ANALYTIC FUNCTIONS AND MANIFOLDS IN INFINITE DIMENSIONAL SPACES
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ANALYTIC FUNCTIONS AND MANIFOLDS IN INFINITE DIMENSIONAL SPACES
This Page Intentionally Left Blank
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:
AMERICAN ELSEVIER PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE NEW YORK, N.Y. 10017
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 [ ( <
for
sup
xeT
rn
a12
(,)
,
E.
X w i Z Z be
of
T
f E
r,
q k N(Z/
.
Definition 2.2.- This set, r i s caZZed locaZZy bounded i f f , f o r a l l 3: E X , each f E r i s bounded i n a neighbourhood w of x j we say r i s uniformly bounded whenever t he previous w can be chosen independantly of f
.
Remark. When
i s normed,
Z
Fx(X,Z)
i s always l o c a l l y bounded.
D e fin itio n 2 . 3 . - A vector subspace r of F x ( X , Z ) , wi t h a locaZZy convex linear topology i s called natural whenever %' i s stronger than t h e topology of pointwise convergence. The s e t of semi-norms for?? i s denoted by N(I'l. Proposition 2.1.bounding s e t f o r for each q
Given 'l a natural Frechet space i n F x ( X , Z ) and T a f + [lq 0 f l l i s a continuous semi-norm of
r , then
e "2)
Proof. The s e t
(f
,
I
Ilq
fll
c o n t i n u i t y of t h e semi-norm s i n c e
is closed
r,
l}
r
I'
is a barrel in
, which
implies the
i s barreled. Actually t h i s previous set
because of c o n t i n u i t y o f t h e e v a l u a t i o n a t any
a b s o r b i n g set b e c a u s e of t h e boundness p r o p e r t y of balanced.
T
x
in
X ; it i s
and i t i s o b v i o u s l y
NATURAL FRECHET SPACES
12
2.1.- A natural Frechet space has a stronger topology than the compact open topology.
COrOllQFy
P r o o f . C l e a r , wi t h
Definition 2.4.-
a s a compact s e t .
T
u
covering of X by open s e t s , and denote by the s e t of f E F x ( X , Z) such t hat fI w ) i s bounded for euery w Q Au We endow A x with t he topology defined by the semi-norms :
llfllw,q =
114
Let
Q
u.
, ueu
O f l l ,
Proposition 2.2.Freche t space.
be
Whenever
u
qe
9
N(Z)
i s countable,
A
*
i s an uniformly bounded,
U
P r o o f . Clear, u s i n g ( B ) .
r be
Proposition 2.3,- Let Q
(f E
r 13
r = rw,
either
natural Frechet space. Given x & X
,j%w)
FEfw,Z) x
r
or
1
Proof. The map
of p a i r s
Q
with P I X ) & w
domain i n E
a,
. Let
be defined
x
sequence some fn
o
?€ =
Px
, define
a v e c t o r subspa c e of
II(f,?)ll qYn a F r e c h e t topology on
Yw ,
in
FE(w,Z)
by p r o p e r t y (B)
i n a domain
.
fn
w'
, and
, such
Yw,
hw)
and w
QS
. Then,
t
FE(w,Z)
[1?11
w,q
denoted
,nGN(i")
A c t u a l l y , g i v e n a Cauchy
fe r
and
Yn
near
i s bounded. Furthermore
p ( x ) G w'
that
r X
= n(f)
converges n e a r some
( f n , ? n)
2n
:
i s unique b ecau s e o f p r o p e r t y ( A ) , and t h e r e f o r e t h e s e t
W e c l a i m now t h a t semi-norms qE N ( Z )
and
QS
bounded, and (f o p , l ) p ( z ) = $ 1 i s Q s e t of f i r s t category i n r.
, f c rwYx , i s
(f,?)
r w,
,
w'C w
, and
w' S i n c e t h e t o p o l o g y of r i s s t r o n g e r t h a n t h e p o i n t wise t o p o l o g y , we have f o p i 1 = 2 i n w ' ; hence ( f , P ) E : Yw
pi1
defined i n
F i n a l l y , t h e map nuous ,
.
(f,?)
-F
f
, from ?w
to
r , is
l i n e a r and c o n t i -
i s i t s r a n g e . Then t h e s t a t e d p r o p o s i t i o n i s now a consequen-
rwYx
ce o f a w e l l known Banach theorem.
Theorem 2.1.- If each poi nt of E has a countable basis of neighbourhood, then euerg tocatly bounded natural Frechet space is u n i f o n t g bounded. Proof. L e t hoods of
xE X p(x)
be g i v en and l e t
. If
r
(wn)
be
such a s e t o f neighbour-
were n o t u n i f o r ml y bounded, we s h o u l d have
13
NATURAL FRECHET SPACES
r %,x
r
#
for all
n
p r o p o s i t i o n 2.3 and t h e Baire p r o p e r t y of -1 ( f o p, i s bounded f o r some n ,
r
r -
fE
and t h e r e would e x i s t some
u rwn
by
; t h i s i s im possible s i n c e
Theorem 2.2.- If i s nomed and E has the countability property above, then every natural Frechet space r can be continuously imbedded i n t o a countable, convenient class AU with
u
P r o o f . Si n c e
Z
r
i s normed,
i s l o c a l l y bounded and t h e r e f o r e uniform ly
bounded by t h e p r e v i o u s theorem ; t h e r e e x i s t s an open s e t w i n that
i s bounded i n
f(w)
we d e n o t e by set
xn =
11.11
t h e norm o f
I
{XEX
Z
. I n t h e f o l l o w i n g argument
f E I'
and by
= N(r)
(rn)
:
defined as
Xn
f o r every
Z
s n . rrn(f)t
llf(x)[[
W e claim t h a t
=
X
fn
fner
Using
gn
x
with
xnEX
+
[rn(fn)
llgn(xn)l[ >
[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 \<
71
by d). The continuity of a
+
c
T o
n>,l
fn(x,a)
71 o
fn(x,a) <
M.E
71;
is a conse-
quence of local boundness of this mapping by a) and b),
Pr_.2JErt_ie_s_ L?f-C!nall! t_i~-rn?PF!i"-gS_ * Definition.- A map f : u + Z is analytic i f f i t is COntinUOUS and Gateaux-analytic. The preaheaf of such maps w i l l be denotes by
oE(Z).
Proposition 3.3.-
Let
f
E &,(w,z)
be given.
..., an)
a ) The maps y ( al,...ra l : x * f " ( x , al, a2, n (ai) Q f and f " ( a ) : x + $2(x,a) belong t o O,(W,Z) a
b ) The map8 are continuous.
+
and
$1(x,al
(al,a2,, ,.,a n )
, all
all a6 Z ,
$2(x,a1,a2,.
+
..,a n
C y ( x , a ) is uniformZy s m a b t e c) Given nc N ( Z ) , the s e r i e near f ( x + a ) in any compact s e t contained in wlx) and uniformly s m a bte f o r a in some neighbourhood VT of the origin in E
.
is closed i n C,fw,Z) dl 6,(w,Z) with respect t o compact open topology.
and f
-+
f " ( a ) i s continuous
As consequence of proposition 3.2.e and 3,3.d, property ( A ) and (B)
.
Proof. a) Given
w
x
and
pc
6E121 s a t i s f i e s
N(Z) : if B and An
(for each integer
ger n > 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 ) \<
o
w
E
sup_ X CK
The c l o s e n e s s o f 6 ( w , Z )
, by
p o f(x) in
.
proposition 3.2.d)
i s a n obvious consequence of
C(w,Z)
t h e same p r o p e r t y i n f i n i t e d i m e n s i o n a l c a s e .
Proposition 3 . 4 . - Any G a t e a m a n a l y t i c mapping which i s locally bounded continuous. The presheaf of locally bounded a n a l y t i c mappings w i l l be Indeed, 6 b ,(Z) = b E ( Z / , wheneve?. E i s normed. denoted as 0 Eb ( Z ) b By proposition 3.3.d 6 s a t i s f i e s property ( B ) ,
i8
.
Proof. -
Let
U
be a b a l a n c e d neighbourhood o f
For any I T EN ( Z )
a
OU
. Then
we have
T(f(xtA)
-
f(x))
t h e r i g h t hand is less t h a n
> 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
<
rl
a
all
n
Then, every f g r h a s a Taylor expension a t any x C; w with R > r( as r a d i u s of convergence. This i s impossible n e a r a s i n g u l a r p o i n t o f t h e boundary o f
w.
