm i n d u c e d by E m ( K E ) ,
w i t h t h e i n d u c t i v e t o p o l o g y coming from
#
K
l i m ME K
HOLOMORPHIC GERMS
315
We have the following diagram
K ME
4
Theorem 2 follows from Lemma 1 and Lemma 2 below. K M i n a LacaLLy m u L t i p L i c a t i u e L y convex a t g e b a a .
LEMMA 1. LEMMA 2.
( g ( U ) ,
T
~
K
= ) proj lim M
K U The proof of Lemma 1 is similar to that of Theorem 1. the
proof of Lemma 2 is straightforward. 3 . BOUNDED SUBSETS OF @(K).
The characterization of the bounded subsets of @(K) is an important tool for obtaining further results about @(K).
The
main result in this section is the following. THEOREM 3 .
G i v e n a bounded n u b b e t X 0 6 %(K),
t h e h e exddb
E
> 0
nuch t h a t (a) X i n c o n t a i n e d and bounded i n
Sm(KE).
(b) Ev'ehy n e t (fa)tX w h i c h i n Cauchy i n S(K)
i6
aLno
Cauchy i n Bm(KE).
Me b a y .then t h a t eat
S(K) = ind lim gm(KE) in a Cauchy hegu
inductive Limit. Theorem 3 was first given by Chae
[2]
but the original proofs were incomplete. We
and Hirschowitz [6], give
a
different
proof, based on Theorem A and Lemma 3 below. THEOREM A (Grothendieck [ 4 ] ) .
7 6 X = ind lim Xn LA t h e i n d u c -
t i v e L i m i t a 6 an i n c h e a n i n g nequence union
i d
X , .then X i n
04
noamed ApaCeA Xn
a (DF)-bpace and e v e h y bounded n u b n e t
whobe ad X
316 i6
J. MUJICA
contained i n t h e
X-CbObUhe
06 a bounded b u b b e t 06 bame
L e t X be a bounded b u b b e t
LEMMA 3 .
06
Xn.
S r n ( K E ) and L e t 0 < 6 < E.
Then: (a)
Evehy n e t ( f a ) C X uhich Cauchy i n
(b)
The
Zrn( K 6 )
C k J b U h E 06
i b
Cauchy i n E ( K )
i b
aebo
.
X i n Z ( K ) i b contained
and
bounded
in Z m ( K 6 ) . The p r o o f of Lemma 3 i s s t r a i g h t f o r w a r d . From
theorem
w e see a t o n c e t h a t e v e r y c l o s e d , bounded s u b s e t o f Q ( K ) p l e t e , and h e n c e t h a t g ( K ) i t s e l f i s c o m p l e t e , b e i n g -space;
a
3
is ccp"_
(DF)-
see [4].
4 . COMPLETENESS OF ( % ( U ) ,
TW)
.
The f o l l o w i n g t h e o r e m a n s w e r s a q u e s t i o n r a i s e d b y Nachbin
P2:I THEOREM 4 .
( %(U)
, T ~ )i b
aewayb c o m p e e t e .
E a r l i e r p a r t i a l r e s u l t s had b e e n g i v e n by Dineen [3] ,Chae
121
and Aron
[l] f o r c e r t a i n open s e t s U . F o r t h e p r o o f of-
orem 4 w e u s e t h e n o t a t i o n of S e c t i o n 2, and d e f i n e
2:
= c l o s u r e of
p
=
u
M~K i n
%rn(~E)
%:
E>O
%:
K
i s t h e c o m p l e t i o n o f t h e normed s p a c e M E , and
w i t h t h e i n d u c t i v e t o p o l o g y coming from %K = i n d l i m E >
0
W e have t h e f o l l o w i n g diagram.
'LK ME
GK
i s endowed
HOLOMORPHIC GERMS
317
Theorem 4 follows from from Lemma 4 and Lemma 5 below. LEMMA 4.
"MK
LEMMA 5.
(
i n t h e completion
E(U)
Tu) =
It is clear that M
K
06
proj lim K
U
M
K
.
iXK %K
is sequentially dense in M
,
and that every
continuous seminorm on MK extends uniquely to a continuous semk QJK
norm on M
. To prove that 2K is complete we use Lemma
rem A to show that
BK =
ind lim
3 and
m~
$: is a Cauchy regular induc-
tive limit. This shows Lemma 4. Lemma 5 follows from Lemma 2 and Lemma 4 .
5. OPEN SETS WITH THE RUNGE PROPERTY. We continue using the notation of Section 2 and Section 4. A compact set K
cU
is said to be U-Runge if MK is sequentially
dense in Z ( K ) . U is said to have the Runge property
if
subset of U is contained in another one which is U-Runge.
every When
U is an open set with the Runge property then we can improve+ ma 2 and Lemma 5 as follows. THEOREM 5.
7 6 U i n a n open h e t w i t h t h e Runge p h o p e h t y t h e n (g(U)
T
~
= ) proj lim @ ( K )
K
U
Theorem 5 follows from Lemma 5 and Lemma 6 below.
LEMMA 6 . (a)
Fok K c U we h a v e :
gK =
sequential closure of M~
~ ~ T I ] ~ I K ) .
318
J . MUJICA
(b)
16 K i d U-Runge t h e n ,304 e a c h E > 0
with 0 < 6 < QJK bat, M =
The p r o o f of
E,
%(K)
thehe e x i d t o 6,
Ern( K E ) C QJK M6.
nuch t h a t
an t o p o l o g i c a l
l n pahticu -
u e c t o h npacen.
( a ) i s a n immediate a p p l i c a t i o n o f Theorem3.
To p r o v e ( b ) w e use (a) and Theorem B below. THEOREM B ( G r o t h e n d i e c k [5]).
v e x npace w h i c h
L e t X be a Haundoh6d L o c a ~ L y c o n
in t h e u n i o n a6 a n i n c t i e a n i n g AeqUet'~Ck?odFhhch.et
opacen xn and annume t h a t e a c h i n c l u n i o n mapping
xn
COntinUUUb. Then a n y c o n t i n u o u n l i n e a h mapping T
: Y
+
.+
x
id
X dhom
a
F h z c h e t npace Y i n t o x can be dactohed c o n t i n u o u n l y t h k o u g h dome X n l i . e . t h e h e e x i ~ t hn and a c o n t i n u o u d &ne.ak mapping Tn : Y
+
Xn
nuch t h a t t h e d o l l o w i n g diagxam i n c o m m u t a t i v e . Y
(h) w e
To a p p l y Theorem B i n t h e p r o o f of Lemma 6 %K
, Y = % " ( K c ) I where n of p o s i t i v e numbers d e c r e a s i n g t o z e r o .
X = S ( K )
Xn = M E
any
( E ~ )i s
take
sequence
The f o l l o w i n g t h e o r e m g i v e s s e v e r a l c h a r a c t e r i z a t i o n s
of
t h e compact s u b s e t s o f U which a r e U-Runge.
any compact n e t K
THEOREM 6 .
u
t h e dolLoWing
conditionn
ahe p a i t w i n e e q u i v a l e n t . i d u-Runge, i . e . MK i n n e q u e n t i a l L y d e n b e i n 8 8 ( K ) .
(a)
K
(b)
E u e h y bounded n u b n e t
0 6 B ( K ) in c o b t a i n e d K ~ ( K ) - c l o n u h e0 6 a bounded n u b n e t 06 M .
(c) G i v e n X C % ( K ) K C K6 C
bounded, t h e h e e x i n t n
U nuch t h a t X C % " ( K y ; )
6
K
j
+
the
with
0
and d o h e a c h
t h e t e e x i d t n a AeqUenCe ( f . ) C Mg with f 3
>
in
f in
f
E
X
Ew(K~).
319
HOLOMORPHIC GERMS
Given
(d)
E
c
> 0 with K
K E c
huch t h a t doh each € (fj)t w i$ th f j
+
u thehe exinth
8 d"(KE)
f
The implication (a) *(d)
i n
6
lllith
O ~ < E
t h e h e e x i n t h a hequence
%"(K6).
follows from Lemma 6.
Each of
the implications (d) --j (b) and (b) + (c) folluds from Theorem 3. Finally the implication (c) => (a) is obvious.
6 . OPEN PROBLEMS.
Can we c a n d t h u c t t h e e n v e l o p e
PROBLEM 1.
main U c E an a n u b d c t 0 4 t h e when E i n n e p a h a b l e ? o d Jonedhon [7],
with
fah
ApeCthUm
06
a a t t e m p t nee
E = co(A),
A
0 6 holomohphy 0 6 a d o ( %(U)
[8]
I
T ~ ) ,a t
LeaAt
. By a coun&Y~examp.&
u n c o u n t a b L e , t h i n c a n n o t be
done i n g e n e h a l . Theahem 7 and Theohem 2 hemain t h u e d o h a n y m e t h i z a
PROBLEM 2 .
b l e l o c a l l y convex hpace E. On t h e o t h e h hand, we have been able
t o entend Theohemh 3 t h a o u g h 6 o n l y t o a c e h t a i n CeaAh 0 6 met&& z a b l e l o c a l l y c o n v e x s p a c e d ; he62 [9]
. Do
Theohemh 3 t h h o u g h
6
rremain t h u e doh any metctizabte l o c a l l y c o n v e x hpace ? fah ttkin p ~ r _ pohe it c e h t a i n t y h u d d i c e h t o e x t e n d Lemma 3 . With t h e n a t a t i o n
PROBLEM 3 .
06
S e c t i o n 2 , d o h K c U , doen
MK
have t h e i n d u c e d t o p o l v g y a d Z ( K ) ? T h i n LA c e h t a i n l y t h e cahe when K in
U-Runge, by Lemma 4 and Lemma 6 . Poed t h i s hemainRhue
i h
genehal ?
PROBLEM 4 .
i n
$(K)
.
PROBLEM 5 .
We dedined K t o be U-Runge id MK i n hequenti&y Ih
der~e
K
t h i h e q u i v a l e n t t o haying t h a t M .h d e m e i n 8$(K)?
Vehy dew examplea
p e h t y ahe known; s e e [9]. phave t h e h e b u l t b i n
[131
06
o p e n h e t h w i t h t h e Runge
pho-
Can we dind new examples ? Can We i m t o nhaw, d o h i n o t a n c e , t h a t eveny
phq
J. MUJICA
320
doconvex open AubACt p&y
? Thin
06
a Banach s p a c e w i f h banin hub t h e Runge p h o -
q u e o t i o n wan hained b y S c h o t t e n L o h e h ; n e e
1141.
REFERENCES
[l]
R. ARON; Holomorphy types for open subsets of a Banach space, Studia Math. 45 (1973), 273-289.
[ 23
S.
B. CHAE; Holomorphic germs on Banach spaces, Ann.
Inst.
Fourier 21 (1971), 107-141.
[ 31 S. DINEEN; Holomorphy types on a Banach space,Studia Math. 39 (1971), 241-288.
[ 4 3 A. GROTHENDIECK; Sur les especes (F) et (DF),Summa Brasil, Math. 3 119541, 57-122. [5]
A.
GROTHENDIECK; Produits tensoriels topologiques et espaces nuclgaires, Memoirs Am. Math. SOC. 16 (1955).
[ 6 3 A. HIRSCHOWITZ; Bornologie des especes de fonctions analytiques en dimension infinie
,
Sgminaire Lelong 1970,
Lecture Notes in Math. 275, Springer-Verlag
(19711,
21-33. [7]
B. JOSEFSON; A counterexample in the Levi problem, Proceed ings on infinite dimensional holomorphylUniversityof Kentucky 1973, Lecture Notes in Math. 364, SpringerVerlag (1974), 168-177.
[ 83 M. MATOS; Holomorphic mappings and domains of holomorphy, Centro Brasileiro de Pesquisas Flsicas, Rio de Janeiro (1970).
HOLOMORPHIC GERMS
321
1911 J. MUJICA; Spaces of germs of holomorphic functions,Thesis,
University of Rochester (1974), to appear in Advances in Mathematics.
[lo] J. MUJICA; On the Nachbin topology in spaces of holomorphic functions, Bull. Am. Math. SOC. 81 (19751, to appear. [ld L. NACHBIN; Topology on spaces of holomorphic mappings, EL
gebnisse der Mathematik und ihrer Grenzgebiete
47,
.
Springer-Verlag (1969)
[12] L. NACHBIN; Concerning spaces of holmrphic mappings, Rutgers
University (1970). [13] PH. NOVERRAZ; Approximations of holomorphic or plurisubhag
monio functions in certain Banach spaces, Proceed
-
ings on infinite dimensional holomorphy Universityof Kentuchy 1973, Lecture Notes in Math. 364, SpringerVerlag (1974), 178-185. [14] M. SCHOTTENLOHER; The Levi problem and Oka-Weil approxima-
tion, this conference.
Current Address INSTITUTO DE MATEaTICA,
INSTITUTO DE MATEMfiTICA,ESTATfg
UNIVERSIDAD CATOLICA DE CHILE, TICAECIeNCIA DA COMPUTACAO-TMECC CASILLA 114-D, SANTIAGO,
UN IVE RS IDADE XSTADUAL DE CAMP INAS
CHILE.
CAMPINAS-SP
-
BRASIL.
This Page Intentionally Left Blank
Infinite Dimensional Holomorphy and Applications, Matos (ed.) 0 North-Holland Publishing Company, 1977
ON A PARTICULAR CASE OF SURJECTIVE LIMIT
1.
We s h a l l c o n s i d e r t h e following s i t u a t i o n : L e t E be
non s e p a r a b l e complex Banach space and l e t ( E i ) i
I
be a
a di-
r e c t e d (by i n c l u s i o n ) family of c l o s e d subspaces of E such that:
(a)
E =
u iEI
(b)
,
f o r any i i n I o n t o Ei
(c)
Ei
t h e r e i s a p r o j e c t i o n ui
from E
,
f o r any i and t i o n uij
j
in I , i
from E
o n t o Ei
j d i a g r a m commutes:
5
j , t h e r e i s a projec-
such t h a t t h e follawing
W e c o n s i d e r t h e s u r j e c t i v e l i m i t ( i n t h e s e n s e of of
(EI U i t U i j ) i
,j € 1 i t . A s a s e t , w e have E+
and w e denote by E c E = E+
,
t h i s surjective l$
b u t t h e p r o j e c t i v e topology of
i s always s t r i c t l y weaker than t h e i n i t i a l norm
It i s obvious, v i a t h e open mapping theorem, t h a t E
topology.
i s an open
s u r j e c t i v e l i m i t and, so, any holomorphic f u n c t i o n on E + t o r i z e s trough an Ei
(4))
( i . e . f o r any f 323
fac-
i n H(U) there is i i n I
Ph. NOVEHMZ
324
and
2
i n H(Ui)
f = f o ui)
such t h a t
.
L e t u s c o n s i d e r a f e w examples i n which t h i s
situation
a r i s e s i n a n a t u r a l way: (Ei)
i s t h e s e t of a l l f i n i t e d i m e n s i o n a l subspaces
of a Banach s p a c e E and t h e p r o j e c t i v e t o p l o g y i s t h e weak t o p o l o g y E = Co(A)
u (E,E')
.
f o r A u n c o u n t a b l e , endowed w i t h t h e
(El) a r e t h e s p a c e s Co(A'),A'
norm, and t h e s p a c e s
c o u n t a b l e i n A , c o n s i d e r e d as s u b s p a c e s of
,)
by i d e n t i f y i n g
x = (x
where
if
y,
E = kp(A)
s p a c e s Ei
= x
a
sup
,€A1
1
and
A uncountable, 1 5 p
are t h e s p a c e s
&'(A'),
Y = ( Y c r ) a EA
with
y,
Co(A)
= 0 i f a EA\A'.
5 +
-
and
the
A' c o u n t a b l e i n
A. K
i s a non m e t r i z a b l e compact s e t and E i s
the
space of a l l continuous functions with
metrizable
s u p p o r t . The spaces Ei
#(K')
a r e t h e spaces
K ' C K, compact and m e t r i z a b l e ,
for
t h e mappings u i a r e
t h e r e s t r i c t i o n mappings which are cmto by the Tietze e x t e n s i o n theorem. E
i s a non s e p a r a b l e H i l b e r t s p a c e and ( E i )
the
f a m i l y of a l l s e p a r a b l e c l o s e d s u b s p a c e s and u i t h e orthogonal projections. W e s h a l l a l s o suppose t h a t t h e f o l l o w i n g c o n d i t i o n
is s a t i s f i e d :
(*)
f o r any c o u n t a b l e s u b s e t I ' of I,
t h e r e i s an index i ' i n I such t h a t
32 5
A CASE OF SURJECTIVE LIMIT
T h i s c o n d i t i o n i s s a t i s f i e d i n t h e examples 2,3, 4 and 5 ( b u t n o t i n example 1 ) .
PROPOSITION 1.
Suppode E and E,
bUme t h a t c o n d i t i o n ( * I
conbidehed a b o v e and
ab
b a t i b d i e d , t h e n t h e compact
i b
ab
dubdetb
i n E and E+ atle t h e dame.
PROOF:
If
K
i s a compact s u b s e t of
i s compact i n E i .
i m p l i e s t h a t (x,)
I f (x,)
Ec
t h e n , f o r any i
i s a sequence i n K
,
condition
i s contained i n a subspace E i .
p r o j e c t i o n , i t f o l l o w s t h a t (x,)
i s c l o s e d i n E and s i n c e E
ui
As
i s contained i n t h e
ui ( K ) and c o n t a i n s a subsequence c o n v e r g i n g i n
, ui(K) (*)
a
is
compact
ui ( K )
. Since K
i s metrizable, t h e proposition i s
proved. On E and Ec w e have t h e same h y p o a n a l y t i c functions(i.e. G
- analytic
w i t h c o n t i n u o u s r e s t r i c t i o n s t o any compact subsets)
a n d , a s E i s normed, any h y p o a n a l y t i c f u n c t i o n (i.e. G have
- analytic
and c o n t i n u o u s ) . I n t h e c a s e
E = Co(A)
,
we
H ( E ) = H ( E + ) a s p o i n t e d o u t by J o s e f s o n ( 5 ) . T h i s i s n o t
t h e case i n general, f o r instance consider E = Qp( A ) . The f u n c t i o n f
[p]
analytic
is
,
d e f i n e d by
means t h e i n t e g r a l p a r t of
p
,
15 p <
f (x) =
C
a €A
+ xk]
=
, where
i s p o l y n o m i a l and continuous
f o r t h e norm t o p o l o g y . S i n c e i t c a n n o t f a c t o r i z e s t h r o u g h subspace
Lp(A')
,
A'
and
countable, t h e function f
any
is not continu
o u s f o r t h e p r o j e c t i v e t o p o l o g y . T h i s a l s o shows t h a t , i n
an
open s u r j e c t i v e l i m i t , h y p o a n a l y t i c f u n c t i o n s d o n o t factorize. L e t us recall t h a t a subset B
of
E
i s s a i d t o be bound
i n g i f e v e r y holomorphic f u n c t i o n on E i s bounded on B . I f E
Ph. NOVERRAZ
326
i s a weakly compactly g e n e r a t e d (WCG) Banach s p a c e l i t i s known (8) t h a t t h e bounding s u b s e t s of E
are e x a c t l y t h e canpact s u b
s e t s . The c o n v e r s e i s n o t always t r u e : t h e s p a c e u n c o u n t a b l e , i s n o t WCG
( 6 ) b u t f o r t h a t s p a c e t h e bounding sets
are compact: i f B i s bounding i n € o r any i
,
E = kL(A) , A
ui(B) i s bounding i n Ei
H(E)
,
i t i s bounding i n H(E+);
and t h e n compact s i n c e Ei
i s a s e p a r a b l e Banach s p a c e . The s e t B i s t h e n compact f o r the p r o j e c t i v e t o p o l o g y and a l s o , a c c o r d i n g t o t h e p r o p o s i t i o n cog p a c t for t h e norm t o p o l o g y .
2.
W e s h a l l c o n s i d e r now a s l i g h t l y more g e n e r a l s i t u a t i o n :
(a)
L e t I b e a d i r e c t e d s e t s u c h t h a t f o r any sequence
( i n )i n I
t h e r e i s an index i
in 5 i
such t h a t
€or a l l n . (b)
Let E
l e t Ei ui
: E
be a l o c a l l y convex s p a c e , f o r e a c h i of I , be a m e t r i z a b l e l o c a l l y convex s p a c e +
Ei
a l i n e a r , s u r j e c t i v e , continuous
and and
open mapping s u c h t h a t a b a s i s f o r t h e t o p o l o g y of
( c)
E i s g i v e n by t h e s e t s
;;(V)
any open s e t V o f E i .
Assume a l s o t h a t any
p a c t s u b s e t of
Ei
p a c t s u b s e t of
E
F o r any c o u p l e
i
€or any i i n I and com-
i s c o n t a i n e d i n t h e image of ca"_
. 5
j , let
u i j be a l i n e a r t c o n t i n u
ous and s u r j e c t i v e mapping from E
j t h a t t h e f o l l o w i n g diagram commutes
to
Ei
such
32 I
A CASE OF SURJECTIVE LIMIT
I n o t h e r w o r d s , w e c o n s i d e r an open and compact surjective l i m i t (4) o f m a t r i z a b l e s p a c e s w i t h an e x t r a c o n d i t i o n of cows able s t a b i l i t y on t h e i n d e x i n g s e t . T h i s c o n d i t i o n excludes the
case of a l c s w i t h t h e weak t o p o l o g y . If
i s a l o c a l l y convex s p a c e , l e t us d e n o t e on
E
a s u s u a l by To t h e compact open t o p o l o g y ,
H(E)
Tu t h e t o p o l o g y o f
Nachbin g e n e r a t e d by a l l t h e semi-norms p o r t e d by compact sub-
s e t s , T6
t h e t o p o l o g y g e n e r a t e d by t h e semi-norms
f o r any i n c r e a s i n g c o u n t a b l e and open c o v e r i n g (U,)
i s a c o n s t a n t C and an i n d e x for all f
in
such t h a t
where
of E t h e r e
5
C/fIU "0
t h e b o r n o l o g i c a l t o p o l o g y Tb
f o r instance
Tolb
Tw,b
Or
S i n c e h o l o m o r p h i c f u n c t i o n s on E t i f y each
p(f)
suchthat
H(E).
L e t us i n d e x by b
a t e d with TI
no
p
w i t h a subspace of
H(Ei)
-
f = f a u
i
I
*
f a c t o r i z e s , w e caniden_
H(E)
by t h e r e l a t i o n f
+
T
and i t i s n a t u r a l t o a s k a b o u t t h e i n d u c t i v e
l i m i t of t h e spaces For each i i n
associ-
H(Ei)
, we
H ( E l 1Tg
f o r d i f f e r e n t t o p o l o g i e s on H ( E i ) .
have t h e f o l l o w i n g diagram: H ( E ) ITw
- s +
J
+
l i m H ( E i ) ,T u , i +
H(Ei)
H ( E ) ,To
J
l i m H ( E ) 'Tori -b
I n t h e f o l l o w i n g d i a g r a m , an a r r o w i n d i c a t e s c o n t i n u i t y o f t h e i d e n t i t y mapping: H(E) ,T6
t
l i m H(Ei) ,T -+ 6 ,i
THEOREM
H ( E ) ,Tu
I
-
t
l$m H ( E i ) , T
H ( E ) ,To
I
u,i
-
T
l$m H ( E i ) 'To
16 E i d a b u h j e c t i v e L i m i t ad c o n b i d e t r e d a b o v e ,
,i
abL
328
Ph. NOVERRAZ
t h e t o p o l o g i e s i n t h e diagham have t h e 6ame bounded s e t s .
I t i s w e l l known t h a t two t o p o l o g i e s have thesame
PROOF:
m-
ed sets i f and o n l y i f t h e y have t h e same bounded s e q u e n c e s . I f
(f,)
H ( E ) ,To
i s a bounded sequence i n
each f n
factorizes
and t h e n , by h y p o t h e s i s , t h e r e i s an index
through a space Ei
n i such t h a t t h e sequence (f,)
gn
i d e n t i f y f n with
H(Ei)
t h e n bounded i n
t h e r e i s a compact s u b s e t K '
-
IfnlKl
Sup l f n l K 2 S i p
n
logies
To,i
TUIi
I
fn =
such t h a t
,Tali: <
and
+ T
m
6
l i m H(Ei) ,T + 6 ,i
, which
of
E
i z e thtlough a bpace Ei
fn 0
t h e set
ui,
. AS
such t h a t
(Zn)
is
K C u i ( K ' ) ,hence
i s metrizable, t h e top2
Ei
have t h e same bounded s e t s ( 3 ) . I i
and t h e n a l s o i n
H(Ei)
p r o v e s t h e theorem.
I n g e n e h a l , a bounded d u b s e t
REMARK:
we
l e t K b e a compact s u b s e t o f E i l
i s bounded i n
The sequence
If
f a c t o r i z e s through E i .
06
H ( E ) doeb n o t dactoh-
as t h e dollowing example bhowb:
Let
, A u n c o u n t a b l e , be conhidehed a6 a p h o j e c t i u e l i m i t o d t h e spaced !LP(A') , A ' c o u n t a b l e i n A , and l e t B = ( f a l a A whehe each f a i b dedined b y f a ( x ) = xa , t h e n e t B i b bound
E = kP(A)
ed in
H ( E ) d o t l t h e compact open t o p o l o g y b u t doeb n o t ductoh-
ize. The b o r n o l o g i c a l t o p o l o g y a s s o c i a t e d w i t h t h e s i x t o p o l o g i e s c o n s i d e r e d i n t h e theorem c o i n c i d e s w i t h t h e topologies:
T6
l i m H(Ei) ,To , i , b + PROPOSITION 2 .
,
l i r n H(Ei) ,TgIi +
,
l i m H ( E i ) lTU,i,b +
following and
'
16 E
is a b u t l j e c t i v e l i m i t
ab
conbideted above
,
A CASE OF SURJECTIVE L I M I T
329
we have:
lp H ( E i )
=
(a)
H(E)
(b)
Tg
id
Tw
Oh
i6
H(Ei) iTw,i
(c)
w,i,b
the bohnologicat t o p o l o g y abhociated To
I
i h bohnotogical
H(E) ,Tg = l i m H ( E i ) l T u , i -+
dotr any
kP(A)
,
15 p <
+ -,
.
A uncountable
t a i n e d and bounded i n a hubdpace AO,
on
H(el)
w e have
theotrcm does n o t
3.
I
H(Q: ) ,Tg =
H(CJ)
E = Co(A)
(see ( 2 ) ) .
It 16 knawn ( I ) t h a t any bounded
REMARK:
then
i ,
The h y p o t h e s i s (c) i s v e r i f i e d f o r i n s t a n c e i f or
With
,
bet
J
lim
06
H ( d ) i b co!
6ini-te Ln I H((CJ)
,
and
although t h e
J fiRite J CI apply i n t h a t case.
We s h a l l now c o n s i d e r t h e s p e c i a l case E = Co(A)
,A
un-
c o u n t a b l e , where t h e holomorphic f u n c t i o n s f o r t h e norm a n d f a r t h e p r o j e c t i v e t o p o l o g i e s are t h e same ( 5 ) . L e t us i n d e x by t h e t o p o l o g i e s on instance
Toll
1
H(E) a s s o c i a t e d w i t h t h e norm topology ( f o r
, TUll
and
Ts,l)
and by 2
t h e t o p o l o g i e s as-
s o c i a t e d w i t h t h e p r o j e c t i v e topology ( T o I 2 , T w I 2
and
TsI2).
W e have t h e following diagram:
From t h e theorem, w e know t h a t a l l t h e s e t o p o l o g i e s have
Ph. NOVERRAZ
330
t h e same bounded sets and so
T6 ,1 = T6 ,2
s i b l e t o p r o v e t h a t t h e two t o p o l o g i e s c i d e : l e t p be a s e m i - norm on p is not
as
each K
I
f a c t , it i s pos and
TwIl
U, U 3 K s u c h t h a t
t h e sequence
Tw,2
H(E) c o n t i n u o u s f o r
T u r l c o n t i n u o u s , f o r any compact K
open s e t
T1-
. In
of
,
t h e set
UK = u
Pol0gY and we have
there
E
0
u(U)
is
. For
( f n r K ) factorizes through a space Co(%)
c o u n t a b l e . I s u i s t h e r e s t r i c t i o n mapping from C o(%)
Tur2. I f
"IfnlK I
p(fnIK)
coin-
Co(A)
I
onto
i s open f o r t h e p r o j e c t i v e
Ifn,KIUK = Ifn,KIU
+ to
f o r each n . Since t h e
compact sets a r e t h e same f o r t h e norm and f o r t h e t o p o l o g i e s , w e can deduce t h a t t h e s e m i
- norm
p
projective
is not
T
w,2
continuous. Contradiction. F i n a l l y , on t h e s p a c e
Co(A)
t h e r e a r e two d i s t i n c t t o p 2
l o g i e s d e f i n i n g e x a c t l y t h e same holomorphic f u n c t i o n s and H ( E ) e x a c t l y t h e same t o p o l o g y T6
(resp.
