Analytic functions and manifolds in infinite dimensional spaces

ANALYTIC FUNCTIONS AND MANIFOLDS IN INFINITE DIMENSIONAL SPACES

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NORTH-HOLLAND MATHEMATICS STUDIES

11

Notas de Matematica (52)

Editor: Leopoldo Nachbin

Universidade Federal do Rio de Janeiro and University of Rochester

Analytic Functions and Manifolds in Infinite Dimensional Spaces G. COEURE Universitb de Nancy I

1974

NORTH-HOLLAND PUBLISHING COMPANY -AMSTERDAM LONDON AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK

0 NORTH-HOLLAND PUBLISHING COMPANY - 1974 All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the Copyright owner.

Library of Congress Catalog Card Number: 73 93562 ISBN North-Holland: Series: 0 7204 2700 2 Volume: 0 7204 2711 8 ISBN American Elsevier : 0 444 10621 9

PUBLISHERS :

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NORTH-HOLLAND PUBLISHING COMPANY, LTD. - LONDON SOLE DISTRIBUTORS FOR THE U.S.A. A N D CANADA:

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PRINTED IN THE NETHERLANDS

Preface This book is written from a course given by author at the Federal University of Rio de Janeiro during the spring and summer quarters 1972. The aim has not been to write a complete review in any direction of works about infinite dimensional complex analysis, but to provide a systematic approach for analytic continuation of analytic mappings in infinitely many variables. For this reason many interesting results are not developed here, nevertheless the author hopes that the bibliography is almost everywhere complete.

115)

and K. Stein t 8 2 1 used by According to a way of H. Cartan M. Schottenloher [ 7 9 ] , chapter I develops the properties of spread manifolds endowed with an analytic sheaf which are needed. Chapter I11 starts with the basic properties of analytic mappings. They are shortly proved, they could be found in M. HervB's 1401 and Ph. Noverraz's books [ 7 0 1 . The second section gives somme examples of analytic continuations which show new facts arising from infinite dimensions. For instance, the classical Riemann extension theorem and analytic continuation in a product

Q: I .

In chapter 11, IV, V, we develap the properties concerning the simultaneous continuation of some natural FrBchet spaces of analytic mappings which generalise some earlier results about functions of bounded

Fz 119,601

type, and regular classes 1791

[25]

and

. Theorems of Cartan-Thullen

type are obtained and also the holomorphic convexity of their envelope of holomorphy when the underlying space satisfies a Grothendieck's approximation property. Chapter I1 takes place before because no vector structure on the underlying space is useful in this chapter. Chapter IV provides an imbedding for the maximal extension of some analytic algebra in its spectrum for suitable topologies. The method is successful when the underlying space E

is a metrizable locally convex vector space. But the indentifica-

tion with the spectrum is an open problem which is solved when E product d I.

is a

Chapter VII is mainly concerned with the existence of the following commutative diagram.

2

X -

Here X X

and Y

>Y

.?"

are spread manifolds, X

.a.

is the maximal extension of

f o r the whole set of analytic functions, Y(A)

the maximal entension of

Y for an algebra of analytic functions a n d 'p is a given analytic mapping from X to Y. The general problem is still open, but the used methods are successful when Y is a sequentially complete locally convex vector space, or 'p is a local isomophisme, or A a natural Fr6chet algebra. Chapter VIII mainly generalises the Runge theorem in finite dimensional Stein manifolds X toward some particular open sets of the product of X with a complex locally convex topological space. I amgreatlyindebted to Professor Leopoldo Nachbin who has encouraged the improvement of the original notes during and after myvisiting time in Brasil. Thanks to him, this book can appear in his collection "Notas de Mathemhtica"

.

I am also indebted to M. Hamadi f o r her typing work.

G6rard Coeurk

Federal University of Rio de Janeiro July 1972 and University of Nancy I Septembre 1973

CONTENTS Chapter I.

5 5 5 5

Spread manifolds 1. Morphisms

p.

5 - 1

2. Maximal extensions

p.

7-9

3. Separation properties

P.

9

4. Univalent extensions

p.

9-10

Chapter 11.

p. 11-14

Natural Frechet spaces

Chapter 111. Analytic mappings

5 1. Basic properties 5 2. Remarks about analytic extensions Chapter IV.

p. 15-20 p. 20-23

Frechet spaces of complex analytic mappings

5 1. Strong invariance by derivation 5 2. A general type of strongly invariant by derivation

p. 24-26 p. 26-29

Frechet spaces

5 3. Analytic extensions 5 4. Characterisation of maximal analytic extensions Chapter V.

p. 29-31 p. 31-36

Holomorphic convexity

5 1. Bishop's lemmas

p. 37-39

5 2. Holomorphic convexity

p. 40-44

Chapter VI. § 1.

Spectrum and maximal extensions Manifold structure and spectrum

5 2. Topological spectrum and maximal extensions 5 3. A particular case : (c' Chapter VII.

Extension of vector valued analytic mappings 3

p. 45-48 p. 48-53 p . 53-56

4

§ 1. Vector valued extensions § 2.

Zorn's theorems

p. 57-58 p. 58-59

§ 3. Functorialproperties

p. 59-61

5 4. Extension of products

p . 61-66

Chapter VIII. Polynomial approximation

5 1. Hilbertian operators

p. 67-69

5 2. A H6rmander's result

p. 69-70

§ 3. Runge theorems

p . 10-15

Index

p . 16-77

Bibliography

p. 78-85

5

CHAPTER I : SPREAD MANIFOLDS

4 1.- I n t e r s e c t i o n o f morphisms. In t h i s s e c t i o n , f o l d spread over

a map from that El'.

into

X

component o f

X

E

i s a Hausdorff l o c a l l y connected s p a c e . A mani-

is a pair

E

(X,p)

with

a Hausdorff s p a c e and

X

p

which i s a l o c a l homeomorphism. Every connected

E

i s open and s p r e a d o v e r

by

E

, so

p

we s h a l l assume

i s connected ; we s h a l l s a y "manifold" f o r "manifold s p r e a d o v e r -1 W e s h a l l demote px as t h e i n v e r s e o f p which i s l o c a l l y d e f i n e d i n X

a neighbourhood o f

p(x)

by

= identity.

p-l o p X

BGpGSitiGn 1.1.- A subspace w of X , homeomorphic wi t h a domain by p i s an open and connected component of p - ' ( V ) . P r o o f . Let R be t h e connected component of suppose w # R

. Then

x0 = (PI,)

[p(xo)]

-1

(pIw)-' IIp(xo)]

a boundary p o i n t

p

-1(V)

V

of

E

which c o n t a i n s w and

o f w a s subspace o f R v e r i f i e s

x

XI

in w

u s i n g a n e t which converges t o x

cannot belong t o a neighbourhood o f

, but

where t h e

r e s t r i c t i o n of p i s a homeomorphism by u s i n g a n e t w i t h converge t o

x

i n n - w .

Corollary.- Each f i b e r

p-'(a)

is di s cr et e.

Proposition 1 . 2 . - I f there is a countable basi s B f o r the open s e t s of X has a l s o a countable basis such t hat every open s e t of t h i s b as is is homeomorphic by p w i t h a domain in E . E, then

Proof. Since to

E

i s l o c a l l y c o n n e c t e d , we can assume t h a t s e t s which b e l o n g

a r e domains ; we d e n o t e by

B

holeomorphic by

, which

x E X

%

p

,-b

B ; let

can be j o i n e d t o some

. Each

t h e se t o f domains i n

B

w i t h a domain o f

Xm

x

X

which a r e

m

by a c h a i n o f

domains belon-

m

Xo is t h e union of

x,

s i n c e X is m c o n n e c t e d , t h e r e f o r e w e have j u s t t o prove t h e p r o p e r t y f o r e a c h X ,

ging to

i s open, and

X

be t h e s e t o f p o i n t s

By p r o p o s i t i o n 1.1,

xoE

'L

w

=? px

-1

X1

(w) QJ

, with

0

; t h e s e t o f such w i s c o u n t a b l e and

p

-1 'L

(w)

0

s o it i s proved f o r

x1 '

w

X

,w

' L ' L

= p(w)

%m, and

v e r i f i e s t h e property,

SPREAD MANIFOLDS

6

Now, we assume t h a t

w i t h :{ E % X ; t h e union o f X m m ; by p r o p o s i t i o n 1.1, t h e r e i s o n l y one

13

ni ent b a s i s of contains

'L

be a conve-

(wn)

:

w i t h :nC

c8

%

such t h a t w

, I ,

c o n t a i n s some domain

wn * i s a c o u n t a b l e union o f domains w i t h a c o n v e n i e n t b a s i s , Xm+l has a l s o a convenient b a s i s .

Therefore and

v e r i f i e s t h e p r o p e r t y and l e t

Xm

'm+:l

Corollary.- If E

has a countable basis of open s e t s , then each f i b e r

i s countable or f i n i t e .

p-lfa)

Definition 1.1.-

fX',p')

A map

from

u

i s a morphism i f

t o t h e space

X

o f an other manifold

X'

p =

i s continuous and s a t i s f i e s

u

p'

0

u.

A morphism i s an isomorphism i f i t i s one-to-one.

proposition 1 . 3 .

x

from

- A morphism i s always open,

and two d i f f e r e n t morphisms

X' s a t i s f y : u ( x ) # v f x ) for a l l x

to

E

x.-

Proof. C l e a r by c o n n e c t e d n e s s ,

Theorem 1.1.-

Let

1*

be a family of morphisms from

n Xi , a

e x i s t s a manifold denoted by by

n

, morphism '0 i

ui

(i)

0

from

n Xi

, and morphisms ui from X

there e x i s t s a morphism and

yi

=

1.4;

o p '

The morphism

.

n Xi

A l l the manifolds

ui

'9'

Xi

, and

such t h a t

from

X'

denoted

morphism

u' from X t o o u for any i, 2 such t h a t : A ui = '0 u

t o 0 Xi

ui = u !

which v e r i f y the above properties are isomorphic. the i n t e r s e c t i o n of morphisms

n Xi

we c o n s i d e r t h e s e t

f o r which t h e p r o j e c t i o n s

e x i s t s a neighbourhood

t o f l Xi

There

.

(X',p') Xi

X

(Xi,pi).

such t h a t :

.)

to

w i l l be called

Proof. I n t h e p r o d u c t s p a c e

x = (xi)

to

( f 7 ui) = ui for any i

( i i l For every manifold X'

morphism from

to

X

V

of

p(x)

pi(xi)

Y

a r e one p o i n t

u

i '

of p o i n t s

p(x)

and t h e r e

which i s homeomorphic w i t h a neighbou-

.

wi o f xi by pi f o r each i n d e x i From such w . , w e d e f i n e a b a s i s Tt wi n Y f o r a Hausdorff topology on Y and s o (Y,;) is

rhood

spread over Let (ui(x))

.

E.

f i ui

be t h e map from

Since

pi

ui

= p

X

into

n Xi

d e f i n e d by : ( A u i ) ( x )

t h e r e e x i s t s a neighbourhood w of

x

such

SPREAD MANIFOLDS

i s homeomorphic w i t h

u.(w)

that

morphism from

( nui)(X)

in

and

pi

and t h e r e f o r e

X i o n t o Xi It s a t i s f i e s

i s t h e canonical projection of (u!(x')).

contains

, is

( nui)(X)

X1

and

X2

be two i n t e r -

s e c t i o n s w i t h t h e p r o p e r t i e s o f p r e v i o u s theorem, we d e n o t e by t h e a s s o c i a t e d i n t e r s e c t i o n o f morphisms

nXi = X 2 . From ( i i ) we

pi

u1 =

u1 o u2

u1

and

u2

, We can p a r t i c u l a r i z e theorem

yi ,

=

- 'Q;

u2

get

and we a l s o have

X2

u.

, u!

= u1

1.1 i n t h e f o l l o w i n g way : u '

to

connected

i s i n c l u d e d i n nXi ,

Y'(X')

Only u n i q u e n e s s remains t o be proved. Let

X1

is a

ui

t h e connected component o f

i s t h e one d e f i n e d by L+?'(x')

n ui , t h e r e f o r e \ 9 ' ( X ' )

=

u'

,

by

n Xi

Y , We t a k e f o r

to Y

'9'

and t h e map

y e to

X

p(w)

I

,

X ' = XI

,

n u i = u2

'4; is a morphism from y ; from X 2 t o X1 .

where with

Checking up t h e s e r e l a t i o n s , we can e a s i l y v e r i f y t h a t y ;

i s a n isomor-

phism.

5

2.-

Maximal e x t e n s i o n s . Let

Z

be a n o t h e r l o c a l l y connected Hausdorff s p a c e and

s u b p r e s h e a f of s e t s of c o n t i n u o u s f u n c t i o n s d e f i n e d on

Z

. The

set of s e c t i o n s o v e r a n open s e t

and t h e germ a t

x

of

f

e FE(U,Z)

U

as

in

fx

. In

E

xo g U

,

: F o r e v e r y domain

fx

= gx

(X,p)

and

.

f

w i l l be n o t e d

FE(U,Z),

FE(Z) : and

g

in

(Y,IT) be two m a n i f o l d s s p r e a d o v e r

We i n t r o d u c e t h e p r e s h e a f X are continuous, all x E U

.

= g

f

,

E

in

with values i n

FE(U,Z)

,

0

0

Let

imply

U

a

t h e f o l l o w i n g s e c t i o n s , we

s h a l l assume t h e f o l l o w i n g a n a l y t i c p r o p e r t y o f (A)

E

FE(Z)

F (Y)

x

and s a t i s f y : (IT o f o p

Y-valued,

Proposition I. 4 . - For every

E

and

whose s e c t i o n s o v e r any open s e t

u E

FE(E, Z)

a

u

0

-1

)p(x)

Z ,

U

FE(z)

in

for

p 15F X ( X J 2) ,

Proposition 1.5.- Given a morphism u from X t o X' then t h e transpose mapping u* : f + f 0 u maps F ~ , ( x ~ , Y )i n t o F ~ ( x , Y ) , Proposition I. 6 .

- The presheaf

Fx(Y)

s a t i s f i e s property

The above p r o p o s i t i o n s are o b v i o u s . L e t now manifolds spread over

Zi

,

FE(Zi)

(A).

( Y . , I T . ) be a f a m i l y of 1

1

a family o f p r e s h e a f s with p r o p e r t y

SPREAD UNIFOLDS

8

, which

(A)

i s indexed by t h e same s e t

Definition 1.2.- Let r = f i be u from X t o X ' i s called a f: € F X , ( X ' , Z i I with f i = f; o r' = (f!! i s unique because of 2

X'

and denoted by

.

u y (I')

a family with fi g F X ( X J Z i l . A morphism r-extension f o r FE(Zil i f there e x i s t s u f o r each index i The f a m i l y (A) and i s cal l ed the extension of r t o

.