We can t a k e f o r w singular point f o r
exp
, the
h a l f plane
Re z < 0
, where
the origin is a
which belongs t o 'I ,
W e study t h e e x i s t e n c e o f some type o f Frechet spaces The a d j o i n t space
E'
logy induced by t h a t of We say
of
i n @x(X,C).
equipped with s t r o n g l y topology ( r e s p . topo-
E
r)
r
i s denoted by
contains E l i f
6
o
p
Eb
( r e s p . Ei, ) ,
belongs t o
r ,all c €
El ,
FRECHET SPACES OF COMPLEX ANALYTIC MAPPINGS
26
Proposition 4 . 2 . - I f there e x i s t s a natural, invariant by derivation, Frechet space r in O,(X, d: ) such that 'I contains E' , then :
.
in E
a ) there e x i s t s a countable, fundamental, system of bounded s e t s
.
b ) E b is a Frechet space f i n e r than E b Proof. F i r s t in
we prove t h a t
i s such t h a t
Ef'
(Sn o
4.1.a), En(a)
r
p
, all
f'(x,a)
belongs t o
b a s i s o f neighbourhoods o f
0
in
r a l ; then proved b )
Bo
Bo
.
is a b a r r e l i n
Ef,
E;
is closed i n
; given
B
r, V i
a d E ; that is s i n c e t h e topo-
(Vn)
be a c o u n t a b l e
a bounded s e t i n
B
s i n c e t h e topology o f
V,
n (Ei,
contained i n
, E)
E
,
i s natu-
E;
0 , W e have
and so a neighbourhood o f
E;
. Noting the d u a l i t y
V;
is contained i n
. By p r o p o s i t i o n
a rS E
. Let
El
On t h e o t h e r hand, t h e r e e x i s t s some t o p o l o g y of
,all
, all
xE X
f
r
in
f
f'(x,a)
i s n a t u r a l . Thus
t h e p o l a r set
5,
i s a F r e c h e t s p a c e , A Cauchy sequence converges t o some
is c o n v e r g e n t t o
p)l(a)
i s convergent t o
logy o f
Ef o
, that
Bo
is
because o f n a t u r a l
i s weakly bounded and a l s o bounded.
Corollary 4 . 2 . - If E is infrabarrelled and there e x i s t s a natural, Frechet space r which contains E' in , then E is d 2 . F
O,(X,C)
space and E' = E; j furthermore, among the metrizable spaces, only the B nomabte spaces have the above properties.
Proof. I t
i s known [51,77]
t h a t an i n f r a b a r r e l l e d space w i t h t h e p r o p e r t y
08
a ) o f p r o p o s i t i o n 4.2 i s a
E;
Lastly since Now, i f c511
E
.v
i s a Frechet space f i n e r t h a n
is metrizable, then E
s i b l e , countable covering of
5
2.-
B
r
i s normed, t h e r e e x i s t s p a c e s
corollary. Actually, we take f o r X
El
B
, we
is metrizable i f f
E'
a
When
i s a Frechet space.
Eb
space t h e n
E
have
E
- B ;El
i s normable
'
with properties of t h e
r a s p a c e of t y p e , such t h a t p(w)
AU
, with u
a n admis-
i s bounded, a l l w E
,
A g e n e r a l t y p e o f 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 s p a c e s .
I n t h e f o l l o w i n g , t h e n o t a t i o n below w i l l b e used : Given a set
in
E
, we
write
a neighbourhood o f
T T
in
X
+
V C X
X
onto
and
a b a l a n c e d neighbourhood of t h e o r i g i n
V
i f f for every
p(x)
+
V
, and
x
eT,
p is a homeomorphism of
T t V = {pi1 [p(x)+V],
xL
TI;
27
FRECHET SPACES OF COMPLEX ANALYTIC MAPPINGS
we write a l s o previous
+
T
Q
with Q a p a r t of
E
when
Q i s c o n t a i n e d i n some
V ,
Definition 4.2.i f , for any w E such that w
A coveringg of
e
+
u
V
by open s e t s w i l l be called admissible
X
there e x i s t s V balanced neighbourhood of t h e o r i g i n i n is contained i n some ~ ' 6
u .
Proposition 4 . 3 . - Whenever8 i s an admissible covering of ciated class Au i s strongly invariant by derivation, P r o o f . Given -
and w
q E N(Z)
sup
q
--
6
o fn( x , a )
ea
2n
X € W
6 sup q o f ( x ) , f o r a l l X& w'
w'
Here,
and
V
X
, t h e asso-
; we have, by p r o p o s i t i o n 3 . 2 . d )
2n
-1
q o f o p , 0
a € V
[p(x)+a.e
ie-- 1 . e - n i e
dB
.
a r e t h e sets a s s o c i a t e d w i t h w by t h e p r e v i o u s
definition.
Proposition 4 . 4 . - Given a covering o f X by open s e t s and a basis of balanced neighbourhoods of the o r i g i n i n E ; there e x i s t s another covering u r of X which i s admissible and f i n e r than ,
u
P r o o f . Given with
Ur can be chosen countable.
i s countable,
If
V
e
x t r(x1.V
v
and
C
w)
u
w
.
It i s easy t o v e r i f y t h a t in c l u d e d i n w ( 1 / 2 . V ) X by w(V)
when
V
w e d en ote
w(V)
w(V)
i s open and
w
{XI=
w(V)
+
[ 3 r(x) >
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\ <
41;
for
( E ~G ) E
E
n > N
with
1 1 ~ 1 1<
;i;
; there
.
Then, w e have :
the origin.
.
such t h a t a n / c -for a l l n > 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
<
o u)"(a)]
<
T[gn(a)]
small enough, we have :
.
u )n ( a )
o
and a neighbourhood
entails
a r e continuous t o g e t h e r
i s strongly invariant by derivation, so i s
Proof. By p r o p o s i t i o n
I
j be t h e cano-
j i s also continuous.
by t h e c l o s e d graph theorem,
Proposition 4.6.-
X
t h e r e s t r i c t i o n mapping
k
and
k
is natural
a F r e c h e t s p a c e and
E , Actually, l e t
* U ~ , ( X ~ , Zand )~
u3c(r)
Z
with
s p r e a d over a f i n i t e d i m e n s i o n a l s p a c e nical injection
by
t h a t we c a l l t h e extended F r e c h e t s p a c e .
isomorphic w i t h
r
*
G N ( r ) d e f i n e s a semi-norm on
E
E
On t h e o t h e r hand, t h e r e e x i s t s
OE E
of
,all )
gn(a) o u =
such t h a t : E1(g
n , Then
o u ) \< r) and
u 6 o 'mr,(g) \< rl
and
all n ; s o , the p r o o f i s c o m p l e t e .
Theorem 4 . 6 . - Whenever r is a strongZy invariant by derivation, natural Frechet space, so i s u'* l r ) , Proof. L e t XI
W be d e f i n e d by
i s continuous i n
W
uic(r)i
.
{ x ' g X'
I
t h e e v a l u a t i o n mapping
2'
at
FRECHET SPACES OF COMPLEX ANALYTIC MAPPINGS
30
i s _ - ~ ~: eg i~v e n q 6 N(Z)
W with TI
2
TI^
6 n1(g
q [g(xt)]
E N(T)
u)
o
and
for any
g g u*
and a b a l a n c e d neighbourhood f o r any
[fn(a)Il 6 n 2 ( f ) We choose V
Then,
x' t 1/2 V
Now, g i v e n
, we
f 8
contains
x
and
Given a sequence set
{g E u*(r)
I
of
2'
have
and
W
x ' e X'
.
, they
is closed s i n c e c l e a r l y XI
q [g(x;)] g
in
W f l Y
a r e joined together
6 1
x
, all
n }
0 x(X,
2)
4.6.-
Let
, if
E
by a
whicil
q o 2''
x"
W
in
.
Y
, the
i s a b a r r e l because o f
; t h e r e f o r e t h i s s e t i s a neighbourhood o f
Corollar
Y
belongs t o
which converges t o
on t h e r e l a t i v e l y compact
q o 2'' ,c , l; t h e n t h e mapping
+
X'.
and i s s p r e a d over a f i n i t e d i m e n s i o n a l s p a c e , We
x' W f l Y
boundedness o f e a c h
such t h a t
w i l l be i n c l u d e d i n
polygonal l i n e and s o , t h e r e e x i s t s a connected submanifold must prove t h a t
exists E
r .
x' t V
Tl E N ( r )
have :
i s included i n
x o E u(x)
of t h e o r i g i n of
V
and
s m a l l enough such t h a t
a E 1/2 V
Then for any
a & V
, there exists ( r ) ; now, t h e r e
x' 6 W
(x:) 0
and c o n t i n u i t y in
r
on which we
i s continuous.
r , a s . i. d., natural, Frechet space be given i n is metrizable then every r-extension f o r 6 b,(Zl is
uniformlg bounded, Proof.-
It i s a n obvious consequence o f theorems 4,6 and 2,1,
Theorem 4.7.zable.
