COMPLEX ANALYSIS IN BANACH SPACES
NORTH-HOLLANDMATHEMATICS STUDIES Notas de Matematica (107)
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COMPLEX ANALYSIS IN BANACH SPACES
NORTH-HOLLANDMATHEMATICS STUDIES Notas de Matematica (107)
Editor: Leopoldo Nachbin Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro and University of Rochester
NORTH-HOLLAND-AMSTERDAM
NEW YORK
OXFORD
120
COMPLEX ANALYSIS IN BANACH SPACES Holomorphic Functionsand Domains of Holomorphy in Finite and Infinite Dimensions
Jorge MUJICA UniversidadeEstadualde Campinas Campinas, Brazil
1986
NORTH-HOLLAND-AMSTERDAM
NEW YORK
0
OXFORD
Elsevier Science Publishers B.V., 1986 Allrights reserved. No part of this publication may be reproduced, storedin a retrievalsystem, or transmitted, in any form orbyanymeans, electronic, mechanical, photocopying, recording or otherwise, without the priorpermission of the copyright owner.
ISBN: 0 444 87886 6
Publishers: ELSEVIER SCIENCE PUBLISHERS B.V. P.0. Box 1991 1000 BZ Amsterdam The Netherlands
Sole distributors forthe U.S.A. andCanada: ELSEVIER SCIENCE PUBLISHING COMPANY,INC. 52 Vanderbilt Avenue NewYork, N.Y. 10017 U.S.A.
Library of Congress Catalogingin-PublicationData
hjica, Jorge, 1946camplex analysis in Banach spaces. (North-Holland mathematics studies ; 120) (Notas de aia&tica ; 107) Bibliography: p. Includes index. 1. Holmorphic functions. 2. DanauLs ’ of holomorphy. 3. Banach spaces. I. Title. II. Series. 111. Series: Notas de m a d t i c a (Rio de Janeiro, Brazil) ; no. 107. QA1.N86 n0.107 [QA33i] 510 s [515.91S] 85-20922 ISBN 0-444-87886-6 (U.S. )
PRINTED IN THE NETHERLANDS
T o my t e a c h e r ,
Leopoldo Nachbin
This Page Intentionally Left Blank
FOREWORD
Problems arising from thestudy of holomorphic continuation and holomorphic approximation havebeen central in the development of complex analysis in finitely many variables, and constitute one ofthemost promising lines of current research in infinite dimensional complex analysis. This book isdesigned to present a unified view of these topics in both finite and infinite dimensions. The contents of this book fall naturally into four parts. The first, comprising Chapters Ithrough 111, presents the basic properties of holomorphic mappings anddomains of holomorphy in Banach spaces. The second part, comprising Chapters IV through VII, begins with the study of differentiable mappings, differential forms and the a operator in Banach spaces. Polynomially convex compact sets are investigated in detail, and some ofthe results obtained are applied tothe study of Banach and Frzchet algebras. The third part, comprising Chapters VIII through X, is
de-
voted to the studyof plurisubharmonic functions andpseudoconvex domains in Banach spaces. The identity of pseudoconvex domains and domains of holomorphy is established in the case of separable Banach spaces with the bounded approximation property. These results a r e e:it.r.!ided to Riemann domains in the fourth part, i.n w k i . i c : h envei.c>pes of holomorphy are also studied in detail. lived from a course taught at the Uni~ d : j dr!e Campinas, Brazil, in 1982. It presupposes versidade familiarity wl t h C n e triieory of Lebesgue integration, with the iis:ornorphi.c functions of a single variable, vi i
vi ii
and
MUJ I CA
with
of
t h e basic p r i n c i p l e s
Topics suchas vector-valued approximation property,
Banach
and H i l b e r t s p a c e s .
i n t e g r a t i o n , S c h a u d e r bases a n d t h e
a r e p r e s e n t e d i n d e t a i l i n t h e book.
The p r e s e n t a t i o n h e r e h a s b e e n a f f e c t e d by c o n v e r s a t i o n s a n d c o r r e s p o n d e n c e w i t h s e v e r a l f r i e n d s a n d c o l l e a g u e s , who w e r e n o t a l w a y s aware t h a t t h e y were s p e a k i n g f o r p o s t e r i t y .