Definition 1.3.- A r-extension u : X + X' i s called maximal i f every r-extension v : X + X ' can be factored through a morphism w : X" -+ X' ( t h a t i s u = w o vl , Proposition I. 7 . - A l l

P r o o f . If t h e p r e v i o u s from

X'

to

position 1.3

v* ( r l -

r-maximal extensions are isomorphic and are

eztensions of any other u

and

X"

such t h a t :

w o

p

and

9

are maximal, t h e r e e+sts a morphism yl

v

u = w w

o

isomorphism. On t h e o t h e r hand

.

u

r-extension

,

o

'4

o

u

and

=

v

Y Ow

, By p r o -

o v

are t h e i d e n t i t y and t h e r e f o r e

i s an

w

i s a v+)(r)-extension.

w

Theorem 1.2.- We assume t hat t he previous f a m i l y i s the union sub-families r Let u : X + X be a maximal r.-extension, then A u i s a maximal j' j j J j r-extension. Proof. W e r e c a l l t h e e x i s t e n c e of morphisms

yj

u

j fj

0

p

o

( n u.) 1

. Every

f

E

r

i s an e x t e n s i o n t o ( I xj ' L e t u s now prove t h e maximality of

r-extension,

t h e r e s t r i c t i o n of

'4

: j h a s an e x t e n s i o n

u'

to

.

n Xj r

n Xj f

j

. Given

+

to

X

j X

with

, therefore

j

u' : X

-+

a

X I

c a n be f a c t o r i e d t h r o u g h a mor-

j w i t h u = u! o u' We can a p p l y ( i i ) of theorem 1.1 u! : X + X 1 j j i and t h e morphism y ' o f ( i i ) g i v e s t h e m a x i m a l i t y .

phism

Theorem 1 . 3 . -

There e x i s t s a maximal r-extension.

P r o o f . By t h e p r e v i o u s theorem, we have o n l y t o c o n s t r u c t t h e maximal e x t e n s i o n of e a c h f u n c t i o n

f € FE(X,Y)

. F i r s t w e endow

s t r u c t u r e of a manifold i n t h e f o l l o w i n g way, F o r e v e r y domain i n

E

, we

consider

N(h,U) = {hx

I

x € U1

. It

FE(Y)

f eFE(U,Y)

-

p : hx

+

x

i s a homeomorphism from

N(h,U)

onto

,

U

can be s e e n w i t h o u t

d i f f i c u l t y t h a t we have b u i l t a b a s i s f o r a Hausdorff t o p o l o g y on and

with a

FE(Y)

U , Further,

,

SPREAD MANIFOLDS

9

i s a morphism from X t o p-I x +(XI t h e connected component o f u(X) i n t o FE(Y) ;

u : x +(f

?

a c t u a l l y t h e map satisfies

f

therefore

f

: hx

hx(x)

+

from

into

X

i s a s e c t i o n of

over

F-(Y)

X

We s h a l l now prove t h e maximality o f f'

it i s c l e a r t h a t

x'

which f a c t o r s

[f'

0

t h e extension of

u through

X

i s c l e a r l y continuous,

v : X

Let

,

.

f

o u

X' be a

-f

i s a morphism from

-1 = ( f ' o v o px ) p ( x ) 0

?

x& U

X

to

?

v . We have :

(P' p'

Y

X' , By t h e n e x t computation

to

f

( f ' o pA-l)pl(x)

-+

.

-

be

i s a f-extension,

and

X

?

2

= h on U f o r a l l h e FE(U,Y) and

o (phX)-'

f-extension with

. Let

(FE(Y),pf

0

-1

= (f

= u(x)

px ) p ( x )

O

.

v(x)

§ 3 . - Separation properties.

r

We s h a l l a p p l y t h e p r e v i o u s theorem t o a s u b s e t

2

what f o l l o w s ,

i s t h e r-maximal e x t e n s i o n o f

r

t h e extension of

.

?

to

for

X

in

FE(X,Y)

FE(Y)

, In

, and f

is

Renever r separates X and the morphism u from X t o i s i n j e c t i v e then X i s isomorphic with the domain u(XI in ?

Proposition 1.8.-

(2,;)

Proof. Clear by -

.

I - =f

the relation

.

o u

Proposition 1.9.- I f r contains a s e t o f functions g 0 p , g E F E ( E , Y ) and E is separated by t h e function g then ? is separated by

.

P r o o f . Let

Xs t h e s e p a r a t e d q u o t i e n t s p a c e o f X a s s o c i a t e d w i t h t h e r e l a t i o n : ?(XI = F ( x ' ) , a l l f E 7 , Since t h e f u n c t i o n s u s d e n o t e by

are i n g o j e c t i o n on E

? , all

t h e p o i n t s of some c l a s s i n

. Then t h e i n d u c e d mapping

phism, and t h e q u o t i e n t map from

2

s i o n . By p r o p o s i t i o n 1 . 7 ,

I

4.-

and

? 2,

to

p

zs

gS

on

gS i s a

have t h e same p r o -

i s a l o c a l homeomor-

morphism and a

?

exten-

are i s o m o r p h i c ,

Univalent extensions.

X

i s a domain i n

Propoeition 1.10.

E

and

r

a subset o f

FE(X,Y)

which s e p a r a t e s

- The fo I lowing properties are equivalent

(i) There i s no pair

(U,VI

,

U and

:

V domains in

E

,

X

.

SPREAD MANIFOLDS

10

r

X , V - X # /a , such that f o r every f 7 6 F,(V,Yl such that Flu = U

c V f7

(iil

X i s isomovphic w i t h i t s maxima2 : Let

Proof. ( i ) - ( i i ) i n j e c t i v e morphism boundary p o i n t of W

.

flu

u : X u(X)

?

be t h e maximal

. We m u s t

*

, then

f Cr. Then T

o p;'(a)

= f(a)

r - e x t e n s i o n of

prove

u(X)

-

is l i k e ( i )

where

for a l l U

x

-+

(fx)f

~

i s a morphism from

cannot be isomorphic w i t h

X

.

(U,V) V

to

px

a Ep(W)

i s a domain i n

( i i ) q ( i ) : Given a p a i r

.

= ?

--I

f

with

x

0

(U,p(W))

I'-extension.

x

E

nu

0 -

, with

X

If xo

i s a boundary p o i n t o f

p(xo)

be a connected neighbourhood of

there e x i s t s

X

is a

. Let

for e v e r y

FE(W),Y) '(W)

and t h e p a i r

p ( W ) / I u-'(W)

.

l i k e ( i ) , t h e mapping

.

Since

V

-

its

X # 0

,

x

11

CHAPTZR I 1 : NATURAL FRECHET SPACES i s a l o c a l l y convex F r e c h e t s p a c e whose set o f semi-norms

Z

Here,

t h a t d e f i n e s i t s t o p o l o g y i s d e n o t e d by connected s p a c e functions

E

E

j

i s a Hausdorff l o c a l l y

i s t h e p r e s h e a f o f c o n t i n u o u s and Z-valued

C (Z)

j

rn

i s t h e subspace o f

CE(U,Z)

j

N(Z)

C (U,Z)

whose f u n c t i o n s a r e

E

bounded,equipped w i t h t h e uniform t o p o l o g y . Throughout t h i s c h a p t e r ,

FE(Z)

i s a sub-presheaf o f

C,(Z)

which

s a t i s f i e s (A) and t h e n e x t p r o p e r t y (B).

(B)

FE(U,Z)

n Ci(U,Z)

i s closed i n

C;(U,Z)

, for

a l l open s e t s

U

in

E.

I n what f o l l o w s ,

Definition 2.1.-

Given

is a manifold spread over

X

r

a set i n Fx(X,Z)

r, i f f 11. 11

called a bounding s e t f o r We are using the notation

; a part

114 o f [ (


[lf,[I ,

n

-

Xn

with l[fn(xn)]l > n

# 0 and t h e r e f o r e llfnl[

must be

W e consider t h e

u in , which i s enough t o c o n s t r u c t

which converges n e a r

there would e x i s t s

.

, all f E r

llf[l J

A c t u a l l y , i f t h a t i s n o t t r u e , t h e r e would e x i s t (x,)

-1

rn(g,)

. fn

.

# 0

, we

x t X

. [ rn(fn)

.

Proposition 2.4.given two points

I

{f E I'

+ Ilfll,

f

Let

r

when

with

(x,)

by p r o p o s i t i o n

r

+

AU

, we

, and

check

r

X ;

is s t r o n g e r than t h e point-

i t i s everywhere dense s i n c e t h e r e e x i s t s

g ( x ) / # g ( x t ) ; t h e sequence

f(x) = f(x')

j

as i n p r o p o s i t i o n 2.1, w i t h w any bounding

P r o o f . T h i s set i s open s i n c e t h e t o p o l o g y o f gQ

1

n

\< 1 ; t h e n gn i s a bounded sequence

r

, f u r t h er mo r e,

.

+ Ilf,II

be a natural Frechet space which separates x and x ' i n X , x # x ' , then the s e t i s e v e w h e r e dense and open. f(x) # flx'l}

wise topology x'

?,(

should have :

To prove t h e c o n t i n u i t y o f t h e imbedding map

r

.

n , For each

2.1.

set f o r

1

and a sequence

for a l l

which must be bounded on t h e r e l a t i v e l y compact s e t

the continuity of

such

X

converges n e a r

g, = f f ,

+ n g s e p a r a t e s x and

NATURAL FRECHET SPACES

14

Theorem 2.3.- Let us assume F E (Zl t o be locally bounded with E having a countable basis of open s e t s . Then f o r every natural Frechet space r in FX(X,Z) such that X i s separated by I' and i s a maximal r-extension, there e x i s t s fE r such t h a t X i s f-maximal. Proof. L e t for e a c h

e

( w ) be a c o u n t a b l e b a s i s of domains for E ; g i v e n an wn n p-'(an) i s c o u n t a b l e by t h e c o r o l l a r y o f , the set Q

n

( x ) now a n everywhere d e n s e sequence i n X , whose n e d e n o t e by N t h e s e t o f p a i r s e x i s t e n c e i s g i v e n by p r o p o s i t i o n 1 . 2 . W p r o p o s i t i o n 1 . 2 . Let

E

r

r

.

# With t h e Wn'xm n o t a t i o n s of p r o p o s i t i o n 2 . 3 , we know by p r o p o s i t i o n 2.3 and 2 , 4 t h a t t h e r e of i n t e g e r s

(m,n)

fE

exists

r

X

isomorphic t o extension o f

f

a)

x1

w

and

n

which d o e s n o t b e l o n g t o any

. We s h a l l prove

separates Q

p(x ) m

such t h a t

.

u

Let

to

.

X'

t h a t every f-extension

be t h e morphism from

and

X

to

u(S2,)

=

x

u(x,)

,

f'

o

and

x1

=

=

u(x,)

, x2 , x '

x'

f(yl) # f(y2)

=

u(yl)

.

b)

u(y2)

u

is onto :

Now, w e can assume

If

X # XI

cannot b e a

w,

X

xm ...

f(y2)

would e x i s t

u(Y,)

X' x

and

u

. There

which are homeo-

on t h e o t h e r hand u(y2)

f' t h e

p;;'(an)

u(R,)

=

f(yl)

=

; thus

t h e imbedding map.

b e l o n g i n g t o t h e boundary of

0

x0 which i s homeomorphic w i t h some wn by P ' w /7 X S i n c e X i s a maximal r - e x t e n s i o n ,

r - e x t e n s i o n and

is i m p o s s i b l e s i n c e

, since

i s a domain i n

, there

a neighbourhood w of bounded on

=

f'

j

.

I'

#

is

X

a = p ' ( x f ) = p ( x 1)

, by t h e p r o j e c t i o n s p and p ' ; f u r t h e r m o r e w e have -1 p x , ( w ) a and t h e r e e x i s t s some an b e l o n g i n g t o w.

a r e b o t h i n Q which i m p l i e s

x1 = x 2

X'

G N and of

(X',pl)

morphic t o some connected neighbourhood w o f t h e p o i n t

= p(x2)

(m,n)

i s one t o one :

u

be two p o i n t s o f X w i t h 2 i12 # R' of e x i s t some neighbourhood sll Let

with

,xm

r

; therefore

f

4

X

,

, f'

Xu w

,

15

CHAPTER I11 : ANALYTIC MAPPINGS

1.- Basic p r o p e r t i e s . Here, some d e f i n i t i o n s and p r o p o s i t i o n s are g i v e n w i t h s h o r t proofs,

. The

we refer f o r them t o [40,70:1

space

E

(resp. Z)

l o c a l l y convex v e c t o r s p a c e ( a b r e v . c , v. s . ) w i t h f i e l d (resp. sequentially-complete be an open set i n

C.

v. s. with

K

i s a Hausdorff

=

m

as t h e

or

dl as t h e f i e l d ) . L e t

E.

Definition 3.1.- A mapping f : w + d: i s Gateaux-analytic i f f f o r any i n the adjoint space 2 , any x g w and a 6 E , the mapping

t

-+

.z I

o

f ( x + at)

w

i s holomorphic i n a neighbourhood of the o r i g i n i n

z'

6: ,

Proposition 3.1.- Given al ,,, ,, an i n E , f o r any f i x e d x , there e x i s t s c ?:n 2 such that f o r any T ~ . N ( Z ) the s e r i e s C c t" i s sumc1 mable f o r E x e a r f ( x + t l a l + , ,, + tnan) and f o r t small enough j here t = ( t , ,,, ,, tn) and a i s an n-multi-index. For a = (1, 1, 1 ) , the associated c o e f f i c i e n t i s denoted by

"

,.., ...$an

) j f o r n = 1 and a = k f n ( x , a l , a2, cient i s denoted by p ( x , a ) ,

Proposition 3.2.-

a ) f " ( x , a,

, the

..., a ) = n! . f " ( x , a ) ,

b ) The mapping

(a,

c ) m e mappings x Gateaux-analytic.

d ) $1(x,a) =

--1

211

,.,.,

+

;1

an)

f"(x,a)

+

.

f"(z, al ,, ,, an) i s n-linear.

and x

-+

f " ( x , al

f ( x + a e i e ) e*iede

the biggest balanced open s e t with x

associated c o e f f i -

,, ,., an)

, for

all

are

a Eu(x)

,

as center and contained i n w.

e ) Any Gateaux-analytic mapping s a t i s f i e s property ( A ) , f ) Let ll a continuous semi-norm on 2 be given. If R o f i s continuous a t x e w , then o f i s continuous i n a neighbourhood o f x and

ANALYTIC MAPPINGS

16

each map a

+ T o

fnfx,al

.

is continuous on E

P r o o f s . The p r o p o s i t i o n 3 . 1 i s a consequence of Hartogs theorem and t h e

.

d:

e q u i v a l e n c e o f weak-holomorphy and holomorphy i n

S i n c e t h e s t a t e m e n t i s obvious f o r n = 1 , w e n-1 ( n - l ) ! fn-'(xtta;a) = f (xtta; a,..,,a 1 whenever

P r o E------osition 3.2.a) --may assume t h a t xtta

. But

6 w

(n-1) times fn-'(xtta;..

i n t h e l e f t hand s i d e ( r e s p . : t h e r i g h t

,)

un-l ( r e s p . : ul,. . u

hand s i d e ) i s t h e c o e f f i c i e n t o f expansion i n

u

( r e s p . : ulT...,u

of

+

) i n t h e Taylor

n-1 (ttu)al

(resp.: n-1 which i s a n a n a l y t i c f u n c t i o n o f f[x

f [x t ( t t u1 t

. .. + un-1 )a])

[resp.:

, u ~ - ~ on some neighbourhood o t he o r i g i n i n

(tyZl,.

(resp.:

C

)

11

. Therefore,

,

f o r s u f f i c i e n t l y small

,

It[

Q:

2

( t ,u)

f"-'(xtta;a)

( r e s p . : t h e r i g h t hand s i d e ) i s t h e sum o f a poHer series i n t , w h e r e n-1 t i s t h e c o e f f i c i e n t of t u (resp.: tul,,.tu ) n-1 i s t h e expansion o f f [x t ( t t u ) a ] ( r e s p , : f [x t ( t t u1 t . . . t u n-1 )a]), namely n f n ( x ; a ) I r e s p . : f n ( x ; a , . ,a)] the coefficient of

.