We s t i l l asurnme t hat
Z
i s a Banach space and E 2s metri-
a ) Let r be a natural Frechet space i n 0 I X , Z ) and u : X + X' a X r-extension f o r O,(Z) , then t he extended Frechet space u* (r) c m be continuously imbedded i n a cl as s A a w i t h Zp' an admissible and countabZe covering of X'. Thus, u* (r) i s al so natural, b ) [797 I f r moreover i s a cZass of type A2( , wi t h an admiss i b l e and countable covering of X , V ' can be chosen such t h a t
=
u*
(ri
as topological spaces.
FRECHET SPACES OF COMPLEX ANALYTIC MAPPINGS
31
The first p a r t i s an o b v i o u s consequence o f t h e n e x t sequence o f
Proof.-
p r e v i o u s r e s u l t s : Theorem 2 . 2 , p r o p o s i t i o n 4 . 4 , theorem 4,6, theorem 2 . 2 , p r o p o s i t i o n 4.4.
2
On t h e o t h e r hand f o r t h e second p a r t t h e r a n g e o f i s a n a d m i s s i b l e and c o u n t a b l e c o v e r i n g o f
u
A
belongs t o
X'
=
v'
u(
u
f'c AUl
w i l l be i n c l u d e d i n
AU
i s t h e extension of
F i n a l l y , t h e Frechet space
f'
u
which b e l o n g s t o
which
u(
) ;
.Further, A u
and t h e e x t e n d e d F r e c h e t s p a c e o f
AUl
s t r o n g e r t h a n t h e t o p o l o g y induced by t h a t o f
AUl
is
and by a p p l y i n g t h e
AU
open mapping theorem, t h e p r o o f is complete.
5
AUl
have t h e same t o p o l o g y ; a c t u a l l y , t h e t o p o l o g y o f
AU
f
i s a n a d m i s s i b l e and c o u n t a b l e c o v e r i n g o f
)
such t h a t t h e e x t e n s i o n of
any
by t h e morphism
and any
which i s bounded on any s e t of
h a s an extension
u
therefore
u(X)
4.- A c h a r a c t e r i s a t i o n f o r maximal e x t e n s i o n s of n a t u r a l 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 , F r e c h e t s p a c e s . 1.
I.*l
Here,
Z
i s Banach s p a c e normed by
[I. [I
c. v. s . For any open, b a l a n c e d , neighbourhood in
X
, the
and
E
is a metrizable
OE
of
V
E
, and
any s e t
f o l l o w i n g 'tboundaryt' f u n c t i o n s are d e f i n e d :
V
d (x) X V d (TI
X
SUP
{r >/ 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
<
IIf(xn)II
, then
bounding s e t f o r I'
m
f E
all
r , the
sequence
(xn) i s a
theorem 4.8.a) e n t a i l s ( i i ) .
(if.
2
r
be t h e maximal extension of
By p r o p o s i t i o n 1 . 8 ,
Given a boundary p o i n t converges near
x
in
0
I
i s i n j e c t i v e and s o ,
u
x
of
0
in
X
X
.,
X
i s a domain i n
, then
sequence is a bounding s e t f o r
.
r
X
.
+
.
any sequence which
X ; however, t h i s
reaches t h e boundary of
X
u : X
which i t s morphism
( i )+ ( i i i ) . It i s t h e theorem 2 . 3 ; t h e converse implication i s obvious.
Definition 4.5.-
Given a f i n i t e dimensional, a f f i n e , subspace H
,
in E
p-'(H) is a m n i f o l d spread over H which w i l l be denoted a s X H , The m n i f o l d IX,p) w i l l be called pseudoconvex whenever X H i s a Sbein m n i fold f o r any H L(E, 2) i8 the s e t of continuous endomorphism8 porn E i n t o 2 ,
.
Proposition 4.10.- I f X is the maximal extension of a s e t A i n and A is invariant by derivation and contains t h e mppings u o p u
E L(E, 2) , then X
any a i n t h e a d j o i n t space describes
1.
Z'
, we
k
contained i n Finally,
f(x') # a
XH
L(E,Z)
, and
X
of
Z
and
H
x E
[d(x,,
, we
f € A
and any
2
C)] have
k [ 6 (X,,dl I]
for
:
k [ 6 (XH,d:I]
a is
cannot reach
and x ' i n
E
XH
and s o by p r o p o s i t i o n 1 . 9 , A separa-
,
x
# x'
, there
f E A
exists
by Hahn-Banach theorem t h e r e i s some a € Z '
f(x')
, then
i s holomorphically convex. separates
x
. Then
[A]
and by c o r o l l a r y 4.8,
X ; t h e n given
f(x) # f(x')
Z'
in
g e t : x belongs t o
t h e boundary and s o tes
K
; t a k i n g t h e upperbound o f both s i d e s when
f[l
[A]
with
is pseudoconvex.
Proof.- Given a compact s e t -
la o f ( x ) [ 4
bx(X,Zh
. The manifold
XH
with
with
is s e p a r a t e d b y O ( X H , C )
, so
it
i s a Stein-manifold. Comment.- We do not study h e r e t h e converse i m p l i c a t i o n i n p r o p o s i t i o n 4.10.
A r e s u l t of Gruman and C.O.
[41]
says t h a t any pseudoconvex manifold
space and i s
Kiselman [36,37] (X,p)
, completed
E with a b a s i s i s t h e maximal extension o f some
b , & X , 6 )-convex
(cf. definition 5.2).
by Hervier
spread over a Banach f
6 ,(X,&)
,
35
FRECHET SPACES OF COMPLEX ANALYTIC MAPPINGS
Moreover, an i m p o r t a n t example o f Josephson ( U p s a l a Univ., n o t p u b l i s h e d ) shows a domain i n
.- I f
, with
= !Lm(A)
u n c o u n t a b l e , which i s
A
OE((i ) - e x t e n s i o n .
pseudoconvex w i t h a p r o p e r
Proposition 4.11
E
is the maxima2 extension of some s . i . d . ,
X
extension of the associated c2ass Proof
.- By
AZX in
c o r o l l a r y 4 . 4 t h e r e e x i s t a such
extension of t h e associated c l a s s a sequence
(xn)
there e x i s t a
X
is t h e maximal,
U
(Z)
bXtX,c ) ,
u
f(x )
i s n o t weakly bounded i n
that
sup
15
f(xn)l
sup
such t h a t
=
+m
Z'
5
. Now
o f
i s t h e maximal l e t us given
and by ( i i ) i n theorem
X
. The
Ilf(xn)ll = +m
, therefore
. Since
X
such t h a t
in ~ , ( x , z )
which r e a c h e s t h e boundary i n f E A (Z)
4 . 9 a function
A
natural,
, then
Frechet space r i n bXtx,2) , which separates countable, a h i s s i b l e coverings o f X such t h a t X
there exists
sequence
C E Z'
A
belongs t o
such
t h e announ-
ced r e s u l t is a consequence of ( i i ) i n theorem 4 . 9 .
With the assumptions o f the previous proposition, if more
Corollary 4.11.-
E has a countable basis of open s e t s , there e x i s t s
f
c
0 tx,g
)
such t h a t
X i s f-rnadmaZ. P r o o f . - Clear by
iii) i n theorem 4 . 9 .
A p a t h o l o g i c a l example.
E
If t h e c o u n t a b l e n e s s i n ( i i i ) is n o t s a t i s f i e d by t h e maximal e x t e n s i o n of some s . i . d . , and n e v e r i s f-maximal f o r a l l
f
t:
,
n a t u r a l , Frechet space i n
6 (X, 6 )
instance
T
be a d i s c r e t e compact s p a c e such t h a t c a r d T >
{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,<
E
A
E
> 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
<
,
p(K)
k
M <
IIXkII
Mt4
ad
E is
t EU
m
j
we p i c k
chosen such t h a t i s a bounding
i s uniformly convergent t o t h e i d e n -
is contained i n
t E'U
.
6?iT
T = W ( S ~ + E ~ . for U )
Ilrk
, 5,
n 2 N2
n>,N2
-
pII
P
-
PI1 K t E l I U 5 ( M t 2 ) E "
, we
*
o b t a i n for a l l
The p r e v i o u s computation g i v e s t h e uniform convergence o f -1 f o px T~ p ( x ) on T (KtE"U)
u
P ( r k o p , gp)
.