Tw
I
on
So, f r a n the
To).
holomorphic p o i n t of view, i t seems t h a t t h e r e i s no r e a s o n t o d i s t i n g u i s h between t h e s e two t o p o l o g i e s . N e v e r t h e l e s s , for the norm t o p o l o g y t h e Levi problem h a s i n g e n e r a l n o s o l u t i o n w h i l e f o r t h e p r o j e c t i v e t o p o l o g y t h e Levi problem h a s always a sol; tion.
BIBLIOGRAPHY
[l]
J. A. BARROSO AND L. NACHBIN,
-
Sur c e r t a i n e s proprigtgs
bornologiques des espaces d ' a p p l i c a t i o n s
holomor-
phes, Colloque de Lisge ( 1 9 7 0 ) .
[ 23
S.
DINEEN,
-
Holomorphic f u n c t i o n s on (Co,X1)
Math. Ann. 1 9 7 2 , t . 1 9 6 , p. 1 0 6
- 116.
-
modules,
A CASE O F SURJECTIVE LIMIT
[3
1
S. D I N E E N ,
-
331
Holomorphic f u n c t i o n s o n lcs, Ann. F o u r i e r ,
1973, t . 23, p . 1 9 - 5 4 .
14
1
S . DINEEN,
SMF
[5 ]
-
On s u r j e c t i v e l i m i t s , t o a p p e a r i n t h e Bull.
.
B . JOSEFSON,
-
A c o u n t e r e x q l e to the Levi problem, Lexington
c o n f e r e n c e , S p r i n g e r L e c t u r e n o t e s nQ 364.
[6
1
J . LINDENSTRAUSS,
-
Weakly compact s e t s , Symposium o n 111
f i n i t e D i m e n s i o n a l Topology, A n n a l s of Math. S t u d -
i e s nQ 6 9 , 1 9 7 2 .
[7 ]
Ph. NOVERRAZ,
-
S u r l a t o p o l o g i e t o n n e l g e e t l a topologie
bornologique associge
l a topologie
de
Nachbin,
CR. Acad. Cs. t . 2 7 9 , 1 9 7 4 , p . 4 5 9 - 463.
[8
]
M.
SCHOTTENLOHER,
-
Uber a n a l y t i s c h e
Banachraumen, Math. Ann.
Fortsetzung
in
( 1 9 7 2 1 , t. 1 9 9 , p. 313-336.
U n i v e r s i t g de Nancy Mathgma t i q u e s Case O f f i c i e l l e 1 4 0
54037 FRANCE
-
NANCY CEDEX
I
This Page Intentionally Left Blank
Infinite Dimensional Holomorphy and Applications, Matos (ed.) @ North-Holland Publishing Company, 1977
THE CONNECTED FINITE DIP33NSIONAL LIE SUB-GROUPS OF THE GROUP Gh (n,C)
BY
DOMlNGOS P I S A N E L L 7
With this work we continue our researches about the infinite Lie group of germs of holomorphic invertible transformations that preserve the origin in C", which in [l] we called Gh(n,C).
A natural problem is the research of it's sub-groups. In [ 2 ] we found the one parameter sub-groups and we showed that these are given by cr.
x
E
E
C
+
exp a a, where exp x is the exponential
gh(n,C) the Lie algebra of Gh(n,C), of germs of holomorphic
transformations around the origin that preserve the origin, id
a Though x1 atl a + + xn atn exp is not invertible around the origin of gh(n,C) and the ide"
the unity of Gh(n,C) and Xx
tity of Gh(n,C) ( [ Z ]
r
- ...
=
-.
we show here that it is locally injective
when restricted to a finite dimensional vector sub-space.
This
is possible because a finite dimensional vector sub-space of a Hausdorff locally convex vector space has a topological Supplement. We can then construct M the connected finite dimensional sub-group associated to a finite dimensional
sub- algebra
gh(n,C) (theorem 1), and we give the inverse of a chart 333
of
around
D. PISANELLI
334
the identity of M . This is equivalent to the construction
of
the local finite dimensional transformation's group. This is the content of 91. In order to obtain a self-contained paper we give concisely in $ 2 the construction of a Lie group theory
in
open set of a locally convex vector space, including the
an
cons-
truction of the local canonical Lie group in a Banach space and some properties of the group Gh(n,C)
. This result will
appear in
[l] and [2].
THEOREM 1:
Thefie cxinbn a ? - I
naL c o n n e c t e d Lie nub-ghoupn
mapping dhom t h e d i n i t e d i m e n b i o
04
Gh(n,C) o n t o t h e d i n i t e dimemio -
naL Lie nub-algeb&an 0 6 gh(n,C). T h i d mapping a h n o c i a t e n t o each c o n n e c t e d d i n i t e dimennianaL Lie nub-g/roup t h e Lie d g e b m 04 t h e t a n g e n t n p a c e at t h e u n i t y . I)
If M is an r-dimensional connected Lie sub-group of Gh(n,C), there exists a chart $e around e=id($,(e)q) -1 1 ' ( 0 ) is 1-1 (linear) map with values in Cr. $e = (4e from Cr onto Te
(the tangent space at M in e)
can define a local Lie group Ge in Te
( I ; ' 0
.
c1
We
will
be a local homomorphism (definition (52,II)) from in Gh(n,C), and its
Ge differential at the origin=iden
tity, will be an homomorphism of the Lie algebra
in
gh(n,C) (92,II). Then Te i n a Lie nub-dgebha ad gh(n,C)
and
i t d
d i m e n n i o n in
r.
The local Lie group Ge is locally isomorphic to a local in Te , with operation $1, infinitesiml e transformation L1(xl), the inverse of this E,(x,) and same al-
canonical Lie group C
LIE SUB-GROUPS OF THE GROUP Gh(n,C)
335
gebra of Ge
(§2,11). The isomorphism (local) between Ce
Ge is given
by an exponential mapping from Te in
(52,II).8=c1 o $ e l o exp
:
exp4
is a local isomorphism between Ce and
9($1(x1,y1) 1 = 9 (8 (x,)
M,. i.e.
Te
and
8 (yl)1
I
(xl,yl around the or&
gin in Te). Then 8' (91(xllyl))(91)'(xl,o)hl=9' (8(x1),0(o)) @'(o)hl.
Y
But 8' (0)hl = hl
{
hl
(W
Te) then:
E
8' (xl)kl = J ( 8 ) dl(xl)kl
(1)
8
(0) =
e
(see (g2,II) f o r the mean of
J(0)).
We will see later (81,II) that 8 = exp.
M is connected, then is a neighbourhood of e in
hood of zero in T e M ah a b e t .
u Un = n,l 18 (Ul)ln, where U n , l M, U = U-' and U1 is a neighbour-
M
=
. We have then t h e
If in M we have another atlas
group with the same tangent space
u n i q u t n e b b o d t h e bub-ghoup
-
9, and ( M , 6 ) is a Lie sub
Te at the identitv e l wewill
have obviously:
6-l e
o
o exp- = exp =
9
c+e
-1 o
-1
9e o exp 9 ,
then
and
6e
0
';9
lomorphic
holomorphic. The identitv mapping iM of at
e with holomorphic inverse
.
M is hg
But iM is
the composition of x
E
( ~ ~ $ 1a l x
E
-1 ( ~ ~ $-t 1a x
E
(M,+)
x
E
(M,;)
a-lx
E
(M,;)
a-1 x
E
(M,+)
-f
-t
-f
This gives the holomorphy of any
a
E
-f
-+
-1 a(a x) = x
E
(M,G)
,
a(a-1 x) = x
E
(M,+)
.
iMand of its inverse
M. (M,$) in t h e n i b o m o a p h i c t o
(MI$).
at
D. PISANELLI
336
11) Conversely let T
be a finite dimensional sub-algebra e of gh(n;C). Let Ge = ( ~ l ~ U l ~ v l be ~ W the l ) local canoE ical group in T
L1 (x,) be Its its
{
whose algebra is Te (52,II). Let e infinitesimal transformation and 6,(x,)
inverse. Let be the system:
8' (xl)hl = J ( 0 ) &,(x,)
(2)
hl
6 (0) = e
In order to solve the last system we must solve the ordinary system:
whose solution is obviously g(t,xl) = exp tx
1
A)
(52,II).
The right member of ( 2 ) satisfies the condition:
integrability
L I E SUB-GROUPS OF THE GROUP Gh(n,C)
337
0 (91(xl,yl) 1 = 9 ( 9 (x,) r e (y,) 1
when
x1 and y1
are around (0,O)
proves B 1 . C)
8
is locally injective.
E Te
X
Te.
This
D. PISANELLI
338
Te h a s a s u p p l e m e n t a r y t o p o l o g i c a l s u b - s p a c e Se.
L e t p1 and
se
then
el
and Let
8'
ei(o) el
and p 2 t h e p r o j e c t i o n s from = p1 o 8 ,
+
x1
:
O 2 = p2 o 8
e se
y1 E T~
+
. We
in Te,
Te)
i
is locally
in-
i s l o c a l l y injective,
y1 = 0
on
C.
By r e s t r i c t i o n of t h e open s e t s
i s i n j e c t i v e i n W1.
0
and W1 , w e c a n
U1,V1
Then by B) and C )
(vl) ,e ( w l ) )
( o l e (ul) ,e
is a l o c a l t o p o l o g i c a l g r o u p i n pological group ( U = 8 (U,)) a r e open i n
y1 E T~
too:
0 t h e r e s t r i c t i o n of
suppose t h a t
el + e2. e se
( o ) h l = 8 ' (0) hl = hl (v hl E Te) , x1 & ' ( o ) h l = 8 i ( o ) h l + B;(0)hl (V h i x1 = identity.
v e r t i b l e and 8
Te
obtain 8 =
e (x,) +
a n a n a l y t i c mapping from Te
t h i s proves
gh(n,C) onto
G h ( n , C ) . GU = n L l Un
and t h e a U
(a E U n - l l
i s a to-
nLl,Uo= {el 1
GU(52,1).
We w i l l d e f i n e i n GU a L i e sub-group s t r u c t u r e ($2,111):
b)
ea
:
x
E aU +
(a1x )
e-'
Ba
i s 1-1 s u r j e c t i v e .
c)
e a ( a u (1 b u ) =
i s open d)
x1
E
E
is a n homeomorphism t o o .
e-1 (a-l( b U ) )
GU.
ea(a u
0
b
u1
u)
eb
-
ti-'(b-l
0
-4 x l )
ea
+
i s o p e n because a"(b
=
6 '
e (ax,)
U)
e (ax,))
L I E SUB-GROUPS OF THE GROUP Gh (n.C)
339
i s holomorphic. e)
i n G h ( n l C ) i s holomog
The i d e n t i t y mapping from G U -1 p h i c b e c a u s e so i s B a = a0
( e -1 a
f)
hl E Te
9)
eab ( e -1 a (x,)
+
-
' (xl)hl I
e;l(y,)
h
= O - l ( b - l - e(xl)
=
(v a
J(a)el
(xl)hl
= 0 - 5 (ab1-l
0 (y,))
i s holomorphic i n ( x l I y l )
E MI.
is injective.
a e(xl) (x,)
= ;-'(b-'
U1
E
X
U1
e -1 a
([O;l(xl)i-l)
e
= e-l(a[a
-
b
(y,))
e (Y,) 1
(V a l b E GU).
T h i s g i v e s t h e holomorphy of t h e p r o d u c t i n h)
b
GU.
(Xl)]-')
lomorphic. T h i s g i v e s t h e holomorphy of t h e i n v e r s e i n The t a n g e n t s p a c e of
G
e l (o)T,
U
a t e is
= T ~ .
By r e s t r i c t i o n w e c a n s u p p o s e t h a t U1 U = e(Ul)
t o o . W e t h e n have t h e c o n n e c t i o n of
i s c o n n e c t e d and Un
and
of
GU
too. 52 I)
LOCAL TOPOLOGICAL GROUPS Let
W be a t o p o l o g i c a l s p a c e l U
c V t W
mapping :
0: ( x , y )
E
v
x
v
-t
xy E
w
U ' U C V
continuous such t h a t
There e x i s t s
e E U
such t h a t
x e = ex = x
vx E
u.
open and
the
D. PISANELLI
340
To each
x
E
and let the mapping
corresponds x-1 E U x x l = x-1 x = e U
x
E
U
+
x1 E
such that
U be continuous.
As a consequence we have the uniqueness of the element
x1 of
x
E
U
and
We will use the symbol
(x
=
x
(W
x
E
inverse
U). Then U = U-l.
($lU,V,W)in order to indicate a
local
t o po Lo gi c a l g h 0 u p .
When
we will have a tapolagical ghoup and we
U = V = W
will use the symbol ($,U). $
and
($,U,V,W) is a local topological group ( U , V , W C G ) , we can
de-
When G is a group (algebraic) with
fine a topological group in (V
a
E
un-',
n,l,
GU = UO
;il =
operation
Un, where aU is open
{el).
A l o c a l hamamahphibm of the topological
local
group
($l,UllVl,Wl) into the topological local group ($,U,V,W), is
a
continuous mapping f defined in an open set of W1 with values in W I such that f($l(xl,yl)) = $(f(xl),f(yl)) when (xl,yl) is around (ellel). A local idamohphidm
of
the
topological local
group
($,Ul,Vl,Wl) into the topological local group (@,U,V,W) is homomorphism fthatis invertible in it$
image that
we
an
suppose
open and €-l continuous. An homomonphibm from the topological group (@,,U,)
into
(@,U) is a continuous mapping that preserves the operations and
@1
0. An i b o m o h p h i b m from the topological group
the topological group
(@l,U1) onto
($,U) is an homomorphism that has a con-
LIE SUB-GROUPS OF THE GROUP Gh(n,C)
34L
tinuous inverse on U . 11) LIE LOCAL GROUP
Let T be a complex locally convex space. Hausdorff
and
sequentially complete. A eocae L i e g h o u p is a local topological -1 group (@,U,V,W) where U,V,W are open in T ,@ and x E U x E U -f
are G-analytic ([5]
)
.
The mapping, linear in h, (x,h) E V x T + L(x)h, where d L(x)h = I-da @ (x,et ah) }a=or is called the i n d i n i t e b i r n a e t h a n n dohmation of the Lie local group. There exists L(x) the inverse of L(x) the linear mappings of T) for any
x
E
E
L(T) (the set of
U. When U = V = W
we
will have a Lie ghaup that we will indicate (@,U). Let H(U) be the complex vector space of G-analytic pings in U C T H(U)
map-
with values in T.
is a Lie algebra when we define the bracket
[fr(r!
= f'g
-
g'f
The mappings x E U
+
L(x)h
H(U) and there exists [h,k] [Lh,Lk]
=
frg E H(U).
E
T
E
T are a Lie sub-algebra
in
such that
Lrh,k]
We have then a Lie algebra isomorphic to T when we define in T the bracket [h,k] = L'(e)kh -L'(e)hk. The operation
i
@
satisfies the L i e e q u a t i o n
@;l(xry)h = L(@) rC(y)h
@(xre)
EXAMPLE:
XIY =
E U
x
Let gh(n,C) be the complex locally convex spce of g
of holomorphic transformations that preserve the origin of
m
c" (;I.).
D. PISANELLI
342
Let Gh(n,C) be the open sub-set of invertible germs, with group operation x o y (x,y with unity i d uehne
x
06
E
06
(-the gehm
Gh(n,C)).
E
Gh(n,C)
ghoup
i b a Lie
t h e i d e n R i - t y m a p p i n g ) , and t h e
Gh(n,C) in t h e gehm 0 6 t h e i n v e h b e
thUMb6OhmUtiCJM
T h e i n ~ i n i - t e b i m a lt h a n 6 6 o h m a t i o n i b L (x)h = J (x)h, i b
t h e Jacobian
in-
.
06 a tephebentative 06 x J(x)
the
06 x
E
Gh(n,C). T h e L i e a l g e b h a
06
whehe
gh(n,C)
i n 2he c l a n n i c a l : ih,k] T h e bybtem
J(k)h
=
-
J(h)k
f
han t h e b o b u t i o n g(t,x) holomohphic i n C
x
gh(n,C) and
exp x = g(1,x) i n an e x p o n e n t i a l mapping, i . e . exp ax o exp Bx = exp(a
(v
a,
B
E
C, x
E
+
@)x
gh(n,C)).
we h a v e t o o : exp x
whehe
Xx
=
a + x 1 atl
=
C - Xz(id) m20 m!
- ...
+
xn
a
%
(i21)
.
A l o c a l homamohphibm of the local Lie group (I$,U,V,W) intois
an holomorphic (G-analytic and continuous) mapping € defined is
an open set contained in U1 containing el EUl,with values in W such that: f
($l(xl,Yl)1
=
0 (f (x,)
If
(yl) 1
around (ellel). A coca.! i n o m o h p h i b m from the local Lie g r o u p (I$ll U,,V,,W,)
into the local Lie group ($~,u,v,W)I is a local homomorphism of the
first into the second that is invertible, it's
image is open
L I E SUB-GROUPS OF THE GROUP Gh ( n ,C)
i s holomorphic.
i n W and f - 1
When
i d a Local? homomohphism
f
~Ol,Ul,Vl,Wl~
homomohphibm A
t h e LacaL
Lie
ghaup
i n t o t h e L a c a l L i e ghaup (O,U,V,W), f ' ( e l )
i o an
06
t h e cohhenpondent L i e algebhan
06
=
Evehy L o c a l ghoup i n a Banach bpace T p h i c t o a Local? canonical? L i e ghoup i n i b
.
L o c a l canonical? L i e ghoup i s a l o c a l L i e group (~l,Ul,Vl,Wl)
where t h e u n i t y i s t h e o r i g i n and L1(x1)x1
phidm
343
given b y t h e exponential
T
.
x1
i b
( x l E U1).
lacaCLy ibomoh-
T h i n Local
idVmVh-
exp x = g ( l , x ) , whehe g ( t , x )
in t h e n o l u t i o n o d t h e s y n t e m
we have t o o
t
g(o) = e
(exp'(0) = identity.
T o each c a n t i n u o u n L i e aLge6ha i n a Banach b p a c e ,
thehe
e x i n t b anRy a c a n o n i c a l L i e ghoup whane aLge6ha i n t h e
given,
([6]
pg. 251).
111) L I E SUB-GROUPS
L e t ( O I U )b e a L i e g r o u p i n T a l o c a l l y convex,
complex
v e c t o r space, Hausdorff, s e q u e n t i a l l y complete. A
a)
L i e nub-ghoup o f
(+,U) i s a p a r t
M C U
such t h a t :
M i s a n a n a l y t i c manyfold: A f a m i l y of p a r t s (i E I )
Oi C M
i s given:
I1
a)
M = iEI
O i
b)
Mappings
Oi
( i E I ) a r e g i v e n , 1-1, o n a n open set of
a l o c a l l y convex, complex, H a u s d o r f f s e q u e n t i a l l y corn plete c)
X
@ i ( ~ l0
(V i E I ) .
oj)
i s open i n
x
(W i , j E I ) .
D. PISANELLI
344
oj)
(v i,j
d)
Q j o * ;$
e)
The canonical mapping id from M in U is holomorphic.
f)
The differential of id is injective, i.e. -1 hl E x ($i ) (xl)hl E T is injective (xl = $i(x),
is holomorphic in
$,(O,
I'
I).
E
-f
'dxEOi, iEI). 6)
M is
g)
The product of U when restricted to M is
a
sub-group of U
(in the algebraic sense). holomor-
phic. h)
The inverse of U when restricted to M
is
holomor-
phic. IV)
Let X and Y be locally convex space, complex, Hausdorff
and sequentially complete. A connected and open in X in Y
,f
LF-analytic in
linear ('d (x,y) E A
x
B)
A x
B
. Let us
x
X (1:5]),
h
E
X
-f
I
B
open
f(x,y)h
E
Y
suppose the integrability condL
tion: fk(x,y)h symmetric in THEOREM:
k
(h,k) E X
+ f' (x,y) * f (x,y)h k Y
x
X
(V(x,y)
E A x
B)
.
The Aybtem
y' (xih = f (x,y h
Y(X0) = Yo
han o n l y o n e A o L u t i o n L F - a n a l y t i c i n A , phouided t h a t t h e d i n a h y di.6 6 e h e n t i a & e q u a t i o n
han a n o l u t i o n LF-anaLytic i n t h e c o n n e c t e d n e t D = {ft,x) E C x X I Xo + t(x - Xo) E A ) .
ak-
LIE SUB-GROUPS O F THE GROUP Gh(nrC)
345
The uniqueness is a consequence of the Taylor expansion
PEOOF:
and the connectedness of A . Let us prove the existence. We may suppose without loss of generality that
xo =
O r
Y , = 0. Let us differentiate the solu-
tion g(trX) with respect to x with increment h : d ( 7 ) dt g&(trX)h = fA(txrg(trX))thax + f'(tx,g(trX))gA(trX)h.~ + Y
+ f(txrg(trx))h.
(8)
d
(tf(txrg(t.X)Ih) = f (txsg(t*x) )h + f;((tx.g(t,X) 1 tx
'h+
+ t f; (tx.g(t.xl )f (tx.g(t.x))x h. Subtracting (7) and (8) and using the
integrability
condition
we have : d dt = f'
Y
(gi(trX)h - t f (tXrg(trX)h) =
(tXrg(trX)) (gk(trX)h
-
t f (tXrg(t,X))h)X.
The ordinary differential equation:
{
a(t) dt
=
fl (txrg(trx) )a(t) x
Y
a(o) = 0
has the unique analytic solution gA(o,x)h = 0 gi(trx)h
(W
-
x
E A,
a(t) = 0. But g(orx) = 0 then
h E XIr
t f (txrg(trx))h = 0
(''(tpx)
E
D) r
and gA(lrx)h = f(Xrg(1rX))h d g(t,o) = 0 From (6) we have dt
(v (W
t
X E E C)
A).
and
(10) g ( l r 0 ) = g(or0) = 0. (9) and (10) show that
In an other work ([3])
g ( l , x ) is the solution of (5).
we made the assumption that X
Y are Banach scale and we gave sufficient conditions
order that (6) has a solution.
on
and f
in
346
D. PISANELLI
BIBLIOGRAPHY
[ 1.1 PISANELLI D., An example of infinite Lie group. 1975. To appear in "Proceedings of the American Mathematical Society".
[
23
PISANELLI D. , An extension of the exponential of a mtrix and. a counter-example of the inversion theorem of an holomorphic mapping in a space H(K)
. To
appear in
"Rendiconti di Matematica", of 'Istituto di Alta Matematica".
[ 31
1975.
PISANELLI D., Thgorsmes d'ovcyannicov, Frobenius, d'inveg sion et groupes de Lie locaux
dans
un'khelle
d'espaces de Banach. C. R. Academ. Sci.
Paris,
S6r. A-B 277 (1973). A943-A946.
[ 41
PISANELLI D. , Solutions of a non linear abstract CauchyKovalewsky system as a local Banach analytic m y fold. To appear in "Proceedings of Colloquium of functional Analysis", Campinas, Brazil, 1974
[5
PISANELLI D. , Applications analytiques en dimension Infinie. Bull. Sc. Math. 96 (1972), 181-191.
61
MAISSEN B., Lie Gruppen mit Banachraum als Parameterraum, Acta Math. 108 (19621, 229-269.
I'NSTITUTO DE
MATEMATICA
DA UNIVERSIDADE DE
E ESTAT~STICA SAO
PAUL0
Infinite Dimensional Holomorphy and Applications, Natos (ed.) @ North-Holland Publishing Company, 1977
MAXIMAL ANALYTIC EXTENSIONS OF RIEMA" DOMAINS OVER TOPOLOGICAL VECTOR SPACES
By K . RUSEK AND J. S I C I A K ( K R A K d W J ( * J
1. INTRODUCTION Let E be a Hausdorff topological vector space over M
(t.v.s.)
( M = Q: or M = R ) . Given a Riemann domain (X,p) over E
- the space of all
let A denote a linear subspace of @(XI
plex-valued M-analytic functions on X. Given a locally
com-
convex
topology T on A let A* denote the topological dual o f (A,T) en-
*
dowed with the topology of pointwise convergence. Let cb:X+A denote the evaluation mapping defined by @(x) (f) = f(x), x
E
X, f
E
A.
Under suitable assumptions on E, A and T we prove that is M-analytic and, if (X,,j,,
cb)
cb
denotes the canonical maximal
analytic extension of X with respect to c b , the triple (X4,jcb,AJ is a maximal analytic extension of X with respect to A. The s e
............................................................... (*)
The present version of this paper was partially done by the
second
author during his stay (February-March, 1976) at
Uppsala University as its guest. 347
the
K. RUSEK AND J. SICIAK
340
bol A@ denotes the set of all functions
;(XI:
=
i(x) (f), x
E
XQI f
I.
E
E
A.
o(X,)
Let us mention here two examples where this
defined by
construction
of the maximal analytic extension of X with respect to A may be applied. EXAMPLE 1
Let E be an arbitrary Hausdorff t.v.s. over CIA- an
arbitrary linear subspace of@(X),
and
-
T
the strongest local-
ly conveX topology on A. Then (X@,j@,A@) is a m.a.e.
of
the
pair (X,A) (see 5). EXAMPLE 2
Let X be an open connected subset of E = IRn
and
let A be a linear space of complex-valued 1R-analytic functions such that (A,T~)is a Frechet space,
denoting the
T~
topology
of uniform convergence on compact subsets of X. Then X@
is
Riemann domain over Cn and (X@,j@,A@) is a m.a.e. of the
a
pair
(X,A) (see 9 ) . In particular we may take A = H(X)
-
the Frechet space of
all complex-valued harmonic functions on X. Then the
"harmonic
envelope of holomorphy" of X is identical with the domain (over Cn)
-
X@ the canonical domain of existence of the evaluation
mapping 0: X+
H(X)* (compare with [ 4 ]
Our results presented in
6-7
I
151 I
[6]).
improve some of the results
due to Coeur6 [2] and Schottenloher ([lo], if E is a complex Baire or metrizable t.v.s. endowed with the bornological topology
iCb
[ll]).
In particular,
and A = @(X)
is
associated with
the
topology T ~ then , a comparison of our construction with Bishop construction of the envelope of holomorphy to the conclusion that =;
i(X,),
x of X
i.e. each point of
5
may
the leads be
MAXIMAL ANALYTIC EXTENSIONS identified with a linear multiplicative functional h
349
: @(X)+C
that is rcb-continuous. Moreover, if (Y,j ,&Y) (x,O(X)) then j*
:
(@(Y)
,
)
is any analytic extension
Tcb)
(@(x),
of
Tcb) is a topological
isomorphism. The corresponding results by Coeur6 and Schottenloherwere obtained under the assumption that E is metrizable. In
10 we prove a theorem on natural Frechet spaces
A
analytic functions in X which is an improvement of Theorem of [ 2 ] ; namely we do not assume that A separates the points
of 2.3
of
A.
All the notions of the theory of topological vectorspaces, used in this paper, may be found 191. Concerning the notion of an analytic function defined open subset of a Hausdorff t.v.s.
E with values in a
on
locally
convex sequentially complete t.v.s. F, along with the
corre-
sponding notation and theorems we follow [l]. All the
vector
spaces considered in
2
-
7
are complex.
K. RUSEK AND J. SICIAK
350
CONTENTS
1.
Introduction
2.
CateTory of Riemann domains over a t.v.s.
3.
Analytic extensions of a Riemann Domain ( X , p ) with
respect
to a family A of analytic functions on X. 4.
Admissible topologies on a linear subspace A of 6 ( X ) the analyticity of the evaluation function
5.
and
0 : X d A * .
Maximal analytic extensions as the canonical domain of exis tence of the evaluation mapping
6.
Analytic extensions of Riemann domains over metrizable
or
Baire topological vector spaces. 7.
Comparison with the Bishop construction.
8.
Holomorphic continuations of IR-analytic functions defined on open subsets of real Banach spaces.
9.
Analytic extensions of open connected subsets of real-ch spaces.