..

n times

.

fn-'(xtua;al..

b ) c ) By a similar argument : f o r s u f f i c i e n t l y small

i s t h e sum o f a power series i n

i t s first o r d e r d e r i v a t i v e w i t h r e s p e c t t o fn ( x ; a , a l , .

u

, for

u

, hence

u

=

0

+

. . ,an- 1 ; t h e r e f o r e a 1 fn-'(xtua;a1,. . . ,an-1) ...,a l i n e a r and b ) w i l l f o l l o w from t h e . n-1 +

Consider t h e T a y l o r e x p a n s i o n , around t h e o r i g i n i n f ( x t ua t u ' a ' ) setting

u'

= o

linear entails l i n e a r i t y of

6:2 , o f

u ( r e s p . : u ' ) i s u n a l t e r e d by 1 1 i t s value i s f ( x ; a ) [resp.: f (x;af)];

: since the coefficient of

(resp.: u

then, f o r constant h fI:x

c ) , and

, is

al fn(x;a,al, [a -+ f 1( x ; a ) ]

1. ,

=

0)

,

, h f Ec , the is

t u( a t i ' a l ) ]

coefficient of

u

i n t h e expansion o f

h f 1( x ; a ) t h ' f 1( x ; a l ) .

d ) i s a consequence o f t h e summability o f T a y l o r expans i o n of t h e map

t + f(xtta)

, for

all

[ t i < 1 and

.

a 6 ~ ( x )

e ) i s a consequence o f c o n n e c t e d n e s s o f

E

by p o l y g o n a l

l i n e s and p r o p e r t y ( A ) coming up i n f i n i t e d i m e n s i o n a l case, f ) T h i s i s a l o c a l p r o p e r t y and we can assume t h a t lr

i s bounded i n

W

by

M

.

o

f

ANALYTIC MAPPINGS

Let x c w all

a E

be given, then E.W(X)

IT

.

f(xta)

17

f(x)l \
,l

fn(x,a)

71 o

fn(x,a)
0 ) are balanced neighbourhood of the origin in E such that An + t A n C B C W - x and p o f \< M on x + B , then po fn(a) C M n times on x + for a , p fn(a,a2,, ..,an) 4 M on x t A~ for a 6 A ~ ,

...

.

a19...,an 6 An+1 For any given a or al,

...,a n d E

, by

n-homogeneity or n-linearity we have p o fn (a) or p fn(a,a2, an ) locally bounded on w ; since fn(a) and fn(a,ap,. ,an are Gateaux-analytic by proposition

..

3.2.c), they also belong to

6 (w,Z)

...,

,

b) is a consequence of proposition 3 . 2 . f ) .

c) Let K be a compact set in w and A the unit disc in 6 . A U a e K, and A in E such that o(x) 2 TA ;

1 and a have neighbourhoods T in

if a finite union of sets Ai

contains K and the corresponding

neighbourhoods Ti of 1 contain the disc then w(x)

contains (l+a,A).K,

ANALYTIC MAPPINGS

18

(l+a)A.K

hence c o n t a i n s t h e compact s e t p 6 N(Z) j p

Given Va

, hence

(1ta)A.K

f 4 M

o

p

on

x

since

+

(l+a)A.K

f n ( x ; a ) 6 M/(1+aIn

o

w(x)

Va

i s balanced.

implies p o fn(x;a)gM € K

,

n d

IN , and

t h e uniform summability. d ) Given a compact s e t

K

c

sufficiently near the origin i n

,

compact s u b s e t o f w sup x G K

p

,

c N(Z)

p

and

: if

{xtta : x 8 K, t

a

i s chosen

c A} =

is a

and :

f " ( x , a ) \
0

with a s u i t a b l e

Proposition 3 . 5 . - [SS]

.

Let

f

E

w

x

Q

X

where IT o

f

fn(x,a)

n t l

> 0

i s weakly a n a l y t i c ( t h a t i s 5 and E i s rnetrizable then f belongs t o 6,1w,z)

n E

X

fn(x;a)

p c N(Z)

be g i v e n

i s summable,

g(a)

g

i s 1.s.c.

0

for a l l

b E ( w , 61 a l l

f

Z

5 6 Z'l

,

; a E w(x) , s i n c e t h e e x p a n s i o n = sup p o f n ( x ; a ) i s f i n i t e .

n b O By t h e a s s u m p t i o n , each f u n c t i o n a

, therefore

X,U

be a Gateaux-analytic map from w t o

f

P r o o f . a ) Let

, all

f o r a belonging t o

be given. a ) If the maps a + $2(x,a) are continuous a t some x e w n E IN , i f E i s a B a b e space then f i s continuous a t x , b) If

i s bounded.

-P

and f i n i t e on

n

p o f (x;a) w(x) ; s i n c e

i s c o n t i n u o u s on w ( x ) i s a Baire

E w(x) , a b a l a n c e d neighbourhood B of t h e o r i g i n i n E and a number M , such t h a t g 6 M on a + B , or p , f n ( x ; a ) \< M a t a. t B , n E /b/ ; w e s h a l l g e t t h e same i n e q u a l i t y space, t h e r e e x i s t

d aE

B

, and

a

t h e proof w i l l be over.

ANALYTIC MAPPINGS

1.

for p

=

1

. But,

by n-homogeneity, for

t a) 4 M i f a E B. o u b ) The property i s known when

]uI = 1 : p

0

n f (x;a+uao) =

fn(x;a

i s f i n i t e dimensional, then

E

ded 0

entails that

f

Gateaux-analytic. Then weak c o n t i n u i t y of R

19

f(K)

f

is

i s boun-

be given, s i n c e E i s m e t r i z a b l e i s l o c a l l y bounded and we can apply t h e proof o f p r o p o s i t i o n 3 . 2 . f ) .

f o r any compact s e t . Let T € N ( Z ) f

, a)

E

Remark.- Without assumption concerning

and b ) i n p r o p o s i t i o n 3 . 5

are false.

a)- We t a k e f o r vanishes f o r

n

E

t h e space

C

020

of sequence

l a r g e enough with t h e uniform norm and

f(x) =

(xn) 2

=

i n C which

. The

(n xn)n i s Gateaux-analytic on E and C n > l 1 1 f n ( o , x ) = (nxn)n Nevertheless f ( - R ) = 1 and ;Rn converges t o n n with (R,) as t h e canonical b a s i s of C function

.

b)- W e take logy, and

f

E

=

, that

2 ,

is

Z

0 $0

.