P
E'W
\< ( M t 2 ) ~ ' and
By t h e Cauchy i n t e g r a l ( P r o p . 3 . 2 . d )
Now, g i v e n a polynomial
K t
.
o p
1I.k
t h e mapping
,
G[,+E'.u
.
l a r g e enough,
By (1) w e g e t
.
--- , where E , and K
El1
ITk
sup
, aZI x
such t h a t :
which i s i t s e l f c o n t a i n e d i n Let
<
El
n K+cl.U a l l
ft A
E
such t h a t f o r a l l
such t h a t
Xk
all
T ~ ( E )x d : ,
> 0
N2,€'
in
Q
41
on
nk(E)
belongs t o
A
X
, so
, by
fc
g P
A
near
t h e assumption (H2),
we g e t ;
HOLOMORPHIC CONVEXITY
42
When
p
d i v e r g e s t o i n f i n i t y we g e t
Now, w e have j u s t t o prove t h a t { p
5
compact set i n
when
Given a sequence
x x
describes in
KtE"U
-1
rk'0 p ( x ) )
.
KtE".U
, since
is a relatively
K
i s a compact s e t , t h e r e
e x i s t s y 6 K and a subsequence ( x n ) such t h a t xn y t 2.E".U -1 = p - l i n p(y) t E.U Moreover w e have p r e v i o u s l y o b t a i n e d : PXn Y ]ITk o p ( x - p ( x n ) l l less t h a n ( M t 2 ) E " and t h e r e f o r e w e have ;
.
- p(y)ll
l/rko p ( x n )
Now, s i n c e quence
(xn)
TI
4 (Mt4)E" <
E:
. Then,
and:
(E) i s f i n i t e d i men s iona l, t h e r e e x i s t s a n o t h e r subse-
k su ch t h a t
'Tk
p(xn)
i s conve rge nt i n
[p(y)
t E.U]nTk(E)
and t h e p r o o f i s complete. The c o n v e x i t y theorem.
oXfX,c1
Theorem 5.1.- Given a 8 . i. d , natural Frechet aZgebra A i n which s a t i s f i e s (HZ), where X is a manifold spread over the Banach space E with property (HI). I f X i s the maximal extension of A f o r 6 E ( ~ then ) X i s A-convex and separated by A .
Proof.
E
is s e p a r a t e d by t h e a d j o i n t s p ac e
E'
, so by
proposition 1.9
HOLOMORPHIC CONVEXITY
and (H1),
X
i s s e p a r a t e d by
Given a compact s e t convex h u l l o f
p(K)
K
in
5
since
.
A
43
, p(?(A))
X
is contained i n t h e l i n e a r
p belongs t o
,5 E
A
El
. Hence
p(?(A))
i s r e l a t i v e l y compact. Given a sequence such t h a t
p(xn)
(xn)
in
e x i s t s a subsequence
.
E E
5, E
Now, w e use t h e p o i n t s lemma 5 . 4 f o r
, there
?(A)
a
converges t o
p - l ( a ) r\ (Ixn t E' .U]
(x,)
introduced i n
large enough,
n
5,
If i s t h e r e an i n f i n i t e number o f t h e n t h e r e e x i s t s a subsequence
(xn)
which a r e e q u a l a t
which converges n e a r
y e X
y
,
and t h e
proof i s o v e r . We prove by c o n t r a d i c t i o n t h a t t h i s c a s e
5,
the
always happens. Suppose a l l
are d i s t i n c t s . For k
.
l a r g e enough, v k ( E ) c u t s p ( a ) t E'U -1 For such a f i x e d k , t h e p o i n t s (Ink(a)] a r e a l l d i s t i n c t p 'n,k 'n and t h e r e i s some f E A which s e p a r a t e s them by p r o p o s i t i o n 2 . 4 and by
t h e Eaire property of The mappings
u
A
n,m
. Now
we c o n s i d e r
= f
's,
-
u n Y mC n k ( a ) ] = f ( c n , , )
,.@a nk(E)
n [p(a)
o u t s i d e a set v a l u e s on
t
Z
p-'(z)
O
-
f
0
# 0
f(Cm,,)
.
such an f -1 bqlong t o @E [p(a)+E'U, P,'
. Thus,
given a polydisc A i n
v a n i s h e s a t any p o i n t o f A not e v e r y u n rm o f measure z e r o . Thus f t a k e s an i n f i n i t e number of
E'.U]
A [u(cn
, whenever
t E'U)]
z C A*-
Z
W e now prove t h e c o n v e r s e . By lemma 5 . 4 , f o r a s u i t a b l e n s u f f i c i e n t l y l a r g e , t h e r e e x i s t s a compact s e t
By lemma 5.2, f o r any nomial of d e g r e e
(d,
.. , , d ,
x
v a l u e s on
A*
describes p-'(z)
u [En
+
[LJ
w i t h a nonzero measure.
, we
d >, e >/ e o
Now, we a p p l y Lemma 5 . 3 when
a]
e)
with
E'.U] (5,
in
t
IT
T
k
(El
in
Xk
k
and for
such t h a t ;
c a n c h o o s e a normalized polyx
6
= En,
. Thus,
E'U)]
Q
.
f
, when
such t h a t
-1 p ( x ) , f o px Tk
p(x))
t a k e s a f i n i t e number of z
belongs t o a s u b s e t o f
44
HOLOMORPHIC CONVEXITY
Remark : If t h e manifold h a s a f i n i t e number of s h e a v e s , i t s always A-convex when it is t h e maximal e x t e n s i o n o f any set which c o n t a i n s t h e mappings 5
p
, cc
E'
, and
A
in
q(X,
)
i s i n v a r i a n t by d e r i v a -
t i o n . Only t h e first p a r t of t h e p r e v i o u s proof i s n e c e s s a r y u s i n g c o r o l l a r y 4.8.
45
CHAPTER VI : SPECTRUM AND MAXIMAL EXTENSIONS
E, Let A be a unitary subalgebra of 0 (X, @ ) which is invariant by derivation. We denote by Ef p the set of mappings {s o p 1 5 6 E f Since ( 5 o p)'(a) is the constant C(a) which belongs to A then (El o p) t A is also invariant by derivation. (X,p) is a manifold spread over a complex c, v. s .
.
Q 1.- Manifold structure and spectrum. Given a linear mapping h from (E' the mapping f
hn(a)
+
h [fn(a)]
nient to notice that hn(a) Let
S(A)
from
,
6: , we
p) t A
6:
(E'
vanishes on E'
be the set of such h
p) t A to
h # 0
o
p
, with
, all
to
denote by
. It is conve-
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)
<
fa A
,
i-
, there exists a neighbourhood b C (a
o
E (necessarily unique by Hahn-Banach p) = 6 o r(h) , all .E t S E'
.
- The restriction to
of h is a homomorphism.
A
Comment about (s3). Any h E S(A)
has a restriction to
which is weakly continuous for the pairing <E',E> 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 <
, for a
r
suitable r > 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
<
m
,
b
describes
(s2)
x &X
belongs t o S(Al
b a l a n c e d open neighbourhood of
fia = p,-1 [-p(x)+a]
i s d e f i n e d . Then
, (s3) are
Theorem 6.3.-
, any
V2
W e can t a k e for -1 p,
where
,
h a s p r o p e r t y (s3) s i n c e
Proposition 6.2.- The evaluation 4 a t any point and n(4) = p ( x ) , Proof.-
U'
a E Vp
all
p(x)
and t h u s
obvious.
a ) Given h E S ( A ) ; the s e t s NV = {ha I a € V ) are a basis SIAl , when h describes S(A) and V the
f o r a Hausdorff topology on
-
s e t s Vh with (a1) (s,) properties, This topologEl i s f i n e r than pointwise convergence on A ,
For t h i s topology (S(A),nrl i s a manifold which may, not be connected, spread over E The mapping x -t P i s a morphism u from X t o S(AI b ) Let % ( A ) be the connected component of u(X) Then,
.
.
.
the morphism u : X .+ i ( A / i s the maximal extension of an the function : h + h l f ) i s the extension of f
7
c)
Proof.-
ha
f o r any
aE V
and y e t by (1) t h e mapping n(ha)
A
for
6 E (C
i s separated by the extended algebra
i(A)
, TI
n(h) t a
mapping r e s t r i c t e d t o
NV
below t h a t t h e mapping
a
have d e f i n e d a t o p o l o g y '&on
a
i s c o n t i n u o u s from
+
ha
. Hence
+
in
, so w e
i s c o n t i n u o u s and
h,(f)
TI
{a
+
)
i s a neighbourhood
a ) By t h e r e l a t i o n (1) of p r o p o s i t i o n 6,1, NV
o f each Since
.