10. A theorem on natural Frechet spaces.
MAXIMAL ANALYTIC EXTENSIONS
351
2. CATEGORY OF RIEMANN DOMAINS OVER A TOPOLOGICAL VECTOR SPACE Let E be a Hausdorff topological vector space (t.v.s.)ovw
c. 2.1 DEFINITION
A Riemann domain o v e t E i d a p a i h (X,p)
A E
X i d a Hauadotdd c o n n e c t e d t o p o l o g i c a l Apace and p: X a l o c a L homeomohphidm
a ( E ) and c a l l e d t h e c a t e g o h y
Oveh E. 7d (X,p) and (Y,q)
06
wit&
Riemann domainb
ahe e l e m e n t d ( o b j e c t d l o h & ( E ) , t h e n
any COntinUOUb mapping j: X j Y duch t h a t p = q o j w i l l
06
c a l l e d a mohphibm
2.3 DEFINITION
be
t h e o b j e c t (X,p) i n t o t h e o b j e c t (Y,q).
We s a y t h a t a d u b n e t U c X, whehe (X,p)E @(El
id plU
i b AchCicht,
i d
.
T h e d a m i l y o d all Riemann domains o v e h E
2 . 2 DEFINITION
be d e n o t e d b y
whete
i d
a homeomotphidm 0 6 U
,
o n t o p(U).
One can easily check that any morphism j is a local home2 morphism. 2 . 4 LEMMA exidt
E
an open b a l a n c e d n e i g h b o u t h o o d B(x,X) o h 0 E E n e i g h b o u t h o o d G(x,x)
d c h l i c h t open
(i) P(G(X,X)
(ii) 7 6
u
=
thehe
and
a
buch t h a t
oh
04
x and
U
i b
0 E E w i t h p(u) = p(x)
+
U,
i d a b c h l i c h t open n e i g h b o u h h o o d
- . . t h e n U c B (XrX) and U Let
04 x
X,
P(X) + B(XrX) ;
a n open b a l a n c e d n c i g h b o u h h o o d
PROOF
x
G i v e n (X,p) EG(E) and a { i x e d p o i n t
c B (x,X).
ex denote the family of all pairs
-
(U,U) satisfying
the condition of (ii). Since p is a local homeomorphism,
qx#B.
K. RUSEK AND J, SICIAK
352
-
Put ;(x,X)
:
=c]U, B(x,X) : = u U , where the unions are
over all the pairs (6,U)
E
21,. It is obvious that
taken
p(i(x,X)) =
.
+ B(x,X) So it remains to show that pi B(x,X) is inject-
= p(x)
ive. This will be done if we show that pI (i,U), (V,V)
?lX.
Put
-
I
E
-1 tu: = (PlU)
- -
Since U /7 V #
B
nected, the set
z
:=
(2
E
-
-
n
Is identical with p(i) sequently
PI 6 u
2.5 DEFINITION
injective, if
-
-
~(XI+
U 0 V is
-
,. -1 p(U)np(V), tv:= (pl'u) lp(U)np(V).
and the set p(6)fp p(;)
p(U)
6 lJ 3 is
=
con-
-
p(V): t"(2) = tv(z)l
n p6).
- . -
So pi U p7 V is injective, and c o ~
is injective (see Lemma 1.7 in [12]).
The
anted n e i g h b o u h h o o d
bet
06
f;(x,X) w i l l b e c a l l e d t h e maximal b a l -
-
x . We d e d i n e px := plR(x,X).
Observe that if j: (X,p)+ (Y,q) is a morphism,
then
jx := jli(x,X) is injective and
- -1 -1 qj(x) o jx. In particular q-1 in P(X) j(x) - jx o px The fallowing statements 2.6 - 2.10 are direct
+
px
-
B(x,X). conse-
quences of the corresponding definitions or may be found in the first chapter of [2J. 2.6
16 (X,p) E @ ( E ) ,
t h e n p d e t e h m i n e s (uniquely] t h e
t u h e 0 4 an E - a n a t y t i c m a n i d o l d b u c h t h a t p
i A
bthUC-
a l o c a t b i h o l o m oh
phiAm. 2.7
Each m o a p h i b m LA a h o l o m o a p h i c ( a n d l o c a l l y b i h o l o m o h p h i c l
m a p p i n g . IAOmOhphibmA o d t h e c a t e g o h y R(E) a h e b i h o l o m o h p h i m A . 2.8
( I d e n t i t y p h i f l d p l e ] . Two mohphibmd h a v i n g t h e bame v a l u e
MAXIMAL ANALYTIC EXTENSIONS
353
at one p o i n t ahe i d e n t i c a l . 16 t w o a n a l y t i c mappingd
04
(X,p)
i n t o (Y,q) ahe i d e n t i c a l i n an open non-empty A u b d e t 0 6 X t h e n
x.
they ate identical i n
7 6 (Xrp),
2.9 $I :
Xj Y , $
(Y,q)r (Z,r) E a ( E ) and t h e h e ahc idomohphidmd
XI
:
Z
and mohphidmd u
Y+
:
2,
v
Z+Y
:
duch
t h a t $I = u o $ a n d $ = v o 0, t h e n u and v a t e idomohphidmd -1 u = v
and
.
3. ANALYTIC EXTENSIONS OF A RIEMA" DOMAIN (X,p) WITH
TO A FAMILY
A
RESPECT
OF ANALYTIC FUNCTIONS ON X
Given (X,p) E @(E)
and a locally convex sequentially corn the vector space of all an5
plete t.v.s. F, we denote by@(X,F) 4 F. If F =
lytic functions f: X
Q1,
we shall write
(X)instead
of U(XrC)
Let A be a subset of @(X,F).
W e nay t h a t a t h i p l e (Y,j,B)
3.1 DEFINITION
t e n b i o n (a.e.1
06
t h e p a i h (XrA) i6 (Y,q)
a mohphidm, D C @(Y,F)
ping j*:B
3
g
,g
o j
E
A
,(OIL
i d
an a n a l y t i c ex j : X+Y
i d
e v e h y q € B and t h e
map-
E
@(El,
4 g o j € A i d b u t j e c t i v e ( t h e n j*
-
b y t h e i d en
tity p h i n c i p l e -
i d
3.2 DEFINITION
A n a n a l y t i c e x t e n d i o n (X,j,A) 0 6 (X,A)iA
bijective).
---
l e d maximal (m.a.e.1, id d o h e v e y a . e . (Y,j,B)
--
exidta
-
04
(XrA)
a mohphidm u: ( Y , q ) 4 (X,p) Auch t h a t j = u The domain
(x,,) .
0
calthehe
j.
will be called the A(X,F)-envelope
holomorphy of (X,p) If F = C and A =@(XI
of
, then the @(X)-enve-
lope of holomorphy of (X,p) is called shortly the envelope
of
354
RUSEK A N D J. S I C I A K
K.
holomorphy of ( X , p ) and i s d e n o t e d by
(2,;).
The f o l l o w i n g lemma and t h e P r o p o s i t i o n s 3.4
-
3.11
are
simply consequences of t h e d e f i n i t i o n s o r may b e found i n
the
quoted r e f e r e n c e s .
(Coeuhg [ 2 J , Nahabimhan [S], S c h o t t e n l o h e h
3.3 LEMMA
[ll]).
Thehe e x i b t b a m a x i m a l a n a l y t i c e x t e n d i o n d o h e w e t y ( X , p ) E @ ( E l and d o h evehy
dubbet A C @(X,F).
3.3a REMARK
T h i b lemma a l b o
dO.&?OWb
and Theohem 5.1 and i n t h i b wag we o b t a i n 3.4
Any t w o A ( X , F ) - e n v e l o p e b
(X,p)
E @(El
3.5
16 (Y,q) E
06
holomohphy 0 6 ( X , p )
holomoaphy
3.10
ph004.
a
06
domain
@(E).
i b o m o h p h i c w i t h an A ( X , F ) - e n v e l o p e
6?,(E),
E
new
i t b
ahe i b o m o h p h i c i n t h e c a t e g o h y
R(E) i d
-
d h o m Remahkb 3.9
then (Y;q)
i b
06
a l n o an A(X,F)-enwe-
l o p e 0 6 holomohphy o d ( x , p ) . 3.6
16 ( Y , j , B ) . i 6 a n a . e . 0 6
(X,A)
a mohphibm
( X , A ) buch t h a t t h e h e e X i b t b
then (Y,j,B) 3.7
7 6 (Y j , B )
( a n algebha j* : B j
3.9
atbo
i d
Let
,
A i b
i b
then D
06
a m.a.e.
an a . e . i b
aLbo
04
--u: X-Y
and id ( X , j , A ) i b a m . a . e . 0 6
-
with j = uoj,
(X,A).
( X , A ) , whehe A i b
a v e c t o h bpace
a v e c t o h bpace ( a n algebhal
and
an a L g e b h a i c ibomohphibm.
bk,F d e n o t e
t h e covehing dpace
06
t h e dhead
0 6 gehmb
MAXIMAL ANALYTIC EXTENSIONS
06 a n a l y t i c d u n c f i o n b 6hom open IT :
@E,F-tE
355
06
bubbetb
E
F.
Let
@(X,F),
the
t o
denote t h e canonical phojection.
G i v e n (X,p)
bZ(E) and a jiixed d u n c t i o n f
E
E
mapping -1 (f o px )p(x)
jf: X 3 x i b
E
06 a
COntinUOUb ((f), d e n o t i n g t h e gekm
%,F
dunction f a t a point
a E El.
Let xf d e n o t e t h e c o n n e c t e d component 06 t h e Apace
E,F
c o n t a i n i n g t h e b e t j, (X). F i n a l l y d e d i n e
?(x)
: = x(IT(x)),
x
06
whefie x(a(x) 1 denoted t h e v a l u e Then (X,,p,) tfiiple
(X,,jf,f)
i d
t h e gekm x a t t h e p o i n t nQ).
(pf: = a l X f ) ,
@(E)
E
Xf'
E
06
a m.a.e.
1
E
@(Xf,F)
and
the
t h e paih (x,f).
The domain (Xflpf)is called the natural domain of tence of f and the triple
exis(Xf,jf,f) is called the canonical m.
a.e. of the pair (X,f). 3.10
When l o o k i n g d o h an
a.e.
n e h a l i t y , one mag abdume t h a t
06
(X,A), w i t h o u t
o d ge-
[Odd
A c @(X,F) i d a v e c t o h b u b b p a c e .
Indeed, given any subset S C @(X,F), let A denote the lin ear span of S; then any analytic extension of (X,p) with respect to S is an analytic extension of (X,p) with respect to
A
and
reversaly. 3.11
I d (X,idX,A)
then theke exibtb f
id E
a m.a.e.
0 6 (X,A) and id x,y
A b u c h t h a t j,(x)
# jf(y).
E
X, x # y,
K. RUSEK AND J. SICIAK
356
AND THE
4. ADMISSIBLE TOPOLOGIES ON A LINEAR SUBSPACE A OF @(X) ANALYTICITY OF THE EVALUATION FUNCTION @:
4.1
G i v e n a domain (X,p)
pace
06
ax).Denote b y
E
a ( E ) l e t A be a a i x e d l i n e a h bub-
(A,?=) t h e l o c a l l y c o n v e x dpace o b t a i !
ed 6hom A b y endowing it m i t h t h e t o p o l o g y g e n c e on compact b u b b e t b ofi X . l e t
T~~
06 p o i n t w i b e convehgence on
4.2 DEFINITION
gy
T
unidohm conwe:
54
T~
denote t h e
t o p o l o g y i n A abbociated w i t h t h e topology topology
X+A*
T
boanological
L . e t T~ d e n o t e t h e
~
A.
We b a y t h a t a l o c a l l y c o n v e x Haubdoh66
on A i d a d m i d b i b l e ( a n d W h i t e
T E
TA) i d
is
topolo
c T and
(A,?)*
i b s - c o m p l e t e , (A,T)* d e n o t i n g t h e t o p o l o g i c a l d u a l endowed w d h t h e topology
0 4 p o i n t w i b e conueagence.
Observe that the family TA of a admissible topologies A is not empty, because the strongest locally convex
on
topology
on A is admissible. (a) T ~ : =
4.3 PROPOSITION
(b) 7 6 d i m E < (@(XI
,fC)
then
E
T~
w
06
adminbible.
i b
and ( A , T ~ )i n a c l o b e d
m e t h i z a b l e o h Baihe and
(@(X),iC),
then
icb i d
q u e n t l y b y Ganack-Steinhaub theohem PROOF
bubbpacc
06
TA.
(c) 1 6 E i n t . v . b .
c l o b e d bubbpace
IT
T ETA
T~~ E
b a h a e l l e d and
i 6
a
conbe-
TA.
The statements (a) and (b) are direct consequences
of
the Definition 4.2. Ad (c). Since the topology bornological and
is
icb
c i Ccicb, it
of A is (by its definition)
remains to show that
T~~
is
MAXIMAL ANALYTIC EXTENSIONS
357
barrelled. In order to prove that
T~~
is barrelled it is enough
to
show that the space (A,rc) is complete. In order to show (A,T~) is complete we may assume, without any loss of
that
general-
ity, that X is an open connected subset of E. Let (fA)AEAbe a generalized Cauchy sequence in the space (A,rc). Then in view of the completeness of the field 6, there exists a function f: X+
Q:
such that for every x
X
E
lim f A ( x ) = f(x). XEA
Therefore fX-+ f uniformly on all compact subsets of
X.
This implies that f is G-analytic in X. NOW, if E is Baire,then
there exists an open non-empty subset of X on which the
func-
tion f is bounded. On the other hand, if E is metrizable, then f is
locally
bounded in X. So in both cases we may conclude that f is analyL ic in X (see Th. 6.1 in [I]). The following lemma and its Corollary are basic for
the
considerations of this paper.
L e t u be an open d u b d e t
4 . 4 LEMMA
huch t h a t
V.A.
i d
a
(G,T)* iid
i d
b
x
E and ( G , r )
s - c o m p l e t e . Addume t h a t
d u n c t i o n duch t h a t d o h e a c h f
9,: u
06
+ $(XI
(f)
E
E G
+:
-
a l.c. U+
t.
(G,T)*
t h e dunction
c
analytic. Then
PROOF
J, i d
analijtic.
The continuity of J, follows from the equation IJ,(x)(f)-J,(xo)(f)
(xo)I , f
E
G, X,X 0 E U.
So it is sufficient to show that J, is G-analytic, this
follows
K. RUSEK AND J. SICIAK
358
from the Morera theorem (see Th. 3.1 in [l]). 4.5 COROLLARY
14 T
i b
a n y a d m i b b i b l e t o p o l o g y o n A,
then t h e
e v a l u a t i o n mapping 4 : X+
(A,T)*
(whehe @(x) (f) = f(x) d o h x E X and f E A]
i d
analytic.
5. MAXIMAL ANALYTIC EXTENSION AS THE CANONICAL DOMAIN OF
EXIST
ENCE OF THE EVALUATION MAPPING cb In this section (X,p) is a fixed Riemann domain over E l A is any fixed linear subspace of @(X),
T
is any admissible topo-
logy on A and (A,T)* denotes its topological dual with the pinL wise convergence topology. By Corollary 4.5 the evaluation mapping 0
:
X+(A,r)*
is analytic.
Let (X,,j,,;)
be the canonical m.a.e. of the pair
The function f: X,+ A
C
defined by
A
(1) f(x):= @(x)f, x is analytic,because €if:
!i
= €if
{l:f
f
X,,
E
o
(A,r)* 3 u
A@:=
(XI@).
E
A,
and the linear mapping
j u(f)
E C
is continuous. Define
E A).
Then we have the following 5.1 THEOREM
The t h i p l e (X,,j9,A,)
i b a m.a.e.
04
the
paih
(XiA)* PROOF
Since j,:
X-X,
is a morphism and the mapping j$:&+A
is surjective, the triple (X,,j,,A,) the pair (X,A).
is an analytic extension&
MAXIMAL ANALYTIC EXTENSIONS
359
Given an arbitrary a.e. (Y,j,B) of (X,A)define i: Y-t
(A,T)*
by the formula i(y) (f): = j*-'(f)
(y), y
Y, f
E
A.
E
We claim that 0 is well defined and analytic. Indeed, put
f:
In view of Lemma 4 . 4 the
= j*-l(f), f E A.
only thing we have to do is to show that every y
func-
Y the
E
tional A 3
(*)
f
4 f(y)
E Q:
is continuous. With this aim in mind observe that the set 2 := {b E
Y: the functional
is continuous for every
(*)
y belonging to a neighbourhood of b) is open and non-empty (because j(X) C
Now given any fixed b
n! Since
2n ( q ) = (A,T)*
-n lim t
t+O
n
E Z
2).
we have
(-l)n-jh(f;
C )(; j=O
A
tq),
is s-complete the functional A
continuous for every q
E
9
q E
E.
-
fn(q)
f+
E C
E and n 2 0, because Aof-+h(f;$rl)E
is C
is continuous for every t sufficiently small. This implies,again by s-completeness of
(A,T)*,
is continuous for every 17 =
q(y)
-
rl E
that the functional A a f j h ( f ; t l ) S(b,Y). Therefore,
q(b), the functional
(*)
by
is continuous for
putting every
*
y
E
B(b,Y). Hence
2 = Y.
-
It is obvious that 0 o j =
@. Hence
..
(Y,j,O)
is an a.e. of
( X r O ) . Thus by 3.2 there exists a morphism u: Y+XO j = u o j0' The proof is concluded.
such that
K. RUSEK AND J. SICIAK
360
6. ANALYTIC EXTENSIONS OF RIEMANN DOMAINS OVER METRIZABLE
OR
BAIRE TOPOLOGICAL VECTOR SPACES In this section E is a metrizable or Baire t.v.s. E
6? (E),
and ( A , T ~ ) is a closed linear subspace of
We already know that the topology
T~~
(
(X,p) E
@(X)
,T~).
is admissible. Therefore,
the evaluation mapping 0 : X+(A,T~~)* is analytic.
-
Let (X,,j,,
@)
denote the canonical m.a.e. of the
pair
(XI@). Define A @ : =
{f: f € A )
I
A
@(x) (f), x E Xol f
where :(x):=
is a m.a.e. and (X~,jolA@)
E
A. By Theorem 5.1 A o C @(X
of (X,A).
We shall write j instead of j,.
PROOF
)
We already know that j*
is an algebraic isomorphism. Since the topology T~~ is bornological it remains to show that j * and j*-l map bounded sets onto bounded sets. It is obvious that j* is bounded. In order to show that j * - l is bounded let B be a
bounded
subset of (A,rcb). Let K be a compact subset of Xa. Then any function f
E A
we have
U:= {f
E A:
1 1 f I Ik
I If 1 I K
<
m.
for
It follows that the set
5 1)
is absorbing. It is obvious that U is absolutely convex
and
closed. Therefore U is a barrel. Thus there exists a positive
COG
361
MAXIMAL ANALYTIC EXTENSIONS stant r such that B C r U , i.e.
I Ij*-l(f) I l K This means that j*-'(B) 6.2
b
i b a
:
f
E
B.
a Riemann domain o v e h a m e t h i z a b l e oh
Baihe bpace E and . i d (Y,j,B)
j*
5 r,
is bounded.
7 6 (X,p) i
THEOREM
I I;I l K
=
i b
(BrTcb)*
an a . e . o d t h e p a i h (X,A), t h e n (A,Tcb)
t o p o e o g i c a l inomohphibrn.
First observe that ( B , T ~ ) is a closed subspace of @(Y).
PROOF
Indeed, if ( g h ) x E A is a generalized sequence of elements of converging to g gx o j+ g o j
E
E
o(Y) uniformly on compact subsets of Y, then
A, and consequently g @:
E
B.
( A , T ~ ~ )and * I:Y + (B,rcb)* denote the eval
X-
and (Yy,jyrBy)be the m.a.e. of
uation mappings. Let (Xo,j,,A,)
(X,A) and (Y,B), respectively. Then there exists
u: Y-X,
Therefore
g o j uniformly on compact subsets of X.
Let
B
such that j, = u
0
j. Since (X,,u,A,)
(Y,B), so there exists a morphism v: X,+Y,
a
morphism
is an a.e.
of
such that jyr=vOu.
Finally, since (Yyrj, o jrBy) is an a.e. of (X,A) there exists a morphism v': Y , +
Xo such that j,
=
Hence, in view of 2.8, v: yY,,X
v' o j, o j. is an isomorphism
and
moreover we have the equation v o j, = j, o j which implies that j$ o v* = j* o j$
.
By Lemma 6.1 the mappings j$ and j$ are topological morphisms. Since the mapping v*: (By,'rcb) 3 f -+ f o v
is a topological isomorphism, so is the mapping -1 j* = j* o o v* o(j$)
.
E
iso-
(Aor'rCb)
K. RUSEK AND J. SICIAK
362
7. COMPARISON WITH THE BISHOP CONSTRUCTION F i r s t l e t us r e c a l l t h e f o l l o w i n g well known
1
f n ( x , y ) :=
d n
(x)[ ( f
-1
ODx
1 (P(x)+ tY)]t,O~X
E X r Y E Ern,().
Then:
10
fn(.rY)
20
Foh e v e h y x
E U(X), E X
y E Et n
2
0.
c o n t i n u o u b homogeneoub p o l y n o m i a l -1
30
( f op,
f n ( x , y ) E Q:
t h e ,junction E 3 y +
1 (p(x)+ y)=
06
degxee n.
E fn(xIy),xEX, Y EB(x,X),
We b a y t h a t a d u b d e t A C @ ( X )
doh evehy y
d o h evelry f
n > 1 fn(.,y) E' o p:= { [ 06
o
E
f
nL0
7.2 DEFINITION E E,
E A
doh
and
a
i b
i d
€@(XI.
d-htable,
Qvehy
integeh
A. We b a y t h a t A c o n t a i n b c o o h d i n a t e b ,
p: 5 E E'I c A ,
irc
whehe E' LA t h e t o p o l o g i c a l
dual
E.
Following [ 2 ] , p. 4 5 , w e s h a l l now f o r m u l a t e t h e following
7.3 DEFINITION
L e t A be a b u b a l g e b h a
06
@(XI
containing coohdinateb. L e t S denote t h e b e t the bet
06
and
d-Atable
06 ale
h E Spec A-
all non-zeho l i n e a h - m u e t i p l i c a t i v e , j u n c t i o n a & o n A-
duch t h a t : 10
Thehe e x i b t b a b a l a n c e d open n e i g h b o u t h o o d V h
06
0
E
E
Auch t h a t t h e A e h i e b n>O h yn ( f ) , whehe h;(f):
-
i b
= h(fn(.,y))
a b d o l u t e l y c o n v e h g e n t d o h evehq y
E
Vh and d o h all fEA
MAXIMAL ANALYTIC EXTENSIONS
20
i d
30
t h e d u n c t i o n Vh 3 y 4 h (f):=
E A
F O R eWehy f
363
Y
n20
( t h i d element
i d
analytic.
Thehe
element n ( h )
e x i b t b an
E E
h y ( f )E C
unique
b y t h e Hahn-Banach theohem1 huch t h a t h ( S o p) =
7.3
S ( I T ( ~ ) d) o h ewehy 5
L e t now CP: X - ( A , T ) *
El.
E
d e n o t e t h e e v a l u a t i o n mapping,
'I
b e i n g any admiAAibLe t o p o l o g y o n A. Then o n e may p h o v e t h e d o l eOMiRg
(il
4aCtA [Ace
Id h
[?I, pp.
46-47] :
doh a l l y
E S t h e n hy E S
(ii) ( h a ) b - ha* d o h a l l a , b za + z ' b
[iii)@(XI c (iw]
and n o 0 = p
S
06
The d a m i l y
Auch t h a t
I z l + Iz'I
id
E Vh
E Vh
E Vh;
5 1, z , z '
E Q: :
;
A C ~ A{ h
Y:
y E VhlhEs
a
don.
a bahib
i b
flaubdohdd t o p o l o g y o n S [.the Bidhop t a p o L o g y ) .Moheoveh, LA a L o c a l homeomohphibm i n t h i b t o p o l o g y and CP: X-S
71
i b
continuouh; (w]
Let
S@
b e t CP(X).
b y :(h) ib
Mated
Let ;:=
and
the pointd
1:
: f E A),
whehe
f:
S
04
containing t h e
So+@
= h ( f ) , h E S C P . T h e n ( ( S C P , p ),@,$
a m.a.e.
7 . 4 REMARK
06
d e n o t e t h e c o n n e c t e d component
LA dedined
,whehe
06
i A
also,
(X,A) ;
16 a d - A t a b L e a l g e b h a A d o e b n o t c o n t a i n
- (X,J,A)
p:=
a m.a. e .
x. Foh i n b t a n c e ,
o d (X,A)
,
then
.-
A
may n o t A t p a h a t e
L e t X = 6! and l e t
baa g e n e h a t e d b y o n e element ez. Then
cooedi-
A
be t h e a l g e -
= A doed not
bepahate
K. RUSEK AND J. SICIAK
364
-
x
= c.
We shall now state the main theorem of this section. 7 . 5 THEOREM
7 6 A in a d-otabte nubatgebha o { @XI
c o o h d i n a t e b then z ( X , )
= S,
and the mapping
G:
containing
XQ+
in
S,
an
ibomohphibm i n t h e categohy @(E). PROOF
19 First we shall that given a fixed z
ear-multiplicative functional h:=
-
check that the conditions 19
E X,,
-Q ( z ) belongs to
S.
the
lin-
We have to
3Q of Definition 7.3 are
satis-
fied by h. To this aim observe that
(*I
; ( z ) (fn(.,Y)) = i,(z,y),
f E A,
2 E
X,,
y
E
E.
Indeed, the functions on both sides are analytic respect to z
E Xo,
and f o r every y
E
with
E they are identical for z
belonging to the open non-empty subset j , ( X )
of X,.
Hence
by
the identity principle we obtain ( * ) . The equation
(*)
implies that
hn(f) = fn(z,y), y E E, f E A , n 2 0. Y n Then for every f E A the series C hy(f) is convergent at every point y
E B(z,X,)
O , n to the sum
absolutely
that is analytic in B ( z , X o ) . A
Therefore h:= Q(z) satisfies the conditions 19 and 2 9 Def. 7.3, if Vh is any open balanced neighbourhood of 0
E
of
E cog
tained in B ( z , X , ) . In order to show that 3 9 of Def. 7.3 is satisfied by observe that
-@(z) ( 5
o
PI
= S(P,(Z)),
E X,,
5
E'
h,
MAXIMAL ANALYTIC EXTENSIONS (because, given 5
E E',
365
the functions on both sides of the equa
tion are analytic in X0 and identical on the open subset j ( O ( X ) ) . Therefore it is sufficient to put T(h) = a ( g ( z ) ) : = p,(z).
(**)
We claim that the mapping
29
Pix z in
S
E
XQ and let
a =(hY :
y
E
i:
XQ+S
is
Vh) be a basic
continuous. neiqhbourhood
of the point h:= $ ( z ) . We may assume that Vh CB(z,X@). Now observe that
-
-1
= f(qz (q(Z)+ y))= cP(q(z) + Y))(f),
f E A, Y E vh
t
where we have put q = po. Therefore ;-I(%) 3 qil(q(z)+Vh)),
so
is a neighbourhood of z .
that ;-l(Q)
Thus we have obtained the required result. 3Q
i(X,) C
gory 6 ? . ( E ) .
So
and
i:
X Q j S o is a morphism in the cate-
Indeed, it follows from the equation
n So
# $. But i(X,) is connected because Now the equation therefore (O(XQ)C - 0SX.o is a morphism. i(X,)
(**)
S@.
(0
= G o j Q that
is continuous, implies
that
(0:
4Q
In order to show that
6
is an isomorphism it is e n o m
to observe that by the maximality of the extension there exists a morphism u:
SQ-
(XQ,j(O,A(0)
X# such that j , = u o @.By 2.3
we get the result. 7.6 COROLLARY
ALL t h e t i n e a ~ - m u L t i p t i c a t i vj~i u n c t i o n a l b
be-
L o n g i n g .to S Q ahe COntinUOUb w i t h h e b p e C t t o a n y a d m i d b i b l e t o poeogy on
o~).