0

equipped w i t h t h e weak s t a r topo-

= identity.

~~~Ee_rrie_s_-~f_re_al_analY :ic_maE?eings E

has

as i t s f i e l d .

Given an open set w i n xE w

E

f ;w

and a map

-+

d:

such t h a t €or any

t h e r e e x i s t s a sequence of continuous, homogeneous polynomials

=

, and

C f n ( x , a ) i s uniformly n)O convergent t o f ( x t a ) , a l l a i n a s u i t a b l e convex, balanced, neighbourk hood U o f 0 E W e d e f i n e f n ( x , a , b ) = f"(x,a(n-k)times, b(k-times)) and 4p+l 1f"(a,b4P) - (4;t2) fn(a,b4P+2) + i (4ptl).fn(a,b n Zn(a+ib) = C a + f n ( x , a ) , with degree

.

n

the series

.

n i (4p+3) fn(a,b4P+3)]

, The previous map

polynomial on t h e complexified space

E

zn

i s uniformly convergent f o r a and b i n n i n e g a l i t y [32] : sup If (al,.. ,a ) [ , ai (5 U n

.

is a continuous, homogeneous C ?(x,a+ib) n>cO 1/2.e U because t h e elementary

of E and t h e series

6

:i

sup I f n ( a )

I ,

a

a LI .

C " f ( x , a t i b ) has a sum which belongs t o n>,O ) , and t h e following d e f i n i t i o n i s convenient.

Therefore t h e s e r i e s

O$c

Definition 3 . 2 . - A mapping f : w + 2 is analytic on the open s e t w , i f there is a neighbourhood of w in the compledfied space of E

ANALYTIC MAPPINGS

20

and f

EDE(8,Z) such t hat

to w

i s the r e s t r i c t i o n of

f

I

The presheaf of r eal anal yt i c mappings w i l l be denoted by A E ( Z 1 . Proposition 3.6.- The propositions 3.2.a-b-e are s a t i s f i e d by A E ( Z I instead of 0 , l Z )

.

and 3.3.b ( c , second p a r t )

This i s an obvious consequence of the d e f i n i t i o n . Remark.- It i s well known t h a t

AE(Z) does n o t s a t i s f y p r o p e r t y (B). Never-

t h e l e s s , some sub-presheaf c o u l d be s a t i s f y it ; f o r i n s t a n c e , t h e s p a c e o f

,

harmonic f u n c t i o n s s a t i s f i e d (B)

5

2.-

to

Z

=

).

Some remarks about a n a l y t i c e x t e n s i o n s .

Let u s assume

, let

set

x n and

(E =

AE(w,

dl

x F

Uw)

d e n o t e by

c . v. s . ; f o r any open

a real

F

with

t h e set of f u n c t i o n s

and are such, t h a t

)

wX , X C F

sections

= 6

E

z

.

-+

f(z,x)

f(z,x)

which b e l o n g

i s holomorphic on t h e

Theorem 3.7.- Let us assume F t o be i n f i n i t e dimensional. Given a compact s e t K i n E such t h a t Kx would be compact in wX , a l l then rlwl i s an extension of r ( w - K I ,

P r o o f . Let

f

e

r(w-K)

X E

F

,

be g i v e n .

il i s p i c k e d f o r t h e r e l a t i o n : R 2 c w and t h e r e e x i s t g2 Gr(i12) ,

1) F i r s t l y , a maximal open s e t

Ql

< il2

gl

E

whenever

r(il,)

with

g2 g E

Now, g i v e n

C

w-K

Ql

c

= f

gl

r(Q) , w i t h

0 d 6 and

e x i s t neighbourhoods A of

wx , a l l

in

such t h a t

i s contained i n

c Kx0

+A C

f

w-K

t 2A

Kx

V

---

0 €5F

of

for a l l

.

C wx , a l l x € v

g(T,X)

on

Ql

(zo,xo)G w

and

. Furthermore

0

=

g1

, there

wx

Kx

such t h a t

A

, with v(z,x)

t

. Then w e have

x E V

let

t 2A

Kx

c a n be cgosen

V

. --a;a '4 (TI . --- . d-r . d;

KF

:

Kx t 2 A ,

support i n

be d e f i n e d byo:

, all xE

T-Z

On t h e o t h e r hand, l e t

-n,

0

i n d e f i n i t e l y d i f f e r e n t i a b l e , is p i c k e d o u t v(z,x)

on

xo

Now, a f u n c t i o n Y ( z ) w i t h v a l u e 1 on

(1)

g2

i s a compact s e t i n

t L

K

=

and

0

x E xo t V

is contained

Kx

=

g

Kx

when t h i s set i s n o t empty. S i n c e

Kx

w-K

on

be t h e c a n o n i c a l p r o j e c t i o n o f

K

on

F

,

V

.

21

ANALYTIC MAPPINGS

Let

h(z,x)

( a x (V

-

(V - K ~ ,)

be d e f i n e d by

This

h

h(z,x)

n

KF))

(1

(cx

i s co n t i n u o u s i n

h = g

we have

ax

(z,x)e

all

by ( 2 ) ; s i n c e

in

V)

g(z,x)

nw

+

v(z,x)

h = g

and

KF

(zo+A)

h = v =

---

X

, the

V

-

(Kx t 2 A )

in

chosen maximal, t h e n

t h e proof i s complete a f t e r ch eck i n g t h e a n a l y t i c i t y o f In

.

ks compact and so h a s no i n t e r i o r ,

. S i n c e , R h a s been

V)n R

( (c x

-y(z))

h

at

.

(zo,xo)

f o l l o w i n g r e l a t i o n i s coming up :

+

(Kx

g(T,x)

A)

. -1 a aT

'p ('I). ---

, d r

.d?

2-T

.

0

0

Noting t h e a n a l y t i c i t y o f

(T,z,x)

-+

1 ---

g(.r,x)

in

2-'I

+

x (zo

(zo

+

A) x V

Comment.- If

A) x V

, then

i s also a n a l y t i c i n

v

. i s f i n i t e d i men s i o n al , t h e p r e v i o u s r e s u l t can b e f a l s e .

F

--I

For i n s t a n c e ,

for

Z'X

are necessary about

< 1 and

121

d i m e nsiona l case, ccf. 8 1 ,

in the finite

K

1x1 < 1 , Some more r e q u i r e m e n t s

The n e x t r e s u l t shows how a new f a c t a g a i n a r i s e s i n t h e i n f i n i t e d im e n si o n a l case, I n

E

, any

= (Rn

AE(C 1, b u t i t i s n o t t r u e i n fR mapping from

# into

IR"

.

lN

open s e t i s a maximal e x t e n s i o n f o r

. We

IN

Theorem 3 . 8 . - [ 4 5 , 7 5 1 , Given a domain w in (R = there e x i s t s an i n t e g e r p such t h a t each 3: ew w i t h (g A (IT(V.J,C) w i t h s a t i s f i e s f = g TRP p

Proof. F i r s t l y , assume and

z

E

, such

t, the

c

and f AE(u, has a neighbourhood IT on ! I

E

.

P

, V

a local s t a t e m e n t w i l l b e e s t a b l i s h e d ; t h e r e f o r e , w e c a n

b e l o ng i n g t o

f

i n some C

nn t h e c a n o n i c a l

s h a l l de note by

&c,,,( 6 )

, There e x i s t s a neighbourhood

f

function

z

+

ITP ( a )

bounded ; s o it i s c o n s t a n t . T h er ef o r e Now, d e n ot e by

statement is t r u e at

n(x)

. For

is bounded i n a-l(V)

that

+ z(a

-

rp(a)J

f o IT ( a )

P

t h e smallest i n t e g e r

x ; n(x)

a

E

V of 0 -1 II (V)

i s e n t i r e and

= f(a) p

any

.

f o r which the l o c a l

is l o c a l l y c o n s t a n t and t h e proof is

ANALYTIC MAPPINGS

22

complete.

Corollary 3.8.- Let I a s e t and f 6 6 ( a I ) be given. For a l l f i n i t e subset A o f I , a l l neighbourhood V of t h e o r i g i n i n & A such t h a t f i s bounded i n V x , we have f f x ) = f [ T r , ( X I ] , a l l X E d: I , T~ A is the canonical projection from a I onto d:

.

Proof. The f i r s t p a r t of t h e proof i n t h e previous theorem i s t r u e f o r a

. Therefore we

I

general s e t

The both hands belongs t o

@(

have I

.

=

f(x)

6

xE V x

f ~ I T ~ ( X f)o]r a l l

then t h e e q u a l i t y i s t r u e f o r a l l

)

x

.

Theorem 3.9.- [ 4 5 3 For any open s e t w and any a f f i n e , closed, i n f i n i t e , w is an extension o f w L f o r codimensional subspace L i n

-

m"

Proof. F i r s t l y , -

w can be assumed connected and w

l i n e [xl,

,,

Actually, given

not f i l l

x1

al,

'm

and

., an,

-

in w

x2

. After

s o t h e r e i s ' a; n

-L

a l s o connected.

can j o i n them by a polygonal

near

a

1

such t h a t

@ L j therefore

in w

- L , with

a' ntl

near

x

does

@ L

i s contained

Cxl,ai]

cxl,ai]

s t e p s , we have constructed a polygonal l i n e

..., an' + l]

[xl,ai,

, we

i n w , The a f f i n e space [x1,x2]

x2]

i n w and not contained i n [xl,al] L

L

does n o t c u t

:

2 '

- .

L p given by theorem 3 . 8 a p p l i e d f o r w Using a Zorn argument as i n theorem 3.7, given x o c w n L w e must cons-

Now, we t a k e t h e i n t e g e r

t r u c t a neighbowhood

-

V (1 (w

L)

; f

of

xo

and

being given i n

Avw

V

There e x i s t s an i n t e g e r

an with

rrrn(xl) = r n ( x 0 ) in

T,'(v~)

in

V

- L , 6) .

(w

(V')

, then

n (w -

and

f

,

IT?"

g E A~

with

and a neighbowhood neighbourhood

[ T T ~ ( V ' ) , C such ]

of

W

.

,a

g

V'

=

f

in

rn(x0) of

that ' f = g

O.T

P

,

Furthermore, P

n >/ p

gBARw(w,d))

V = a-'(W) contained i n w n By theorem 3.8, t h e r e e x i s t s m >/ n

in

IT

,

g

L) = V

= g

o a

P

-

W L

a P

,

near

can be chosen such t h a t belongs t o f

x1

= g

AQR,N

TT

P

(w)

i s contained i n

(V, @ 1 , Noting t h a t

-

-

o TT in V L since V L i s connected P which belongs t o V L , The proof i s complete.

-

Before extensively studying a n a l y t i c extension i n t h e next c h a p t e r ,

ANALYTIC M A P P I N R

23

b we p o i n t o u t t h e f o l l o w i n g f a c t . Given a s e t i n 0 E(X,Z) b r - e x t e n s i o n u : X + X f f o r 6 ,(Z) is a r-extension f o r

r

, each

OE(z) , but

t h e converse is g en er al l y false. Example.-

E

Z

i s t h e F r e c h e t s p a c e (chsequipped w i t h t h e p r o d u c t t o p o l o g y ,

i s t h e s p a c e o f sequences

x

=

(x,)

which converge t o z e r o , equipped

w i t h t h e uniform norm. Let

f

each i n t e g e r the unit b a l l llxll > 1 not for

be d e f i n e d i n

P p

, t h e sequence f B , b u t n o t bounded

. Setting r

6 ~ ( Z I.

= if)

6

b(E,

; B

)

by

f (x) P

=

C

(x:,x~)~ for

n>O

i s i n 6 ( E , Cm) , bounded i n P i n any neighbourhood of e a c h x which (f

+

E

i s a r-extension f o r

0 ,(Z) , b u t

24

CHAPTER IV : FRECHET SPACES OF COMPLEX ANALYTIC MAPPINGS i s a complex F r ech et c . v . s. and

Z

a complex c . v . s .

When

with

E

E

X

.

6,(x,z)

is f i n i t e d i men s i o n al ,

u a countable covering o f

i s a m a nifold s p r e a d o v e r

i s a class of t y p e

n a t u r a l F r e c h e t s p ace f o r t h e compact open topology. When d i m e n si o n a l , it i s a n o t h e r matter. I n a Banach spa c e sequence i n t h e a d j o i n t s p ace

E'

quence, t h e bounding sets f o r

O E ( ~ d: , )

hence,

E

is i n f i n i t e l y

such t h a t e v e r y

E

c o n t a i n s a p o i n t w i s e c onve rge nt subsea r e r e l a t i v e l y compact

11241

;

1 i s n ev er uniformly bounded and s o i s n e v e r a n a t u r a l

bE(E,

F r e c h e t sp a c e by theorem 2 . 1 .

6,(x,z)

l y thin i n

AU

X by r e l a t i v e l y compact sets and s o i s a

T h e r e f o r e , n a t u r a l Fre c he t s p a c e s a r e g e n e r a l -

; n e v e r t h e l e s s t h e i r p r o p e r t i e s a r e n i c e enough t o

b e d e s c r i b e d now.

I 1.- S t r o n g i n v a r i a n c e by d e r i v a t i o n , Definition 4.1.- A linear topological vector space r in 6xrx,z) w i l l be called strongly invariant by derivation (Abbrv. 8 . i . d.) i f : - T ( a ) e r , a l ~f

cr,

~

Q

E

- The mappings (a, f ) + ? ( a ) from E X r i n t o r are equicontinuous when n describes the integers. The next proposition i s established t o j u s t i & the above d e f i n i t i o n . Proposition 4 . 1 . - Let r be a natural Frechet space in invariant by derivation ; then

-

the mppings f n and a € E ;

-if

E

cmtinuoue f o r a l l

+

i s a Frechet c . v . n from

Proof .- By t h e p o l a r i z a t i o n

E

x

r

from

?(a)

r

8,

into

into

r

are COntinUOU8 for any

t h e mappings

r

o,(X,Z) which i s

(a,f)

+ ?((a)

are

,

i d e n t i t y between

.

f" (a )

and

fn(al,a2,.

..,an),

t h i s l a s t mapping b el o n g s t o r W e can a p p l y t h e c l o s e d gra ph theorem t o prove t h a t t h e ( n + l ) l i n e a r mapping ( a l , . . . , a ;f) + f n (al,a2,...,a is n n s e p a r a t e l y c o n t in u o u s from E o r r i n t o r

.

25

FRECHET SPACES OF COMPLEX ANALYTIC MAPPINGS

i s metrizable and

E

Further, since

c o n t i n u i t y implies c o n t i n u i t y , [77)

r

.

i s a Baire space, t h e s e p a r a t e and

Now, t o prove t h e closed graph p r o p e r t y , t a k i n g sequence

,

( f k ) which converge t o a i n E and f i n r l e t us suppose t h a t n f ( a ) converges near g and f n ( % ) near h i n k

.

r

Since t h e r-topology is s t r o n g e r than pointwise convergence and t h e mapping

a

* fn(xaa)

is continuous from

into

E

Z

, we f

We know, by p r o p o s i t i o n 3 . 3 . d , t h a t t h e mapping

have +

ger than compact topology by c o r o l l a r y 2 . 1 ; then we have E

, the

=

fn(a).

is

fn(a)

r-topology i s s t r o n -

continuous f o r compact convergence, f u r t h e r t h a t t h e Hence, f o r t h e usual space

h

=

g

fn(a)

.

previous d e f i n i t i o n i s conve-

n i e n t and introduces a s t r o n g e r property f o r n a t u r a l Prechet spaces. The following example shows t h a t t h e r e a r e n a t u r a l Frechet spaces which a r e i n v a r i a n t but not s t r o n g l y i n v a r i a n t by d e r i v a t i o n .

, and

Example : Given a domain w i n c

let

r

be t h e Frechet space o f holo-

morphic f u n c t i o n s i n w such t h a t a l l d e r i v a t i v e s of any

r

on w :

i s equipped with t h e semi-norms

=

Pn(f)

a r e bounded

f g

I(dn/dzn. f l l w

,

If t h e boundary of w has a s i n g u l a r p o i n t f o r simultaneous continua-

r , then

t i o n of

l' i s not s t r o n g l y i n v a r i a n t . Actually, suppose t h e conver-

se ; we should have : Given

for all

1.

E

, there

> 0

< rl

a

all

f

E

r

and 17 > 0

N

exist

with

c

such t h a t :

; ; 11 f(lw .k

k

1/2.V

1

is

; t h e n we have c o n s t r u c t e d a n a d m i s s i b l e c o v e r i n g o f describes

.

Corollary 4 . 4 . - I f Z i s Banach space and E i s metriaable, any natural Frechet space i n b x ( X , Z l can be continuously imbedded i n t o a natural 8 . i. d . Frechet spaceJ that i s a c l a s s o f type Aa with an admissible and countable covering of x

u

.

P r o o f . It

i s a n o b v i o u s consequence o f theorem 2 . 2 , p r o p o s i t i o n s 4.2 and 4.3.

Proposition 4 . 5 . - Given a bounded s e t B i n @./X,Z/ equipped d i t h compact topotogy, and assume 2 i s a Banach space and E i s metriaable ; then B i s contained i n t o a convenient class of type A with as an admissible and countable covering o f X

.

-

u

u

FRECHET SPACES OF COMPLEX ANALYTIC MAPPINGS

28

Proof. Since

