Infinite dimensional holomorphy and applications: [proceedings]

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 0 such t h a t i f < 6

-

,

I[y'

-

y

11

< 6

:jf(x') (y') 1
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~
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


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 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.

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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 -


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
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


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
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


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

BIBLIOGRAPHY

[BMN]

J . A.

BARROSO, M.

nets

C. MATOS AND L. NACHBIN

-

On

baunded

h o l o m o h p h i c m a p p i n g s , P r o c . 1973 I n t e r n a t .

06

Conf. on I n f i n i t e D i m e n s i o n a l Holomorphy,

Lecture

Notes i n M a t h . , v o l . 3 6 4 , S p r i n g e r - V e r l a g , B e r l i n and N e w York, 1974.

[

Bl]

P . J . BOLAND

-

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

nucleah spaceb, Proc.,

1 9 7 3 I n t e r n a t . Conf. on I n -

f i n i t e D i m e n s i o n a l H o l o m o r p h y , L e c t u r e N o t e s in Math., v o l . 364, S p r i n g e r - V e r l a g ,

B e r l i n and

New

York,

1974.

[ B2] P. J.

BOLAND

-

Holamahphic Fuflctions an n u c l e a h

TAMS, v o l .

[ B3]

P . J . BOLAND

-

npace~,

209, 1975.

An example

06

a n u c e e a h Apace i n i n d i n i t e

dimi?nniond hoLomotphy, Arkiv €or matematik, 1 5 : l (1977).

[

G

]

A.

GROTHENDIECK

-

Sufi CehtaiMA edpaced d e d o n c t i a n s h o l g

mahphes, I and 11, C r e l l e s J o u r n a 1 , v o l . 192, 1953.

[

K

]

G . KOETHE

- Qualitat i

n detr F u n k t i a n t h e o h i e , C r e l l e s J o u r -

n a l , v o l . 1 9 1 , 1953.

[

M

] A.

MARTINEAU

-

S U La ~ t o p o d o g i e deb U p a c e d d e 6 o n c t i o n s

h o l o m o h p h e s , Math. A n n a l e n , Band 163, Heft 1, 1966. [Muj]

J. M U J I C A

-

Spaces

06

gehmb

06

hoLamohphic 6 u n c t i o n n , t o

a p p e a r i n Advances i n M a t h e m a t i c s .

P. J. BOLAND

J 38

IN

1

L. NACHBIN

-

T o p o L o g y on npacen

06 halomohphic

Ergebnisse der Mathematik und ihrer

mappingn,

Grenzgebiete,

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
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 - 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


n f o r a l l n. -~ (x)I 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' =

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


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

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 >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
-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 _m = (m n 1 < Pn

$-
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


n, g ' = + a vn -+ >F ,

as well as >F = >

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 ) = >

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


> F

in

(c) >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

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