I n pahRicuLah, i d E i b m e t h i z a b t e o h B a i h e , t h e n all
the
K. RUSEK AND J. SICIAK
366
6 u n c t i o n a l A a { SQ a t e tion
04
cb- c o n t i n u o u A . Th.46 giVeA a g e n e h a e i z a dome he6uQtA due t o Coeuh; [Z] and S c h o t t e n t o h e t [ I I ] . T
8. HOLOMORPHIC CONTINUATIONS OF IR-ANALYTIC FUNCTIONS
DEFINED
ON OPEN SUBSETS OF REAL BANACH SPACES In this section E stands for a Banach space over IR, R is an open connected subset of E, F is a topological vector
space
over C and F* denotes the space of all continuous linear
map-
pings from F to 4: with the topology of pointwise convergence,znd finally
= E
8.1 THEOREM that
Id P
Baihe and I : Sl-+F*
i d
i d a
E.
r(unction
duch
F t h e {unction
W f E
f* i 6
+ iE denotes the complexification of
: sl 3
X
d
W(x) ( f )
E Q1
IR-analytic, then
(i)
W
a
R 3 r > 0 V f
E
E
F
p(Taf*) > r,
w h e t e p(Taf*) d e n o t e d t h e h a d i u 6 a { t h e p o i n t w i d e c o n v e h g e n c e the Tayloh
6 e t i e 6 01(
(ii) Y
n-anaeytic.
i 6
f*
a t a;
Moaeoveh, t h e c o n d i t i o n b ( i ) and ( i i )ahe a t w a y 6
l e n t (F b e i n g Baihe PROOF
(i) Observe that
F
3
f4 f:(X)
equiva-
oh n o t ) .
W
a
E R,
V
x
E
ear mappings
(1)
06
:=
are continuous. Indeed
1! n
(6”,*)
(X) E C
E and
W
n
2 0 the C-liq
MAXIMAL ANALYTIC EXTENSIONS
36 7
and t h e components o f t h e sum are c o n t i n u o u s l i n e a r mappings o f F i n t o C; by t h e Banach-Steinhaus theorem t h e f u n c t i o n a l s
(1)
a r e continuous. Given a f i x e d a
E
n, t h e set
[ f i ( x )[
Fk:= I f E F:
2
0 , x E S),
11x1 I = 11, i s c l o s e d and F =
where S = {x E E:
k = 1,2,...,
u
Fk. I n d e e d ,
kll i f f E F, then t h e r e e x i s t s p > 0 such t h a t
c f;(x), IJxIl < P , nO , whence, E h a v i n g t h e Baire p r o p e r t y , t h e r e e x i s t s p = p f > 0 such t h a t C I If:\ I pn < m. T h e r e f o r e f E F k r i f 1 < p. NOW, using nLv t h e Baire p r o p e r t y of F, o n e c a n f i n d k so t h a t i n t Fn# B.Herce, f * ( a + x) =
t h e r e e x i s t s a n open neighbourhood U o f 0 E F s u c h t h a t
I If;
I _<
So f i n a l l y Wf
2 kn, f E U , n > 0:
I If;
E
1 5
F3Mf > 0 s u c h t h a t
Mf kn, n
2
0.
Therefore
and moreover
n C I I f A l [ rl < =, if 0 c rl < r . n>0
(i i ) F i x a E
and p u t
Yn(x) ( f ) : = f i ( x ) , x Then In: E + F *
E E,
f E F.
i s a c o n t i n u o u s p o l y n o m i a l of d e g r e e n and
in
view o f ( 2 )
-
Thus I i s R - a n a l y t i c i n R. Prom Theorem 8.1 and from
following t w o Propositions.
7 of [l] o n e c a n d e r i v e
the
K. RUSEK AND J. SICIAK
368
8.2 PROPOSITION
Undeh t h e a d n u m p t ~ o n ~ oTh. 6 8.1 t h e h e
an open
duch t h a t
C
bet
o(i) d o t h a t - d u n c t i o n Y : S2xES2, f E F . f
E
8.3 PROPOSITION
d o h evehlj f
one
E F
can
fllsl = f . Moheoveh, t h e h e e x i d t d a F* b u c h t h a t
Let
YIR =
d(S2)d e n o t e
t h e Apace
ued I R - a n a l y t i c d u n c t i o n i n S2. L e t F C
dind
holomokphic
and y ( x ) ( f ) =
Y
f (x)
~ O R
a l l c o m p l e x - v a-l
06
be a Baihe
&(Q)
exidtd
X.V.A.
Oveh 6! d u c h t h a t 604 e a c h a E R t h e C - t i f l e a h mapping F 3 f-f(a) i d
E
C
continuoud. Then t h e d o l l o w i n g d t a t e m e n t d ahe t h u e
(4)
06
W
a
E S2
3
r > 0 duch t h a t W f
f a t a i d nohmally convehgent i n t h e b a l l
(ii) T h e e v a l u a t i o n d u n c t i o n (iii)Thehe e x i d t d an open E
t h e Tayloh
E F
(6)w i t h
8 . 4 EXAMPLE
f =
:In,
0:
iC 6
Q+F*
i d
dekied
I Ix-al]
L e t E = IRn and F = H ( R )
E F3f
with @ =
E #(;,F*)
-
< r;
n-analytic;
d u c h t h a t R c;, W f
and t h e h e e x i d t s
Taf
t h e space of a l l
E
ilS2. com-
plex-valued harmonic f u n c t i o n s i n Q. By t h e Harnack theorem H(Q)
i s a F r e c h e t space i n t h e topology of uniform convergence
on
compact s u b s e t s of R . S i n c e each f E H(S2) i s I R - a n a l y t i c , t i o n @:
0-F"
(F = H ( Q ) ) i s a l s o I R - a n a l y t i c .
e x i s t s an open s u b s e t E
so t h e e v a l u a t i o n k -
&)(El such t h a t f =
c Cn ?In.
such t h a t sl C
Moreover, t h e r e
'i and
W f E H(R)
3 ?E
MAXIMAL ANALYTIC EXTENSIONS
369
9 . ANALYTIC EXTENSIONS OF OPEN CONNECTED SUBSETS OF REAL BANACH
SPACES
I n t h i s s e c t i o n E s t a n d s f o r a r e a l Banach s p a c e , R i s an
complex-
i s t h e s p a c e of a l l
open connected s u b s e t of E , & ( R )
a(Q).
v a l u e d l R - a n a l y t i c f u n c t i o n s on R and A i s a s u b s e t of
The thiple ( ( X , p ) , j , & ) w i t - ! b e c a l l e d a n
9 . 1 DEFINITION
lytic extendion
04 the
p a i h (X,A)
ana-
, id
-10
(X,p) E
20
j : R--1X
LA a C O n t i n U O U A m a p p i n g 6 u c h that i d Q = p o j ;
30
&t(C)'X),
g o j E A
j*
g E A_
d o t evehq
the
and
mapping
4 g o j E A i6 bijective.
113 g
T h e m a p p i n g j * i6 b i j e c t i v e id a n d o n t y id it
9 . 2 REMARK
id
6 u h j ectiv e +
PROOF a E
n.
L e t j * be a s u r j e c t i o n . Suppose g1 o j = g2 o j and f i x I t f o l l o w s from t h e e q u a t i o n i d R = p o j t h a t t h e r e
i s t s an open j (U) C
subset
6 ( j ( a ), r )
where
U
of
E
such
6 (j ( a ) , r ) d e n o t e s
a E U C R
that
the "ball with
j ( a ) and r a d i u s r " i n X. Then pa:= p l r ? ( j( a ) , r ) i s a phism o f
ii(j ( a ) , r )
onto t h e b a l l B ( a , r ) C
p o j (U) = U C B ( a , r )
. Since t h e
E.
f u n c t i o n s g1 o pa
(Prop. 6 . 6 .
I1 of
center
homeomor-
-1
and g2 o pa
so by t h e i d e n t i t y
[lJ) t h e y are i d e n t i c a l i n
T h e r e f o r e g1 and g2 a r e i d e n t i c a l i n q u e n t l y , i n X.
U,
and
particular
-1
are holomorphic i n B ( a , r ) and i d e n t i c a l on principle
In
ex-
B ( j ( a ), r )
and,
B(a,r).
conse-
3 70
K.
RUSEK AND J . SICIAK
((X,p) j , A ) ,
We d a y t h a t an a n a l y t i c e x t e n d i o n
9.3 DEFINITION o d t h e paih ( R , A )
((X',p'),j',&')
id
06
m a x i m a l , id d o h e v e h y a n a . t y t i c
extendion
(Q,A) t h e h e e x i d t d a m o a p h i d m u: X'+X
huch
t h a t j = u o j'.
06
Dbdehve t h a t (X,u,&) i d a m . a . e . 9 . 4 Let A be a vector subspace of &(Q)
-
(X',&').
such that there
-
-
-
exists
Z E and a vector subspace A C @ ( a ) an open connected set R C
so
that the restriction mapping r:
i3 g
qlQ E A
is surjective (and therefore an algebraic isomorphism by
the
principle of analytic continuation). Consider ( A , r ) as a t.v.s. ly
convex topology T
= T
endowed with the maximal local
- -
The ~ function ~ . 9: Q-A*:=
~
defined by
-@(z) (f):= r-1 (f)( z ) ,
f
E A,
z
-
R
E
-
@ I n is R-analytic in
is C-analytic and its restriction O:=
(0-
Let
,a) denote the sheaf of germs of
-
E,A*
(ArT)*
R.
holomorphic
functions defined on open subsets of E with values in
A*.
The
mapping (*)
j,
:
n
3 x+
is continuous. Let a,
(GlX
E@-
E ,A*
denote the connected component of
containing j , ( Q ) . One can easily check that if
i, 2
0E,A*
and
are
MAXIMAL ANALYTIC EXTENSIONS
371
is surjective. 9.5 THEOREN
Undeh t h e a b b u m p t i o n b
9.4 t h e Rottowing b t a t e -
06
mentb a t e t h u e : I
(i)
i b a m . a . e . o h (Q,@) :
(Q,,j,,@)
.
(ii) (Q,,j,,A)
i d
is continuous, q =
a m.a.e.
06 (n,~).
u, j, = u o k. Hence u ( Y ) C n,
and
the
(ii) Given any a.e. (Y,j,B) of (RIA) it is enough to
ob-
( B , T ~ )is continuous (because j*
is
1~ o
proof of (i) is concluded.
serve that I*-':
(AI?)-
an algebraic isomorphism and
and next to repeat
T = imaX)
the
proof of Theorem 5.1. 9.6
Let E
=
IRn , and let A C &(Q)
be a Frechet space
when
endowed with the topology T ~ . Let A* denote the topological dual of ( A , T ~ )with the topology of pointwise convergence. Then by Proposition 8 . 3 evaluation function @: an open set f = :In,
R+
is IR-analytic and there
A*
C en such that R c
and there exists
6,
E @(E,A*)
W
f
E
A
3z
with 0 =
the exists
E @(;I
with
(PlQ.
Let (Y,j,B) be any a.e. of (i2,A). Then (B,iC)is a Frechet is a Cauchy sequence then there
space. Indeed, if {gn} C B ists g
€
@(Y) such that qn+
Y. So gn o j+g fore q o j
€
CJ
ex-
uniformly on compact subsets of
o j uniformly on compact subsets of R. There-
A, and so g
€
B.
K. RUSEK AND J. SICIAK
3 72
The mapping j*: B +
A is, by Banach theorem, a topologi-
cal isomorphism. Therefore we are lead to the following 9.7 PROPOSITION
7 6 t h e a b b u m p t i o m o d 9.6 a l ~ eb a t i d d i e d
then
ti) and (ii) 0 4 Theotrem 9 . 5 h o l d t t r u e .
t h e btatementb
10. A THEOREM ON NATURAL FRECHET SPACES
Let E be a complex t.v.s. admitting a countable
w= (wn}
of open sets, Z a Banach space, (X,p)
A C @(X,Z)
basis and
E a ( E )
- a natural Frechet space (i.e. the topology of
A
is
stronger than the topology of pointwise convergence, and consequently, stronger than the topology T ~ ) .
19 If w is any connected open subset of E and x is
PROOF
point of X such that p(x) A
w,x
-
-
:={f
E
A: 3 f
-
(f)p(x) - (f AWIX =
0
-1
p,
then the set
E w,
E @(w,Z)
a
,f
is bounded and
)p(x)}is either a set of 1 category or
A. Indeed, given any continuous seminorm q on A define
seminorm q on A
WIX
a
by the formula
q(f):= q(f) + sup YEU The vector space Aw
IX
I
If(y) 1
I
I
f E Awl,
with the topology
T
defined by
seminorms
is Frechet. Indeed, if ifn} is a Cauchy sequence in the space
MAXIMAL ANALYTIC EXTENSIONS
x'
3 73
then it i s a l s o a Cauchy sequence i n t h e space A.So t h e r e
exists f
such t h a t fn+f
E A,
( i n A ) . I n p a r t i c u l a r f,(y)+fb)
f o r e v e r y y i n a neighbourhood of x. The sequence ed a n a l y t i c function
?.
i s uniformly convergent i n w t o a b o m i -1 neighBut z n ( y ) = ( f n o px ) ( y ) i n a
-
1. 1. T h e r e f o r e f ( y ) = ( f o px-1 ) (y) -1 neighbourhood o f p ( X I . Hence ( f ) bourhood of p ( x ) , n
-
in
a and
(f O px ) p ( x ) '
A*,X.
I t i s obvious t h a t t h e imbedding
3 f-
A,,x
f
E A
i s continuous. By a w e l l known Banach theorem w e A w t X = A o r A, 29
,X
obtain
that
i s of t h e f i r s t c a t e g o r y .
Given x , y E X I x # y , p ( x ) = p ( y ) , d e f i n e
A(x,y) = {f
E A:
(f
0
-1 px ) p ( x )
-
(f
-1 PY ) P ( Y ) ) '
W e claim t h a t A(x,y) is a c l o s e d subspace of A and
A(x,y) # A,
so t h a t A(x,y) i s nowhere dense. L e t ( f n ) C A(x,y) t e n d t o f E A . S i n c e
(fn
0
-1 -1 px )(z) = ( f n o py )
(2)
,n
21
f o r every z i n an open neighbourhood w o f p ( x ) . Therefore -1 -1 (f Px ( 2 ) = (f 0 Py ) ( 2 ) f o r y E w . Hence f E A ( x , y ) . L e t ( X f , j f , f ) denote t h e c a n o n i c a l m.a.e.
of ( X , f ) , where -1 j f ( x ) = ( f o px ) p ( x ) . Because t h e t r i p l e (X,idXIA) i s a m.a.e. of ( X , A ) , w e may i d e n t i f y any p o i n t x E X w i t h
the
family
( j f ( x ) I f E A . Therefore i f x , y E X and x # y t h e n t h e r e
exists
f E A such t h a t j f ( x ) # j f ( y ) (see 3 . 1 1 ) . So A(x,y) i s nowhere dense.
A(x,y) # A
and
K. RUSEK AlJD J. SICIAK
374
39
For any w
p-'(z,).
Q =
E
Wchoose
a point z w
in w
and define
By the Poincari-Volterra theorem Q is a
coun-
wEW
table dense subset of X. Define
and
Then by 19 and 2 9 the set
@:=
A
w'x
(w,x)EA
of the first category. So the set A\@is
U
u
(x,Y)
A(x,y)
is
not empty (it is
of
the I1 category). 4Q
We claim that for every f
is a m.a.e.
E
A \ Q the triple (X,idx,f)
of (X,f), 1.e. the mapping j, :
x
3
x--$
(f
0
P;l)p(x)E
Xf
is bijective. a)
of
jf is an injection. This is a direct consequence
the fact that f
B
bourhood w
of p(a)
A(x,y) for every (x,y) E A . Indeed, If j,(a)= -1 -1 in a neigh = jf(b), a # b, p(a) = p(b), then f o pa = f o pb E
. We may assume that there are
connected neighbourhoodsw', w "
schlicht
of a and b, respectively, such
-1 -1 Then that w c p(w')fl)p(w''). Put x = pa ( z w ) , y = pb ( z w ) . -1 -1 -1 (x,y)E A and (f o P, )p(x) - (f 0 pa )p(x) = (f Pb lp(y) - (f 0 Py-1IPtY)' This, however, contradicts the definition of f.
Thus jf(a) # jf(b). b)
jf is onto. jf beinq injective, we may consider X
a subdomain of Xf. If X # Xf, there would exist a point xo belonging to the boundary of XI a schlicht neighbourhood U x
0
such that w = pf (U)
E
w
and
?
is bounded on U, and a
E
as
Xf of
point
z E Q fl U. The domain X being a m.a.e. with respect to A, we have Aw,X # A. Further we have (2 o (pf);l)p(x) - (f 0 P;l)p(x)
MAXIMAL A N A L Y T I C EXTENSIONS
and
?:= 1 o
(p,);'
375
is analytic and bounded on
L
W.
Thus f
AWIX'
what contradicts the definition of f. The proof is concluded. L e t (X,p) E a ( E ) , whehe E i d t h e d a m e
1 0 . 2 COROLLARY
Let
Theohem 10.1.
F = {fl,...
fs}C @(XI
CIA
i n
b e a d i n i t e Aybtem
06
06
h o l o m o k p h i c d u n c t i o n d o n X d u c h t h a t (XlidX,F)i d a m . a . e .
.
(X,F) T h e n t h e a e
e x i d t numbekd
i b a m , a . e . 0 6 (X,f), urheke f = PROOF
By putting A = {zlfl
10.3 COROLLARY
kl,..
. hs
f
+...+
)i
1 1
+...+
E C Auch t h a t (&if+f )
Asfs.
s s : zi E C , fi E F } .
z f
16 0 C IRn i d a d o m a i n , A C & ( a )
i d
a Fnechct
.
d p a c e when endowed w i t h t h e t o p o l o g y rC, and i d ( X , j , A )
maximal a . e . o d ( R , A ) ,
(X,j,j*-'f)
then theae e x i d t d
f
E
A
duch
that
,HI
denotes
i n a m . a L . e . o d (0,f).
In particular, if R is a domain in Rn and (XIj the m.a.e.
a
i b
of (n,H(R)), where H ( R ) is the Frechet space of com-
plex-valued harmonic functions in R, then there exists such that fX,j,?)is a m.a.e.
6
fi
E
of (a,; o j).
In other words, the harmonic envelope of holomorphy
of
any domain R C lRn is a natural domain of existence of a ha?Xno& function f
E
H(n) (see [4], [5], [6]).
REFERENCES
111 J.BOCHNAK and J . S I C I A K , Analytic functions in
topological
vector spaces, Studia Math. , 39 (1)(1971), 59-112. 121 G . C O E U w , Analytic Functions and Manifolds in Infinite
Di-
mensional Spaces,Nort Holland/American Elsevier,l974.
3 76
[3]
K. RUSEK AND J. SICIAK
R. C. GUNNING and H. ROSSI, Analytic functions of
several
complex variables, Enqlewood Cliffs, N. J.: Prentice-Hall 1965.
[ 41
M. JARNICKI,
Analytic continuation of pluriharmonic func-
tions, Zeszyty Naukowe UJ, Prace Mat.
18
(to ap-
pear).
IS]
C. 0. KISELMAN, Prolonqement des solutions aux deriv6es partielles ii
d'une gquation
coefficient
constants,
Bull. SOC. Math. France 97(4) (19691, 329-356. [6]
P. LELONG,
Prolonqement analytique et singularit&
com-
plexes des fonctions harmoniques, Bull. SOC.
Math.
Belq. , 7 (2) (1955), 10-23. 171
E. LIGOCKA and J. SICIAK,
Weak analytic
continuation,
Bull. de 1'Acad. Polon. des Sci., 20(6)
(19721 I
461-466. [81
R. NARASIMHAN, Several Complex Variables, University
of
Chicago Press, Chicago and London, 1971. [9]
H. H. SCHAEFFER,
Topological vector
spaces ,
Macmillan
Company, N. Y., London, 1966.
[lo] M.SCHOTTENLOHER,
Uber analytische
Fortsetzung in Banach
raumen, Math. Ann. 199 (19721, 313-336.
1111 M. SCHOT!CENLOHER, Riemann domains: Basic results and open questions, Proc. in Infinite
Dimensional
Holomor-
phy 1973, Springer Lecture Notes 364 (19741, 212.
196-
MAXINAL ANALYTIC EXTENSIONS
317
[12J M. SCHOTTENLOIIER, Das Leviproblem in Unendlichdimensionalen Raumen mit Schauderzerlegung, Habilltationsschrift, Munchen 1974.
Mathematics
Institute,
University Jaqielloiiski,
ul.
Reymonta 4,
Krakbw, Poland.
This Page Intentionally Left Blank
Infinite Dimensional Holomorphy and Applications, Matos (ed.) @ North-Holland Publishing Company, 1977
POLY NOE I 1PL AP P ROS I ElRT I ON 3 N COE 1PACT S CTS
By I I A R T I N S C H O T T E N L O H E R "
I n T h i s n o t e a s i m p l e p r o o f of t h e f o l l o w i n g r e s u l t g i v e n : A pseudoconvex, f i n i t e l y Runge open s e t U i n a
is
locally
convex Hausdorf f s p a c e w i t h t h e a p p r o x i m a t i o n p r o p e r t y i s p o l y n o m i a l l y convex. T h i s g e n e r a l i z e s r e s u l t s of
Dineen
141
and
of
the
Noverraz [13]. W e a l s o p r o v e a n a p p r o x i m a t i o n theorem Runge t y p e f o r s u c h domains.
F i n a l l y , i n t h e l a s t s e c t i o n of t h i s n o t e , we d i s c u s s
a
s t r o n g form o f t h e Oka-Weil a p p r o x i m a t i o n theorem, which i s u s e f u l i n t h e s t u d y o f t h e Nachbin t o p o l o g y T~ on t h e s p a c e
@ (U)
o f holomorphic f u n c t i o n s on U (see blujica [ll]).
1. NOTATIONS AND PRELIE~IINARIES
L e t E b e a l o c a l l y convex Hausdorff s p a c e o v e r 6
s h o r t : l c s ) , and l e t c s ( E ) d e n o t e t h e set of c o n t i n u o u s norms on E . F o r a
€
cs ( E ) , x
€
E and r > o t h e " a - b a l l "
( for
semiabout
............................................................... *
Research s u p p o r t e d by t h e Brazilian-German C o o p e r a t i o n
Aqre-
- Gesellschaft fur
Mathe
ement (Conselho N a c i o n a l d e P e s q u i s a s m a t i k und D a t e n v e r a r b e i t u n g )
. 379
M.
380
SCHOTTENLOHER
x with the radius r is (x,r) = {y
B~
E EI
The "a-boundary distance" dz
cr(x-y) < rI :
U-
. for an open set U t E
[o,m]
is defined by :d
For B
(x) = sup {r >
c U we
" put dU
tance function
:
Ba(x,r)
( B ) = inf {d:
U
x E+
~ ~ ( x , a= )sup {r >
1x1
01
01
C
U}
(x)
I
,x x
U.
E
. Another
E B}
dis-
[o,m] is given by
x + ha
u
E
c r}, (x,a) E U
x
x
for all
E C,
E.
d" is continuous, while 6 u is in qeneral only lower semicontinu U
ous
.
An open set U C E is called pseudoconvex if -1oq to
plurisubharmonic, i.e. if the restrictions o f -1oq 6u plex lines in U
E are subharmonic. Let @ ( U )
x
is
(resp.
denote the set of plurisubharmonic (resp. continuous
com-
@c(U)) plurisuh-
harmonic) functions on U, and let @4U) clenote the vector space of holomorphic functions on U. For
n
C
@(U)
and
I< C
U
the
"Q-convex hull" of K is defined by
KO
=
{x
UI
E
and for A c @ ( U )
-
KA
whereby
=
{x
I I fI I
v(x) 5 sup v(y) for all v YEK the "A-convex hull" of K is
UI
E
=
If(x)I 5 IlfllK for all f
sup
[ f (y)I Iy
E
E
01,
E A),
K l . Prom the characterization
of pseudoconvex sets in finite dimensional spaces one can
de-
duce (cf. [17]): PROPOSITION 1 a4e
Fo4
an o p e n s e t U
C
I.: t h e . ~ o ~ l o w i n gp t o p e a t i e n
equivalent 1? 20
U i b pbeudocanwex. a -log du i 6 p l u t i h u b h a a m o n i c
o n {x
E U I d$
(x)>o}
404
POLYNOMIAL APPROXIMATION a E cs(E)
euehy
30
d:
381
.
F o h e v e h y compuct
K C U
With
t h e h e i6
a E cs(E)
604 CVV.hY
compact K~u.Hehe,
( ? Q ( ~ ) ) > 0.
4Q
ii
phecompucf i n U
Q ( ~ )
L C U iA c u t l e d p4ecompuct i n U i.6 L iA phecampuci uvld i d Rhehe a E cs(E) with
d;(L)
exinto
> 0.
I t i s an open q u e s t i o n whether t h e above e q u i v a l e n c e s h o l d
if
@(U)
.
i s r e p l a c e d by G>c(U) A partial answer is given in section 2.
U i s c a l l e d h o l o n o r p h i c a l l y convex i f
k
(iT) is p r e m p a c t
i n U f o r e v e r y compact K C U . A h o l o m o r p h i c a l l y c o n v e x open s e t
U cE
i s p s e u d o c o n v e x . The c o n v e r s e i s t r u e f o r A a n d a!
p r o b l e m ) , f o r a! (IN) [ 7 ]
spaces E with a basis (cf. L O ] ,
[2]
,
E =
an
(Levi
and f o r c e r t a i n s e p a r a b l e
[S!, 1 1 7 1 ) . I t i s a n o p e n q u e s -
t i o n whether t h e c o n v e r s e h o l d s i n g e n e r a l . A p a r t i a l answer i s given i n t h e n e x t s e c t i o n .
u
c E i s polynomially convex i f
compact K C U, whereby
TI
C. B ( U )
Gn
u
i s precmpact i n
for all
denotes t h e space of a l l
con-
t i n u o u s p o l y n o m i a l s f r o m 1: t o a!. S i n c e n i s d e n s e i n Z ( E ) w i t h r e s p e c t t o t h e canpact o p n topology on
KT
kTr
( E ) , w e have
k@(E)
=
*
i s contained i n
2
=
{x
/ p ( x ) I 5 1Ip
F EI
lK
for all
p
n).
E
-
-
W e d o n ’ t know, w h e t h e r f o r a p o l y n o m i a l l y c o n v e x U, K = K
TI
is
t r u e i n g e n e r a l . I n t h e case of a F r 6 c h e t s p a c e E w i t h t h e a p p r o x i m a t i o n p r o p e r t y t h i s w i l l b e p r o v e d i n s e c t i o n 3. Closely r e l a t e d w i t h p o l y n o m i a l c o n v e x i t y o f a Runge o p e n s e t
U
E.
U i s c a l l e d Runge i f
s ( U ) w i t h r e s p e c t t o t h e compact open t o p o l o g y .
same a s t o s a y t h a t
8 (E)
is
the TI
notion
i s dense i n
This is
the
i s d e n s e i n z ( U ) . F i n a l l y , rJ i s c a l -
l e d f i n i t e l y Runye ( r e s p . f i n i t e l y p o l y n o m i a l l y c o n v e x )
i f for
382
M.
SCHOTTENLOHER
a l l f i n i t e d i m e n s i o n a l v e c t o r s u b s p a c e s F of E l U f l P i s
Runqe
( r e s p . p o l y n o m i a l l y convex) i n p .
2. POLYNOFIIAL CONVEXITY
Throughout t h i s s e c t i o n l e t E b e a l c s w i t h t h e approxima f o r e v e r y compact F
t i o n property, i.e. every
E
and a ( x
m
Let
PROPOSITION 2 €3.
Then
doh
- $(x))<
K63(U) l e t
e(E)
for a l l x E K.
open
U I Xo&
XoE
K6qU)
-
with
@(U)
E
with
e u c h y compact R C U :
.
Then t h e r e i s v
E
U be a pbe.udoconifcx, d i n i t e l y Runge
I n o r d e r t o show K
v(xo) > sup v ( y ) . Y EK Due t o t h e s e m i c o n t i n u i t y of v t h e r e a r e a s >
e v e r y a E C S ( E ) and
> o t h e r e e x i s t s a c o n t i n u o u s l i n e a r map 4 : E+E
dimc $(El <
A e t in
cz. i f
cs(E), q > o and
o such t h a t f o r a l l x E K a B a ( x , 2 s ) ~ i land v ( y ) < v ( x0 1- rl f o r y E R ( ~ ~ 2 s ) .
Because E h a s t h e a p p r o x i m a t i o n p r o p e r t y t h e r e e x i s t s a c o n t i n u
ous l i n e a r
+:
x E K . Now
IJJ
E+E =
4
+
w i t h dimc $ ( C ) <
xo
-
and u ( @ ( x ) - x) < s f i r
m
@(xo) s a t i s f i e s n ( @ ( x )- x ) < 2 s
x E K , hence j (I;) C
I.'
f ? I'
where f = s p (@(I?)
v o @(x) < v(xo) I t follows f o r
VI
-
= vlr>
rl
1
f o r a l l x E I<.
e,:.f)1 ' )
:
and
for
POLYNOMIAL APPROXIMATION
Consequently, xo
4
.