X h a s c o u n t a b l e b a s i s o f neighbourhoods o f e a c h p o i n t , and

i s cbmplete,

C X (X,Z)

i s complete f o r t h e compact open t o p o l o g y , and

is a l s o complete by p r o p o s i t i o n 3 . 3 . d ) . T h e r e f o r e t h e s p a c e

bx(X,Z)

which i s spanned by t h e c l o s e d convex h u l l

B , and

f o r t h e Minkowski-norm a s s o c i a t e d w i t h

t h a n compact t o p o l o g y ; h e n c e , Now, s i n c e

Z

of

B

, is

E-B y

a Banach s p a c e

i t s t o p o l o g y is s t r o n g e r

is a p a r t i c u l a r n a t u r a l F r e c h e t s p a c e .

EB

i s a Banach s p a c e , we c a n a p p l y t h e p r e v i o u s c o r o l l a r y .

Z

for the

Remark. By t h e p r e v i o u s c o r o l l a r y , a bounded s e t i n O , ( X , Z ) compact open topology is l o c a l l y u n i f o r m l y bounded when Z b we can ask t h e same p r o p e r t y f o r a bounded s e t i n (X,Z)

6X

is normed. Then,

Z is not

when

normed. But it i s n o t t r u e as t h e f o l l o w i n g example w i l l show. An example :

Z

which converge to

,P

(x) =

C

n

>

~A 2 t p 2 p

.

-

For each

p

which b e l o n g s t o

d e f i n e s a mapping

there exists

and e v e r y

a € K ; d e n o t e by

describes

K

Each

Fk

Given

a

fk YP

by

6 ( ~ , d l. )

Fk

M

N

which b e l o n g s t o

P

of

K

E

P

k

and a l l

a 6 K ,

is l o c a l l y bounded.

= (a,) C E and

e x i s t s N such t h a t

[an\
N

with

1 1 ~ 1 1
N P 2P P n t h e upperbound o f I C p(a,) when a n< N

1 ( a ) / 6 Mp t - ~ j -f o r a l l 2 P

Then we have :

-

us define

i s bounded on e v e r y compact s u b s e t

Fk

.

.

is t h e s p a c e of sequences

'

+ fkYP

The sequence

; E

, let

(k,p)

(-k x )n

The sequence O(E,Z)

,&

w i t h t h e norm o f

0

F o r each p a i r of i n t e g e r s

fk

6"

i s t h e product space

N e v e r t h e l e s s , t h e sequence

Fk

i s n o t u n i f o r m l y bounded around

29

FRECHET SPACES OF COMPLEX ANALYTIC MAPPINGS

5

3 . - Analytic extensions.

r

Given a F r e c h e t space sion for

bE(Z) .

U ~ Oi ? ( f f )

=

Ti(fto

Any

71

. Thus,

, all f t cu"(r)

U)

, and

in DX(x,z)

u : X

u"(r)

and I?-exten-

X'

ur(r)

i s a Frechet space

u*(r) is n a t u r a l with

However, we do n o t know, f o r t h e g e n e r a l c a s e , i f

t o g e t h e r . N e v e r t h e l e s s , we have t h i s p r o p e r t y when

r

is s t r o n g l y i n -

v a r i a n t by d e r i v a t i o n . To b e g i n w i t h , we p o i n t o u t t h a t u*(r) whenever

i s a F r e c h e t space b , ( X , Z ) ,

6 x , ( ~ 1 , ~ ) c* o x ( X , Z ) c

; then

fienever

fr)

U*

.

r

j

o k

3 . 2 . d ) for a € E

-&2n i

u ( x ) t a eie7 ewnie d0

g o pitx,

[p

o

That i s : g n ( a ) o u i?

E

N(r)

1 a€ V a

= (g

[(g

G v e n t a i l s u*

0

V


/ b

inf

dX(x)

[

%*+-rV

V

x c T

is c o n t a i n e d i n

X

1

.

Definition 4 . 3 . - Let r be a s e t i n ~,(x,z); t h e s e t T f o r r i s defined by :

3 r i = { X E x [ [ [ f ( z ~ lll lf l ~ l

, aZZ

T-huZZ o f a bounding

f cr 1

.

Let 'l be a e. i. d, natura2 Frechet space i n 6xt~,~). Then T i n the I'-maximaZ extension ? f o r the extended space has the fozlowing properties :

Theorem 4.8.-

wry bounding s e t

F

a ) For eome neighbourhood

for

F , contained i n li. ,

V

of

0

E

,

T

+

V i s a bounding s e t

b ) For any balanced, open, neighbourhood V of O G E , we get : [i'fr)] 3 eup { r > 0 I T + r A a i s n bounding s e t contained i n ? all a E V 1 , Here, d i o t h e u n i t disk i n d: ,

V di

. .

,

FRECHET SPACES OF COMPLEX ANALYTIC MAPPINGS

32

a(?)

c) Given V o i t h property a ) , then + h.V i s a bounding s e t contained i n X , a l l < 1 Furthemore, Ff?) + V is contained i n A T t V ( ? ) , whenever r i s an algebra and 2 = d?

.

Proof. a ) The set 52

=

k" I

{f

a c t u a l l y , f2 i s a b a r r e l s i n c e

.

.\< 1) -llfi s11 normal,

r

d e r i v a t i o n , w e get some neighbourhood hood

?

in

such t h a t

Ilfn(a)

11 4

a c 2

.V

, all

Ilf

a11 11

.

x 6T

Since

*

of

V

O€

all

1

t h e T a y l o r expension of any f 6 -1 < 1) and o p, [p(x) t

i s a neighbourhood o f

0 E

f

;

Now, by s t r o n g i n v a r i a n c e by

, and

E

, all

52'

f

some neighboura

r

V , Thus,

i s u n i f o r m l y convergent i n X . V

- andXI-'X- i satlhle f

< (1 -

n'

E

r

spans

all

r-maximal e x t e n s i o n ,

w e have g o t t h e p r o p e r t y a ) .

b ) There i s n o t h i n g t o be proved, when t h e r i g h t hand o f ( b ) vanishes.

r > 0

Thus, w e can t a k e some contained i n

X

, all

T t r , A

such t h a t

a t: V , Given

,

a 6 r'.V

.a

i s a bounding set

a ) e n t a i l s t h e e x i s t e n c e of a b a l a n c e d neighbourhood of

T

+

r/r'.A.(a

is

t W)

, the

r' < r 0

-X

first part such t h a t

C E

a l s o a bounding s e t c o n t a i n e d i n

, By Cauchy

i n t e g r a l , ( p r o p o s i t i o n 3 . 2 , d ) , we g e t :

C

The s e r i e s

n >,o which b e l o n g s t o Q ,(W,Z)

a c

,

T^ , d e f i n e s

x E

; we d e n o t e by

a function

( f a l a t h e germ d e f i n e d by

fa(a) fa

at

w . a'& a t W

When fa : from

f n ( x , a t a)

(fb)a=a'-a . X , since

into

r!V

, the

c ) When

b

=

0

(fb=o)a,o n

recall t h e construction of

X

, the

f o l l o w i n g r e l a t i o n e x i s t s between

; h e n c e , t h e mapping

r

i s t h e germ o f

f

at

and

x , W e must

same computation g i v e s :

u EX.Y

,

[XI < 1

i s an a l g e b r a , t h e same i n e q u a l i t y f o r powers o f

If(x+a)[ 6 ( 1 - X I

fa'

i s continuous

+

i n c h a p t e r I , and ( b ) i s j u s t p r o v e d .

, all When

a

-1

i;

llfll

TtV

all

ci

E X.W

,

all

f

k ,

. gives :

FRECHET SPACES OF COMPLEX ANALYTIC MAPPINGS

Then, llf[l +tv

Corollary_ 4.8.and A

If

< [ [ f l l TtV , and X

$(?)tV

33

A -

is contained i n

i s the maximal extension of some s e t

TtV(r) A

n

i n b,CX,Z),

i s invariant by derivation, then t he following relai5on e x i s t s for

K i n X and any balanced, open, neighbourhood W of

any compact s e t OG E :

= d;(K)

dy L i l A l ] Remark : Here, then

d!(?(A))

Proof.- Since

,

can be chosen such t h a t

W

> 0

, for =

?(A)

such

t(i)

f c A , we

I- ; t h u s ,

r

f

belongs t o

f belongs

rq

A

which

t o g e t h e r , s i n c e t h e r-topology i s

compact convergence.

Denote a s X ; then

of

i s c l o s e d f o r t h e compact open

A

know by p r o p o s i t i o n 4 . 4 t h a t

is a l s o a s . i . d , Frechet space with

Xr

X

i s a l s o i n v a r i a n t by d e r i v a t i o n by

and

t o a n a t u r a l , s . i. d , F r e c h e t s p a c e stronger than

i s contained i n

W ,

p r o p o s i t i o n 3 . 3 . d , w e can assume t h a t t o p o l o g y . Now, g i v e n

K t W

ur

t h e morphism from X W ur [;(A)] t d" (K1.W

'r

4.8.b.

to the

r A A-maximal e x t e n s i o n

i s contained i n

-

Xr

by theorem

-

S i n c e X i s t h e i n t e r s e c t i o n of t h e m a n i f o l d s Xr by theorem 1 . 2 , W W ?(A) t dX(K).W i s c o n t a i n e d i n X ; t h e n t h e i n e q u a l i t y d: ;(A) 3 dX(K)

is proved, and t h e c o n v e r s e i s o b v i o u s .

Definition 4.4.- A sequence (xnl i n X say t o reach t he boundary whenever d W l x I converges near 0 f o r a l l balanced neighbourhoods X n 0 in E .

W

of

Theorem 4.9.- The following properties of t he manifold X are equivalent, related t o a natural, s . i . d, Frechet space r i n b X t X , Z ) which separates

X

.

.

(i) X i s t he maximal extension of r (ii) For any sequence ( xn) i n X which reaches the boundary, there e x i s t s f E r with sup l l f ( x n ) [ [= +m , ( i i i l I f E has a countable bas i s o f open s e t s , then X is t h e maximal extension of some f E r

.

Proof.-

(i)-(ii)

.

FRECHET SPACES OF COMPLEX ANALYTIC MAPPINGS

34

When

sup

(ii) Let




{O,l}

d

, and

E

c o u l d be

O(X,c)

as t h e f o l l o w i n g example w i l l

be showing. Let

X

t h e Banach subspace o f

C(T,&)

x , for p r o v i d e d by

c o n t i n u o u s f u n c t i o n s whose s u p p o r t s a r e c o u n t a b l e w i t h uniform norm. The unit ball in

E

i s denoted by w .

Proposition 4.12.a ) There e x i s t s a proper, d i r e c t subspace

f= f

0

p

, a l l f g fi

bl Let

g

(w,

C) , for a

H

of

E

be a holomorphic function i n the u n i t disk

proper continuation. Then the mapping

such t h a t

convenient projection :x

+

g o x

p

onto

H ,

which has no

belongs t o

6 (w,E)

FRECHET SPACES OF COMPLEX ANALYTIC MAPPI-NGS

36

and w i s 'Q -maximal. Proof.

-

0 (u,c ) ; s i n c e is n - l i n e a r fn(O,xl, ...,x s u c h t h a t f n ( O , x l , ...,x ) = n f E

a ) Let u s g i v e n

(xl,...,x

)

+

uric[@: E I I '

t h e mapping and continuous, t h e r e exists

.,. 8

p (x, 8

xn) ; here

@TI

is the t e n s o r product equipped with t h e p r o j e c t i v e topology. L e t E,Le

t h e Banach s u b s p a c e o f C(Tn,

c ) provided

by c o n t i n u o u s f u n c -

t i o n s whose s u p p o r t s a r e c o u n t a b l e and fi t h e n - l i n e a r c o n t i n u o u s mapping : (x,

,... , x

*

)

... x

x1

from

En

into

. The

En

universal property of t h e

t e n s o r product provides a

t o p o l o g i c a l is o mo r p h i sm between t h e c o m p l e t i o n

gnIT

p r o v i d e d by f u n c t i o n s w i t h c o u n t a b l e su p -

an d t h e s u b s p a c e E

E

ports , therefore

pn

Let u s d e n o t e by

In = { t

and f o r an y of

x

x g E

8: E

is the restriction t o Tn ;

#

[pnl ( { t } )

with support o u t s i d e

of a m easu r e- o n 0)

, pn(x)

In

,

In

Tn

.

is countable

since t h e support

0

is countable.

Let

I

be t h e u n i o n o f

In

,

f u n c t i o n s whose s u p p o r t d o e s n o t c u t

K t h e subspace of I

,

H

E = H @ K

and

fn(O,x)

i s t h e p r o j e c t i o n of

x

...

x(tl) onto

H

p r o v i d e d by

t h e subspace o f I

by f u n c t i o n s whose s u p p o r t i s c o n t a i n e d i n

E

, we

E

have t o p o l o g i c a l y

x ( t n ) dp ( t ) = f n ( O , x H )

. Therefore

n

f

provided

, here

has a continuation i n

x

H

w + K .

b ) The r n a p y i s c o n t i n u o u s a n d G a t e a u x - a n a l y t i c ,

6 (w,E) . with

L a s t l y i f o were n o t 'Q -maximal,

I I X ~ / / ~ =1

ne i g h b o u r h o o d Ixo(t)l

1

U

, we

Corol2ary 4.12.-

such t h a t 0 : A

+

g [A

t h e r e might e x i s t xo(t)]

. When

we ch o o se

t

f i n d a c o n t r a d i c t i o n with t h e choice of

g

of

1 for a l l

t C T

belongs t o

xo 6 E

i s holomorphic i n a such t h a t

.

With the above notation, w is maxima2 f o r a s . i . d . , 6(w,61, but is never f-maximal f o r a22

natura2 Frechet space i n

feO(w,&!. P r o o f . - Clear by p r o p o s i t i o n 4 . 1 0 a n d p r o p o s i t i o n 4 . 5 . Comment.-

t h u s it

An o t h e r example c a n b e found i n [47]

I

37

CHAPTER V : HOLOMORPHIC CONVEXITY $ 1.- Some B i s c h o p ' s Lemmas.

Let

z = (zi)

U

be t h e u n i t b a l l o f

(r

. The d i s t i n g u i s h e d boundary

Definition 5.1.-

A polynomial

value of i t s c o e f f i c i e n t s i s of degree a t most

llzll = s u p lzil

normed by

is

U

of

U*

with

,

i s normalized if t h e maximum (d, dn) if it i s

P in C

,...,

1. I t is of degree

i n the j t h variable.

d.

3

Lemma 5.1.- Given r , 0 r < 1 , there e x i s t s a constant Y such t hat for a l l t , 0 t < 1 , the Lebesgue measure of Ad defined as

1

polynomials

IP(z) I 6 td 1 is l e s s than -M/Logt P in of degree Id, d d)

Proof.- L e t

zq

{z g r.U

be t h e monomial o f

hence t h e r e e x i s t s

(Elnad4

P

11

U*

with

z o e r / 2 U'*

whose c o e f f i c i e n t i s 1.

rU*

at

zo

IP(zo)I >,

. Since

(Ad)

).Iz

0

. [d.Log.

When

zo

- Log 2 ) t

-

-

I

r/2.U*

Then, t h e r e e x i s t s a c o n s t a n t

MI

r.U"

,

such t h a t

(w)

dp,

Log.2)

t h e Poisson measure

e q u i v a l e n t w i t h t h e Lebesgue measure o f

, the

is negative i n

(dtl)n

-

and

0

lECW2l

Log

r/2.U4

u,

and

(dtl)"

n Log(dt1) 6

.

r nd (5)

-l-pl--

n.Log ( d t l ) ] 5 n . d . ( L 0 g . r

describes

,< llpll

r.U*

Log

we obtain

nd(Log r

$1

P(;.w)w-'

Now, w e use t h e mean v a l u e i n t e g r a l i n P o i s s o n k e r n e l of

a l l normalized

.

,.. .,

an

T h e r e f o r e , we g e t

, for

, and

0

-

r.U,

n.Log(dt1)

.

is u n i f o r m l y

pz 0

HOLOMORPHIC CONVEXITY

38

Finally, for a suitable M

Ad C

mes.

Log t

Lema 5.2.-

an

d

integers

and

e

o f degree

P

Proof.- Cover

and

Q be a compact subset o f a manifold

Let

and l e t f E bx(X,O;I. There i s an P < 1 and an i n t e g e r

mial

n

with (d,

d

..., d,

0

, there

e b eo el

e > 0

in

we have

spread over

fX,p)

, such

t h a t f o r a21

e x i s t s a normaZized polynovariables such t h a t

nfl

by finitely many polydiscs

Q

r

.

M -----

-

which only depends on

1 xits EU

, where

E

is-chosen

such that Q t EU i s contained in X xi ( 1 6 i d N) are in * c be given such that C > SUP. [IY IIPill QtEU Y q+EU]

Q , Let

llfll

Let of degree

k = (kl

L be the vector space of all polynomials in ,n+l variables (d, ..., d, e) . For each i 1 4 i \< N , and each

,...,

kn) let Wik be the linear functional on k Now the dimension of L uik(P) = D P(pl ,...) pn, f)(xi)

L

.

and there are fewer than Ntn

functionals w

with

is

defined by (dtl)"(e+lIy

Ikl < t ; so if t

ik is chosen to be the greatest integer less than N-l(dtl)(e+l)l'n

is a nonzero We may take Let

P

P

P

P(pl

G L such that uik(P) =

jl

aP1

P '

-

*

,. . . , pn, f) . Then by

is a sum of at most jn fjntl

Pn

o u r choice of

Q+E.U \< (d+l)"(etl)cndte

,

, there 1 $ i \< N

(d+lIn(e+l)

,

terms of the form

.