V
ha)
in
S(A)
(S(A),f
).
is its inverse
i s a local homeomorphism. W e prove belongs t o
6E(V,c
)
and t h e n
is
SPECTRUM AND MAXIMAL EXTENSIONS
48
finer than pointwise convergence on A
since
haze =
h
and so
is a
Hausdorff topology. Using proposition 6.2, it is easy to check that u is a morphism from X to
(S(A),r)
b) Now we prove that a where V
a€ V
+
, we
ha+tb(f) = (ha)tb(f) =
belongs to 6 , ( V ,
ha(f)
is the domain in E
in E
enough and b
.
6
)
, Given
associated with (sl) ; for r small
get by (1)
C tn ha[fn(b)] n>,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) <
with
A
2
=
V
Now, w e have o n l y t o prove t h a t be a morphism from
u
ded F r e c h e t sp a ce
f
+
X
* u (A)
to
a
t%
, all
V
i n t e g e r s n.
!!
= X.V
( A < 1)
.
is contained i n
X(A)
sp. A
. Let
know by theorem 4.7 t h a t t h e e x t e n -
i s a l s o s . i . d . and n a t u r a l . That i s t h e mapping
i s c o n t i n u o u s from
l(h)
, we
?(A)
, all
i s an a b s o r b i n g s e t , t h e r e q u i r e m e n t s
E}
U
E
49
A
to
6: , for
any
h E ;(A)
. Since
f ( h ) = h ( f ) ; t h e p r o o f is complete. Remark.- When
p
E’
i s contained i n
A
, spy*
A
sp? A ,
u
.
Proposition 6.5.- [17,8011 Suppose E nomed and l e t be a countable, The admissible covering o f X , such t h a t p ( w i i s bounded a l l we &ma2 extension of the space f o r 6 E ( ~is) a connected component Aa of BP? A f o r the topology of theorem 6.3. Whenever E is r e f l e x i v e , the , same r e s u l t is true f o r sp. A
Proof.-
.
It i s a p a r t i c u l a r case of theorem 6 . 4 provide d by p r o p o s i t i o n 4.3.
Proposition 6.6.- Suppose E is a strong dual of a r e f l e x i v e Frechet space. Given A a natural, 8 . i. d, Frechet algebra i n d:) which contains E l 0 p and is invariant by derivation, then the m a x i m a l extension o f A for oEco) i s a connected component o f sp. A f o r t h e topology induced by S ( A )
o,(X,
.
Proof.since
By c o r o l l a r y 4 . 2 , w e know t h a t E
E’ = EL and
B i s r e f l e x i v e . Then spf*A = s p .
Remark.- When
E
EfS
is r e f l e x i v e
A ,
i s f i n i t e l y d i men s i o n al , b X ( X , & )
is a s p a c e of t y p e
A
24
as we have s e e n i n Chapter I V . More is known : s p . O ( X , C ) i s c onne c te d and
t h e r e f o r e i s t h e maximal e x t e n s i o n [38]
.
Now, w e are concerned w i t h t o p o l o g i e s on t h e whole s p a c e
=
bx(X,6:
W e d e n o t e by
on
0 (XI
such t h a t
6 (XIs
(resp.
b(X)
spy 0 ( X I i s r e l a t e d w i t h ? = ? c& ( X ) ] . 6 (XIc , r e s p . 6 (X)p.c) t h e p o i n t w i s e topology
( r e s p . compact t o p o l o g y , r e s p . precompact t o p o l o g y ) .
S i n c e p o i n t w i s e convergence i m p l i e s uniform convergence on compact
6 ( X I , t h e set 6 ( X I s and 6 ( X I c . I t
sets f o r e v e r y eq u i co n t i n u o u s s e t i n
of precompact ba la nc e d
convex s u b s e t s are e q u a l f o r
w i l l b e de note d by
SPECTRUM AND MAXIMAL EXTENSIONS
50
K(X).
If
is complete and any set of
is metrizable, 6 ( X I C
E
K(X)
is
compact.
.
Proposition 6.7.- [SO, 1 4 1 There i s a locally convex, Hausdorff, topology on 6(X) , denoted by 0 ( X J E such that : a ) i t i s the f i n e s t one t h a t induces on any T € K(X) t h e pointwise convergence, b ) given a c . v . s . G , a linear mapping from 6 (XjE t o G i s continuous i f f i t s r e s t r i c t i o n a t any T E K f X ) i s continuous. Proof.-
6 (X) 6(xIS
7;,
7; b e t h e f ami l y o f convex, b a l a n c e d , a b s o r b i n g
Let
su c h t h a t
T
at
7; is
i s a neighbourhood o f
M r\ T
XM
> 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
<
for all
a
i n a convenien-t
EXTENSIONS OF VECTOR VALUED ANALYTIC MAPPINGS
neighbourhood
F
of t h e o r i g i n of
U
trp*(%) t U
proved t h a t
t h e assumption ( i ) o f theorem 7 . 5 . We d e n o t e by
and f o r a l l
i s contained i n
-
t h e p r o j e c t i o n of
TI
proposi t i on 7.6 t h a t
-
into
S(A)
IT^^ ( G )
?[ty*(%)]
and
TT
assumption (iv) o f theorem 7.5 i s s a t i s f i e d by
. Thus
= g y y [%I
[tIp'(G)]
Lastly, assumptions
I
We have j u s t
2 E w
; t h a t is
we have proved i n
F
o v g6(g,F)
, Then t h e
and f o r any
E
we have
t h e assumption ( i i ) o f theorem 7 . 5 .
ttfris continuous i n
by p r o p o s i t i o n 7 . 6 . b ) . A l l t h e
u(X)
theorem 7 , 5 a r e s a t i s f i e d by
of
E A
tL+u.
i s s e p a r a t e d by t h e e x t e n s i o n
S(A)
f
for a l l
S(A)
63
'Yw
and t h e p r o o f i s
complete a f t e r u s i n g a n obvious c o n n e c t e d n e s s argument w i t h theorem 6.4.
Corollary 7.6.a.-
If
i s a space of type
A
b2e and countable covering o f
Corollary 7.6. b.
- If
Y
, then
A2(
y=
, which a
i s f i n i t e dimensiona2 and A =
F
as an a h i s s i -
~ y s o 2 v e sthe diagram ( 1 ) .
6 (YJ CC )
then the
~
same conc2usion i s true. They are particu2ar cases of theorem 7.6. Theorem 7.7.
- If
i s a Banach space with. a Schauder basis and
F
= 0 (Y),
A
then t h e same conczusion is true.
Proof.- By t h e Gruman-Kiselman-Hervier s i o n o f some
f6
.
0 (Y)
theorem
is t h e maximal e x t e n -
b e l o n g s t o a space o f t y p e
f
By theorem 4.5
Y(A)
AU
and w e can a p p l y c o r o l l a r y 7 . 6 . a . Comment.- The theorem 7.6 c a n b e g e n e r a l i s e d [48] ?(A)
such t h a t f o r any sequence
there exists
ie
(j[?(A)]
(yn)
such t h a t
in
?(A)
sup
[
t o maximal e x t e n s i o n which r e a c h t h e boundary,
z(yn)(
=
. If
t m
is
A
a s . i. d . n a t u r a l F r e c h e t s p a c e , t h i s assumption i s s a t i s f i e d by theorem 4 . 9 , b u t y e t no o t h e r example does e x i s t , § 5 . - Extension
Let
E
F
and
from
X
X
of p r o d u c t s .
and
. For
be two m a n i f o l d s r e s p e c t i v e l y s p r e a d o v e r c . v . s .
Y
any
i n t o @(Y)
f e 6 ( X x Y)
f * i s t h e map
x
-c
{y
+
f(x,y)}
,
(resp. E and FI
Theorem 7.8.-
If
(resp : 0 [xJ
0 (Ylb]
E
and the map
is metrizabZe f *E O [ X , O ( Y f + f * i s onto 0 [xJ 6 ( Y J C ] )
)~]
.