A
~ ( K ) Q ( ~ ~ Since ~ ) U n F is pseudoconvex and
Runve, there exists a polynomial 9 19(xo) I
'1
383
:
F+@
with
141 I$(K) (cf. [9, p. 5 3 1 ) . Now 9 o JI E
19
(xo)I =
$
9(xo)l
'
11"1$(K)
= 114
and
T
$11,
I
-
hence x
0 !$ %CE) From the propositions 1 and 2 we deduce:
COROLLARY 1
A p b e u d o c a n v e x , I j i n i t e L y Runge open b e t
UC E
i b
hoLomotphicaLly convex. REPIARK
Since for domains R C C : " the main step in solving
Levi problem is to show that pseudoconvexity implies
the
holomor-
phic convexity, corollary 1 is a certain contribution to the s o lution of the Levi problem. However, in infinite
dimensional
spaces E a holomorphically convex domain need not be a
domain
of holomorphy. In fact, Josefson [lo] gives an example of
a
pseudoconvex domain Q in I? = co(A), A uncountable, which is not
a domain of holomorphy. Ploreover, since this domain can be
de-
it
is
fined by a global plurisubharmonic function v E @ ( E ) , finitely Runge and hence holomorphically convex. But
in
certain
infinite
dimensional
spaces,
for example in Silva spaces [15] , every holomorphically convex domain is a domain of holomorphy or even a domain of existence. TIIEOWH 1
Fo4
a17 o p e n
b e t U C I?, t h e doli!owing p h o p e t t i e s a 4 e
eq u i va i! e n t : 10
U LO p b e u d o c o n v e x a n d d i n i . t e i ! y Runge.
2Q
U
A &
3?
U
i a pvlqnomiaf!Py convex.
h o l c m o 4 p h i c a l l q conL'ex and 17unge.
M.
384
PROOF
According t o c o r o l l a r y 1, U i s holomorphi-
20".
"19
SCHOTTENLOHER
c a l l y convex. T h e r e f o r e , i t s u f f i c e s t o show t h a t a
finitely
Runqe s e t i s a Runge s e t . T h i s was proved i n [I] i n a more g e n e r a l c o n t e x t . I n o u r s i t u a t i o n t h e p r o o f o f [l] i s a s
,
Let f E g ( U )
I<
c U compact,
follows:
> 0. Because of t h e
E
continuity
o f f t h e r e are a E c s ( E ) and s > o s u c h t h a t f o r a l l x
1 f(x) -
Ba ( x , s ) C U and
E
f (y) ] < 2 f o r y E B
There e x i s t s a c o n t i n u o u s l i n e a r 4 : E + E and a ( + ( x )
-
a
(x,s).
w i t h dimc @ ( E l < m
x ) < s f o r a l l x E K . IIence
$(K) c
U
n r,
where F = 4 ( E l
Ilf -
f
411,(?
0
,
and
E *
Since U f l I' i s f i n i t e l y Runqe t h e r e i s a polynomial
IIf
with
u/II.'
I If -
E K
9
ql
0
$IIK 5 I I f -
I+(K)
<
5
. Now g f
0
+I!<
g
k I L jt h)e,r e
is f
E
+
IIf
0
4- g
0
+'I1;
%(U)
with ] f ( x o ) l
<
-
Now I<
- PI/K U{xo)
-
=
kiu, is
<
1
2 (lf(xo)1-1\f\lK)8
>IlfllK. .
xo $
precompact i n U s i n c e U i s
E.
Fc86(L:).
S i n c e U i s Runge, t h e r e e x i s t s a polynomial p E n w i t h
/If
6:
o cp E n and
L e t Y C t' be compact. E v i d e n t l y T'N!.
"2?+3?".
I f xo E U , xo
E
-
g: ??*
'%[C)'
bloanorphically
convex. '13'.'*40"
i s t r i v i a l and " 4 ? 4 1 ? ' " f o l l o w s from
the
f i n i t e d i m e n s i o n a l r e s u l t s [9, p . 531. Applying a r e s u l t o f Noverraz [14, t h . 31 w c see
that
f o r a F r 6 c h e t s p a c e E w i t h t h e appro::imation p r o p e r t y t h e
fol-
REPmRI<
lowing i s t r u e : For any d e n s e v e c t o r s u b s p a c c s e c t i o n of a l l pseudoconvex domains '' C
r c I:
with
r
the
Cii,
intei i s equal
385
POLYNOMIAL A P P R O X I M A T I O N
t o t h e i n t e r s e c t i o n o f a l l domains o f e x i s t e n c e R C E ,
with
I n o t h e r w o r d s , t h e p s e u d o c o n v e x c o m p l e t i o n FQ of P aare
F C R.
of P.
es w i t h t h e h o l o m o r p h i c c o m p l e t i o n
3 . POLYNOMIAL PPPROXIMATION
I n t h i s s e c t i o n l e t C be a m e t r i z a h l e l c s w i t h
ap-
the
p r o x i m a t i o n p r o p e r t y . Moreover, l e t E b e h o l o m o r p h i c a l l y p l e t e , i . e . E = E o ( c f . f o r example [ 1 4 ] )
. We
need t h e
corn
follow-
i n n c h a r a c t e r i z a t i o n [14, P r o p . 101 : F m e t r i z a b l e l c s E i s ho-
se
l o m o r p h i c a l l y c o m p l e t e i f f o r e v e r y n o n - c o n v e r v e n t Cauchy E there exists f E %(E)
v i t h sup
I f (xn) 1
q u e n c e (x,)
in
THEOREM 2
L e t U b e n p e u d o c o n v e x , 4 i n i R e t r r Runge o p e n n e t i n
-
C, a n d l e t I: = I<
B(U)
=
m.
be n c o m p a c t b u h b e t o S 11. T h e n e v e h r r 4iinc-
t i o n f w h i c h i n holornoflphic i n a neiohbofihood 0 4 K can be
p t o x i m a t e d uni{ohmtr/ o n
1 :
brr c o n t i n u o u d po4trnominPd
I?g m
N o t e t h a t f o r any compact K C U ,
on R.
s p a c e s . Hence
BiU)
i s compact s i n c e
K
-
BW) -
accog
i s compact
d i n 9 t o t h e above C h a r a c t e r i z a t i o n of h o l o m o r p h i c a l l y
I<
np-
KB$iEf
complete
r
( ropos i ti on
2).
PROOF Or TIIEOREM 2 L e t E > 0 . T h e r e a r e a E cs ( C ) and
s >
0
s u c h t h a t f i s h o l o m o r p h i c o n T J = IC + n a ( O , s ) IT, and f o r E a x E: I ( , If ( x ) f ( y ) I < 3 i f 7 7 E R ( x , s ) . T*Te f i r s t show
all
T h e r e e x i s t s a f i n i t e r a n k l i n e a r o p e r a t o r rt, : E
d E
-
(*) fi
with .9
& q J
(1:) )
c l !
and n(Q\fx) - x) < s f o r all x E K .
L e t ( a n ) khe a n i n c r e a s i n n s e q u e n c e of c o n t i n u o i i s norms on E , a
5 (L n ' whjch n e n e r a t c s
t h e t o p o l o n v of R .
semiSince
E
386
SCHOTTENLOHER
M.
h a s t h e a p p r o x i m a t i o n p r o p e r t y t h e r e a r e c o n t i n u o u s l i n e a r maps
@n
x
: E-E,
dimc a n ( E ) < A
Assume t h a t $ J n ( K )
E K.
fi
xn
E 4 , n ( ~ )\
v.
m,
a
for all
a l l n E I N . Then t h e r e
are
s u c h t h a t an(4,,(x)-
@ i7 f o r
P u t E~ = s p (I;
u
[ r t l n ( ~ )
denote t h e c o m p l e t i o n of Eo. For a l l q
F
I n
F
x) <
IN)) and l e t E~
I
@(El),
(xn) 1
K), s i n c e $ n ( E l i s a f i n i t e d i m e n s i o n a l , h e n c e p l e m e n t e d s u b s p a c c of E l .
Pccordinn t o c u r choice of
(an)
( x n ) i s bounflinn i n E l a n J , s i n c e El i s a
Hence,
Frgchet space,
(x,)
rectly)
. Now
Wnj Thus xg E E ,
is
that
relatively
t h e proof i n [16] can h e t r a n s f c r r e c l
di-
s a hounrlinq Cauchy s e n u c n c e i n C : ,(I<)+ ! l n \ l I c f o r a11 n E % ( c ) . n1 I: i s h o l o m o r p h i c a l l y c o m p l e t e . P i n a l l v ,
< lim 1 1 0 o + n j I 1 1 < = I J n l Icr(xo) I = l i m I n ( x n j ) I j +m j +m T h i s i s a c o n t r a d i c t i o n t o xn $ V f o r a l l n E: I?!. TIC t h u s proved ( * )
anc?
xo.
xnj
(This is a straiqhtforward oeneralization of t h e r e s u l t
compact: f o r
cnm-
se p ar a b l e
h a s a converrrent suhseciuence
e v e r y bounilinq s e t i n a s e p a r a b l e Banach s p a c e
5
for
have
.
Now f
i s h o l o m o r p h i c i n T ' f-l I', whcrehv n hence i n a neinhhorhood o f L = 4 (I<) C L e t q:
%.)
=
(c),
r+
be
r
POLYNOMAIL APPROXIMATION
p 4 o p e h t i e b l? - 40 i n t h e o h e m 1 atie
The
COROLLARY 3
387
eqiiiva-
PetiR R c 5?
K
c I1
404
a!P c o m p a c t
=>
moor
Cvidentlv " 5 ~
K C [J.
k% ( E l
3911, s i n c e
.
I
C
ic.
To s h o v
a r c compact s i n c e E i s l e t I: C [J b e c o m p a c t . Then K and 1:. &YE) holomorphically complete and s i n c e U is p o lv n o mially convex. A L
k n (E\U)
so, KO =
i s compact. Now a h o l o m o r p h i c f u n c t i o n f can
-
-
u KO
be d e f i n e d i n a n e i n h h o r h o o d of K = Kg(r,
equal t o
0
i n a n e i n h b o r h o o d o f KBe(c)
and
f
by p u t t i n s
eoual t o
1
f
a
in
n e i a h b o r h o o d o f KO. C o r o l l a r v 2 i m p l i e s t h e e x i s t e n c e of a con t i n u o u s p o l y n o m i a l p: E + C with I - pi 1~ < 71 T h u s , I IpI 1 i.e. KO = d , < 2 < Ip (x)1 f o r a l l x F KO. I t follows
If
- . .
K = I<
%(C)
IK<
.
= "-
4 . Theor rn 2 c a n be f o r m u l a t e d i n t h e f o l l o w i n n way.For a
compact
T C I1
r
4
d e n o t e t h e s p a c e o f T e r m s of
%rL)
holomor-
p h i c f u n c t i o n s on L . Then, i f U i s a p s e u d o c o n v e x ,
finitely
PunFe open s e t i n a h o l o m o r p h i c a l l y c o m p l e t e , m e t r i z a b l e
Ics
w i t h t h e a p p r o x i m a t j on p r o p e r t y , t h e f o l l o w i n a h o l d s : ( 1 ) T o eue4lj c o m p a c t I , C J,
c o n t a i n ~ n gK ,
map"
@ (11) LA
denbe
412
%(L)
(Take L =
1'
c1
A U C ~t h a t
-
thehe
c o ~ ~ e n p o nad ~
compact
t h e imnqe undeh t h e " t e n t t i c t i o n
g ( L )
4enpec.t
1 ~ 4 t h
k,
,UI
).
t o ,the & u p n n h m
t o p o ~ o n r ro n
The same a p p r o x i m a t i o n r e s u l t
shown f o r a pseudoconvex domain U s p r e a d over a r r 6 c h e t
%a).
can 01
be
Cilva
3 88
M.
SCHOTTENLOHER
[IT]
s p a c e w i t h a f i n i t e dimensional Schauder decomposition A
1.
(aaain with L = i n t h e case of C = Q:
n
F\
,a
s t r a i n h t f o r w a r d r e a s o n i n n shows
that
s h a r p e r v e r s i o n (1) holds f o r a p s e u d o
convex domain U (acrain w i t h L = K): ( 2 ) T o evehq
L
C
compact I.: cc
t h e f i e c o t 4 e ~ p o ~ dan
U, c o n t a i n i n g I<, ~ u c ht h a t t h e i m a g e u n d e t t h e
conip.c.t
“fiesttiction
map”
%$(u)id
%(Lf
A e y u e n t i a L L y d p n b e w i t h 4ebpec-t t o t h e n a t u f i a t i n d u c t i v e t i n-1
it R o p o L o g y o n % ( L ) , The i n d u c t i v e l i m i t t o p o l o n v on g ( L ) i s d e f i n e d by = l i m inr1 V Ff(L1
@(L)
where &(L)
8%
(v) ,
i s a base of open n e i q h b o r h o o d s of L and i%”(V)
is
t h e Banach s p a c e of bounded h o l o m o r p h i c f u n c t i o n s on V w i t h t h e s u p norm. PROOF OF
( 2 ) FOR A PSEUDOCONVEX DOFIRIN
compact and L = I<
Let f E %(L).
t h a t f i s h o l o m o r p h i c on L ( s ) = L
hence I q ( x n ) l 5 xnj
j xo E
Ilnl IL(+)
+
for a l l
L e t I<
CU
he
There c x i s t s s > o
such
Assume
that
B(o,s)
E
Cr”:
1’
c I I .
%(u).
€ 0 11o v r s
It
L f o r a subsequence (x
) o f (x,). Contradiction. nj Now l e t 1 8 ( t \C L ( s ) f o r o < t < s . A c c o r d i n q t o t h e approximaA
t i o n t h e o r e m o f 01:a-Vcil
\Ifm-
flIL(t)+o,
striction
mdr)
[ 9 , p. 911, t h e r e e x i s t fn
i.e. fm j
8%”(Iz( t ))-+
?%(I,)
t h e qerms o f fm c o n v e r n e t o f in
f in %‘(I(t)).
6
Rfll)v i t h
Since t h e
rc
i s c o n t i n u o u s , it follows that Bil,,.
W e d o n ’ t know w h e t h e r o r n o t ( 2 ) h o l d s for
all
open
POLYNOMAIL APPROXIMATION
s u b s e t s of Cn.
389
T h e r e e x i s t domains i n Cn s a t i s f y i n 9 ( 2 ) h u t n o t
b e i n ? pseudoconvex. ror e x a m p l e , i f 0 i s pseudoconvex and K C 2 i s compact s u c h t h a t U = Q \ K
connected, then U s a t i s f i e s ( 2 ) .
P r o p e r t y (2) i s p a r t i c u l a r l y i n t e r e s t i n n i n t h e
infinite
d i m e n s i o n a l case: T f an open s e t U i n a norme?. s p a c e E satisfies
(2), t h e n t h e Nachhin topo1oc.y j e c t i v e l i m i t of a11
,
(%(u)
T
w
T~
s(~ , K )c u
pro-
[12] i s o b t a i n e d a s t h e
compact:
) = lir, r ? v J T’C cpt.
(K).
T h i s i s d i s c u s s e d i n [ll] and [ 3 ! . T h e r e a r e o n l y few e x a m p l e s of o p e n s e t s i n i n f i n i t e m e n s i o n a l s p a c e s €or which i t i s known t h a t ( 2 ) i s e.n.
r
f o r a b a l a n c e d open s e t i n a n o r h i t r a y l c s
di-
satisfied, and for
cer-
t a i n R e i n h a r d t open s e t s i n a n a n a c h s p a c e w i t h a n u n c o n d i t i o n d b a s i s [ll]. U n f o r t u n a t e l y , t h e methods p r e s e n t e d above as as t h a t of [17!
proof of
we1 1
do n o t p r o v i d e more e x a m p l e s . A l s o , t h e
( 2 ) f o r a pseudoconvex U C C
n
above
cannot be t r a n s f e r r e d t o
it can
domains i n i n f i n i t e d i m e n s i o n a l n o r n e d s p a c e s . IIowever, b e t r a n s f e r r e d t o t h e case of a pseudoconvex c7olnain U
in
an
a r b i t r a r y p r o d u c t CpA of l i n e s , b e c a u s e s u c h a domain i s t h e p r o duct R
c”,
x
CA’ o f a pseudoconvex
R in a certain C
n
an? t h e
space
A ’ = ~ \ { l , . . . n ~~ 2 1 .
ACKNOWLEDGEMENT:
I want t o t h a n k M.C.
Matos for h e l p f u l c a m n e n t s .
REFERENCES
[l]
R . ARON
-
PI.
SCIIOTTENLOHER, C o n p a c t h o l o m o r p h i c
mappincrs
o n Banach s p a c e s and t h e a p p r o x i m a t i o n p r o p e r t y . appear i n J . runct. Enalvsis
.
To
(Pnnouncement i n : R u l l .
Amer. rqath. Soc. 80 (19741, 1245
-
1249).
390
[Z]
M.
SCHOTTENLOHER
V . A U R I C I l , C h a r a c t e r i z a t i o n of d o n a i n s o f h o l o m o r p h y
over
a n a r b i t r a r y p r o d u c t of c o m p l e x l i n e s . D i p l o m a r l ~ e i t . !Tiinchen 1 9 7 3 . [3]
S. B.
CIIAE, I I o l o m o r p h i c nerms o n B a n a c h s p a c e s . P n n . I n s t .
F o u r i e r 2 1 ( 1 3 7 1 ) , 107 [4J
- 141.
f u n c t i o n s on locallv c o n v c x v c c t o r
S . DINCEN, I r o l o m o r p h i c
spaces IT. Ann. I n s t . F o u r i e r 2 3 ( 1 9 7 3 1 , 155
[5]
- 18s.
S . D I N E E N , A q r o w t h p r o p e r t y of p s e u t l o c o n v e x d o m a i n s
in
l o c a l l y c o n v e x t o p o l o u i c a l vcctor spaces. P r e p r i n t . [6]
S . DINEEN
-
P h . NOVERPJZ
- IY. SCIIOTTKNLOVJ?R, Le
prokl6me
de L e v i d a n s c e r t a i n s e s p a c e s v e c t o r i c l s t o p o l o n i o u n l o c a l e m e n t c o n v e x e s . To a p p e a r i n B u l l . P'ath. 171
L.
GRUKAN, T h e L e v i problem i n c e r t a i n i n f i n i t e d i m e n s b m l
v e c t o r spaces. I l l . J . Pqath. 18 ( 1 ? 7 4 ) , 20 [8]
L.
CRUtIAN
-
C.
KISELEIAN,
ces dc B a n a c h ( 1 9 7 2 ) , 1236 [9]
Trance.
L. HOPYANDER,
-
L e p r o h l k m e de L e v i d a n s l e s espp
base. C . P . ?.cad. S c i . P a r i s
-
26.
274
1299.
"An I n t r o d u c t i o n t o Complex D n a l y s i s i n S c v e
r a l V a r i a b l e s " . Van N o s t r a n d , P r i n c e t o n 1 3 6 6 .
[lo]
B.
JOSEI'SON, A c o u n t e r e x a m p l e t o t h e L e v i p r o b l e m . "Proceedinas on I n f i n i t e Dimensional S p r i n q e r L e c t u r e Notes 264 ( 1 9 7 4 ) 1 f j P
[ll] J . PILIJICA,
In:
ITolomorphv"
-
177.
S p a c e s o f oems of h o l o m o r n h i c f u n c t i o n s . T h & ,
U n i v . of P o c h e s t e r 1 9 7 4 . To a p n e a r i n F d v . i n M a t h . . See also t h i s proceedinns.
.
POLYNOMIAL APPROXIMATION
[12]
L . NACMBIN,
39 1
"Topolonv on s p a c e s of h o l o m o r p h i c
nappincrs"
.
S p r i n n e r - V e r l a n , New York 1 9 6 3 . [13] P!!.
S u r l a p s e u d o - c o n v e x i t 6 e t l a c o n v e x i t s poly
NO"ERP?Z,
n o n i a l e e n d i m e n s i o n i n f i n i e . Pnn. I n s t . F o u r i e r ( 1 9 7 3 ) , 113
[14] PII. NOVCR?Z\Z,
-
23
134.
Pseudo-convex c o m p l e t i o n o f l o c a l l y
convex
t o p o l o n i c a l v e c t o r spaces. P a t h . Pnn. 208 ( 1 9 7 4 1 , 5 9 69. [15]
~ € 7 . M ~ ~ V E R P . A Z ,? s e u d o - c o n v e x i t g
e.1.s.
I n : S6m. L e l o n n 13/74.
4 7 4 (19751, 6 3
[16] br.
e t h a s e de Schaucler clans les
-
S p r i n n e r L e c t u r e Notcs
82.
SCIIOTTENLOIIER, Boundinn s e t s i n Ranach s p a c e s and r e q u l a r c l a s s e s of a n a l y t i c f u n c t i o n s . I n : A n a l y s i s and A p p l i c a t i o n s
(Symposium
runctional R e c i f e , 1972) ,
S p r i y e r L e c t u r e Notes 284 ( 1 9 7 4 ) , 109
[ 171 M. SCHOTTENLOHER, P a s
-
122.
Leviproblem i n unendlichdimensionalen
Raumen m i t S c h a u d e r z e r l e q u n a .
IIabilitationsschrift.
Nunchen 1974. The L e v i p r o b l e m f o r domains
spread
o v e r l o c a l l y convex s p a c e s w i t h a f i n i t e d i m e n s i o n a l Schauder decomposition. T o appear
in
Ann.
Inst.
Fourier.
rlathematisches I n s t i t u t der U n i v e r s i t a t Munchen Theresienstr.
D
8
39
E'liinchen 2
This Page Intentionally Left Blank
Infinite Dimensional Holomorphy and Applications, Matos (ed.) @ North-Holland Publishing Company, 1977
T~
=
f o r Domains i n C
T~
IN
by MARTIN SCHOTTENLOHER
Let
a ( U ) be t h e space of holomorphic f u n c t i o n s
on a
domain U & E , w h e r e E i s a F r g c h e t space o v e r C , a n d l e t 'lo (resp. T d e n o t e t h e compact open t o p o l o g y ( r e s p . p o r t e d t o w
pology o f Nachbin [ 6 ] ) on
H(U). I t
i s clear t h a t T~ -c0 i f E is n o t a Monte1 s p a c e : C o n s i d e r t h e semi-norm f -t s u p { l D f ( a ) Ix E B } , f E @ ( U ) , where a E E and B c E i s a bounded, c l o s e d , non-compact subset o f E . However, for i n f i n i t e d i m e n s i o n a l Frgchet-Monte1 s p a c e s it seems t o b e unknown w h e t h e r T~ = T~ o r T~ # - c 0 , e x c e p t f o r a r e s u l t i n [ 3 ] , IN where ( % ( C IN 1 , T ~ = ) ( % ( C ) , - c 0 ) i s shown. The q u e s t i o n for o f w h e t h e r T~ = T~ or T # ' c 0 h o l d s i s o f some i n t e r e s t
.XI
w
F r g c h e t - S c h w a r t z and F r g c h e t n u c l e a r s p a c e s (see
[41
.
I n t h i s s h o r t n o t e w e want t o e x t e n d B a r r o s o ' s r e s u l t [ 3 ] t o t h e following:
PROOF:
Let j : U
(s.a.c.1
of
lo g E
%(U)
%(U)
+
U'
b e a s i m u l t a n e o u s a n a l y t i c continuation
with t h e properties
j * : ( @ ( U ' ) , T ~ ) ->
( B(U),To)
,
,
g + goj
$& (u'), i s a n open m a p ( i . e . j i s a " n o r m a l " s . a . c .
of
i n t h e s e n s e o f [l]). Z0
j i s maximal w i t h r e s p e c t t o lo ( i . e . i f i : U
393
+
U"
394
SCHOTTENLOHER
PI.
i s another s.a.c.
of
such i* : ( @ ( U " ) , T ~ )
8f$(U)
i s open t h e r e e x i s t s a n s . a . c .
k : U"
U ' of
-f
j = k o i ) . Such a maximal normal s . a . c .
of
t o C ( c f . [l]).
s ( U )
with
%(U)
existsr
continuous
it c a n b e c o n s t r u c t e d u s i n g t h e s e t of phisms from
(~(U),T,)
+
8&(U")
and
homomor-
Because of t h i s construction
i t f o l l o w s t h a t U ' i s h o l o m o r p h i c a l l y convex. T h e r e f o r e , U ' isomorphic t o a product R
x
Cw
s p r e a d o v e r some C m l m E IN
I
where R i s a S t e i n
( c f . [5]
t h a t ( @ (R
on
X
(Ul according
jections (+I
% (a
C m ) , T ~ )= (
.
x Cm)
T ~ I )s i n c e
8%
((2
on
= T~
T~
X
T
w
= T
o
c o n t i n u o u s b&
4
cm)
r'c
$$(n
x
' Cm)
continuous
Let p be a
then
(2) :
I11
(
domain
Now it s u f f i c e s t o show
,
t o t h e following diagram of
and homeomorphisms
domain
for a schlicht
U' C E , and 1121 f o r t h e g e n e r a l c a s e )
is
t h e compact s e t K C R
x
(
8% ( u ' ) , T o )
'
(
%(u)
,'lo).
c a n b e p r o v e n i n a s i m i l a r way t o [ 3 ] . semi-norm o n Cm
,
x
GIN)
ported
by
and l e t L b e a compact neighborhod
o f p r l ( K ) c R , where p i l d e n o t e s t h e c a n o n i c a l p r o j e c t i o n
R
Cm
x
sup{lzil
wj
R . Choose c o n s t a n t s ri <
+
( z = (zlrz2,...)
: = {z E
cm1 l z i l
E
m
with
p r 2 ( K ) } < ri f o r i
< ri f o r i = 1 , 2 , . . . , j 1
and
E IN,
set
Since
c C m .
p
i s p o r t e d by K c L x W P(f)
5
CjII
fll
L x
there e x i s t constants c with j j for a l l f E ~ C x RC m ) ( ~ ~ f ~ ~ X : = s u p { l f ( x ) l ~
w j
x 8 X I ) . Hence, f o r sj : = c . r and v = ( v1 , v 2 , . . . , v . ) 3 1 1 V < c j l l a l l L r V 5 jlalILsV have p ( a z V ) 2 c j 1 1 az 1 1 L x w , a E
g(n)
( z v : = zlV1z2 v3
1
...z j
E
IN^
for
we all
V.
1 ) . Now s e t t
1 . Since f
=
( 2 j ) 2sj and
PN)
f o r a l l j E m}. E x j "depends o n l y on a f i n i t e number o f v a r i a b l e s " ( c f . [5] , [ 2 ] ) t h e r e e x i s t s j F IN s u c h t h a t f h a s a power series e x p a n s i o n
M : = {z
E
cm
1zj/ 5 t
-
-
To
FOR DOMAINS I N CmT
395
of t h e form f(x,z) =
c
v E IN]
av(x)zV , (x,z) E
n
X
Cm
,
with av E %(Q).
I l f 1)
By t h e Cauchy i n e q u a l i t i e s l a v ( x ) ItV5
,
v E IN].
Hence ,
Since L REMARK:
x
M i s compact it f o l l o w s t h a t p is r o - c o n t i n u o u s .
The p r o p o s i t i o n a l s o h o l d s f o r domains U s p r e a d o v e r
cm . ADDED I N PROOF: A d i f f e r e n t p r o o f of t h e above r e s u l t s w a s given
by B a r r o s o and Nachbin
( t o appear elsewhere). REFERENCES
1.
H . ALEXANDER, A n a l y t i c f u n c t i o n s o n Banach s p a c e s . T h e s i s ,
U n i v e r s i t y of C a l i f o r n i a , B e r k e l e y (1968) 2.
V . AURICH , C h a r a c t e r i z a t i o n of domains o f holomorphy o v e r
a n a r b i t r a r y p r o d u c t o f complex l i n e s . U n i v e r s i t a t Mfinchen (1973) 3.
.
Diplomarbeit,
.
J. A . BARROSO, T o p o l o g i a s nos e s p a q o s d e a p l i c a q o e s h o l o -
m o r f a s e n t r e espaGos l o c a l m e n t e c o n v e x o s . A n a i s de Acad. B r a s . d e C i g n c i a s
4.
K.
D. BIERSTEDT
-
43
( 1 9 7 1 ) , 527-546.
R. MEISE, N u c l e a r i t y a n d t h e S c h w a r t z
p r o p e r t y i n t h e t h e o r y o f h o l o m o r p h i c f u n c t i o n s on m e t r i z a b l e l o c a l l y convex s p a c e s . P r e p r i n t .
5.
A.
HIRSCHOWITZ , Remarques s u r les o u v e r t s d ' h o l o m o m i e d'un p r o d u i t dgnombrable d e d r o i t e s . Ann. I n s t . F o u r i e r 1 9 (1969)
6.