Since ji \< d , 1 6 i 6 n,jn+l C e at each of the points x. , we have, by Schwarz'

x.t~.U

Since Q

.

c we obtain

with l a ] ,< 1

has total order t lemma in

Ikl < t

to be normalized.

'I' since

0 for

is contained in

{xi

+

1/2 EU)

we obtain :

HOLOMORPHIC CONVEXITY

Now

t + l 3 N - l ( d+ l ) ( e+ l ) l / n

of (1) i s dominated by

There e x i s t s

>/ N-’

delln

, with

( r ( e ) ) de

t h e te rm on t h e r i g h t hand +-1 l/n : r ( e ) = ( 2 ~ ) ~ ’ ~ ’, ~2

r ( e ) < 1 for

such t h a t

e

, so

39

d > e > eo

.

. We choose

r = r ( e 1 and we o b t a i n t h e i n e q u a l i t y of t h e lemma. 0

Lemma 5 . 3 . - Given a part T of 8” , an i n t e g e r e such t h a t f o r a l l 0 integers d and e w i t h d >, e >, eo , there e x i s t s a normalized polynon i l such t h a t : miaZ P o f degree (d , d, e ) i n d, e

,. .,

Then f o r a21 E > 0 , the s ection Tz of T i s f i n i t e when a tetongs t o a subset of E.U w ith nonzero measure.

c

(z,w) = a (z)wP , some c o e f f i c i e n t d,e P4.e a normalized polynomial o f d eg r ee ( d , d ) which we de note

--I’roof.-

Writing

P

...,

b e t h e set { z G EU*I la Let A (z)l d r 1/2d.e1/” 1 . d,e d Ye By lemma 5.1, f o r a f i x e d e , w e have f o r a s u i t a b l e M :

[g.

mes

Then, for e

hdre] 6

[ i mes. ~ A ~ , 8~ ]---------M d w

Lim

l a r g e enough,

. Adye

zo E IEU*

f o r any

l i z e d polynomial of d eg r ee \< e

Since, a sequence

, and

g, 0

we have

gzo s

%

,(w)

gz (w) = 0

(zo,w)

. That

.

.

e

‘d

e(zo9w)

is a normasuplap(zo) I Whenever ( z o r w ) ( T , (w) gz --J--------

0 sd

i s n o r mal i zed wi t h

such t h a t

whenever

0

,

I ,

d,e d en o t ed by

rU 2 . d . e l / n

i s less t h a n

IgzoYd(w)I

-lim A

a

.

bounded d e g r e e , t h e r e e x i s t s

converges t o a nonzero polynom ia l gzO

9%

belongs t o

is,

TZ 0

T

.

h a s a complement w i t h a

nonzero measure. I n t h e f o l l o w i n g we f i x such a n NOW,

r

e%og

a ( z ) is P a (z) d re

and

is finite.

zo

I

E.U*-

1im.h } d te

HOLOMORPHIC CONVEXITY

40

S 2.- Holomorphic c o n v e x i t y . i s a manifold s p r e a d o v e r t h e Banach space

(X,p)

Definition 5.2.- Given a subset A i n i f i ( A ) i s a compact s e t whenever K

= 6,(X,@

A

If

r e s u l t of Gruman, C.O.

i s called A-convex i s compact s e t i n X

.

Kiselman, Y . H e r v i e r , s a y s t h a t

Here, we develop t h e c o n v e x i t y problem when

since

X

i s pseudoconvex, t h e aforementionned

X

Frechet al gebra i n b X ( X ,G )

bE(

.

X

A-convex

is

has a basis.

E

whenever

and

)

DXtx, 6 ) ;

E

and

X

is a s . i. d , natural

A

i s t h e maximal e x t e n s i o n of

for

A

) , The p r o o f does n o t r e q u i r e t h e f i n i t e d i m e n s i o n a l c a s e . O f c o u r s e ,

X

i s A-maximal,

X

i s pseudoconvex by p r o p o s i t i o n 4.10 and t h e n

X

i s 6 x ( ~ , )-convex c by Gruman's r e s u l t , b u t A-convexity i s a s t r o n g e r property. Let

L(E)

be t h e v e c t o r s p a c e o f c o n t i n u o u s endomorphisms o f

E

.

W e need t h e f o l l o w i n g assumptions :

( H I ) For B , there e x i s t s a sequence n k E L ( E ) , wi t h f i n i t e dimensional range, which i s pointwise convergent t o t he i d e n t i t y mapping.

( H 2 1 For A : ( i )- the mappings €J o uop belong t o A f o r a l l 5 E E' , all u E LIE) ( i i )- the mappings 3: * F ( r , u o p ( d ) belong t o A all f c A , all u L(E) ,

for

a

f i l l t h e s e r e q u i r e m e n t s whenever is Au a c o u n t a b l e , a d m i s s i b l e , c o v e r i n g o f X such t h a t p(w) i s bounded, a l l

Comment.- The s p a c e s o f t y p e

w

E

u

; also

A

u

'

i s c l e a r l y an a l g e b r a .

i s a compact s e t i n X ; U i s t h e u n i t b a l l -1 i s t h e manifold p I I ~ T ~ ( E )which ] i s spread over t h e f i n i t e

In t h e f o l l o w i n g , in

E ; Xk

K

d i m e n s i o n a l v e c t o r space that

K+E.U

-rk(E)

is contained i n

. By

and i s a bounding set f o r

X

more h

K(A) t a.U

theorem 4 . 8 , t h e r e e x i s t s

is c o n t a i n e d i n

/\

Km.U(A)

, all

a,
0

such

. Further-

.

The main Lemma Given a sequence

x

h

i n K ( A ) such t h a t p ( x n ) n a € E : For n l a r g e enough, n b N , I1p(xn) p(a)ll -1 t h e n t h e r e i s one p o i n t 5 , in p ( a ) n [ x n + E.U]

-

.

is c o n v e r g e n t t o i s less than

E

and

HOLOMORPHIC CONVEXITY

Lema 5 . 4 . - There e x i s t

k > Nl

integers

there e x i s t s a compact s e t

f o r a l l poZynorniaZs P all

on

.

n E N2

and

Nl

P r o o f . - By t h e Banach S t e i n h a u s theorem, out

E'

and

h

El'

with

0

K t C1.U

is contained i n

set for

A

.

On t h e compact s e t t i t y , we choose

N1

We f i x such a

For

n


1

, all

a Q E,

the following properties ;

- The series

C h:(f) is absolutely convergent for all a n >,O in some open, balanced neighbourhood Vh of O € E , all f E A Then, (sl)

the sum of this series.

we denote by h ( f ) a (s2)

of 0 6 E

.

- For all a Q , such that : Sup.

(s;)

and

l\(f)


true

; any homomorphism from A

which is weakly continuous in (El

to o

with

p)Q A

(sl)

(El

o

p)A A

, The converse is also

-

(a2)

properties,

has a continuation in S ( A )

by the Hahn-Banach theorem.

Pmpoeition 6.1.- Given h € S ( A ) a Vh , and we have (1)

IhJb

=

ha+b

*

U

+ U)AVh ,

- There exists a(h)E

theorem) such that h(E (s,+)

Vh

at1 a

, then

ha

belong8 t o S ( A )

for a l l

and b such that xa + s ' b ' E Vh

,f o r

SPECTRUM AND MAXIMAL EXTENSIONS

46

Proof.- a) The series a

Vh

ha(f)

,

C hn(f) h:(g) is summable, for all n>,O,m>,~ a all f and g in A , Thus, we have :

. ha(g)

=

C

C

n >,O

ptq=n

. h:(g)

h:(f)

=

C

n>,O

h(

C

p+q=n

fp(a) gq(a)) =

= ha(f .g) So,

ha

t

zb'e Vh

.

in E with the requirements of the proposi-

b) Given a and b tion then za

Vh

is a homomorphism, all (a

, all

[zI < r

,

IzlI
l .

C h:a+zlb(f) is the expansion by a series of homogeneous n2O polynomials for the holomorphic function (z,z') + hz,atzI.b (f) in the bi-disc 1. < r , Iz'I < r Therefore

.

The following, relation ( 2 ) , will be proved below ; (2)

fn(a+b)

=

C

p+q=n

cfp(a)]

Using this, we get of hza+zlb(b)

C

P8 9

9

(b)

. 9 (z'b)]

h [[fp(za)]

as the Taylor expansion

on the aforementioned bi-disc. Hence we get :

Now, we prove ( 2 ) by the following computation ; f"(atz'b)

= =

C

Z"

94 n

.

fn(a+r.z'.b) Itl=l

C

f(r.a+T,t.z'.b)

96 n

With the new variable t' =

=

C

ptq=n

---dt $+I

ztq

Tt

, we

[fp(a)f(bl

The proof of (1) is complete.

get ;

.

dt . dr --------qtl ntl

t

.T

'

SPECTRUM AND MXIMAL EXTENSIONS

a E V

c ) Given 0

E

such t h a t

is a suitable

h

z.a

V

+

, there

e x i s t s a b a l a n c e d neighbourhood

is contained i n

U'

47

, a l l 1.

Vh

U of

(s2) associated t o

h

, then

of

. Then

< 1

w i t h (sl) p r o p e r t y by (1). F u r t h e r , i f

ha i n t h e aforementioned

U'

U'

i s chosen

U'

has itself

ha

t h e p r o p e r t y ( s 2 ) by (1). I n fact

= sup

sup. I(h,)b(f)l Finally,

ha

Ihatb(f)l


,O

and this series is absolutely

.

convergent for It1 < r Thus, we have proved the Gateaux-analyticity of the said mapping in is locally bounded and so belongs to V , Further by ( s 2 ) , a + ha(f)

0 E(V, dl

belongs to

by proposition 3.4. We have just proved that

)

.

6 (S(A), d:

) ; indeed extends f Now, we take an A-extension v : X

as v* the extension mapping from A arguments as for 9 for any YE: Y

, we

can easily see that y

to S(A) ; hence the range of w nected and contains u(X)

b,( 63

Y for

and we denote

)

onto bY(y, 61) , With the same

and the mapping w : y

and then w o v

+

+

o

v.* belongs to

S(A)

v* is a morphism from Y

is contained in ?(A)

. Actually : (x o v(x))(f)

since it is con-

= (v*f)(v(x))

u and moreover the maximality of ;(A)

= f(x),

is proved.

c) is obvious. §

2.- Topological spectrum and maximal extensions. Here, A

is a unitary and topological subalgebra of

0 x(X, c

which

is invariant by derivation. We obtain some information about the relationship between the topological spectrum sp. A

of A

and g(A)

, the

last

one being defined above without a topology on A , We denote as sp?A the set of h C sp. A which are weakly continuous on

(E'

is denoted as

sp?*A

Theroem 6.4.-

p)A A

,

,a

. The set of their weak continuation to A+E'op

i. d, Frechet algebra in @ , ( X , C ) ; i s contained i n S(Al and ? ( A ) is a connected component Given A

natural,

8.

then sp?*A of q ? * A f o r the topology induced by Proof.-

First, we prove that the

is contained in S(A)

q E N(A)

S.

S(A)

on sp?*A

,

i. d. assumption entails that sp?*A h € sp?*A , there exists

. Actually, given

a balanced neighbourhood V

of 0 E E

,E

> 0

such that :

SPECTRUM AND MAXIMIL EXTENSIONS

lh[fn(a)]l

< 1

, all

Then, s i n c e

f

{f

E A

Iq(f)

-?

and ( s ) a r e s a t i s f i e d w i t h

(sl)

q(f)
0 ) ; actually

(A

W a neighbourhood of

with

7;

x CM n T / X J A f o r 6 ( X ) s , since

0

and

K(X)

6(xlS

(T-a)

T

at

OCXL

that

TE

, that

K(X)

, ME '%,

n T/X J =

[W

T/X 6

X . W ~T ,

. The re fore

K(X)

and so is Hausdorff.

, then T -

a E T

nM

i s a neighbourhood of

is,

(a t

M)n T

belongs t o

M

f o r a l o c a l l y convex topology

which is o b v i o u s l y f i n e r than b ( X I s

Now, g i v e n to

0

.

7,

nT =

i s a b a s i s of neighbourhoods o f

~(x)E

M B

in

f o r t h e r e s t r i c t i o n of

0

.

, a l l Te K(X) a b a s i s of a f i l t e r and f o r any

M

sets

a

a l s o be longs

i n t h e r e s t r i c t i o n of

0

i s a neighbourhood o f

a

. To prove

i s t h e f i n e s t o n e, w e have only b ) t o c he c k, t h u s g i v e n a

b a l a n c e d convex open s e t

V

in

G

, then

@-'(V)

be longs t o

7 and

the

p r o o f is complete. The t o p o l o g y by

. Let

T€

be g i ve n and

K(X)

AT

b e t h e spa c e spanned

and normed by t h e Minkowski-norm a s s o c i a t e d w i t h

T

mapping

AT

-).

AT,

i s co n t i n u o u s whenever

T

T

, the

6 ( ~ which )

i n d u c t i v e l i m i t of is w r i t t e n 6 ( X) ,

Proposition 6.8.-

We have

.

AT

h

T

.

The c a n o n i c a l

i s contained i n

is d i r e c t e d by set i n c l u s i o n . S i n c e e v e r y

K(X)

some

0,

f 6

T'

O(X,C)

, and be longs t o

p r o v i d e s a b o r n o l o g i c a l topology on

@(X), < @(XlE < 6 (X)p,c

, Here,

means

" f iner than I t , Proof.-

€ o r t h e r i g h t hand w e u s e p r o p o s i t i o n 6.6.b) and the e q u a l i t y of

p o i n t w i s e convergence and precompact convergence on any e q u i c o n t i n u o u s s e t . € o r t h e l e f t hand, w e have only t o check t h e c o n t i n u i t y of e a c h mapping AT + 5 ( X g

. Thus

a convergent sequence i n

AT

is ultimately i n a fixed

SPECTRUM AND MAXIMAL EXTENSIONS

51

6 (Xg

TI G K(X) and i s p o i n t w i s e c o n v e r g e n t , t h e n it i s convergent i n

Let

Proposition 6.9.-

a Frechet c. v .

f " : 6 * 50 f

a ) The mp b) If

F

be given, and J@G Ox(X,F)

6.

.

O I ' X ) ~ is continuous.

from F i i n t o

is topological on i t s

is locally topological, then f"

f

.

range. Proof.- a ) By t h e Banach-Dieudonnk theorem, we have o n l y t o check t h e cont i n u i t y of

i n any convex b a l a n c e d e q u i c o n t i n u o u s s e t

f*

t h e p o i n t w i s e t o p o l o g y . A such belongs t o

K(X)

topology on

is

T

mapped by

f* ( T )

b ) Since

, therefore i s open,

f

F'

in

for

i n t o a set which

induced t h e p o i n t w i s e

is continuous.

f*

f * i s c l e a r l y one t o o n e , L e t

5,

a

f

o

&-topology b e g i v e n , t h i s n e t i s u n i f o r m l y conver-

convergent n e t for t h e

g e n t on any precompact set o f

X

by p r o p o s i t i o n 6 . 8 . Any compact set i n

i s homeomorphic w i t h some compact s e t i n c a l , t h e n 5,

f*

by A s c o l i ' s theorem and S ( X &

T

i s convergent i n

FA

X

since

f

F

i s l o c a l l y topologi-

.

Proposition 6.10.- I f E i s metrizable, @(Xl, is a semi-Monte1 space, 6 ( x ) , is barrelled and is the bornological space associated with f i ( x l C . Proof.-

Given a bounded set

by p r o p o s i t i o n 6.8, t h u s

B

B

,

(5 (XL

in

B

i s a l s o bounded i n

0 (XIc

i s l o c a l l y u n i f o r m l y bounded and e q u i c o n t i -

nuous by p r o p o s i t i o n 4 . 5 and Cauchy i n e q u a l i t i e s . The c l o s e d convex b a l a n ced h u l l B

B

of

in 0(Xlc

belongs t o

, and

i s r e l a t i v e l y compact i n O ( X L

K(X)

so

by A s c o l i ' s theorem, t h e n

0 (XL

W e have j u s t proved t h a t any bounded set i n 0 ( X I C

space

AT

,

T

6

K(X)

, used

t o d e f i n e t h e topology

i s a semi-Monte1 s p a c e . i s bounded i n some

0 (XIb , t h e n

topology is t h e b o r n o l o g i cal topology a s s o c i a t e d w ith b ( X ) each

T 6 K(X)

the s p a c e s

AT

i s compact i n

6 (XIc

are Banach s p a c e s

j

since

0 (XIc

C

this

, Finally,

i s complete and so

t h e i r inductive l i m i t is b a r r e l l e d .

Proposition 6.11.- Suppose E normed ; then the following statements are equivalent : (il 6 (Xi, is bornoZogica1 (iil 6 (X), is barrelled (iii) E is f i n i t e l y dimensional.

SPECTRUM AND WXIMAL EXTENSIONS

52

Proof.-

(i) implies (ii) by proposition 6.8. (ii) implies (iii) : Given xo g X , The mapping f + f'(xo,a)

from

continuous by Cauchy Integral (proposition 3.2.d). from Now, consider the mapping Q : f + f'(xo)

6(XIc

0 (X)

E

1 [[f'(xo)[[

into Q: is

into the

6 1) is a barrel is barrelled, 0 is conti-

adjoint normed space E' ; the set (f E @(XI in b(XL by the first remark. Since O(XL nuous. Now consider the mapping JI : 5 + 5 range of a convergent sequence