64
EXTENSIOIVS OF VECTOR VALUED ANALYTIC MAPPINGS
-
n * ( f 1 ( x ; h t i s a c o n t i n u o u s homogeneous
P r o o f . - To b e g i n w i t h , we show t h a t
E
polynomial from
6 (Y)c
into
( r e s p . @ (Y)b)
n
for a l l
e SrJ , a l l
X E X . be a convergent sequence i n
(hi)
Let
m e t r i z a b l e t h e r e e x i s t s a sequence
= 0
l i m Eihi
. Then
( f n ) (x,Eihi)
l i m Ei =
theorem 6 . 1 0 ) . Using c o n t i n u i t y of
E
t o zero, since
E
is
E
i n fl such t h a t l i m E = t 03 and i i i s bounded i n 6 (YIc ( r e s p . 6 ( y I b by
and homogeneous p r o p e r t y , w e o b t a i n t h e
f i r s t l y a t z e r o and a f t e r everywhere s i n c e ( f n f i x )
(fn)*(x)
i s a polynomial. Now, p b e i n g g i v e n
( p > 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 <
i s t h e a s s o c i a t e d norm. The n e x t Von Neumann theorem w i l l be used :
one from
Dx o n t o H2
Proposition 8.1.-
.
i s one-to-one onto
A'
-
Proof.-
The s a i d r e l a t i o n
A-'(y)
and it i s t h e one i n
R;
=
(Ker A)
i m p l i e s t h a t t h e p r o j e c t i o n of any
Proposition 8.2.11)
is
DA
A'.
x
and
[I .I\
i s one-to-
RA '
, which
x i A-'(y)
is easily verified, on
.
A-l(y)n
Ei
y e t belongs t o
and H 2 Bd E be given suppose :
HI- A E
Let
0
Id t A o
,>
~;i c D-B and [ [ i ( y l [ [ 4 [ [ z l I y ) [ [
.
, at2
y 6 ~;i
B')-'(Y)
11 , a t 2
Then we have : (2)
R~
Proof.-
c R~ Let
t h e sequence
and y e RB (y,)
11 I
A')-'I~)
11
6
11 I
b e g i v e n and l e t defined i n
E
by
x y
e get : e x i s t s by Von Neumann theorem. W
=
(
1
-
E R~
.
-1 ( y ) , We a r e d o i n g w i t h
BT)
yk t A
.-
A(yk)
, this
sequence
POLYNOMIAL APPROXII.IATION
68
c l o s e d , bounded set i n H1 i s weakly compact, t h e r e exists
a
Since
,
i s i n t h e weak c l o s u r e o f xo E H1 I IIxoll 4 11x11 such t h a t x 0 i ( y k ) f o r any n , However A o A(yk) i s s t r o n g l y c o n v e r g e n t t o y k >/n t h e n ( x o , y ) b e l o n g s t o t h e weak c l o s u r e o f a bounded s e t o f t h e graph o f
u
A
. Since t h i s
g r a p h i s c l o s e d we have
y
= A(xo)
and
-
RA '
=
(xo)
A'
X T , Y
Theorem 8.3.- Let
A
and
Z
AY
II
is i n c l u d e d i n
, therefore
xo b e l o n g s t o R i II( Bi )-'(y)Il. and ( [ ( A ' ) - l ( y ) I I ,< IIxoII =
The same argument p r o v e s t h a t y
RB
be given w i t h
S.T=O.
Suppose :
D?"Dsc
(3)
D;
and I I z ( y ) l [ 2 < l l i ( y ) [ 1 2 + I l S C y ) [ [ 2 , p22 Y E D ? n D S .
Then we have : a)
R J c Rr 8 R s and
b)
all
y E RJ
u E D;
A
Es
, there
Let
A
A
2
X
u
e d s t only one
c D,A
%
and
suoh t h a t :
y E Xer.S
c) Furthermore i f Proof.-
= {O)
RT A Rg
Y
v =0 ,
then
be d e f i n e d by
A(x,z)
=
T(x) t z ( z ) , C l e a r l y
is d e n s e l y d e f i n e d .
-
Let a sequence T(x n that
t S(zn)
, Rz> = T and 5
operators
converges t o that is
,
A
(xn,zn)
converges to
y
-
, let
0
such t h a t
. Since
y
p
are c l o s e d ,
, then
p(y)
(xn,yn) S o T
be t h e T(xn)
w e have
=
0
converges t o
, we
T(x)
= p(y)
and
and
-
p r o j e c t i o n map o n t o converges t o
(x,z)
easily verify
RT ._The and S ( z n )
p(y) %(z)
= y
-
p(y)
,
is closed.
Now w e v e r i f y t h a t t h e r e q u i r e m e n t (1) o f p r o p o s i t i o n 0,1 i s satisf i e d by
A
g i v e s (1)
.
Since
i~ x gs
. We have < RT
. Then
D-A =
, R-S
>
Dc;,n DS = 0
I
and
w e have
a ) and b ) are g i v e n by ( 2 )
A(y)
Ei
.
(?(y), S(y)) (Ker A ) '
, then
(Ker T
(3) - 0
X
Ker S )
=
69
POLYNOMIAL APPROXIM4 BION
y E Ker S
L a s t l y , if Then
g(v)
Ker
? niiS =
5 2.-
belongs t o
.
{O}
we have
Ker S
n R;
S
=
o
=
g(v) and
{O]
=
S o T
since
0
v
0
.
belongs t o
A Hbmander's r e s u l t
Let
be a f i n i t e l y d i m e n s i o n a l S t e i n m a n i f o l d , n o t n e c e s s a r i l y
X
X
spread. W e t a k e a r e l a t i v e l y compact open s e t w i n
such t h a t
W
is a l s o
a S t e i n m a n i f o l d , and w i s endowed w i t h a h e r m i t i a n , Riemannian m e t r i c i n v a r i a n t element o f volume i s d e n o t e d by
whose
this construction).
For any
, L2(y ) P
m
6 C (w)
tf
b l e d i f f e r e n t i a l forms of t y p e
. (cf.
do
i s t h e H i l b e r t space of square i n t e g r a for t h e measure
(p,O)
e - Y , do
.
d i f f e r e n t i a l o p e r a t o r o f Dolbeaut cohomology i s w r i t t e n
d (GIN
(ai)
Let-a
for
[SO3
. The
N , W e use t h e
be g i v e n f o r some i n t e g e r
following H i l b e r t i a n , densely, c l o s e l y , defined o p e r a t o r s :
c
T~(U)
N
ai ;ui
from
CLi((p
( a i dv)
from
L 2~ ( Y
s;(v)
=
Ji(w)
= ( C [ail 11'2,
w
from
N
11
into
~ : ( y )
into
CL:(v
2
into
11N ,
L:(
-
The metric of w and '4 can be chosen such t h a t the equation D~ n R? f o r any 11) : $ ( u ) = v , ha8 a unique sotution i n
Lema 8.4.v6
L y ( l p l such t h a t :
dv =
0
I I v 1 2 (Z[ai12)-' do <
and
Moreover, the BOh4tiOn u (21
IIuI2 e-'pdu
Proof.-
For
N
,
v e r i f i e s the following estimation :
I l v 1 2 IXlai121e-C da
4
1 and a = 1
, this
.
lemma i s t h e theorem 5.2.4 o f [ 5 0 ]
,
which g i v e s t h e f o l l o w i n g e s t i m a t i o n : (3)
and
Ilv112
s
6 Ilf (v)1I2 + I I S ( V ) ~,~a~l l v E DT 1 = s1 *
Using t h e assumption
D?
n D~
when
v
a 6 @(;IN
belongs t o
, we
see t h a t
D?a f ) DSa N N
Then ( 3 ) g i v e s t h e f o l l o w i n g i n e q u a l i t y :
.
DS
ai
, v
with
T
= T1 1
belongs t o
'
POLYNOMIAL APPROXIMATION
70
(4)
Il(3,")
11'
(v)
I[(?,")
,<
(v)
11'
11 , a l l
+ IlS:(v)
N
TN
Now t h e announced lemma i s p r o v i d e d by theorem 8 . 3 .
a
When ( 4 ) i s s a t i s f i e d and
D-,a
h a s a unique s o l u t i o n i n and
u
5 2.-
N
verifies :
SN o :T
n
, the
0
for
N
any
.
v E D-a nDsa
a TN(u) = v
equation v E
a
n
%a
N
Ker SN
A Runge theorem
Let
be a c . v . s . whose p o i n t s are denoted by y , L e t A be a com-
E
X be g i v e n , ( i 3 ( X , E )
pl ex manifold
X x E
ned on
such t h a t
x
is the set of functions
0 (X)
belongs t o
f(x,y)
-+
f(x,y)
and
y
+
defi-
f(x,y)
i s a polynomial.