L
. NACEBIN,
,
219
-
229.
Toplogy on spaces of h o l m r p k i c mappings. Ergebnisse der
Mathematik 47 (1969), Springer, N e d York
- Heidelberg.
Mar ti n S ch o t t e n 1oh e r 8024 K r e u z p u l l a c h 5 Germany
(May 76)
This Page Intentionally Left Blank
Infinite Dimensional Holomorphy and Applications, Matos (ed.) @ North-Holland Publishing Company, 1977
HOLOMORPHY OF COMPOSITION
By JAMES 0. STEVENSON (*)
ABSTRACT
I n t h i s p a p e r we s t u d y t h e holomorphy, w i t h
respect
v a r i o u s l o c a l l y convex t o p o l o g i e s , of t h e c o m p o s i t i o n
+
: (f,g)
spaces E
E
X
x Y
and F
* gof
,
E Z
If
X
i n c l u d e s a l l the constant
f u n c t i o n s , t h e n 4 w i l l n o t be holomorphic when Y
i s t h e space
of e n t i r e f u n c t i o n s from F t o G . P o s i t i v e r e s u l t s
c e r t a i n s u b s e t s of a s that on Z ,
map
of holomorphic functions between Banach
and F and G .
r e q u i r e t h e t o p o l o g y on Y
to
generally
to be t h a t o f u n i f o r m convergenceon
F and t h e t o p o l o g y on
s u c h as c o m p a c t - o p e n , To
X
t o be the same type
r Tu
*
.............................................................. AMS
(MOS) s u b j e c t c l a s s i f i c a t i o n s ( 1 9 7 0 )
. Primary
46E10,
58B10. Key words and p h r a s e s . I n f i n i t e d i m e n s i o n a l holomorphy,
composition, G
- holomorphy, amply bounded, h o l g
morphic c o n v e x i t y . (*)
T h i s work i s based on p a r t of t h e a u t h o r ' s d o c t o r a l diss e r t a t i o n a t t h e U n i v e r s i t y of R o c h e s t e r under t h e s u p e r v i s i o n of Leopoldo Nachbin, and w a s s u p p o r t e d i n p a r t by an NSF T r a i n e e s h i p . 397
398
J. 0 . STEVENSON INTRODUCTION
I.
We wish t o c o n s i d e r t h e f o l l o w i n g two problems f o r E l F , G Banach s p a c e s o v e r t h e complex f i e l d CP
H(E;F) , H(F;G),
and
t h e c o r r e s p o n d i n g s p a c e s of holomorphic
H (E;G)
functions between
them ( w e f o l l o w t h e d e f i n i t i o n s and n o t a t i o n g i v e n i n [ 81 ) : (1) X CH(E;F) , Y CH(F;G),
For what v e c t o r s u b s p a c e s
and c o r r e s p o n d i n g l o c a l l y convex t o p o l o g i e s
-rX
, T~ ,
t h e composition
4
be h o l o m o r p h i c ?
( 2 ) I n v e s t i g a t e t h e holomorphy o f
for
U
c
E
,
V
C
F
: (f,g) E
,
(X,Txf
X
Z C H(E;G)
( Y , T ~ ) I+ g
0
will
T~
f E (Z
,T z )
W C G open. We are d r i v e n t o c o n s i d e r non-
,Z
0
is
holomorphic, t h e n i t w i l l be s e p a r a t e l y c o n t i n u o u s , and so
in
normable l o c a l l y convex t o p o l o g i e s on X , Y
p a r t i c u l a r t h e evaluation
f E (H(F;C)
,T)
since i f
* f (x)
c o n t i n u o u s . B u t combining r e s u l t s from Alexander
r51, and J o s e f s o n
[ll]
, we
then i n f e r t h a t
T
E
CP
w i l l be
[I ] , Dineen
i s n o t even f i r s t
we
c o u n t a b l e when F i s a Banach s p a c e . (Note: i n t h e s e q u e l s h a l l o f t e n w r i t e " i f f " f o r " i f and o n l y i f " ) .
2 . PRELIMINARIES
We i n t r o d u c e h e r e some b a s i c d e f i n i t i o n s and results
. Throughout,
1101
Nachbin
IR w i l l d e n o t e t h e r e a l s ,
C
(see the
complexes, and N t h e n o n n e g a t i v e i n t e g e r s . I n t h i s section l e t X
and
(LCS's)
let
Y
be complex l o c a l l y convex t o p o l o g i c a l v e c t o r
and
Ls(mX;Y)
W
c X
a n open, nonempty s u b s e t . F o r
spaces
m = 1,2,
...
r e p r e s e n t t h e v e c t o r s p a c e of c o n t i n u o u s , symmeg
r i c , m - l i n e a r maps from X
m
to Y
,
and
P(mX;Y) the vector space
HOLOMORPHY OF C O N P O S I T I O N
39 9
o f continuous m - homogeneous polynomials from X to
A
E
Ls(mX;Y)
Axm = A(x,.
E
++
.. , x ) .
P(mX;Y) is a linear bijection where (Let LS(OX;Y)
f : W
DEFINITION 2.1
Y
-+
V C W containing
lim
q [f(x)
x E V.
Let
.
7 5 Y i d Hausdoadd
++
,
then
04
f
a t xo
.
The
dm f ( x o ) ( X - xoIm =
m
1 -m C d f ( x o ) (x- x o ) m= 0
.
Y
id
said t o b e G - halomohphic iC (phg
T2) doh e v e h y
xo
E
DEFINITION 2 . 2
z E V
(T2
06
xo in w h i t t e n
Tayloh dehieo od f a t
vided X i d
open
iml
w h i c h h e p h e d e n t t h e d e h i v a t i v e o o d ohdeh m
m=0
an
in u n i q u e and we W h i t e
(A,)
d m f ( x o ) = m! A m l i m f ( x o ) = m!
1
N,ouch
A ~ ( X- x , ) ~ ] = o m= 0 H(W;Y) h e p h e d e n t t h e v e c t a h opace
each f and xo t h e d e q u e n c e
m
i d
E
- c
holomohphic mapd d h o m W t o Y doh
m
id
W
s a i d t o be holomahphic o n
i d
Y).
E
xo d o h w h i c h M
M +m
unidohmly d o h
A(x) =
P(OX;Y) = Y and Ax = A
e v e h y continuoud deminohm q o n Y t h e h e
t h a t doh bubdet
=
Then
0
xo E W t h e h e i d a s e q u e n c e Am E LS(mX;Y)
d o h ewehy
.
Y
f (xo
+
f : W
+
W
x E X
and
, t h e map
zx) E Y i6 haLomohpkic w h a e V = cz
HG(W;Y) w i l l d e n o t e t h e w e c t a h Apace o 5
G -
E Q: : x
0
+zx E W).
holomotlpkic ~uncaXoMd
dhom W t o Y . DEFINITION 2 . 3
f : W
+
Y
ewehy c o n t i n u o u d AeminVhm
W, t h a t i d ,
doh
ewehy
taining x on w h i c h vecttah space
i b d a i d to b e ampLy bounded id q
x E W
qof
on Y , q o f thehe
i d
i d bounded.
i d
doh
l a c a l l y bounded on
a n open d u b d e t
VCW
coy
AB(W;Y) w i L L d e n o t e t h e
0 5 amply bounded d u n c t i o n s 6hom W t o Y .
Notice
J . 0. STEVENSON
400
i d f A e i t h e h continuow o h loc&y We h a v e
HG(W;Y) n AB(W;Y)
DEFINITION 2 . 4 604 e u e h y
f : W
= H(W;Y)
.
in h a i d t o b e weakdy hokbmo4phic id
Y
-+
bounded, then Lt .& a m p l y bounded.
JI i n t h e d u a l o p a c e Y '
$ o f :W
Y,
56
holg
i.4
(I:
+
mohphic.
3.
TOPOLOGIES
W e s h a l l now i n t r o d u c e t h e l o c a l l y convex t o p o l o g i e s
of
s h a l l b e c o n s i d e r i n g i n t h e s e q u e l and d i s c u s s some relationships. L e t E
[ 8 1 . (Throughout o u r d i s
then
1 f IA
i s a bounded map from a
f
d e n o t e s s u p / I f ( x ) I[)
x EA
b y t h e beminohmd
f
I+
p a c t - open t o p o d o g y
If
ah
I x ,K
dedined b y t h e d a m i l y
compact and
m
E
T
c o m p a c t , i b caleed t h e comuni6okm conuehgence o n
56 ~
-
T. h e t o p o l o g y
heminahmb
06
H(U;F) genuated
f
on
T~
ldmf I x
corn
H(U;F)
doh
K C U
N.
I t follows t h a t
2
T~
T-,
dimensional o r
F = 0 (Alexander
DEFINITION 3 . 2
A heminr'lm p o n
t h e compact b u b o e t
x,
.
C U
topology
p a c t Aubnetb and i n d e n o t e d i d
s e t A i n t o a Banach space,
T h e l o c a l l y canIiex t o p o l o g y an
DEFINITION 3 . 1
their
and F be complex Banach s p a c e s a n d U C E
a nonempty, open s u b s e t . S e e Nachbin cussion i f
we
K C U
t h e h e b u 4eaY numbeh
f E H(U;F). T h e t o p o l o g y
and
T~
[l])
-
T~
iff
E
is f i n i t e
.
H(U;F) i b o a i d t o b e
id d o t euehy open c > 0
ouch t h a t
'I@o n
H(U;F)
V C U
p(f)
5 clf
pohted by containing
Iv
doh &
dedined by & Aeminuhmb
HOLOMORPHY OF COMPOSITION
each p o h t e d b y a compact b u b b e t Suppose (that is, K - x o in
+ c
and
xo E U
401
U.
06
K C U
is compact and xo -balanced
is balanced). Then for every sequence
=
E
( E ~ )
(the space of sequences of positive real nmkers tending
0
to 0 1 ,
H(U;F) where B1 is the open unit
defines a seminorm on
on
x -balanced, then the topology
in E . If U is
0
is defined by the seminorms
P, ,K for all
ball
H(U;F)
and K as
E
above
(cf. Remark 4.2 in Aron [ 2 3 ) . Again we have mensional or
whehe
5
and
T~
The t o p o l o g y
the
06
a = (a,)
E
c
+
KCU
and
0
5
‘lW
T~
F = 0.
DEFINITION 3 . 4
CE
helation and
g
r?
Let
K
on
H(U;F) i b d e d i n e d b y
5
T~
,
and
h(K;F) an d a l L o w b :
and d o h aLL
‘cW
=
x
T
iff E
U
h(K;F)
be t h e
f
%
g,
E
whehe W
W, f ( x ) = g ( x )
f
E
H (U;F)
c E buch t h a t
. An
equivalence
cland will b e c a l l e d a gehm 0 6 holamohphic m a p p i M g A ahound and w i l l b e d e n o t e d
[f
1
is
U 3 K . We d e d i n e an e q u i v a L e n c e
H(V;F) , id .thehe i n an open b u b h e t
KC W cU nV
aLL
i n compact.
b e compact and L e t
H(U;F) doh a l l o p e n %
on
T~
finite dimensional or
06
iff E is finite dL
6ohm
If follows that
union
= T~
‘cw
F = 0.
DEFINITION 3 . 3
Aeminohmb
‘rw
whehe f L A a n y h e p h e n e n t a t i u e .
K
Let
J . 0. S T E V E N S O N
402
tuxe
. Then
= h(K;F) /a
H(K;F)
A U C ~t
a u n i q u e vectoh
had
H(K;F)
A)xzce
A-t’LuC
h a t t h e mapn f E H(U;F)
[f]
I+
a t e l i n e a h d o h a l l open
E H(K;F)
give
U 3 K . We
H(K;F)
t h e dinedt l o -
c a L l y c o n v e x t o p o l o g y d o h w h i c h all t h e a b o v e mapn a x e c a n t i n g VUA
with henpect t o
T
on each
w
. Thin
H(U;F)
in
an
inductive
limit t o p o l o g y and w e may d e n i g n a t e t h i n b y H(K;F)
=
(H(U;F)
l i r i
, T ~ ) .
C‘2K
We b h a t l d e n o t e t h i n t o p o l o g y on
t h e name t v p v e o g y id i n n t e a d
H(K;F)
06
(H(U;F)
t i v e limit 0 6 t h e ( B a n a c h ) npacen
u,
tiOnA o n
Let T
i7
on
U C E
H(U;F)
A U ~n
with
HB(U;F),
T ~ .(We
by
, T ~ ) we
get
take the indui
bounded halomohphic dunc-
06
1
om
lul.
b e open and nonempty. We d e d i n e t h e t o p o l o g y
t o be t h e c o a h n e n t l o c a l l y convex t o p o l o g y
-
duch t h a t d o h aLL compact
H(U;F)
a.hV
f E H(U;F)
a t e cantinuoun W i t h t e o p e c t
rfi to
K C U,
on
t h e mapd
E H(K;F)
on each
T~
H(K;F).
T h i n LA a
phajective limit t o p o l o g y and we can w h i t e
W e have
5
T~
Let
L
T~
5
T~
. If
is
U
xo -balanced,tkien
T
~
=
T
(cf. C31). DEFINITION 3.5
be a v e c t o h nubnpace T ~ ( L )i n
l o c a l l y convex t o p o l o g y
open c o u c h
I
nibting
all
06
06
U,
i E L
let
LI
We d c d i n e t h e n a t u h a l t o p o l o g y an
The
H(U;F).
d e d i n e d an 6 v l l v w ~ . F v h evehy
be t h e
nuch t h a t
06
f
vectoh
nubnpace 0 6
L
c o ~
i n bounded on e v e h y V E I . L~
by the
06 neminomn
~
HOLOMORPHY OF COMPOSITION
f
H
If
Iv
V
doh
E
I. Then
403
(L) i b d e d i n e d t o be t h e ~ ~ n e Lg ht
T~
caLLy conuex t a p o L a g y an L b u c h t h a t t h e i n c l u b i a n d 4 & L I
c o n t i n u o u b d o t aLL open c o u e t n tion
open
06
CVUehb
A
.
In duct, t h e collec-
in d i h e c t e d b y h e 6 i n e m e n t n and
becomeb t h e i n d u c t i v e L i m i t 0 6 t h e T
U
06
LI. l n
me
( L , T(L) ~ )
a 6 neminohma,
tthmb
(L) i d d e d i n e d b y aLL beminohmb pohted b y a l e open
U, whehe p i b pahted b y t h e o p e n coueh I a 6 U id t h e h e c > 0
V
and a d i n i t e u n i o n
doh a L L
06
bet4
06
COUehA
ate
i n I Auch t h a t p(f)<_cIfIV
f E LI.
T h e LacaLLy convex t o p o l o g y
T&(L) L A d e d i n e d t h e
dame
way e x c e p t t h a t onLy c o u n t a b l e open c o u e / ~ . ~1 a t e u n e d . If
that
L = H(U;F) , we set
.rA/L5 rA(L)
and
T & IL
rA(L) = T~
5
,
Notice
(L) = T &.
T & ( L ) , that is, the
inclusion
.
(L,T~(L))(=+ (H(U;F), T ~ )is continuous and similarly for We have the following significant properties for It is a bornological topology on L , and since U
T~(L).
is metrizable,
it is in fact the bornological locally convex topology associated with
-r0lL. If
T
w
( L ) is the topology defined by all sem&
norms p on L each 7,orted by a compact subset K C U for every open p(f) 5 clflV
DEFINITION 3.6
V t U
for all
containing K there is f
A nubbet
E
L), then
(that is,
c > 0 such that
-rU(L) 5 T ~ ( L )5 T & ( L ) .
A C U i d caLLed a bounding d u b n e t
06
U .id e u e h y campLex - v a l u e d haLamahphic d u n c t i o n o n U LA bounded
on A . Thid io e q u i u a L e n t t o h e q u i h i n g e u e h y bounded o n A d o h e u e h y
f
H(U;F) t o b e
E
F # 0 . B y hepLacing compact Aubbeh
U w i t h bounding b u b n e t b i n t h e d e 6 i n i t i a n b
we o b t a i n t h e c o h t e n p o n d i n g t o p a l o g i e d
06
'loB,
T T
~ T, ~ and , ~
~and ,
06
Tw '
T LOB'
J . 0. STEVENSON
404
Hence
‘loB
5
5
T~~
‘lwB
.
S i n c e e v e r y compact s u b s e t o f
and
5
T~
’
T o 5 ‘oB
‘ ‘w 5
‘TB
‘wB’ H i r s c h o w i t z
DEFINITION 3 . 7
f 6 H(U;F) w h i c h a t e bounded o n evehy
w h i c h i d bounded i n E and
d(A,aU) > 0 , w h e t e
d i b t a n c e d t o m A t o t h e boundaty
, au
U
06
dined b y t h e b e m i n o t m b (Hb(U;F) ,
T
f
~i s)
~
* If
IA
whete
(ouch A a t e on
for a l l (where
F
of
E
e v e r y bounded sg
Hb = Hb(U;F) )
then
PROOF:
T~
Let p
open covers of
where
Hb(U;F) # H(U;F)
(Hb) #
T~~
T~~
I
and i d U
i h
xo-bd
.
b e a seminorm on
H b ( U ; F ) p o r t e d by a l l countable
U , C o n s i d e r t h e c o u n t a b l e open c o v e r I = ( U n ) n E N
Un = Cx E
is
a
Un
and
.
I n genehat rg(Hb) 5
f o r e , e a c h Un c > 0
C) # H ( E ; Q ) , and so
[ S ] used this
# 0 . One c o n s e q u e n c e o f t h i s r e s u l t i s t h e following
PROPOSITION 3 . 1
anted,
U - bounded.
Indeed, Josefson
q u e n c e h a s a weak* c o n v e r g e n t s u b s e q u e n c e . Dineen Hb(E;
A de
a F r e c h e t s p a c e . W e a l s o h a v e t h e useful
showed t h a t i n t h e d u a l s p a c e E ’
p r o p e r t y t o show
Aaid
Hb(U;F)
LA
A C L‘
F # 0 , Hb(U;F) # H ( U ; F ) .
r e s u l t t h a t f o r every
A C U
d(A,aU) LA t h e
U - bounded). The natutral topoLogy
[ll]
5
‘ I ~‘lwB( ‘ c 6 .
T h e ~ p c oe d dunctiond a d bounded type,Hb(U;F!
COnAibtA a d aLL
t o be
have
[ 6 ] showed t h a t T~
h a v e t h e same bounded s u b s e t s , s o t h a t
T~~
we
i s bounding,
U
bounded on e v e r y
u
: d f x ,aU)
>
U - bounde? and E I
Uk
such t h a t E I,
1 n i l ’
!j x
11
< n i11
. There
U, ci’n+l. Hence, t h e r e are
p(f)
2 clfl
for
and s o i n t h i s case f o r a l l
all
f
f E %(U;F).
HOLOMORPHY OF COMPOSITION
Thus p
is
T
~
c o~n t i n- u o u s , s o t h a t
Now s u p p o s e t h a t
T*
(Hb)
=
t h e bornological topology associated H(U;F) # Hb(U;F)
,
choose
.
T~~
(and t h e r e f o r e with
. Then
g E Fi\Hb
5
‘*(Hb)
‘rob
405
T~
g
.
(Hb) )
is
T~~
Since
h a s a T a y l o r se
-
m
k C P k ( x - x o ) where Pk E P( E ; F ) (k E IT) ,which k= 0 n c o n v e r g e s u n i f o r m l y o n canpact subsets of U. L e t Sn(x) = C Pk(x-xo). k=O
ries a t
xo
,
~~
Sn E H b ( U ; F ) , and s i n c e Sn c o n v e r g e s t o g u n i f o r m l y on
Then
ISn : n E N
compact s e t s , w e h a v e bounded a l s o ( s i n c e U
- bounded,
T * ( H ~ )=
Igl 5 c
g E Hb(U;F). Hence,
T*
~
~
c > 0
then there is a
B u t t h i s means
T
3.
is
T~
- bounded,
T) h .e r e f o r e , i f
s u c h t h a t f o r a l l n,lSnlA <_c.
also, which y i e l d s t h e (Hb)
#
I n summary, t h e n s i n c e
contradiction
.
T~~ T~
A
am‘: so T obC U is
/Hb
5 l&Hb) , we
have t h e f o l -
all
lowing d i a g r a m of c o n t i n u o u s i n c l u s i o n s , where i n g e n e r a l but
4.
j
are b i j e c t i o n s .
BASIC SETTING FOR THE PROBLEM I n o r d e r t o e s t a b l i s h t h e most g e n e r a l s e t t i n g i n
which
t o i n v e s t i g a t e o u r t w o p r o b l e m s ( s h o r t of m a n i f o l d s ) , w e
t o consider whether vector
H ( U ; V ) C H (U;F)
and
s u b s p a c e s or open s u b s e t s , where
H, ( U ; V ) C %(U;F) are U tE
and
V C F are
open and nonernpty. S i n c e t h e y b o t h c o n t a i n t h e c o n s t a n t t i o n s , t h e y w i l l be v e c t o r subspaces i f f
wish
func-
V = F . Hence, w e
sume V # F and t u r n t o t h e q u e s t i o n of w h e t h e r t h e y
as
a r e open
J . 0. STEVENSON
406
subsets.
Id
LEMMA 4 . 1
U c E
upen c o v e t I a d V =
3
u ui , t h e h e
thenein a
i d
open and nunempty, t h e n t h e h e .
e x i d t o an
xo 8
ulv
g E H ( U ; I E ) ~ batiddying
(Recall
= { f E H(U;(II)
H(U;IE)I
ed. Indeed, i f
{ a t wkich given any
: f
E
H(U;Q) which i s
ed f
,
E .
-case
u s i n g t h e Cauchy i n e q u a l i t i e s , e v e r y n o n c o n s t a n t e n t i r e
U f E
> 0
E
bounded oneachWEI}).
U = E l then arguing a s i n t h e classical
t i o n i s unbounded. I f
an
with
C I
g ( x o ) = 1 and j g l v <
F i r s t w e assert t h e r e i s an f
PROOF:
.. , U n 3
buch t h a t 6 u t e v e t y EU,,.
i d
func-
and i f t h e r e were no s u c h unbo~~@
t h e n U would b e a n open bounding s u b s e t of E w h i c h would
imply t h a t
= H(U;Q)
Hb(U;IE)
( c f . Dineen
[S])
,
an i m p o s s i b i l
ity. Now f o r e a c h ing x
x
t h e r e i s an open
E U
contaiz
such t h a t
u n i f o r m l y on
V(x)
. In particular,
f
i s bounded on
I = {V(x) : x E U } i s an open c o v e r of
which f
.
V ( x ) Then
U on e a c h member
of
i s bounded.
Let V
But s i n c e f that
V(x) c U
be t h e u n i o n of any f i n i t e subset of 11then
i s n o t bounded on U
If ( x o ) I > If
Iv
. Let
h E H(U;IE)Il
,
t h e r e i s an
h ( x ) = f (x) / f ( x o ) .
h(xo) = 1
,
IflV<m.
xo E U L ; ~ such Then
Ih V < 1 .
T h e r e f o r e , f o r e v e r y E > 0 , t h e r e i s an n E N s u c h t h a t n n n s a t i s f i e s t h e conclu / h I v = l h l V < E , and s o g ( x ) = I h ( x ) I s i o n t o t h e lemma.
HOLOMORPHY OF COMPOSITION
H(U;V)
i n nat apen i n
PROOF:
Let
y
E V
0
(H(U;F) ,-rX).
f o ( x ) = yo
and t a k e
N
neighborhoods
Then t h e r e e x i s t a
4.1. of
of zero. L e t
I
such t h a t
2
p(f)
implies t h a t
fo
+
N
(cll Y 1
. Then #=H ( U ; V ) .
-
y,)
P fE
W e s h a l l now see
I
for
> 0 and
E
form a b a s e
of Lemma
and f i n i t e u n i o n W o f s u b s e t s
and c h o o s e g a s i n Lemma 4 . 1 s o t h a t
D e f i n e f (x) = g ( x ) (y,
1 ,
f o r a l l f E H(U;FlI. L e t ylE F\V
clfIw
E /
E
I b e t h e open c o v e r i n
c > 0
1gIw <
E U.Then
-interior point.
= ( f E H(U;F) : p ( f ) < PIE p a seminorm p o r t e d by a l l open c o v e r s of U The s e t s
x
for all
b u t w e s h a l l show i t i s n o t a
f o E H(U;V),
then
U t E , V F F ake a p e n and n o n e m p t y ,
76
PROPOSITION 4 . 1
40 7
Hb(U;V)
g(xo) = 1
-
-
Y O N
f E N
and
PIE
a n d ( f o + f ) (xo) =ylE V
i s n o t open i n
(Hb(U;F) I ~ O b )
under c e r t a i n r e s t r i c t i o n s on U .
DEFINITION 4 . 1
Foh
A
5'-c o n v e x
hue1
06
the
&&.=
ix E
U (open, nonempty)
c U
and
A t o
be
u
v e x [ i n p a h t i c u l a h , aLb i b
Dineen
Hb(U)
-
[4]
I.
LEMMA 4 . 2
5
i n naid t o be
U - bounded bu6beA: A 0 6
U
: If(x)I
FC
U,
06
lflA
-doh aee
Hb(U)
-
iHb(u) i b U E ),
then U
c a n v e x i6 it i b a domain
06
convex
dedine
we
H ( U ) = H(U;a!)
f
€%=I.
id h a t
bounded. 16 U
i b Hb(U)
Hb-
eveky i b
- conuex.
haeomohphy
7 6 d o h e u e h y U - bounded d u b n e t A a 6 t h e
con h
a
( ~ 6 .
nonempty
J . 0. STEVENSON
408
# U, t h e n d o t each U - bounded hubhe2 A
open h e t
.them i d an
PROOF:
xo
U \ A huch t h a t
E
g(xo) = 1
with
an
xo
and
lglA <
By h y p o t h e s i s i s and
E ‘L\A
60rr
f E Hb(U)
~
> 0 we have a g E %(U)
E
.
E
c U
A
aLL
c
1s
i s U-bounded, then t h e r e
such t h a t
If (xo)I > If
IA .
The
r e s t o f t h e p r o o f f o l l o w s a s i n t h e p r o o f o f Lemma 4 . 1 .
16
PROPOSITION 4 . 2
U in
- convex, t h e n
Hb(U)
arre open and nonemptg and id
U c E l V 4F Hb(U;V)
i h
no2 open in (%(U;F) ,
T ~ ~ ) .
The argument i s a n a l o g o u s t o t h a t o f P r o p o s i t i o n
PROOF:
4.1
u s i n g Lemma 4 . 2 i n p l a c e of Lemma 4 . 1 . Hence, t h e most r e a s o n a b l e s e t t i n g i n which t o discuss the holomorphy o f t h e c o m p o s i t i o n f u n c t i o n I$ X C H(U;F) , Y C H(F;G)
is t o take
.and
from
X
x
2 c H(U;G)
Y
into
as
2
vector
subspaces.
5.
G-HOLOMORPHY
4
OF
A s indicated i n section 2 , we s h a l l investigate t h e holg
morphy o f
I$ by examining s e p a r a t e l y when i t i s G
- holomorphic
and amply bounded. W e may r e d u c e t h e problem by u s i n g a t h e o -
[ll]
rem o f Nachbin
an open s u b s e t of
M
which i m p l i e s t h a t i f
,
and
rl(N)
t o p o l o g i e s on a v e c t o r s p a c e N of every dition
T,
(N)
- bounded s e t i s
5
M
i s ,a
LCS, W i s
. r , ( N ) . a r e two l o c a l l y convex
such t h a t t h e T, ( N )
- bounded
(A)), t h e n H ~ ( w ; N ~n ) AB(W;N~)= H(w;N~)
-
T ~ ( N )
closure
( d e s i g n a t e d COG
HOLOMORPHY OF COMPOSITION
where
i = 1 , 2 . C o n d i t i o n ( A ) i s i m p l i e d by
= ( N , T ~ ( N ) ) for
Ni
(B); e v e r y
T ~ ( N -) b o u n d e d s u b s e t o f
(C): r Z ( N ) i s
rl(N)
- locally
is
N
a r e v e c t o r s u b s p a c e s , and l e t
T~
sets a s
.
T ~ ( N )= -c0
Now l e t
5
X C !I (U;F) and Y C H(F;G)
T ~ ( N )l i s t e d
T ~ ( N )5
T ~ )
in
section
3
and so h a v e t h e same bounded
T&
,
N l = (Hb(U;F) ,
a
has
(N)
= ( H ( U ; G ) , T ~ ) . Then candition
Nl
(B) h o l d s f o r a l l t h e t o p o l o g i e s s i n c e they s a t i s f y
T~
or
rl(N) -closed).