(5,)

o

p from E'

of E'

into &(XI K(X)

belongs to

E

; the

and

5, o p is convergent in Q (XIc ; then @ is continuous, in the identity ; therefore Q Finally, it is clear that Q o JI onto El By proinduces a topological isomorphism from 0 (X) /@-'(O) position 6.8 the quotient space E'

.

E

is a semi-Monte1 space and is finitely

dimensional by Riesz's theorem, (iii) implies (i), since is finitely dimensional.

6 (XIc

is a Frechet space whenever E

Theorem 6.12.- Let u : X + Y be a 6 (X)-extension for 6E(6).Then the extension mrp u* from Q ~ x ) , (resp. 6 1 onto b t ~ ) ,(resp. O ( Y L I is topological whenever E is metriaable.

(xi

Proof.-

a) b ~ t _ ~ ~ [o17,~48~ ] g y, By theorem 6.10 the b-topologies are bornological and we have to

verify that the range of any bounded set by

u* and

( ~ ~ 1 -is l bounded

for compact open topology. This property is obvious for (u*)-' , Let T be a bounded set in b(X)(= , so T is contained in a suitable s . i. d, natural Frechet space by proposition 4.5 ; theref.ore u*(T) is locally uniformly bounded by corollary 4.6 and so is equicontinuous by Cauchy inequalities. Fipally u*(T) is closed in 6 ( y I S since T is closed in o(X)s O(YIc

.

, and by

*

Ascoli's theorem u (T) is a compact set in

.

b) g:t_gpo&o_a ' [ 8 0 ] The continuity of from b(Y), onto &(XIs implies that any T e K(Y) is mapped in a set of K(X) ; using theorem 6.6.b), we

.

get the continuity of (u*)-' from O < Y & to O(X& Now, let T 6 K(X) be given, the first part a) has shown that $(TI

-

53

SPECTRUM AND MAXIMAL EXTENSIONS

belongs t o b(Y)

. Since

K(Y)

K(Y)

a r e compact i n @ ( X I c

and

i s continuous s i n c e

K(X)

u* r e s t r i c t e d t o

by theorem 6.10,

E

and

K(X)

i s continuous. To complete t h e proof we apply theorem 6.6.b).

' ) u (

Theorem 6 . 1 3 . - The mcCimzl extension [Q(X)] sp" [@ (X)€ ] , whenever E is metrizabZe. Proof.- Given fore

h

2 [@(x)] , then

h 4

belongs t o

sp*

h

belongs t o

[ o ( X L 1 = spy [@

(iL]

CoroZtaw 6.13.- Suppose E a Frechet c. v . ned i n sp [&(XI,]

.

Proof.-

By p r o p o s i t i o n 6.9,

8 (X)€

is contained in

8.

induces on

topology which i s t h e compact topology s i n c e

sp*

[o(%),]

; there-

by t h e previous theorem.

Then

x [o ( X ) ]

E' o p

is contai-

t h e precompact

i s complete. Then w e have

E

sp

6 (XI,

.

j u s t t o apply t h e Arena-Mackey theorem t o g e t

sp*b ( X I E

-.-

which a r e u s e f u l . The main

Comment

There a r e o t h e r t o p o l o g i e s on

one o f them is t h e Nachbin topology [62] c a r r i e r s of f u n c t i o n a l s on @ ( X I [18,29]

6' ( X )

which i s used t o study t h e

.

Josephson has constructed (not y e t published) an open s e t w i n t h e

= E, I non countable,

Banach space &-(I) for

P

6 E(c ) , and

he has found a p o i n t

is not continuous f o r 6((w),

. Then

which has a proper extension x6

~ ( W L

-w

such t h a t t h e e v a l u a t i o n

can be s t r i c l y f i n e r than

0 (w), by t h e previous theorem 6.12. Moreover t h e theorem 6.12 i s false f o r compact topology and such

.

8 3.- A particular

case,

E

=

CC' .

Nothing i s known about

sp*

[ (3 ( x ) ] -

[6

(XI]

f o r a general

c. v. s. Nevertheless it is p o s s i b l e t o complete theorem 6.13 when E t h e product space

, with

set o f f i n i t e subsets o f map from @ I o n t o by

(zi)

with

zi

I

is

as a g e n e r a l s e t . We denote b y % (I) t h e

I ; f o r any A of I ;( I )

, mA

is the projection

4:A ; we i d e n t i f y g A with t h e subspace o f (rl defined = 0, i g I-A The u n i t d i s k o f d: i s w r i t t e n A

.

.

Theorem 6 . 1 4 . - Let f X , p ) be a m n i f o l d spread over d; I and suppose the mdmt extension of bfX1 i s X Then there e d 8 t s A o 6 T f I ) and a Stein mznifold i 8pread over Ao such that X is isomorphic w i t h

i

x &-A0

.

c

.

SPECTRUM AND MAXIMAL EXTENSIONS

54

Proof. We claim the existence of A e ' F i I ) such that for all x EX

there

of x and V of ITA o p(x) in a and W is I-A. 0 homeomorphic by p with V + 6: Clearly this property is satisfied exist neighbourhoods W

.

at some given xo€ X table at any x g X

and now we verify that

.

A.

xo

obtained at

is sui-

Let H be a finite dimensional subspace of C1 which contains some I-A. given a Q and such that x and xo belong to a connected spread over H [541[

, with

. By

of p-'(H)

component X'

, hence -

da(x')

proposition 4.10, X'

Log da

= sup

is a Stein manifold

is a plurisubharmonic function on X'

{r 2 0

I

+

p(x')

.

r.A

is homeomorphic by p

with a subset of X' which contains X I ) This function vanishes in Won X' , then da vanishes everywhere on X' c 5 4 ] Thus the connected component of x is homeomorphic by

p with p(x) t I.-A

.

p-l [p( x) t

C

I-A

2-A0] which contains

by proposition 1.1. This

. Let V be a connected neighbourhood of x which is homeomorphic by p with p(V) , We claim that p is one-toI-A. one on w = LJ (y + CT Actually, suppose : p(y') = p(y*') , Y E V I-A. I-A. d= , then Y,,€Y1 t 63 , Y' component is written x

+ Q

0

.

c

belongs to the connected component of p -1.[p(y,) + I-"] contains y2 and y' = y" by proposition 1.1. Since p I-A

W is homeomorphic with

TA

0

p(v) t

which is an open map,

6:

0

Let

(91 I-A.

be the equivalence relation defined on X

by x: xl(Q ) iff

.

x'E x t 63 Each coset is closed then the quotient space i is separated. The projection map defined on 2 by IT(;) = TIA p(x) is 0

continuous and, with the above notations, TI restricted to W is an homeoo p(V) Then (;,IT) is a manifold spread over 43 , morphism onto

.

I-AOTAo

c1 . Let

and x 43 is clearly a manifold spread over canonical imbedding map from X into ;, the map u u(x) =

[ IQ(x),

Clearly

t

TI^-^^

o

p(x)-l

-

be the defined on X by

is an isomorphism from X

is a Stein manifold spread over

6hO

.

'4

onto

i

I-A.

X

6

.

SPECTRUM AND MAXIMAL EXTENSIONS

55

Proposition 6.15.- Let X be a manifold spread over a f i n i t e l y dimensional space. For each f 6 6 (X x C I I with I an:arbitrary s e t :

a ) There e x i s t s a countable subset N of I = f ( x J 71N f y ) ) a l l (x,y) 6 X X b :

such that

I

f(x,y) =

b ) T h i s s e t Idis written uIn b i t h In an increasing sequence of f i n i t e s e t s . Then t h e sequence gn(x,y) = f ( x = rrrl( y ) ) is convergent n t o f in 6 (x x CrjE

.

Proof.-

of

= f(x,

f(x,y)

IY

be given, By c o r o l l a r y 3.8 t h e r e e x i s t s a f i n i t e xo & X I and a s u i t a b l e neighbourhood V o f xo such t h a t

a ) Let

A.

subset

, for

%( y ) )

by a countable covering o f

b ) The sequence

X

.

6

i s l o c a l l y equal t o

gn

l a r g e and t h e r e f o r e is precompact i n The d e f i n i t i o n of

,y

a l l -x (V

. Then,

we c o n s t r u c t

for

n sufficiently

f

&'Is

6(X x

and i s equicontinuous.

E-topology g i v e s theannounced r e s u l t s .

Theorem 6.16.- Let (XJp) be a mznifold spread over the maximal extension of O(X1 Then X = sp. 6 ~

CI 8 W h x ) ~

Proof .- By theorem -

with

.

.

2

X =

6.14, we can suppose

cJ

X

P

that

X is

a S t e i n mani-

O(X&

f o l d spread over a f i n i t e l y dimensional space. Let

h 6 sp.

For every f i n i t e subset h o f

hA(g) = h [ g ( x , 'rrA(y)]

6 (X

i s an homomorphism on

J x

defined by

li ) hA . It

h,,(g)

=

exists

A

x Q! )

Now l e t c i a t e d with x

f f

(xA, 2,) The s e p a r a t i o n of ?

= x,,,

xA

(xoayo)

g g o(X

&(X

.

g(xA, zA)

implies t h a t

and

lrAI(zA)

j,

=

X

by p r o p o s i t i o n 6.15.b).

cJ& we

n

CA

cA by

such t h a t

analytic functions

A'C A

for

zAI

. Thus,

there

Since

h

i s continuous i n

have :

= lim.

h [ f ( x , vI (y)] n

Moreover, t h e previous argument g i v e s

r1 (yo)]

X

2 X C A such t h a t h,,(g) = g ( x o a I T ~ ( Y , ) ), f o r a l l , a l l f i n i t e subset A of I . J E (2 x 1 be given and t a k e t h e sequence In asso-

h(f)

f [xoa

t h a t hA i s an

i s known [38]

e v a l u a t i o n and so t h e r e e x i s t s

be given.

, using

h [fCx,

. 'In

(y)]

=

y e t p r o p o s i t i o n 6.15.b) t h e proof i s complete.

56

SPECTRUM AND MAXIMAL EXTENSIONS

Comment.- The theorem 6 . 1 6 i s proved

m. Aurich

[S]

f o r t h e b-topology

by a more d i f f i c u l t way t h a n h e r e . I n t h i s way some r e s u l t s have been o b t a i n e d i n [6l]

by M.C.

Matos.

57

CHAPTER V I I : EXTENSIONS OF VECTOR VALUED ANALYTIC MAPPINGS

5 1.- Vector v a l u e d e x t e n s i o n s . L e t a complex s e q u e n t i a l l y complete c . v . s .

E

s p r e a d o v e r a n o t h e r complex c . v . s .

Proposition 7.1.-

.

OE(W

for

P r o o f . - Given

f

be g i v e n .

,

a # 0

an e x t e n s i o n

in

6 , ( ~ , & ) and

a 6 F

f

= f.a , w i t h f

Theorem 7.1.-

x

sions of Proof.-

a s a n e x t e n s i o n of

Suppose

[17,48,80].

for

. The mapping (6y(Y,F) . For

i s contained i n

v a n i s h e s and t h e r e f o r e t h e r a n g e o f

thus,

is an e x t e n s i o n

Every OX(X,F)-extension Y f o r W E ( F )

6 x ( ~ ,and ~ )h a s

belongs t o

5

f

and a m a n i f o l d (X,p)

F

0 y(Y,6?

in

f

f.a

any

d: .a =

.

5 E

aoY

aoo ;

m e t r i z a b l e then t h e s e t o f exten-

E

6 , ( @ ) and G E ( ~ ) a r e e q u a l .

The c o n v e r s e of p r o p o s i t i o n 7 . 1 m u s t t o b e proved.

F-_~s_a-Ernchet_nEace L e t u : x + Y be a n e x t e n s i o n

a)

. 5( . The mapping

for

O E ( a: 1

of

6 , ( x , c ) , and

F' , t h e f u n c t i o n 5 o € 6,(Y,d! ) 5 + i s c o n t i n u o u s from FA i n t o O,(Y, 6: Is by p r o p o s i t i o n 6 . 9 and theorem 6 . 1 2 . Now by t h e Arens-Mackey theorem, t h e r e e x i s t s f ( y ) g F such t h a t 6 o f ( y ) = 5 o f ( y ) , a l l 5 F', f 6 6 x ( ~ , F ) For a l l in

f

thus

h a s an e x t e n s i o n

.v -

i s weakly a n a l y t i c and i s a n e x t e n s i o n o f

-

f ; w e have j u s t t o

a p p l y p r o p o s i t i o n 3 . 5 . b ) and t h i s p a r t o f t h e p r o o f i s complete.

F--~z_s~¶uest_lallY-~~~~l~~~ For e a c h c o n t i n u o u s semi-norm p on F , F P normed by p , f is t h e completed s p a c e , 71

b) F/p-l(o)

.f.

map

F +

f.

u =

TI

P

.

P By t h e first s t e p , t h e r e e x i s t s

..

-

is t h e space

i s t h e imbedding P , 6 (Y, F 1 such t h a t P P y E Y f o r which t h e r e e x i s t s

f

f , Let fl be t h e set o f p o i n t s P such t h a t f ( y ) = nP o f ( y ) f o r a l l c o n t i n u o u s semi-norms p P 0 W e have j u s t t o prove fl = Y Obviously fl c o n t a i n s u(X) , t h e n we must 0

,P f(y)(

F

.

is c l o s e d . Let

prove t h a t

and a b a l a n c e d neighbourhood ned i n

0

and

yo E yl

+

2U

,

I

yo U

Q

fi

be g i v e n , t h e r e e x i s t s

o f t h e o r i g i n such t h a t

. Let

a€ U

be g i v e n ,

y

1

+

y1 6

.

fi

U i s contai-

EXTENSIONS OF VECTOR VALUED ANALYTIC MAPPINGS

58

C $.zn w i t h z n d ? , t h e seriesis c onve rge nt f o r P n>,O 1. < 2 But f o r 1. < 1 ?D (yl t z a ) = vp ?(y, + z a ) S i n c e F i s s e q u e n t i a l l y complete, u s i n g Cauchy i n t e g r a l , we get some a n C F such n t h a t ap = 71 ( a ) f o r a l l p and n , Moreover t h e serie C an z n P n n>,O i s convergent i n F f o r a l l < 2 s i n c e F i s s e q u e n t i a l l y com-

P

(y

I

+

=

za)

.

.

.

1.

p l e t e ; then there e x i s t s

v

P

+

?(y

o

1.

=

A p p l i c a t i o n t o a f u n c t o r i a l . p r o p e r t y o f t h e maximal e x t e n s i o n s .

§ 2.-

(X,p)

Let

w e d e n o t e by

for

for all

za)

(y + za) z a ) e F such t h a t P 1 < 2 , So, t h e p r o o f i s f i n i s h e d .

+

?(yl

6 E( d 1

.

and

u : X

E ;

be man i f o l ds s p r e a d o v e r a Banach spa c e

(X',p')

-+(X,p)

u' :

and

- -

+ (X',p')

XI

t h e maximal e x t e n s i o n s

.

Theorem 7.2.- [ 7 9 ] Let '4 a locally bi-analutic mzpping from X t o X' be given. There e x i s t s a locallg bi-analytic mzpping from i t o ? ' such that the following diagram i s c o m t a t i v e .

Proof.-

(X, p '

bi-analytic

.y

t h e e x t e n s i o n of morphism

0(XI)

Now l e t

p ' .'p

.y

from

i s a morphism from

fe 6 (XI) , i t s

(f

.'p

( X , p'

extension

'p )

(X, p '

t h e maximal e x t e n s i o n o f t h e f a m i l y

from

(xl,,')

(x',?') . Moreover t h e

1 to

is

when

0)"

)

to

($,,I

,

t h e extension of Dx

p'

oy

from

(X,p)

be t h e d i f f e r e n t i a l o p e r a t o r a t

is l o c a l l y bi-analytic,

(X,p)

s i n c e 'p i s l o c a l l y to

g iv e n by theorem 7 . 1 . L e t p'

E

7

0'9

.'f i s

u'

describes

f

'p

u'

; on t h e o t h e r hand

(it, p ' ) and f o r each

to

f

i s a man i f o l d s p r e a d o v e r

x

i n t o t h e Banach s p ace

D;'(p'

+

L(E)

j

o'p )

be

x ; since

i s a n a n a l y t i c mapping

by theorem 7 . 1 ,

D-'(p'O

'p) h a s

-

a n e x t e n s i o n t o (2,;) and it i s eas y t o check t h a t D-'(p',p o D(p'oY) i s t h e i d e n t i t y o f L(E) ; t h e n p ' o'p i s l o c a l l y b i - a n a l y t i c by t h e i m p l i c i t f u n c t i o n s theorem. Thus, E

and t h e morphism

the functions

f

u : (X, p '

~ ' pwhen

f

.'f m) (2,

)

i s a m a nifold s p r e a d o v e r

+

(2, p .'p ') is

an e x t e n s i o n f o r

d e s c r i b e s (9 (XI) ,

N o w , u s i n g t h e first p a r t of t h e p r o o f , t h e r e e x i s t s a morphism

59

EXl'EmSIONS OF VECTOR VALUED ANALYTIC MAPPINGS from

(x, p '

such (XI, i t )

to

.if)

'-p

locally bi-analytic,

(p

that

= u'

o u

o

. S i n c e 'p i s

'p

h as t h e same p r o p e r t y .

Remark.- To e x t e n d t h e p r ev i o u s r e s u l t t o more g e n e r a l c , v, s . E

, it