Theorem 8.5.-
Suppose
a f i n i t e dimensional S t e i n manifold, l e t
X
holomorphically convex compact s e t in X given.
and
a
K
U a neighbourhood o f
Then there e x i s t s a r e l a t i v e l g compact neighbourhood
W of
K be such
K
that
(i) K C W C Z C U
(ii) For any f E 0 / IU,E/
and
E
> 0
such th a t : Ilg(x,y) -
-5
ffX,Lf)IIK
Furthermore, continuous i n X
X
g
E * IIffX>Y)IIW
9
can be chosen continuous in
v e l y compact and c o n t a i n s
g
c 6(xlN
K
W
K C W L:
p o i n t s a r e denoted by
. To b e g i n w i t h ,
K
Cw(w)
Firstly,
y
X
x
, with
whenever
E
X
, such
is
f
z
w
g
N C
in 0 I (w,E). IN ,
C U ; h e r e A i s t h e u n i t p o l y d i s k of
. For a l l
support i n
i s fixed i n
that w is relati-
we c o n s t r u c t
= g - l ( A ) A U i s a r e l a t i v e l y compact open p
]O,llI
i s compact and p can be chosen such t h a t
X 6
.
YE E
i s h o l o m o r p h i c a l l y convex, t h e r e e x i s t s
such t h a t
which v e r i f i e s
all
E ,
P r o o f . - Let w b e a n open S t e i n sub-manifold o f Since
g E 0 1 fX,E)
there e x i s t s
E
.
K
the set
M
is contained i n
6
set
. Let
whose
g -1( p . A ) f l M
W and v a l u e s 1 i n a neighbourhood o f
W
M ,
W e a p p l y t h e lemma 8 . 1 on t h e S t e i n manifold
POLYNOMIAL APPROXIMATION
w x p.A
;IX
; since
. . [C
jzi
f
-
71
gi(x)l 2]-1/2
belongs t o
t h e previous lemma g i v e s a unique s o l u t i o n of t h e equation
-& . f
=
c
-
[zi
gi(x)]
For t h i s s o l u t i o n t h e f u n c t i o n
dG =
fies when
and s o i s a n a l y t i c i n
0
.f p.h .
x
G
w
X
C Ezi - gi(x)] ui
t
Now, t h e uniqueness o f s o l u t i o n s implies t h a t
u
y
X
describes
and t h i i s
E
belongs t o
G
c, / (w
CIA),
x
2
2
i n a convenient space [L,(Y)]
dui
Cm(W
veri-
i s polynomial l i k e f
pA,E)
.
Using mean i n t e g r a l values and (51, we g e t t h e following e s t i m a t i o n s G :
for
IG(x,z,y)
(7)
Here, k w x p.A
,
1 and k 2 are
f u n c t i o n s which a r e l o c a l l y bounded i n
The Taylor expension of ,y) e
CN x
(w,
E)
-ni0
. Since
A > 1 such t h a t
(8) IIG(x,g(x),y)
-
(Aqt1
A')-'
do
i s contained i n
Gn(x,g(x),y) q
=
Gn(x,y,z)
and
i s a compact set o f
Gn
p.A
belongs t o
, there
exists
and we g e t by ( 7 ) :
p.A
Gn(x,g(x),y)llK d
C nsq
.
Ilk2(x,g(x) 111,
Pointing o u t t h e e q u a l i t y of with
gives
p.A
z 6 p.A
for a l l
g(K)
A.g(K)
-
in
G
belonging t o
(w,E)
s u f f i c i e n t l y l a r g e and
o
G(x,g(x),y)
, the
and
f
in
K
and
i n e q u a l i t y (1) i s v e r i f i e d i n w
g(x,y) =
Moreover t h e Cauchy i n t e g r a l f o r t h e following e s t i m a t i o n f o r g
( I f ( x , y ) 11
Gn
I
Gn(x,g(x),y)
.
ns4 and t h e e s t i m a t i o n ( 7 ) give
,
POLYNOMIAL APPRDXIMATION
72
Then t h e r e e x i s t s a l o c a l l y bounded f u n c t i o n i n w such t h a t :
open sets wn
such t h a t
sub-manifold o f
.
f E f/ , (w2,E)
(i)
=
fl
fn
(iii)
,
=
wn c u wn
X
w
x E
. There e x i s t s a sequence o f
En c wn c untl , , K i s contained n
.
wn is a n open S t e i n
i n W1 and By induction, we a r e c o n s t r u c t i n g a sequence f n such X
X
and :
f
E T(mntl,EI
W
, in
I[f(x,y)llw
(1) i n t h e whole
Second s t e p . - We extend
that
.
[ g ( x , y ) [ 6 k3(x)
(9)
and
-
IIfntl(x,y)
is a s u i t a b l e neighbourhood o f
bounded i n
w
fn(x,y)IIP < wn
2-"
'E
w1' and each 4
is l o c a l l y
kn
nt2
Actually t h e first s t e p g i v e s f 2 and W , Now suppose be constructed. Then, t h e first s t e p g i v e s f n t 2 c, ~ ( ~ n t ,E) 3
I
- fntl(x,y)[[$
t h a t [[fnt2(X,y)
wnt2
set contained i n
I\fCx,y)\\
the (ntlIth order f o r
ntl
<. rl
.
\lfnt1(x,y)[Iw
n
. The e s t i m a t i o n ( i i ) a t t h e nth
, here
f 2 ' ' * ,fntl such 9
Wn
is a
order gives ( i ) a t
small enough. By ( 9 ) we g e t t h e following :
, with k l o c a l l y bounded i n wnt3 ; n then a f t e r using ( i i ) a t t h e nth order we check ( i i ) a t t h e ( n t l I t h o r d e r .
&nt2(x,y)I $ k ( x )
IIfntl(x,y)IIw
L a s t l y , it is c l e a r by ( i ) t h a t
f n ( x , y ) i s uniformly convergent on g belongs t o 6 I (X,E) and satis-
. Then i t s l i m i t fies t h e i n e q u a l i t y (1) . any compact s e t o f
Third s t e p . - If l y bounded i n Ilg(x,y)
-
bounded i n
f
wntl
X
is l o c a l l y bounded i n x E
f n ( x , y ) I I ^ <.
w
n
x E
wn
U x E
, each
fn
is a l s o l o c a l -
by ( i i ) . Furthermore (ii) provides 2'
.
f o r each
E
n
2-" and
Ilf(x,y)llw g
; thus
i s continuous with
is locally
g f
together
.
C o r o l l a q 8 . 6 . - For any open s e t w i n X we denote by T c ( w J E ) the s e t f w ? C t i G ? 2 8 which belongs t o 9 ( w J E ) and i 8 continuous in w x E , If w is an open Stein s u b m n i f o l d of X Tcc(X,E) is sequentially
of
POLYNOMIAL APPROXIMATION
.
dense i n P c ( w J E I b Proof.- Let
be an i n c r e a s i n g sequence o f compact holomorphically
Kn
, and
convex s e t s whose union is w
f 6 Tc(u,E)
E
llfll
= 2-" '
E
-
2"(f
Iff(x,Y)[[
f
i s bounded i n
=
f in Pc(u,E)b
Proof.-
Un
of t h e
fnE Pc(X,E)
such t h a t
. Hence t h e sequence
.
fn
K be a compact s e t i n X , we introduce the space when w is running through the neighbourhoods
Let
2 9 q(w,Ejb
K. If K is holomorphically convex, q ( X , E I %(KJEjb
> 0
Wn x Un , We choose
i s bounded on any compact s e t o f w x E
fn)
E
*
; t h a t provides a sequence
;;x"n
Corollary 8.7.-
of
'
such t h a t
i s convergent t o
qc(KJEib
<
n' i s continuous t h e r e e x i s t s a neighbourhood
f
Since o r i g i n of
Wn which s a t i s f i e s
g 6 4),(X,Ef
- g(x>Y)llKn
llf(x,Y)
be given. By theorem 8.2
i n w such t h a t f o r any
t h e r e e x i s t s a r e l a t i v e l y compact set there exists
73
.
is sequentially dense i n
It i s an obvious consequence of t h e c o r o l l a r y 8.6 a f t e r p o i n t i n g
out a fundamental system of S t e i n neighbourhoods of
K
.
We a s s m e E i s a complex c . v . 8. An open s e t B i n is X-equilibrated iff X x {Ol is contained i n 8 and I x , y I e $2
Definition 8.1.X x E
implies
(x,t.y)E
n , at1 It[ 8
I
.
Theorem 8.8.Let X be a f i n i t e dimensional S t e i n manifold, 8 a X a q u i l i b r a t e d open s e t i n X x E , w an open S t e i n sub-manifold of X be given. m e n Pctx,E) is sequentially dense i n ((w x E ) Q 81b
.