( Y , T ~ ) where
W =M = ( X , T ~ )x
T ~ ( N -) b o u n d e d ,
closed ( t h a t i s ,
b a s e of n e i g h b o r h o o d s o f z e r o which a r e Set
409
and
r2(N)
.
= T~~
In
this
c a s e , P r o p o s i t i o n 3 . 1 shows c o n d i t i o n (B) f a i l s i n g e n e r a l . l h e n e x t r e s u l t shows c o n d i t i o n (C) w i l l h o l d , however.
PROPOSITION 5 . 1
PROOF:
T~~
Hb(U;F) i d
on
I t s u f f i c e s t o show i f V = { f E Hb(U;F) : If
then
be a n e t i n V
IA
do E D such t h a t f o r a l l
f E
2 11 i s
T
c
0
do ,
d E
cloded.
- closed. compact
> 0, K C U
E
x E A . Then f o r e v e r y
- locally
i s a U - b o u n d e d s u b s e t of U,
converging uniformly
f E Hb(U;F). Then f o r e v e r y
where
A
T~
kt ( f d ) d E subsets
<
E
. Take
If ( x )
1 5
1
+
> 0,
is a
compact, t h e r e
I fd - f I K
E
,
to
K
=k)
so t h a t
v. Hence, w e need o n l y s t u d y t h e holomorphy of
t o p o l o g y on t h e r a n g e s p a c e i s
T ~ .W
following equivalent condition f o r G
PROPOSITION 5 . 2
Let
X
nonempty, open d u b n e t 0 6
and Y
e s h a l l make u s e
the
of
the
- holomorphy.
be camplex,T2, LCS'd and
X . 16 Y
t h e d o l l o w i n g afire e q u i v a l e n t .
I$ when
i d
W
a
d e q u e n t i a l l y compL&te,then
J. 0 . STEVENSON
410
*
f : W t X
(i)
LA G - h o t o m o t p h i c .
Y
( i i ) T h e h e LA a dunction E
W
f(xo
+
x
0
and x
-
zx)
f(xo)
( i l l . Let
( i ) =>
PROOF:
* 0
z
-
+ zx.
L(xo,x)
* 0
in
d u c h t h a t doh ! . & a
'
a.
xo E W , x E X , and V = { z E
g = foh : V * Y
Thenfrom D e f i n i t i o n 2 . 2 , h ( z ) = xo
* Y
E X,
Z
i n Y a6
L: W x X
i s h o l o m o r p h i c where
By t h e Cauchy i n e q u a l i t i e s ( c f . Nachbin
w e have f o r e v e r y c o n t i n u o u s seminorm q o n Y
[ (g(z)
q
where
-
-
g(0)
] 5
dg(O)z) / z
5
{ t E C : It1
setting
p
II,
+
Z)
- JI o
E
Y' , w e have f o r
f o h(zo)
Z
,
-
zo E V,
CL(xo + zoxrx)1 *
* 0 . Hence f o h i s weakly h o l o m o r p h i c on V . But
q u e n t i a l l y complete i m p l i e s t h a t [9]).
ST 4 ( t ) l
ItI'P
) C V . This y i e l d s t h e desired r e s u l t
( i ) .F o r e v e r y
II, o f o h ( z o z
)
, sup
P(P-/zl)
[9]
L(xo,x) = d g ( 0 ) .
( i i )=>
as
a : xo+zxE W}.
Thus f
is G
Y
o se-
f o h is holanarphic (cf. Nachbin
- holomorphic.
The r e q u i r e m e n t t h a t Y
be s e q u e n t i a l l y c o m p l e t e and
i s no r e s t r i c t i o n i n o u r case, s i n c e a r e c o m p l e t e and T2
(because E
(H(U;G),
T ~ ) and
(I-$,(U:G)
i s m e t r i z a b l e and G i s
T2 , T ~ ~ )
com-
p l e t e and T 2 ) .
Now s u p p o s e
X c H ( U ; F ) , Y c H ( F ; G ) , and
v e c t o r subspaces (such t h a t go, g E Y,
and
wz(fo,go,f,g) = 4,
0
is defined).
Z C H(F;G) a r e
Take
for f E X
z E C , and s e t
I (forgo) + z(fh-11 - ~ ( f o , g o )- Z L 1 (fo,go),
(f,g)l
r
411
HOLOMORPHY OF C O M P O S I T I O N
Now
where
y o = f o (x)
PROPOSITION 5 . 3
hpaceh. Then dotr
and
Let
X
.
y = f (x) H e n c e , f o r
c H(U;F) and
@ : (X,Tx) x
(Y,Ty)
-+
Y
r > 0
w e have
c H ( F ; G ) be w e c t o k d u b
(H(U;G)
a n y l o c a d l y convex Hauodahdd t o p o l o g i e n
,To)
i n G-holomonpkic
r X , -ry
.
J . 0. STEVENSON
412
PROOF:
T
0
=
I
is
t f (K)
i s compact,
/tl=r
Ko
Hence,
f0(x) +
U
where K C U
above w e have
c o n p a c t . From e q u a t i o n ( * )
Since
IK
I
i s g e n e r a t e d by t h e seminorms
1 ~ ~ ( f ~ , g ~ I ,K f +, g0 ) a s
z
0,
-+
so t h a t $
is G-hg
l o m o r p h i c by P r o p o s i t i o n 5 . 2 .
6.
AMPLE BOUNDEDNESS OF
Let
I$
X CH(U;F), Y c H ( F ; G ) ,
s u b s p a c e s f o r which
$
Z c!'(U;G)
and
i s d e f i n e d , where
be
i s open
U C E
nonempty, and l e t ?@ b e a c o l l e c t i o n of s u b s e t s of
X t o b e t h e LCS o f a l l
t o p o l o g y d e f i n e d by t h e f a m i l y of s u p seminorms likewise f o r $I : Xm x
Z
(Y,T~)
w E V, ( $ ( - ) I w
U.
f E X bounded on e a c h s e t i n
m
vector
(1
and Define
@ with
IW)WEvand
(cf. s i m i l a r notation i n Definition 3.5). -+
2%
is
amply
bounded
iff
Then
for
every
xW<
i s l o c a l l y bounded,that i s , f o r e v q ( f o r g o ) E
t h e r e a r e n e i g h b o r h o o d s of z e r o M i n X m
and N
in
Y,
(y;Ty)
such t h a t SUP
fEM,gEN
19(f0 + frgo +
9)
Iw
<
*
I
N.B.
From now on assume X
c o n s t a n t f u n c t i o n s on U t o F
LEMMA 6.1
Fa& a l e
contains t h e set ( t h e n so d o e s
Xm)
f o , f E X, g E Y , W E %!, a n d
F
of
all
. E
> 0
,
HOLOMORPHY OF
PROOF:
zo = yo
t h a t is, fine
f ( x ) = yo
+
f o ( x o ) where
f o r all
I] y o ] ]
x E E . Then
) g o ( f o + f) SUP f=constant /FIE-
PROOF:
<
E
and
lflE <
E
xo
E W.
and
: w 2 I1 q ( z o ) I1
,
f o E Yq
1 is a
N = { g E Y : 1glWl < q
i s bounded on W'
, we
T~
- neighborhood
SO
$
t o be
.
Then
of z e r o .
Since
W
go E Y and l e t
De-
-
F i r s t w e show t h e c o n d i t i o n i s s u f f i c i e n t f o r
amply bounded. L e t
go
Now t a k e zo 6 BE ( f o (W)),
5 are clear.
The i n e q u a l i t i e s
%??
E
have
s o t h a t $ i s amply bounded. C o n v e r s e l y , i f $ i s anply bourded, then f o r foE and
W
E
???+ , w e have a n
of zero N
E
> 0, V E @
, and
'rY
-
s u c h t h a t , w i t h t h e a i d of Lemma 6 . 1
xn,
go=O E Y,
neighborhood and
letting
W' = B E ( f o ( W ) ) , w e have
T h i s means t h e r e i s a n No = ( g E Y :
lgIw, <
rl
> 0
such t h a t
1 ) . T h e r e f o r e No
NCqNo
is a
T~
-
where
neighborhood
J . 0 . STEVENSON
414
of z e r o and s o i n p a r t i c u l a r i s a b s o r b i n g . Hence, e a c h g
in Y
and so i s bounded on W' .Therefore Y CH(F;G)CW,l,
i s a b s o r b e d by N o
is a
and t h e i n c l u s i o n i s c o n t i n u o u s ( s i n c e N o
T
Y -neighborhocd
of z e r o ) .
,
W e c a n combine t h e lemmas t o y i e l d t h e f o l l o w i - . g r e s u l "
letting where
= { B c ( f , W )( f ( W ) )
JE = J , ( X m , m ) E
m
: Xnz
.
IR+
+
( N 3 t e J~
: f E
xq, w
QnI
E
i s a n open cover of
F
since X contains a l l t h e constant functions).
PROPOSITION 6 . 1
The d o l l o w i n g ahe C q u i u a L e n t .
4 : Xm
(i)
zm
( Y , T ~ +)
x
in ampLy bounded.
( i i ) Thehc i d an open caueh JE
06
t\z&
F nuch
Y
C
R(F;G) E
cvntinuounly
(.SO t h a t
(iii) ( Y , T ~ ) = Y
w
1C W '
BE ( f o ( W )
Now l e t Z
Z
by
T~~
I
t h a t is, p
ing) s u b s e t s of
PROPOSITION 6 . 2
m2
U and fo E
with
x
w1
if
U
U
, we
Let
P- E /;nz
AC W c W ' ,
A E
?%-
if
containing A , there is a c > 0 I n general, we
denote
%?. i s t h e c o l l e c t i o n of compact (resp. M write t h e
V2 and
a coLLection I
nuch t h a t
E
i s p o r t e d by
p(h) 5 c l h l W for a l l h E Z.
. But
w'
and
*
f o r e v e r y open s u b s e t W of
T~
fo E X m
and
> 0
E
(Y)).
h a v e t h e t o p o l o g y d e f i n e d by a l l seminorms on
p o r t e d by sets i n
such t h a t
T~
whehe d o h e u e h y
t h e ' i e i n an
E
T~
an
ml be
06 n u b n e t h
, we h a v e E
usual
a
> 0 , and
W'
T~
(resp.
collectionn 06
E
V E
F
~
~
06 n u b n e t n
nuch t h a t d o t
m,- ,
~
an open
bet
06
evefly WCU
d u c h t h a t BE(fO(W))CV.
1
.
415
HOLOMORPHY OF C O M P O S I T I O N
Then
'xm, Y R.2 +.
PROOF:
L e t p be p o r t e d by
Wt
,W
E
c > 0
,
i.r5 ampty bounded.
(Z,.ruW2)
A E
331.- For
every f o E X ~
l
,
c
and V a s i n t h e s t a t e m e n t o f t h e theorem. Ncw take p(h) 5 clhlW for all
such t h a t
.
h E Zm
Then f o r ex
+ f , g 0 + 9)
so t h a t fo +
figo
.t
The t o p o l o g y d e f i n e d by t h e f a m i l y of seminorms ( 1
- Iw)wE
n2
i s c a l l e d t h e t o p o l o g y of u n i f o r m convergence on s e t s i n r/z
$?
We s h a l l be most i n t e r e s t e d i n t h e c a s e s where
(a)
is
one of t h e f o l l o w i n g c o l l e c t i o n s of s u b s e t s of
3 - set @
=;
d: =
of compact s u b s e t s of
I
0
w )=
, U
I
,
= c o u n t a b l e open c o v e r of
I 6(
U
s e t o f U - bounded s u b s e t s of U
U :
U,
s e t of bounding s u b s e t s of
I = open c o v e r o f
.
U
{B6 (w) ( W ) (1 U : W E
,
a'
G9'2'1 where
one of t h e above c o l l e c t i o n s and 6
:
-+
1R+
is
,
and
(b)
one of t h e f o l l o w i n g open c o v e r s of
JE(X,
(f ( W ) )
=
X i s a v e c t o r s u b s p a c e of
: f E
H(U;F)
F :
X, W
E
: X x
rib'
-+
%?'I where
containing
c o n s t a n t f u n c t i o n s , %' i s o n e of t h e i n ( a ) , and
E
the
collections
lR+.
W e now d i s c u s s by c a s e s t h e ample boundedness (and therefore
~
e
J . 0. STEVENSON
416
t h e holomorphy) of
f o r t h e topologies given i n 53,
IC$
using
P r o p o s i t i o n 6 . 1 and 6 . 2 . But f i r s t w e m e n t i o n o n e g e n e r a l n e g a t i v e r e s u l t . Identify F with t h e vector subspacr U
t o F. Then F and
Fm
6
C
X
of a l l
constant functions f r a n
a r e homeomorphic a s w e l l . The
same
holds f o r G . This y i e i d s t h e following.
PROPOSITION 6 . 3
The e u d u d o n map w : (y,g) E F
in n o t t o c a t l y b o u n d e d
-
-f
G
e
. Thus
w e i.ave
a n open s e t
Let
V E J
-t
m
(H(F;G) ,T,)
x
J = J
c i s a b o u n d i n g subset of I
of F
F.
,
so
which i s i m p o s s i b l e .
(Xlrx) a n d (Z,rz) be. LCS' n c o n t a i n i n g F
t r e n p e c t i u e L y an tocully conue.x
(H(F;G)
-
@ :F
f o r some open c o v e r
H(F C ) = Hb(F;a:) ( D i n e e n 1 5 1
COROLLARY
is just.
topoBogy
LCS
4 i s amply bounded, t h e n from P r o p o s i t i o n 6 . 1
H ( F ; G ) c H(F;G)
Therefore that
if
(H(F;G), r y ) q ( y ) E G
( = amply bounded) 6otr a n y
A s s u m e t h e c o n t r a r y . Now w
PROOF:
X
avid G
subspacu and nuch that I$ : (X,.rx)
x
(Z,rz) i b d e d i n e d . T h e n $ in n o t u m p l y b o u n d e d .
From t h e d i s c u s s i o n p r e c e d i n g t h e P r o p o s i t i o n w e have t h a t e v e r y t o p o l o g y of u n i f o r m c o n v e r g e n c e on some c o l l e c t i o n of sets when r e s t r i c t e d t o F
a g r e e s w i t h t h e Banach s p a c e t o p o l o g y on
F. Thus i n p a r t i c u l a r , r o
and
T~~
a g r e e on F w i t h i t s
t o p o l o g y , and t h e r e f o r e so d o a l l t o p o l o g i e s b e t w e e n Likewise for
G , Hence a l l t h e t o p o l o g i e s w e a r e
T~
own
and rob.
studying yield
F and G as l o c a l l y convex s u b s p a c e s , so t h a t t h e C o r o l l a r y a p plies.
Any p o s i t i v e r e s u l t s f o r o u r p r o b l e m , t h e n , c a n
come t a k i n g t h e s e c o n d s p a c e Y a s p r o p e r s u b s p a c e of
only H(F;G)
.
HOLOMORPHY OF COMPOSITION
CASE 1
% =$
T o t 'Iw(
= {compact s u b s e t s o f
From P r o p o s i t i o n 6 . 1 w e g e t ( s i n c e
76
PROPOSITION 6 . 4
417
X C H(U;F) and
1
U
)
= (X,ro(X))
Xx
Y C H(F;G) a t e u e c t o t r
nub
c o n t a i n n t h e c o n n t a n t b u n c t i o n n , t h e n I$: ( x , T ~ ( ~ )y)
npacen and X
( Y , T ~ ) ( H ( U . G ) , T ~ ) i n ampLy bounded L i d t h e l r e i n a J = J E : ( X & ) -f
n u ~ ht h a t
Now i f and
i s any open cover o f
J
w e hri-re
fo(X) c V1
B E ( f O( X ) ) C V l
u . .. u Vn
K E
Hence
continuounly (and n o
Y C H(F;G)
x
,
is a function
E
:
X
x
,&
IR'
+
F
...
u
, u
2
T~
T ~ ( Y ) ) .
then f o r every V,
f o r some
Vk
f o r some E > 0.
fo E X
Thus
E J.
there
s u c h t h a t H(F;G)JCH(F;G)JE(X
c o n t i n u o u s l y , from which w e o b t a i n t h e f o l l o w i n g .
I$ : ( X , T ~ ( X ) ) x ( Y , T ~ ) ( H ( U ; G )
COROLLARY
L5 amply bounded
-f
t h e t r e i n a n open
COV~R.
06
J
F
n u c h .thaA
Y C H(F;G)J c a ~
tinuounlg.
PROPOSITION 6 . 5
I$ : H ( U ; I ? l I
bounded d o t a l l
I = 1 6 ( & )and
PROOF:
q2=
Hence
CASE 2
H(F;G)
+
(H(U;G) , T ~ )
J. L e t
r =
f o E HI 1 -F
K C W C CJ'
Br ( f o ( W ) )
T~~
, -rwB
and
K
E
3,
8 , and take V = B
E ( f o ,K) , t h e r e
--
E: (fo
i s a n open s e t
B 6 ( K ) (K) ( I U E I
cV,
and
i d amply
.
J = JE(HI,&)
We s h a l l use Proposition 6.2 with % =
Then f o r that
x
%l
rK)
= I and
(f,,(K))
k? C L'
such
fo(W) C B r ( f o ( K ) )
and P r o p o s i t i o n 6 . 2 a p p l i e s .
(@=
d3
= {bounding
Again by P r o p o s i t i o n 6 . 1 w e h a v e
E J.
s u b s e t s of
U} )
.
.
J . 0. STEVENSON
418
16
PROPOSITION 6 . 6 npacen and (Y,7y)
-+
X
c Ii(U;F) and
X c o n t a i n n t h e con.s.tant
( H ( U ; G ) , T ~ ~in ) a m p l y
ouch t h a t
Q : H(U;F)I
PROPOSITION 6 . 7
6 u n c t i a n n , t h e n Q: ( X , T ~ ~ ( X ) )
bounded 4 5 6 thetre h a J = J E ( X , @ j
a m p l y bounded d o h a l l
(Hb(F;G) ,
x
I = Is
T
~
~+ ) ( H ( U ; G )
f12 = {bounded s u b s e t s of F 1. F o r E 8 , t a k e W = B6 (A) (A) (1 U E I . Then
each
A
W
,
so t h a t
V = B,(fo
=
letting
and
i s bounded i n F
-rA (Y))
,-rUB)
(W))E
63,
and Y
and
f o E HI
i s open and
m2
t h a n t h o s e g i v e n for -rU and
f o r any
7UB.
s u l t s i n t h i s case are t h e same a s t h o s e f o r TTI
I-rw
and
CASE 4
7TB
'c6
If
I
'
%? =
'LOB
f o (W)
r > 0.
-rU
Hence t h e
take
re-
since
, -rUB
'
open cover ( c o u n t a b l e ) ) .
i s an o p e n c o v e r o f
w e have f o r a n y
in
= I,
T h e r e seem t o be no o t h e r r e a s o n a b l e s u b s p a c e s t o for X
.
(0).
Again w e use P r o p o s i t i o n 6 . 2 ,
PROOF:
-ry
c o n t i n u a u n l y (and n o
Y C H(F;G)
uhe v e c t o h nub
Y C !1(F;G)
J = JE(HI,I)
is amply bounded. S i n c e
U , t h e n from P r o p o s i t i o n
that
H(U;G)I
0
6.1
: H(U;F)IX H(F;G)J+ H ( U ; G I I
( H ( U ; G ) , T ~ )is c o n t i n u o u s ,
w e have
PROPOSITION 6 . 8
eh) I
06
U
and
F o h e v e h y open couch ( h e n p . c o u n t a b l e open ccy J = J (HI,I) E
,
Cp : H(U;F) I x H(F;G)J
-+
(H(U;G) , T )
419
HOLOMORPHY OF COMPOSITION
i n a m p l y bounded when CASE 5
@=d
(
T~~
T
rX
=
(heop.
1;
=
T ~ ) .
= { U-bounded s u b s e t s o f
U} )
,
From P r o p o s i t i o n 6 . 1 w e g e t
Id
7ROPCSITION 5.9
, T ~ ~ i )n
( Y , T ~ ) + (Hb(I':(:)
nuch t h a t
T
Y
C
H(F;G) me u e o t o h nub
X c o n t a i n n b h e canobataltt dutiCticMn, then
n p a c e n and
CASE 6
X C Hb(U;F) and
Y
~
C
,
T
bounded
Xhthehe
continuouaey ( w h u e
H(F;GIJ
~ and,
am&
(U
xo-balanced)
T~
Q
: ( X , T ~ ~ ( X )Y.)
A a
J=J,(X,c )
2
.
- c (~Y ) )
'sW
TO t r e a t t h i s f i n a l c a s e w e need t h e T a y l o r s e r i e s e x p a n
xo
s i o n of gof a b o u t t h e p o i n t
E
U.
T h i s i s given by (cf. Nachbin
PI gof(x) = gof (xo)
+
m
Z
wl where
j=(jl
,...,j n )
[
m C
dng(f(xo))
C
E INn
k-
d J f ( x o ) ( x - x o ) l = ( dJ lf ( x o ) ( x
and
Hence,
- x0)1 ]
( j > l ) , ] j l = j l +. . . + j n , j ! = i
and
so t h a t
d J f ( x o )(x
n=l j j l = m n !
!...'n''
- x o ) J I,. . . , d Jnf (x,) (x-xo) Jn) E ?
J . 0. STEVENSON
420
PROPOSITION 6 . 1 0
4 : (H(U;F) ,T-)
amply bounded d o t e-vehy PROOF:
For
K
,
f
J =
m E N
II(F;G)
X
JE (HI$
+,
1
-+
L .iS
(H(U;G)
.
( f o , g o ) a n d ( f , g ) E H(U;F)
H(F;GIJ,
we have
an
where
set
- I n! 1 ;I"(fo +
f)
= { f E H(U;F) :
N,
IK.
sup
Oiizm
I
wheze V = B p
+
P+6
< 61
+
9lv
-1
iifoIK
and
i!
. Then for
f E , N
I
Therefore i f we choose
(fo(K)).
and
p
6
such
6 < E ( f o , K ) , t h e n t h e r i g h t hand s i d e i s bounded
a f i x e d number f o r a l l g
lzicm
1
= c14,
that
Sup
0 , we g e t
P >
and
M =
Let
such t h a t
f E Nm ( a
by
neighborhood of z e r o ) and
.
gIv <
We s h a l l c o n s i d e r n e x t t h e t o p o l o g y T
0
on
H(U;G).
Recall
t h a t i t i s d e f ned by a l l seminorms of t h e form
for
c1
+
= (arn) E c 0
s i n c e 'for any
a E
and
+ co
0
>i
. In fact,
there is a
( n a m e l y , pm = s u p { a n ] ) n Lrn PROPOSITION 6 . 1 1
K E
such t h a t
B
E
qa,K
we may assume an >_ an+l: c:
2
: H(U;F) , T ~ ) X (Hb(F;G
'n 2 'n+1
with 'BIK
'
421
HOLOMORPHY OF COMPOSITION
i b ampLy b o u n d e d .
Suppose
M = p a1
K
( f o ) and
pa , K ( f ) c 6 where f, and f E
(H(U;F)
Then m
m
C
m= 1 am
am = p a I K ( f 0 + f )
Hence, m
c
m
amam]" =
c
m=n
m= 1
c
I jl=m
a j , ]laj,
m
-< c
(M
+
IM +
6
.
. . .aI nj"a j n -<
6)n
n
n n!
14,
(M
+
+ 6)"
91"
n=O ( p / 2 ) "
n= 0 If
p > 2e(M
+
14,
+
41v
P
6 ) , t h e n t h e r i g h t hand s i d e is bounded
by
a
J . 0. STEVENSON
422
f i x e d number f o r a l l
such t h a t
g E Hb(F;G)
PROOF:
f E H(U;F)
En
Since
,
f,
Then
F
En+l),
K E
,
E H(U;F)
&,
f E (H(U;F) / T u )
p X I K ( f )< 6
and
.
lgIv <
L e t ( f o r g o ) and ( f , ,
c i (with
such t h a t
Hb(F;G)
a n d s e t co =
M = p
take
E,K
1
E
= ( E ~ ) in
(9,
+ 910 ( f o + f ) ( X o ) I
(f,)
a d ~ ; , ~ ( f <) 6 .
m
+
I: a < p E r K ( f , m= 1 m -
fI
2
M + 6
so t h a t
l
W
I: m= 1
amIn =
W
I: m=n
c
1j ~ = m
Hence, W
a JI
... a
Jn
5 (M +
6)"
.
42 3
HOLOMORPHY OF COMPOSITION
BIBLIOGRAPHY
[ 11
ALEXANDER, H .
, - "Analytic
F u n c t i o n s on Banach Spaces", Thg
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121
ARON, R.
,-
" T o p o l o g i c a l P r o p e r t i e s o f t h e S p a c e of Holo
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RDchester,
N e w York ( 1 9 7 0 ) .
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CHAE, S . B .
,-
"Holomorphic germs on Banach s p a c e s " , A n n .
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[
41
DINEEN, S.
-
,
"The C a r t a n
- Thullen
theorem f o r
spaces", Ann. S c u o l a N o r m . Sup. P i s a , 667
[
51
Banach
24
(1970) ,
- 676.
DINEEN, S .
,-
"Unbounded h o l o m o r p h i c f u n c t i o n s on a Banach
s p a c e " , J. Lon. Math. SOC., 4 ( 1 9 7 2 ) , 4 6 1 - 465.
[
61
HIRSCHOWITZ, A .
,-
"Bornologie d e s espaces d e
fonctions
a n a l y t i q u e s e n dimension i n f i n i t e " , Sgminaire Lelong, 1970, S p r i n g e r - V e r l a g ,
[ 71
NACHBIN, L.
,-
P.
Bd. 205 ( 1 9 7 1 ) .
" L e c t u r e s on t h e T h e o r y of D i s t r i b u t i o n s " ,
U n i v e r s i t y o f R o c h e s t e r ( 1 9 6 3 ) . Reprduced i n t e x t o s d e Matemstica, 1 5 , U n i v e r s i d a d e F e d e r a l d e Pernambuco, R e c i f e , B r a z i l (19 6 4 ) .
[ 81
NACHBIN, L . ,
-
" C o n c e r n i n g S p a c e s of HolomorphicMappings,
Dept. o f Math. R u t g e r s U n i v e r s i t y , New
Brunswick,
N e w J e r s e y (1970)
[ 9)
NACHBIN, L. ,
-
" C o u r s e i n I n f i n i t e D i m e n s i o n a l holomor
phy", R i o d e J a n e i r o (19 7 1 ) .
-
424
[lo]
J . 0. STEVENSON
NACHBIN, L . ,
-
"Limites e t perturbation des applications
holomorphes"
,
Colloq. s u r les F o n c t i o n s Analytiques
d e P l u s i e u r s V a r i a b l e s Complexes, C.N.R.S.,
Paris,
Agora Mathernatica, G a u t h i e r V i l l a r s
1972
(to
appear).
[ll]
JOSEFSON, B . ,
-
"Weak s e q u e n t i a l c o n v e r g e n c e i n t h e d u a l
of a Banach s p a c e d o e s n o t i m p l y norm c o n v e r g e n c e " , B u l l . Amer. Math. S O C . , 8 1 ( 1 9 7 5 1 , 1 6 6
-
168.
U n i v e r s i t y of A r k a n s a s F a y e t t e v i l l e , AR 7 2 7 0 1
Infinite Dimensional Holomorphy and Applications, Matos (ed.) @ North-Holland Publishing Company, 1977
THE NUCLEARITY OF
by
o(U)*
L . WAELRROECK
P . Boland h a s p r o v e d t h a t t h e c o m p a c t - o p e n t o p o l o g y i s a nuc e a r t o p o l o g y o n t h e a l g e b r a of e n t i r e
f u n c t i o n s on t h e d u a l
of a F r g c h e t n u c l e a r s p a c e . T h i s r e s u l t h a s b e e n g e n e r a l i s e d i n d e p e n d e n t l y by P . Boland a n d t h e a u t h o r a f t e r w e d i s c u s s e d
the
m a t t e r a t t h e complex a n a l y s i s m e e t i n g i n Cracow, i n 1 9 7 4 .
It
t u r n s o u t t h a t t h e compact o p e n t o p o l o r j y i s a l s o a n u c l e a r t o p o l o g y o n t h e a l g e b r a @(U)
of h o l o m o r p h i c f u n c t i o n s o n a n o p e n
s u b s e t o f t h e d u a l of a F r g c h e t n u c l e a r s p a c e . The p r o o f s of B o l a n d a n d t h e p r e s e n t a u t h o r a r e d i f f e r e n t . Each u s e s d e v i c e s t h a t w i l l p r o b a b l y b e u s e f u l e l s e w h e r e i n i n f i n i t e d i m e n s i o n a l complex a n a l y s i s .