would b e i n t e r e s t i n g t o o b t a i n a g o o d s u b s t i t u t e for i m p l i c i t f u n c t i o n s theorem. For i n s t a n c e , t h e r e a r e some F r ech e t s p a c e s w i t h a n i m p l i c i t funct i o n s theorem

[Ell

; f o r such s p a c e s theorem 7 . 2 i s y e t t r u e .

I

Corollary 7.2.- [ 4 8 ] , The r m x i m l extension f o r (3,((fr

spread over E

p

is independent of

of rmnifold

(X,pl is a

(up t o an isomorphism) when E

Baruch space.

Proof.-

Clear w i t h l/l a

b i - a n a l y t i c mapping.

5 3 . - Z o r n ' s theorems. Here we assume t h a t

has Babe property.

E

be a sequence of continuous homogeneous polynoun m i a l ~with a f i x e d degree defined on E and F-valued. I f un is pointwise convergent t o a homogeneous polynomial u , then u is continuous. Proposition 7.3.-

Let

Proof.-

IT^ N(F)

sup. r 0 u i s semi-continuous n >/o n from below and f i n i t e everywhere. By Baire theorem, t h e r e e x i s t s a n o t For e a c h

empty open s e t w and and a l l n 6 ll4

M

€fl

therefore

given, f o r each

h e E

t h e function

IT o

such t h a t

un(x)

IT

4 M

i s a l s o bounded on w

u

t h e function

z

+

u(h

-+ z

.

xo)

x e w

for all xo 6 w

Let

be longs t o

be

6 (,F) ~

t h e n we have :

Then f o r continuity of

Theorem 7 . 4 . - [7,

F , If

f

Proof.-

Let

s u f f i c i e n t l y small

h 71 o

u

f o r each

68, 8 5 1

. Let

IT

u ( h ) \< M

77

E N(F)

f

be a Gateaux-analytic mzp from X

is continuous a t some point then f IT^ N(F)

. We have proved t h e

and t h e proof is complete.

belongs t o

be g i v e n . By p r o p o s i t i o n 3.2.f),

6 (X,F)

IT

f

.

into

i s continu-

o u s i n a neighbourhood of each p o i n t of c o n t i n u i t y . Then w e c a n i n t r o d u c e t h e n o t empty open set verify that

W

W = {xQ X

I

TI

f

i s continuous a t

x)

h a s a empty boundary, t h e p r o o f w i l l b e complete.

, We

EXTENSIONS OF VECTOR VALUED ANALYTIC MAPPINGS

60

Using a contradiction argument, let xo be given in the boundary There exists a complex line through xo such that d nW has a of W

.

boundary point x1 which belongs to d 0 X , We take a sequence (x:),

.

xf n C d n W , which converges to x1 For all h e E , all p € w, fp(xA,h)

is convergent to fp(x 1,h) since

f is analytic on the subspace (d @ h) n X

tinuous by proposition 3.2.f), then p g IN

71

, moreover

fp(xl)

fp(xl) is conn is also continuous for all

, by

proposition 7.3. Now we verify the continuity of IT

o

f at x1

IT

, the

contradiction

will be complete. Since the series

C fP(xl,h) is summable for h sufficiently pa0 small (prop. 3.3.c), sup. IT 0 fP(x 1) is locally semi-continuous from below and finite, then there exists a not empty open set w and M such that IT

o

, for

\c M

fP(xl,h)

IN

all p

and all h E w , Now using

the same argument as in proposition 7.3, we check the uniformly boundness of IT o fp(x,) in a neighbourhood of the origin. That easily implies the uniformly convergence of a( C fp(x,)) to TI f in a neighbourhood of x1

, and

Remark.-

Pt o

lastly the continuity of IT o f at x1 , This result can be extended from Baire spaces to some other spaces

as, for instance, adjoint of Frechet-Schwartz spaces [46]

by(Y,C 1

Theorem 7.5.- Let A be a s e t i n

,

which s e p a m t e s

Y

. Let

f be

a Gateaux-anulgtic mrrp from X i n t o Y such that : li) A l l

xo

e X , there

t h e o r i g i n i n F and wx Y , for a l l

x 6 wx 0

.

e x i s t s a connected neighbourhoods U

X

of xo

such t h a t fix) + Ux

0

(ii) For eveqi 8 € A

Proof.-

II o €

,

g

belongs t o

belong to

i s contained i n 0

o

f i s Gatwrux-analytic i n X

(iii) f i s continuous a t some point of X Then f

of 0

6x(~,Y).

6x(x,F)

.

. -1

by theorem 7.3, therefore r f t X )o II

o

f

is analytic in a neighbowhood of each x 6 X , defines a germ denoted by , Hence W = { x G X 1 fx = f 1 is an open set which is the set of point X x e X such that f is continuous at x We have just to prove that

Zx W

.

= X , that is W is closed. Let xo g P be given, and V an neighbour-

hood of O e F such that V

+

VCUxo

.

61

EXTENSIONS OF VECTOR VALUED ANALYTIC UPPINGS

Since IT o

x

is co n t i n u o u s , we can choose

f

IT o

f ( x ) E IT

wx A W su ch t h a t

1

, for

f(xo) t V

x 6 wx

all

f(xo)

T

wX

+

such t h a t

0

:

, Furthermore t h e r e e x i s t s

0

i s c o n t a i n e d i n IT

V

o

f(xl)

+

.

Ux 0

,

-lo

o 71 o f b el o n g s t o O ( w x Y) , Thus "f(X1) 0 On t h e o t h e r hand g o f b e l o n g s t o Q ,(x,& ) by theorem 7 . 3 , -1 o 71 f and g f are a n a l y t i c i n wx and e q u a l therefore O ITf(X1) 0 -1 Hence g o IT o IT o f and g f are i n a neighbourhood o f x1

.

f(Xl)

and by t h e s e p a r a t i o n assumption o f A

wX

equal i n

, we

obtain

..

= fx.

fx 0

Remark.-

When

h a s a f i n i t e number o f she a ve s o v e r e a c h p o i n t of

Y

0

,

F

t h e r e q u i r e m e n t ( i ) i s always s a t i s f i e d .

5 4 . - F u n c t o r i a l p r o p e r t i e s of e x t e n s i o n s . (X,p) Let

and

v :Y

and

be an u n i t a r y s u b a l g e b r a o f

A n

+

Y(A)

0 (Y)

, W e de note by

the c a n o n i c a l morphisms from

e x t e n s i o n and from

OF(@ 1

are man i f o l d s s p r e a d o v e r Fre c he t s p a c e s

( Y , IT)

. Let..

Y

i n t o the

into the

X

E

u : X

+

x

6(X)-maximal

o,(C)

A-maximal e x t e n s i o n f o r

and

and

6 O,(X,Y) be g i v e n , t h i s s e c t i o n i s concerned by t h e -r. e x i s t e n c e o f y d 6 i(X,Y(A)) s u ch t h a t t h e n e x t diagram i s commutative 'p

x-y

h

X

Of c o u r s e , whenever

?(A)

t

i s smaller i n (5, (Y)

A

t h i s problem i s easier.

It i s c o n v e n i e n t t o n o t i c e t h a t problem i s s o l v e d by theorem 7 , 2 i f

a

local isomorphism,

E

F

and

For a n y with

+

i s t h e e x t e n s i o n from X

&OpOSitiO?I

are Banach s p a c e s .

'4" t h e map f 2 6 2 , we d en o t e by

W e d e n o t e by

7.6.-

a)

t

to

f ~ ' pfrom

t y * ( E)

fi

O(Y)

, the

map

into f

+

6 (XI /v

f

.'p

.

'p

(fi)

is

,

,

* (2) is an homomorphism from the algebm

6 fY)

i n t o Q; b) v

o p

c) For ar?y

= t ( p * ~ ~

2E

,

v* s a t i s f g

the requirement (a3)

EXTENSIONS OF VECTOR VALUED ANALYTIC MAPPINGS

62

in

pr, 1-J

, that

is :

bE F

there e x i s t s

such that

(5

trf*(&

o

IT) =

, for

{(bl

a l l 5 ISF '

Proof.- P r o p e r t i e s a ) and b ) a r e obvious. By theorem 7 . 1 , lr .'p w

OY

extended according t o a

In

which belongsto

o r d e r t o prove t h a t

7 = t y X is

O ( ~ , F ).

a

can be

Hence, we o b t a i n :

a s o l u t i o n f o r t h e diagram (1)

,

we should v e r i f y t h a t p r o p e r t i e s ( s 1 and ( s 2 ) i n ( V I , l ) must be s a t i s f i e d by

. Unfortunatly,

ty*

a general subalgebra

1

t h i s problem i s not y e t solved f o r

The mp

Theorem 7.6.-

A

.

A

soZve8 the diagram

natural, Frechet subalgebra of

6 lY)

.

(I)

, if

A

= @(Y)

is a

or

i. d ,

8.

Proof*- a ) ThE-E!ngE_of--- t$r_---____----___-________ i s contained i n S(A) The range of t h e r e s t r i c t i o n of with

. Since

A/Kery*

'4'

is natural,

A

to

A

i s a l g e b r i c a l l y isomorphic

Ker cpris closed i n

c a r r i e d topology of Since

E

A/Ker

Y*

i s a n a t u r a l Frechet a l g e b r a i n

i s m e t r i z a b l e , t h e extended Frechet a l g e b r a V*(A)

n a t u r a l by theorem 4.7.

Thus, we have proved t h a t t h e map t h e topology o f sp? A

, that

A

-

is

if 4 % )

t l v

h/

with t h e

0 (X)

,

is also

i s continuous f o r

f + f o'p

belong t o

and

Y *( A )

-

A/Kery* i s a Frechet space f o r t h e q u o t i e n t topology. Then

A

sp.A

and f u r t h e r t o

by p r o p o s i t i o n 7.6. Now, using theorem 6.4, we o b t a i n t h e announced

result. b)

I-~~-pelong-To--~-~~l-8(1112 is a n a t u r a l Frechet a l g e b r a , we know by theorem 2 . 2

Since ?"(A)

h-*

i s l o c a l l y uniformly bounded qnnd g o € 2 b e i n g given, t h e r e e x i s t s a neighbourhood w of such t h a t g + [1g[1, i s a continuous

t h a t Y?A)

.

semi-norm f o r

On t h e o t h e r hand, t h e maps equicontinuous s i n c e i n t o V*(A) (a,f) + fn(a)

er

A

(a,f) + fn(a)

i s s . i . d.

and

from

F x A

into

are

A

i s a continuous map from

A

a s it has been proved a t t h e first s t e p . Therefore t h e maps

ov

imply t h a t

a r e equicontinuous. Hence, t h e previous arguments cogcth-

C

n&O

[[ =fv

11


1) and u s i n g Cauchy i n e q u a l i t i e s , we N G

obtain t h e following majorization f o r a l l

and a l l

h

sufficiently

small : pN

[

n=N

-

f*(xth)(y)

C

(fn?(x;h)(y)l

6 sup

0 (YIc

Then, t h e l e f t hand s t a y s i n a bounded set o f when

h

d e s c r i b e s t h e sequence

Since

(hi)

formly convergent t o b e s t h e sequence

f*(xth)

in

.

(hi)

With t h e f i r s t s t e p t o g e t h e r

O[x,D wb] .

resp.

Let

*

g (x)(y)

and f o r a l l

g*g6[X,b(Y)c]

o3(Ylc

, we

N

C

t e n d s t o i n f i n i t y , t h e series

pN

.

(f(x+th,y)l

Itl=P

n=O

.

when

X

descri-

g(x,y) E and

, whenever

Y

X

h

f*c&[X,o(Y)c]

be g i v e n , we have t o prove t h a t

d e f i n e s an a n a l y t i c f u n c t i o n on

6 (Y),) ,

i s uni-

(fn)*(x,h)

nbO ( r e s p . @(Y),)

obtain :

(resp.

F

are

metrizable.

g

Obviously and

, then

y

g

i s s e p a r a t e l y a n a n a l y t i c function o f each v a r i a b l e i s a n a l y t i c on each f i n i t e d i m e n s i o n a l s u b s p a c e of

. Then

by Hartogs theorem [8]

g is a Gateaux-analytic f u n c t i o n .

Now, we v e r i f y t h e c o n t i n u i t y and w e t a k e two sequences (yn)

which converge t o

a

and

b

g(xn,yk)

i s u n i f o r m l y convergent t o

sequence

(yn)

continuous

.

Corottarg 7.8.-

o (x, O ( Y ) J

Comment.- When

6 [X, b

(Y),]

. Thus If

=

E

in

X

Y

and

g(xo,yk)

. Since

when

we o b t a i n t h e c o n t i n u i t y o f

and

F

yk

g

are rnetrizable, we have

i s n o t m e t r i z a b l e t h e map

. For

i n s t a n c e , suppose

X = E

f

+

g*is

E

g (xo)

is

8tX,

f * cannot

with

(x,) and continuous

describes the

since

6 CX, D(YI, 1 .

F

x X x Y

and

be o n t o B an i n f i n i t e

EXTENSIONS OF VECTOR VALUED ANALYTIC MAPPINGS Y = E'

d im e n s i o n a l Banach s p a c e and p a i r i n g i s n o t c o n t i n u o u s on

x

x

into

E

.

E x EA

(Ed);

but t h e

are metrizable, the m d m u l extension of

If E and F

Theorem 7.9.-

x

t h e weak a d j o i n t spa c e . The d u a l i t y

U

d e f i n e s a co n t i n u o u s l i n e a r map from

Eb>

CE,

65

2

Y for 0 E x F ( d : ) is t h e product of maximal extensions Y for B ~ ( C Cand ) 6,(&) ,

?

and

of

and

Proof.and

rx ( r e s p .

Let

(resp.

Vx

bE(C )

ry)

be t h e p r o j e c t i o n map

t h e e x t e n s i o n morphism

V )

Y

X

X

X

x

for

x Y

(V,

.

0 ExF(C

@(X,Y) t h e b i j e c t i v e mapping from ( j ( X x Y)

Let

* X ( r e s p . Y),

(resp.

-+

( r e s p . G F ( C ) ) . To b e g i n w i t h , we v e r i f y t h a t

i s an extension o f

Y

Y o

+

rX

, Vy

for o

ry)

onto 6[X,@(YL]

as a

d e f i n e d by theorem 7.8 and c o r o l l a r y 7 ' 8 . After p o i n t i n g o u t 6 ( Y &

s e q u e n t i a l l y complete s p ace by theorem 6.10, w e can i n t r o d u c e t h e e x t e n s i o n map I : & [ X , @ ( Y ) E ] * ~ [ ~ , O ( EY]I g i ve n by theorem 7 . 1 and t h e t o p o l o g i c a l isomorphism J between ~ ( Y ) Eand 6(GE) give n by theorem 6.12.

Then t h e e x t e n s i o n map from

@-'(i,?) 0 J

o

I

.

Q(X,Y)

0

O ( Xx

Now w e v e r i f y t h e maximality of be a n e x t e n s i o n f o r

.

0 ExF( &

c

X

Y)

onto -

x Y

.

O(i

u : (X

Let

We i n t r o d u c e t h e f o l l o w i n g e q u i v a l e n c e r e l a t i o n Z :

z

IT-'

(1,)

zt

[IT(=)

if

aE

. Each

z

coset is closed i n

is a Hausdorff and connected man ifold s p r e a d o v e r

Z/q(E)

Z

t h e imbedding morphism Let

f e @(XI

X

Y)

(Z,IT>

+

(resp.

RF) on

b el o n g s t o t h e connected component o f

zt

which c o n t a i n s

t F]

i s g i v e n by :

x

+

.

Z/a(E)

be g i v e n , t h e n

c o n s t a n t on each c o s e t f o r f i i ( E)

.

U*O

ri(f)

Z

, then rE

E , Let

be longs t o b ( Z )

, is

and s o d e f i n e s a n a n a l y t i c f u n c t i o n on

The morphism rE h a s been c o n s t r u c t e d such Z / q ( E ) which e x t e n d s f t h a t rE o u can be f a c t o r e d t h r o u g h 'rX by IT; such t h a t TI(; o IT =

IT^

o

u

. Then -r;(

r i e d through ux o

ri

= Vx

X

i s an e x t e n s i o n of

by a morphism

. Thus w e

3

X

f o r @,(&

: Z/R ( E )

+

, so

X

it c a n b e f a c t o -

which v e r i f i e s :

o b t a i n t h e f o l l o w i n g commutative diagram :

EXTENSIONS OF VECTOR VALUED ANALYTIC MAPPINGS

66

IT

X~

x c

--(X,Y)

_

= (u

Now, i t i s easy t o v e r i f y t h a t )I from *

X

-

X

Y

2

to

IT

Y

X X_ Y

which s a t i s f i e s $

o

o TI

, uy

o

u = (Vx o I T ~, Vy

> Y

IT^) 0

i s a morphism

.

I T ~ )Then

i s t h e maximal e x t e n s i o n s i n c e it can be f a c t o r e d t h r o u g h

2

.

61

CHAPTER VIII : POLYNOMIAL APPROXIMATION § 1.- H i l b e r t i a n o p e r a t o r s

We d e n o t e by

H1-+

from t h e H i l b e r t s p a c e and

DA

A H

a closely, densely defined l i n e a r operator 2 t o a n o t h e r H 2 ; - t h e r a n g e o f A i s w r i t t e n RA

H

1

i s t h e everywhere dense s u b s e t where

The t r a n s p o s e d o p e r a t o r o f H2 t o H

d e n s e l y d e f i n e d from -

denoted by

1

t h e r e s t r i c t i o n of

j

- , it

is defined.

A

i s denoted by

A

i s closely,

A

A

E-A

to

The s c a l a r p r o d u c t o f some H i l b e r t space i s w r i t t e n