Q
Proof.-
By c o r o l l a r y 8.6, we have j u s t t o prove t h a t t / ,(w,E
t i a l l y dense i n
6((w
X
E) n
n Ib
.
C
i s sequen-
That is t h e next lemma.
&ma 8.0.-
Let X a complex manifold not necessarily f i n i t ( l y dimensional, 8 a X-equilibrated open s e t in X x E be given. Then 0 1 .fX,E) if, s e q u e n t i a ~ ~dense y in 6 (5312,
.
Proof.-
Let
f
6(n)
be given ; we use t h e Taylor expension
f
=
C fn n>,O
POLYNOMIAL APPROXIMATION
14
with fn
If(..
1
= 5;:
fn ( x , y )
e
i8
belongs t o Ci),(X,E) , Let
, y) e
-nit3
dt3
I
{(x,ty)
(x,y)
Q
It1 < 1)
a
. Clearly
be a sequence which convergesto 1 and
E~
( E ~ converges ) ~ t o i n f i n i t y , For any compact
6
(x,y)G R
for all
,
in R
Q
i s another compact s e t i n
and
f o r a s u i t a b l e p > 1 we g e t :
of R
C f ) i s bounded on any compact s e t n nS N fn' i s convergent t o f i n O ( R ) ,
C
therefore
a
Theorem 8.9.-
-
N (f cN
Then t h e sequence n$ N
.
Let X be a manifold spread over a Frechet c. 0. 8 . E and be the m a x i m a l extension of a 8 . i. d., natural Frechet space
suppose X i n UIX). Let F b e another sequentially complete c. v. librated domain i n X x F together be given.
8.
and R an X-equi-
Then the b ( i l / - m a x i m a l extension f o r O E X F ( 6 )is an open s e t of X x F .
-
Proof.of
L e t 0 be t h e domain of
which extend 52 f o r
X x F
mal extension o f
ii
to
F
v :y
p
into
E
X
F
v o u
=
q
'f:
Now, w e prove t h a t lity
ii
X x F
Y since
ii
Actually, l e t For any
f E
u : c
from fi t o
x
+
6+
(Y,q) and
X
d
from
Y
u = p
into here
X
X
p
F
, that
v
JI
: (x,a)
p o v
and
q
i s one-to-one,
a
t h e mapping
v e r i f i e s v o u = iden-
on Y
+
in 6 y ( ~ , ~ )
i s t h e p r o j e c t i o n of
i s a morphism from
v
be t h e maxi-
X
X
F
c
u(fi) e n t a i l s t h e
into
X
X
F
.
that w i l l entail that
-.
Y
is an
(up t o an isomorphism) which c o n t a i n s fi and t h e equai s maximal. y'
and
y"
be given i n
such t h a t
Y
v ( y ' ) = vfy").
0 ( h ) , t h e nth d e r i v a t i v e f n ( x , a ) belongs t o Tc(X,F) and gn i n O ( Y ) which can be w r i t t e n gn = f" o v ; t h u s
has an extension we g e t
(x,a)
. The t h e e q u a l i t y of
e q u a l i t y everywhere and open set of
OEX,(c ) . Let
by theorems 7.6 and 7.1. Let us consider
( q ( y ) , $(y))
.+
which i s maximal among t h e open s e t s
F
can be extended according t o 'Q ( r e s p . $)
( r e s p . by(Y,F) t i t y and
X
eExF(t ) .
for
The a n a l y t i c mappings from
X
gn(y')
gn(y")
. The
extension
g
of
following r e l a t i o n by theorem 6.12 and lemma 8.9
f
in
0 (Y)
verifies the
POLYNOMIAL APPROXIMATION
Since
Y
i s s e p a r a t e d by
6 (Y)
we have
75
y" and t h e p r o o f i s
y'
complete. When
i s f i n i t e d i m e n s i o n a l and
E
v i o u s r e s u l t i s improved
Let
CoroZZar~8.9.-
i s a Frechet space t h e pre-
F
i n t h e f o l l o w i n g manner.
X a manifold spread over a f i n i t e dimensional
R a X-equilibrated domain i n X
be given. Then t h e
F
x
with
X
P r o o f . - We have o n l y t o v e r i f y t h a t 61 b e i n g c o n s t r u c t e d i n
X
is an open s e t
extension f o r O E X F ( 6 )
x
maximal extension of
-X
theorem 8 . 9 c o n t a i n s
.
6in X
-
and
to
O(Sl) , we
6 (2)
I[f n ( a )
d e n o t e by
. There ,all
6 (6) , a l l
hood w o f t h e o r i g i n i n
Q i n w w e have
set
F
-
as t h e
X
in
K
X 0
, its
maximality of
*
R
as i n
_ - . For any
K
aE F
X
such t h a t [ l g P ( a ) [ l K ,<
K x w c il. For any b a l a n c e d compact
such t h a t
[/gnllkxQ<
in
sum b e l o n g s t o
[[fllKxQ
0 (K
x -+ f " ( x , a )
. There e x i s t s a b a l a n c e d neighbour.
The l a s t i n e g a l i t y e n t a i l s t h e summability of t h e s e r i e s on
F
O(.X)-
X/R
t h e e x t e n s i o n of t h e mapping
g (a)
e x i s t s a compact set fG
and
a f t e r t h a t a p p l y t h i s theorem.
By c o n t r a d i c t i o n , we t a k e a compact s e t f E
x F
E
6 (Rl-maximal
X
U)
X gn ( x , a )
and i s an e x t e n s i o n o f
f
. The
gives the announcedresult.
Comment.- By t h e same way i t i s p o s s i b l e t o e x t e n d t h e c o r o l l a r y 8 . 9 t o i n f i n i t e dimensional
E
i f t h e Nachbin's t o p o l o g y c 6 2 1 is b o r n o l o g i c a l .
But very few r e s u l t s a r e known [18,29]
is known on m a n i f o l d s .
and n o t h i n g c o n c e r n i n g t h i s problem
76
Index o f Symbols
s e t of i n t e g e r s r e a l ( r e s p . complex) f i e l d spread manifold presheaf o f continuous and 2-valued germs on s e c t i o n over
E
of a sub-presheaf FE(2) o f
U
sub-presheaf o f
CE(2)
provided by complex a n a l y t i c
CE(Z)
germs sub-presheaf of 6 , ( 2 )
provided by l o c a l l y bounded ana-
l y t i c germs sub-presheaf of
provided by r e a l a n a l y t i c germs
CE(Z)
= OE(W,C.) Hausdorff l o c a l l y convex vector space 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 defined i n s e c t i o n IV,1 t o p o l o g i c a l spectrum defined i n s e c t i o n VI,2 defined i n s e c t i o n VI,1 s e t of continuous semi-norms on a c . v . s . t o p o l o g i c a l a d j o i n t space of
E
continuous endomorphisms from t h e c . v. s. other
= Eb
( r e s p . E$
E; ( ')c,resp.pc,resp.s
E E
t o the
Z
L(E,E)
E'
equipped with s t r o n g ( r e s p . weak) topology
E'
equipped with t h e topology induced by a Frechet sub-
space o f (
.I
6(w)
which c o n t a i n s
E'
equipped with compact ( r e s p . precompact) ( r e s p .
pointwise) topology
( . I Ey ( . I b
equipped w i t h E-(resp. b) topalogy d e f i n e d i n s e c t i o n VI,2
f"(a)
nth d e r i v a t i o n o f
fn( x ,a)
nth d e r i v a t i o n o f
h(a)
homomorphism h
f f
at
x
translated t o
VI ,1
%I)
s e t of f i n i t e s u b s e t s of
I
a
, defined
i n section
s e t of bounded mappings from
I
d: equipped
to
with
t h e uniform norm subspace o f km(ld) p r o v i d e d by sequences which v a n i s h
for
n
l a r g e enough
set of convex, b a l a n c e d , e q u i c o n t i n u o u s , precompact p a r t s
of
6 (x)~
llfll,
sup
, when
((f(x)((
X DA ( r e s p . R A ) 5
A
0 IC(X,E)
XO
T
'
mappings from E
T
A
X x E
, (resp.
to
, analytic
on
X
and polyno-
c o n t i n u o u s mappings . . . I
t e n s o r p r o d u c t equipped w i t h p r o j e c t i v e t o p o l o g y c a r d i n a l of card.
transpo-
A
transposed operator of
mial on @IT
describes
domain ( r e s p . r a n g e ) o f t h e l i n e a r o p e r a t o r sed operator o f
card,
x
d e f i n e d i n s e c t i o n IV,4
dV
T(X,E),
T
A-convex h u l l o f
?(A)
...
78
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