I t appears reasonable t h a t
b o t h p r o o f s rjet p u b l i s h e d . T h i s p a p e r w i l l c o n t a i n t h e a u t h o r ' s p r o o f . P. B o l a n d ' s p r o o f w i l l a p p e a r e l s e w h e r e
present [2].
The i n t e r e s s e d r e a d e r i s a l s o referred t o P . B o l a n d ' s earlier paper
[l],
c l e a r when
E
w h e r e B o l a n d shows n o t o n l y t h a t O(E) i s n u -
i s d u a l F r k h e t n u c l e a r , b u t a l s o t h a t a l l reasop
a b l e t o p o l o g i e s c o i n c i d e o n @(U)
if
U i s o p e n i n a d u a l Frgchet
____---__--__--___--__^________^________-----------------------
(*)
To P r o f e s s o r G.
Kothe, on h i s s e v e n t i e t h a n n i v e r s a r y . 425
426
L . WAELBROECK
nuclear space. 1
-
Let: u s f i r s t s t a t e t h e most g e n e r a l form o f t h e main t h e -
orem i n t h i s p a p e r .
1 w i l l b e a l o c a l l y convex s p a c e ,
(El
@
w i l l be a nucle-
a r b c u n d e d n e s s or, E which i s f i n e r t h a t t h e t o p o l o g i c a l bound-
r 3 -1
edness (cf
-
nuclear space
I
p . 6 9 , o r t h i n k o f a d u a l i t y between
( F , b ) where
E
is t h e d u a l of
k i n g s t r o n g e r t h a n t h e weak t o p o l o g i e s
E,
BU
,
u (E , F ) , E =F
be t h e s e t o f e q u i c o n t i n u o u s s u b s e t s o f s u b s e t of
F
X
both r a n d
(J
(F , E) ;
1. If
w i l l be t h e s e t o f e l e m e n t s cf
fi
will
i s a n apen
U
63
a
and
E
which
are
r e l a t i v e l y compact i n U .
IIni;ohtrd
THEOREM.
Q,
cotiuehgence u n ACre efiementn 0 4
c l c a 4 top cloy^^ o n t h e a L g e b h a
@(u)
a nu-
i b
v j hloniu~zphicduncfiuno on
U. The t h e o r e m i s e a s i e r t o s t a t e when s e q u e n t i a l l y c o m p l e t e . T7e c a n t h e n l e t bounded s u b s c t s of sets of
U
I
E,
Bu
0
E
is d u a l nuclearand
be t h e
set
of
w i l l b e t h e r e l a t i v e l y compact
u n i f o r m c o n v e r g e n c e o n t h e e l e m e n t s of
@"
all sub-
w i l l be
t h e compact o?en t o p o l o g y .
COROLLARY 1.
Lp_t U
be a n open h u b b e t
0 6 a dual n u d e a h bequcn
X i a L L y compLete .space. T h e compact o p e n t o p o l o g y
i b
nucLeah
on
@(U).
FrGchet n u c l e a r s p a c e s and t h e i r d u a l s ( d u a l F r e c h e t nu
-
c l e a r , or c u c l e a r S i l v a s p a c e s ) are d u a l n u c l e a r a n d c o m p l e t e .
COROLLARY 2 . 04
let
U
b e a i r o p e n b u b n e t o d a F . ) ~ ~ c h ~ , t n u c L e Apace, ~14
a d a S-iLua nucLeatr o p a c e . T h e compact o p e n t o p o t o g y
nucleatr
NUCLEARITY
on
427
O(U). The main t h e o r e m c a n a l s o b e a p p l i e d t o s p a c e s w h i c h
are
n o t d u a l n u c l e a r . A l l t h a t w e n e e d i s a n u c l e a r boundedness which i s Compatible
with the duality.
A bounded s u b s e t of a l o c a l l y c o n v e x s p a c e i s r a p i d l y d e -
c r e a s i n g i f it i s contained i n t h e closed a b s o l u t e l y convexhull
of a r a p i d l y d e c r e a s i n g s e q u e n c e . The r a p i d l y d e c r e a s i n g bounde d n e s s o n a l o c a l l y c o n v e x s p a c e i s n u c l e a r when t h e c l o s e d a b ( B i s completant
s o l u t e l y c o n v e x bounded s u b s e t s a r e c o m p l e t a n t . when
B i s a b s o l u t e l y c o n v e x , d o e s n o t c o n t a i n any o n e - d i m e n -
s i o n a l s u b s p a c e , and i s s u c h t h a t i t s P4inkowski f u n c t i o n a l is a Banach s p a c e norm on t h e v e c t o r s p a c e EB a b s o r b e d by
CQ?.QI,LAF.Y
B)
.
L e t U be o p e n i n a L a c a L L y convex bpace whose dosed
3.
a b b o l u - t e l q c o n v e x bouiided o e t o a h e compCetun-t. T h e t o p o l o g y
06
u n i Q o t m c o n v e k g e n c e o n -the h e L u L i v e l y compuc-t, h a p i d l y d e c h e a o -
u
i r i g ouboet,$ 0 6
2.
DEFINITION 1.
vex opuce E
Qotm
.
n ~ h t t i ] oQ
O(U),
in n u c l e a t o n Let
X
be
We b h u l l c a l l
d
COnlpUCR b U b A e t 0 6
@ (X)
X
uk? u d l j i n e o u b o p u c e o
CUM-
t h e Runaclz u t g e b h a ( i n t h e u%
corztinuoub Qunctionh o n
Eo - i n - t e h i o h o Q
lOCUlRy
c(
x tuhooe h e o , t t i c t i o n t o t h e
Eo
i n h o l o m o h p l z i c d o h a l l Q i n i t e d&eMnio+
Eo
06
E .
The r e a d e r nay c o m p l a i n t h a t o u r d e f i n i t i o n o f
@(XI
is
n o t r e a s o n a b l e . I t would b e more r e a s o n a b l e t o r e q u i r e holomorphy on t h e
E o - i n t e r i o r of
a n a l y t i c sets
X (1
E o . In t h i s paper,
reason t o consider
@,(XI
compact i n a f i n i t e
- dimensional
Eo
for a l l f i n i t e dimensional
f o r t u n a t e l y , we w i l l have
e x c e p t when
X = X
1
+
X2
,
no
w h e r e X1 i s
s u b s p a c e a n d e q u a l tothe closure
rA.
4-28
WAELBRGECK
o f i t s r e l a t i v e i n t e r i o r , a n d X2 s u c h compact
is compact and convex.
&(XI
X I a l l reasonable d e f i n i t i o n s of
For
coincide.
And by t h e way, a m a t e u r s o f u n i f o r m a l g e b r a s c a n h a v e lots of f u n playing around with such "simple" Let ~
t i o n maps all
be t h e c l o s e d u n i t d i s c .
D
vn
is
a(X).
An D
D N C a!lN
i s c o m p a c t . So
, i f t h e An a r e p o s i t i v e r e a l numbers. R e s t r i g
a ( ~ i"n t o
Q
(vn
AnD)
if
0 < An
2 1
for
n .
PROPOSITION 1.
a l l n ccnd
T h i n h e n t 4 i c t i o i i i n ~ u c l e a hi d
0 < An
< 1
dU4
.
c
W e consider
a(DIN)a n d w r i t e ,
f E
a t least formally its
Taylor series,
The n o t a t i o n s a r e s t a n d a r d , lN ( N ) i s t h e s e t o f s e q u e n c e s of i n t e g e r s such t h a t
k
n
= O
when
kn=O
for
n
large. If
k = (kn)
n > N, k! = ko!
-a k = -
a
...k N ! +
ko
...
+ kN
ko.. . azNkN
a zk azO
k zk = z Ak
0..
.z N k N
kO
...
0
= ho
AN
kN
The Cauchy e v a l u a t i o n shows t h a t t h e l i n e a r form on
a(DW)
(k,) and
429
NUCLEARITY
a(DN)
h a s norm u n i t y . On t h e o t h e r hand, zk i s an e l e m e n t i f whose r e s t r i c t i o n t o % X n D
.
Ak
h a s norm
The r e s t r i c t i o n t o nAnD of t h e f o r m a l T a y l o r s e -
ries w e wrote f o r @(rAnD)
f
w i l l b e a n u c l e a r mapping of
into
&(DN)
i f t h e series
'kE
IN ( N ) Ak
c o n v e r g e s . And t h e s e r i e s d o e s c o n v e r g e
=
vnE qn IN
= 0 Xn
kn
Our a s s u m p t i o n s e n s u r e t h a t t h e p r o d u c t c o n v e r g e s . And t h e mapping
@(Urn)
-+
t h a t w e o b t a i n i s of course t h e
&(rhnD)
re
-
s t r i c t i o n mapL2ing. 3
-
L e t next
S1
and T1
an , e a c h equal
b e compact s u b s e t s of
t o t h e c l o s u r e of i t s i n t e r i o r , T1 b e i n q f u r t h e r m o r e i n t h e i n t e r i o r of T = T1
all
S1
. We
XTXnD . A s
n and t h a t
PROPOSITION 2 .
Q(s)
can c o n s i d e r t h e s e t s
+
&(TI
p r e v i o u s l y , w e assume t h a t
C An
<
< 1
for
a .
in n u c L e a 4 . X DN
t i v e l y t h e s l i c e Froduct of
&(lj(AnD)
0 < An
and
1n t h e h e c i h c u r n a t a n c e n , t h e a e n t a i c t i a n mapping
W e n o t e t h a t @CS,
and
S = S1 X D I N
. The
and a ( T 1 X r h n D) a r e @(S,)
and o f
r e s t r i c t i o n mapping
s l i c e p r o d u c t of t h e r e s t r i c t i o n mapping
respec
@ (DIN) , a d
-
of &(T1)
@(S)
+
@(T)
is the
&(Sl)
+
&(T1)
and o f
4 30
L . WAELBROECK
a(DIN
t h e r e s t r i c t i o n mapping
)
*
@(
r
11 X
D)
.
Both f a c t o r mappings
a r e n u c l e a r . So i s t h e i r s l i c e p r o d u c t . Some r e a d e r s may n o t know w h a t a s l i c e p r o d u c t i s . I f and
E$ F
F a r e Banach s p a c e t h e i r s l i c e p r o d u c t
E*
t i f i e d w i t h t h e l i n e a r mappings t h e u n i t b a l l of mappings
F*
*
E
E*
+
F
E
c a n be i d e n -
to
whose r e s t r i c t i o n
i s w e a k - s t a r c o n t i n u o u s , o r w i t h the l i n e a r
whose r e s t r i c t i o n t o t h e u n i t b a l l o f
F*
is
w e a k - s t a r c o n t i n u o u s , o r y e t w i t h t h e b i l i n e a r f o r m s on E*X F* w i t h a weak
- star
continuous r e s t r i c t i o n to t h e aroduct o f t h e
unit balls. If
E
i n d u c e d norm,
F C C ( Y ) are c l o s e d subspaces with
C ( X ) and X
b e i n g compact s p a c e s . E $ F
and Y
belong s e p a r a t e l y t o
and t o
E
u E C(x x Y ) s u c h t h a t u(x,
)
E F
u(
x
for a l l
i.e.
with
for
E E
*
E
0
-+
E1,F
L1
+
all
the y
space
f: Y
of
while
-+
+
F1
can be f a c t o r e d
El
L1 * F1
F + c0 * where t h e mappings
,
which
E X.
N u c l e a r mappings E + c
F
c a n be i d e n
X X Y
t i f i e d w i t h t h e s p a c e of c o n t i n u o u s f u n c t i o n s o n
the
co * Ll
a r e o b t a i n e d by m u l t i p l y i n g t h e ss
q u e n c e s c o o r d i n a t e w i s e by summable s e q u e n c e s , s p e c t i v e l y . M u l t i p l i c a t i o n by (11,
v,)
(un)
a n d ( u n ) re-
i n d u c e s a n u c l e a r mapping
co (IN X IN) + L1 (IN X IN) . W e c o n s i d e r t h e n t h e c o m m u t a t i v e diagram
w h e r e t h e d i a g o n a l arrow
co @ co
+
Ll
6 L1
comes form identifying
431
NUCLEARITY
these spaces r e s p e c t i v e l y with
co(IN x
and
IN)
IN) and
ll(iN
coordinatewise m u l t i p l i c a t i o n w i t h ( p n v m ) . This d i a g o n a l arrow i s n u c l e a r . I t f o l l o w s t h a t many o t h e r mappings i n t h i s d i a g r a m
a r e , among o t h e r s t h e mapping
4
-
E @ F
-f
El @ F 1
. S1 and
L e t now E b e a l o c a l l y convex s p a c e . L e t
two compact s u b s e t s o f a f i n i t e d i m e n s i o n a l s u b s p a c e
be
T1
of E,
Eo
each equal t o t h e c l o s u r e of i t s i n t e r i o r ( w i t h r e s p e c t t o E o ) , T being a l s o contained i n t h e i n t e r i o r of
a l s o (x,)
S
r e l a t i v e t o Eo.Let
be a r a p i d l y d e c r e a s i n g s e q u e n c e o f e l e m e n t s o f E . W e
let S =
T = T
where a s u s u a l , 0 < An
1
C
(An)
+CAnDxn
1
i s a sequence of r e a l
+
in
&(TI
that
a
+
€Il
numbers
such
that
w .
05
T , the
S and
hebthiCtiOM
nubnucCea4.
linear
subnuclear i f an isometry G
I: A n <
Fo4 ouch c h o i c e b
a ( S )
Recall
tion
CDxn
f o r a l l n and
PROPOSITION 3 .
mapping
+
S1
map of Banach s p a c e s
€I
+
H1
G
+
H
is
e x i s t s s u c h t h a t t h e composi-
i s n u c l e a r . The c o m p o s i t i o n o f two s u b n u c l e a r mp-
pings i s nuclear. When
f
E @(S),
s1 E Sl
ulf(sl s i m i l a r l y , when
,t)
f E &(T)
,
,
= f(sl
s1 E T1
u 2 f (sl ' t ) = f (sl The mappings
u1
: a ( S )
-+
,
t E DN
a(S, X D
N
+ c
,
tn X n )
t ETAnD
+ c )
we l e t
,
tn
u2 :
,
we let
Xn)
Q (T)* Q (T1xWAn
D)
4 32
L. WAELBROECK
are i s o m e t r i c i n b e d d i n g s . W e h a v e t h e c o m m u t a t i v e d i a g r a m
I n t h i s diagram, t h e h o r i z o n t a l arrows are i s o m e t r i c inbeddings. The v e r t i c a l a r r o w s a r e r e s t r i c t i o n m a p p i n g s , t h e r i g h t
-
hand
v e r t i c a l arrow i s n u c l e a r . The l e f t - h a n d v e r t i c a l arrow i s t h e r e
fore s u b n u c l e a r .
5
-
To p r o v e o u r main t h e o r e m , w e m u s t s t i l l p r o v e
Let
PROPOSITION 4 . detn t o
S1 and
T1
oh
. It
X E (J,
i n p o n n i b L e t o Bind compact nu4
d i n i t e dimenhionad b u b b p a c e n Eo,mch e q u d
a
t h e c d o n u h e o d i t n i n t e t i i o r r , T1 c o n t a i n e d in t h e i n R e h i o 5
S1 h e d a t i v e t o toto
cind a
xn
Eo,
ad
t o Bind a h a p i d l y d e c h e a n i n g n e q u e n c e 0 6 vec
ACqUenCt
06
conntantn
with 0
An
An < 1, C A n < a ,
nucl7 t h a t
X
id
T = T1
~
+ CAnDxn , S
T = S1
~
+
S
~
U
CDxn
.
A s i n t h e main t h e o r e m , i n t h e s t a t e m e n t o f p r o p o s i t i o n 4,
U i s a n o p e n s u b s e t of a l o c a l l y c o n v e x s p a c e ,
is a
nuclear
b o u n d e d n e s s o n t h a t s p a c e , whose e l e m e n t s a r e t o p o l o g i c a l l y b o & e d , and
Bu
compact i n
i s t h e s e t 0 s e l e m e n t s of
of a sequence
x n o f e l e m e n t s of
for t h e b o u n d e d n e s s X .
which a r e r e l a t i v e l y
U .
The t h e o r e m of Komura and Komura
contains
4
E
,
p r o v e s t h e existence
which i s r a p i d l y decreasing
a n d whose c l o s e d a b s o l u t e l y c o n v e x
The e l e m e n t s of
a being
hull
topologically bounded,the
sequence i s r a p i d l y d e c r e a s i n g f o r t h e v e c t o r space topology.
433
NUCLEARITY
X i s r e l a t i v e l y compact i n
and U
U ,
i s open.We c a n f i n d
a n o p e n , a b s o l u t e l y convex n e i q h b o u r h o o d of t h e o r i g i n i n E l say V,such that
a
=
X t V. 14e c h o o s e
U 2
10-l w i l l d o , and
a > 0
s u f f i c i e n t l y small,
s u c h t h a t ( l + n4) x n E aV when n > no.
no
Eo b e t h e v e c t o r s p a c e g e n e r a t e d by ( x o , . . . , x n
We let
and
) 0
n X'
We n o t e t h a t
=
there i s an
x
such t h a t
of
t x n ] n > n0
x' E X ' x- x'
co
sn x n E X I
and f o r e v e r y
.
Also
x' E X'
f o r every
there isan
i s i n t h e c l o s e d a b s o l u t e l y convex h u l l
I .
xn E aV
Since
m
1,
1
i s compact and c o n t a i n e d i n Eo
XI
x E X E X
m
{To0 s n xn 1 c o 1 s n
when
n > n
and V i s open
0 '
absolutely
convex, XI
W e choose
T1
and
c x +
av
.
S 1 , compact s u b s e t s o f
Eo
,
T1 i n t h e i n t e r i o r o f
c l o s u r e of i t s i n t e r i o r ,
X l c T I C S I C X +
e a c h e q u a l to the
S1,
such t h a t
aV.
On t h e o t h e r hand, l e t t i n g m
we see t h a t t h e set of
X"
sn xn
Vn:
jsnlz 11
m
=
cno+l
Dx.
c o n t a i n s t h e c l o s e d a b s o l u t e l y convex
xn,n > n X C- X '
+
1'
X" G T 1
+
X"=T1
m
+ cno+l
'n
i t w i l l b e s u f f i c i e n t t o show t h a t
+1 (1 + n 2 ID x n % U
m
+
h u l l of
,hence that
0
To prove p r o p o s i t i o n 4 ,
'n
0
but
1
L
434
. WAELBROECK
hence
c
(1 + n 2
DX&
L + n4)
(1-1
av
l + n and
s1 + ~ ( l +2 n D
2
X ~ Cx
l + n (1 7) av
+ av +
l + n
C X + V ~ U s i n c e a w a s c h o s e n s m a l l enoucjh, and
X
a n d w e know t h a t
6
-
+
V C u.
The t h e o r e m i s now e s s e n t i a l l y p r o v e d . S t a r t w i t h a n y cofz
tinuous semi-norm
v
dominates
,
S
on
where X
r,x ( € 1 Associate
v
and
a ( U ) , f i n d a semi-norm
i s compact i n
= "axxEx
T
px
which
U and
jf(x)\
to X a s d e s c r i b e d i n p r o p o s i t i o n 4 . Propo-
6 (S)
s i t i o n 3 shows t h a t t h e r e s t r i c t i o n mapping
+
&(T)
is
subnuclear. O f c o u r s e , t h e c o m p l e c i o n of
isometrically i n in
&(S),
(
@(U)
w h i l e t h a t of
,
ps
is
contained
( @ ( U , p T ) is contained
TI. The n a t u r a l mapping of t h e c o m p l e t i o n o f ( @ ( U ) , p s )
t h a t of
(
o(U), p T ) i s
into
t h e r e f o r e s u b n u c l e a r . T h i s p r o v e s t h e re-
s u l t s i n c e pT d o m i n a t e s t h e semi
- norm
v weconsidered i n i t i a l l y ,
NUCLEARITY
while
ps
435
is a continuous semi-norm.
REFERENCES. -
7
L 1 _ , P . J . BOLAND. H o l o m o r p h i c f u n c t i o n s on T r a n s a c t i o n s A.1"I.S.
[2
j
J. 209,
nuclehr
spaces
.
1975.
P . J . BOLAND. An e x a m p l e o f a n u c l e a r s p a c e
in
infinite
d i m e n s i o n a l h o l o m o r p h y . To a p p e a r , A k i v f o r Mathem a t i k . 1 5 . 1 , May 1 9 7 1 .
[3
]
H.
HOGHE-NLEND.
T h g o r i e des h o m o l o g i e s e t a p p l i c a t i o n s .
Springer -Verlag.
213,
;4
1
T.
L e c t u r e Notes i n P d a t h e m a t i c s .
J.
1971.
KOMURA AND Y .
KOMURA. Uber d i e E i n b e t t u n g d e r nuklearen
Raume i n ( s ) ~!lath. .
Annalen.
152. 1966. p.
284
288.
U n i v e r s i t z L i b r e de B r u x e l l e s D e p a r t e m e n t de M a t h e m a t i q u e Faculte des Sciences Campus P l a i n e , C P . Bruxelles
-
214
Belgique.
-
This Page Intentionally Left Blank
I N D E X O F TERMS AND CONCEPTS
PAGE
TERNS AND CONCEPTS admissible tooology
356
amply bounded
399
a n a l y t i c a l l y dense
205
a n a l y t i c e x t e n s i o n (complex c a s e )
353
analytic extension property
206
a n a l y t i c extension (real case)
369
ag2roximation property (= a.p.1
1 0 7 , 379
a p p r o x i n a t i o n t h e o r e m s for c o n v o l u t i o n equations
1 8 6 , 189
a u t o n i o r p n i sm
140 82
A $1
Banacn s p a c e
2 31
2 6 7 , 269, 270
cO
9.1
271
9.p
261,
2 6 9 , 270
b o r n o l o g i c a l topolorJy a s s o c i a t e d
327, 403
bounding set
261, 269, 270, 325,403
c a n o n i c a l maximal a n a l y t i c e x t e n s i o n
355
cateclory R ( E )
351
Cauchy r e g u l a r i n d u c t i v e l i m i t
315
chart
76
compactly p r o p e r
79
compact-open t o p o l o g y ,-ro
,
1 5 0 , 381, 3 9 3
TO
c o m p l e x i f i c a t i o n of a r e a 1 B a n a c i . l s p a c e
220
c o m p o s i t i o n map
398
437
403
MATOS
4 38
, EDITOR
convolution operators C o u s i n , problgme d e
174 27
Co ( A )
324,
(DP)- s p a c e
315
@F-H
232
81
W@
81
d i f f e r e n t i a l operators of i n f i n i t e order
175
dispersed
234
d i v i s i o n theorems
180
domaine d ' e x i s t e n c e domain o f Hb-holomorphy domain s p r e a d
184
24
407 76
d u a l of F r i c h e t n u c l e a r s p a c e
131
n
108, 117
L
bor
E(n)
-,
En
I
En
e n v e l o p e of holomorphy E
- product
E
- topology
equi
97 319, 95,
96,
133
- bounded
83
e v a l u a t i o n mapping
358
e x i s t e n c e theorems f o r convolution equations
186
e x p o n e n t i a l mappings
342
exponential
353
- polynomials
179
exponent sequence
120
extension p a i r
156
f a c t o r i z a t i o n ofholomorphic function
323
factor locally
78
6 -convex h u l l
407
190
108, 116
I N D E X OF TERMS AND CONCEPTS
4 39
finitely 2lynomially convex
381
finitely Runge
381
fixed point theoren
217
Fourier-Bore1 adjoints
176, 178
Fourier -Bore1 transformation
173
Frschet Monte1 space
147
Frschetnuclear, Frschet Schwartz space
147,
G - analytic mapping
341
gauge of a convex set
226
genera1 interpo1ating sequence
202
germ of holomorphic mapping
401,
G -holomorphic mapping
399
harmonic envelope of holomorphy
375
Hartogs' theorem
156
Hb-convex
407
EN (U)
133
holomorphically barreled space
51
holomorphically bornological space
33
holomorphically complete
385
holomorphically convex
381
holomorphically infrabarreled space
58
holomorphically Mackey space
68
holomorphic Banach-Steinhauss theorem
56
holomorphic Fock spaces
173
holomorphic of bounded type
404
holomorphic on L'
192
holomorphic germ
313
homomorphisms of Lie algebras homomorphisms of topological groups
343 340
131,
387
393
MATOS, EDITOR
440
inductive l i m i t s :
148,
boundedly r e t r a c t i v e
110
compactly r e g u l a r
116
s t r o n g h y boundedly r e t r a c t i v e , Cauchy r e g u l a r i n f i n i t e s i m a l transformation infra
- 'lontel
98 341
property
64
i n t e r p o l a t i n g sequence
202
i n v e r s i o n theorem
223
isomorphism of t o p o l o g i c a l g r o u p s
340
k
- spaces
L c v i problem LF
- analytic
313
147 381, 344
Lie algebra
342
L i e groups
341
L i e subgroup
343
l i m - s u p - s t a r theorem
259
Lindelof space
147
Lipschitz constant
263
l o c a l canonical L i e group
343
l o c a l homomorphism
340
l o c a l homomorphism of a l o c a l L i e g r o u p
342
l o c a l isomorphism
340
l o c a l L i e group
341
l o c a l l y convex s h e a f
123
l o c a l l y m u l t i p l i c a t i v e l y convexalgebra
314
l o c a l l y uniform convexity
264
l o c a l r a d i u s of boundedness
257
l o c a l topological group
340
383
I N D E X OF TEW4S AND CONCEPTS
12/11
logarithmic convexity
269
LP ( A )
232
maxima 1 a n a 1y t i c e x t e n s i o n
353, 370
m e romorph
20
minimal Ti-domain o f f a c t o r i z a t i o n
73
Mobius t r a n s f o r m a t i o n
140
modulus, i n n e r a n d o u t e r
271 , 272
monotone b a s i s
142
Monzel p r o p e r t y
62
Monte1 s p a c e
1 4 3 , 395
Nachbin t o p o l o q y
389 , 313
n a t u r a l domain of e x i s t e n c e
355
n a t u r a l Frgchet space
3 72
Newmann s e r i e s
260
nuclear holomorphic f u n c t i o n s
172
n u c l e a r h o l o m o r p h i c f u n c t i o n s on L
2
192
n u c l e a r i t y , s .- n u c l e a r i t y , nuclearity types
93, 95, 1 0 1 , 120
n u c l e a r space, (DFN)-space, s nuclear space
93, 1 0 1
-
Oka w
- Weil
approximation theorem
- space
211
-morphism Ti
3 7 9 , 388
77
- topology
132 , 1 3 3
plurisubharnonic function
148
po 1y n o m i a l d u a l i t y
172
p o l y n o m i a l l y convex
362
ported topology,
33.3,
T
w
projective l i m i t
315
pseudoconvex c o m p l e t i o n
3 85
4 00
MATOSr EDITOR
442
pseudoconvex s e t
148,
r a d i u s of b o u n d e d n e s s
251
319
r a d i u s ofboundednesswithrespect t o f (BG)
258
r a d i u s of c o n v e r g e n c e
251,
r a n g e of a n a l y t i c f u n c t i o n
201
Rienann domain
351
Rudin - C a r l e s o n t h e o r e m
203
Runge
381
Runge compact s e t
317
Runge p r o p e r t y
317
Schwartz space,Silva space,(DFS)-space
94,
S c h w a r t z lemma
222
separable range
234
u
- convex
space
163
simultaneous a n a l y t i c c o n t i n u a t i o n
393
Souslin space
148
s p e c t r ura
233,
s t a b l e exponent sequence
123
S t e i n , domain d e surjective l i m i t :
157,
323
countable
81
directed
80
open
T6
26
81
- trivial
81 81
functions
319
19,
compact
symmetric n - l i n e a r
101
211
S i l v a continuous
non
270
150 403
I N D E X OF TERMS AND CONCEPTS
400
m
402
LA
402
T
T
401
0
T a y l o r series
399
Tg t o p o l o g y
152
Tu t o p o l o g y
152
t o p o l o g i c a l group
340
translation invariance
176
U .- bounded s e t
404
uniform convexity
265
uniform C
- convexity
265
variational derivatives
191
variational equations
193
very s t r o n g l y convergent t o 0
86
weak h o l o m o r p h i c f u n c t i o n
154
weakly holomorphic
400
Zorn s p a c e
163
443