In particu-
lar I w o u l d l i k e t o m e n t i o n R i c h a r d Aron, K l a u s - D i e t e r B i e r s t e d t , Roberto C i q n o l i , Jean-FranGois
Colombeau, S e z n D i n e e n , Dicesar I s i d r o , Msrio Matos, R e i n h o l d
F e r n s n d e z , K l a u s F l o r e t , Josg M. Meise a n d M a r t i n S c h o t t e n l o h e r .
One p e r s o n h a s h a d more i n f l u e n c e o n t h l s book t h a n
anybody
else, long before t h i s p r o j e c t w a s evenconceived. N o t o n l y f o r a c c e p t i n g t h i s t e x t i n h i s series Notas d e Matemztica f o r having led
me into research
h e r e a c k n o w l e d g e my
on
,
b u t mainly
this beautiful subject. I
g r e a t e s t d e b t t o Leopoldo Nachbin.
a m p a r t i c u l a r l y g r a t e f u l t o my
wife
a n d c h i l d r e n , Ana
M a r i a , Ximena a n d F e l i p e , f o r t h e i r s u p p o r t
and encouragement
I
while
I was w r i t t i n g t h e book, a n d f o r m a i n t a i n i n g a t home a n
a t m o s p h e r e ideal f o r s t u d y and r e s e a r c h . Finally, I am very pleased
t o t h a n k Miss E l d a Mortari f o r
h e r e x c e l l e n t t y p i n g of t h e m a n u s c r i p t .
Jorge Mujica Campinas, J u n e 1985
CONTENTS
. POLYNOMIALS 1 . Multilinear mappings . . . . . . . . . . . . .
1
.................. 3 . Polynomials of one variable . . . . . . . . . 4 . Power series . . . . . . . . . . . . . . . . .
12 18 27
CHAPTER I
2 . Polynomials
CHAPTER I1
.
5 6. 7 8 9
. . .
.
HOLOMORPHIC MAPPINGS
. . . . . . . . . .... . . .
Holomorphic mappings Vector-valued integration The Cauchy integral formulas G-holomorphic mappings The compact-open topology
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
33
40 45 58 69
CHAPTER I11 . DOMAINS OF HOLOMORPHY
. .
............. . . . . . . . . . . . . . . . . . . . . . . . . .
10 Domains of holomorphy 11 Holomorphically convex domains 12. Bounding sets
79
85 94
CHAPTER IV . DIFFERENTIABLE MAPPINGS
.
Differentiable mappings . . . . . . . . . 14 . Differentiable mappings of higher order 15 . Partitions of unity 13
....
. . . . . . . . . . . . 16 . Test functions . . . . . . . . . . . . . 17 . Distributi.ons. . . . . . . . . . . . . .
. . . .
. . . .
99
. . . .
111 118 122 127
.........
139
CHAPTER V . DIFFERENTIAL FORMS 18 . Alternating multilinear forms ix
MUJ I CA
X
19 . 20 21 . 22 . 23
. .
. . . . . . . . . . . . . . . . . . . .
Differential forms The Poincars lemma . . . . . . . . . . . The 3 operator . . . . . . . . . . . . . Differential forms with bounded support . The 3 equation in polydiscs . . . . . . .
144
153
. . .
156 162
. . .
168
CHAPTER VI . POLYNOMIALLY CONVEX DOMAINS
. Polynomially convex compact . Polynomially convex domains
. . . . . . . . Schauder bases . . . . . . . . . . . . . . . . . The approximation property . . . . . . . . . .
24 25 26 . 27 28 .
CHAPTER
Polynomial approximation in Banach spaces
. . . . . . . . . . . . . . . . .
VII
. 30 .
29
31 . 32 33 .
.
sets in 8 in C n . .
177
185 188
194 202
. COMMUTATIVE BANACH ALGEBRAS . . . . . . . . . . . . . . . .
. . . . .
211 214 219 227 236
. . . . . .
245
Banach algebras Commutative Banach algebras . . . . . The joint spectrum . . . . . . . . . Projective limits of Banach algebras The Michael problem . . . . . . . . .
. . . . .
. . . . .
. . . . .
CHAPTER VIII . PLURISUBHARMONIC FUNCTIONS
.
34 Plurisubharmonic functions . . . . 35 . Regularization of plurisubharmonic
. . . . . . . . . . . . . . . 36 . Separately holomorphic mappings . . . . . . . . 37 . Pseudoconvex domains . . . . . . . . . . . . . functions
38
.
CHAPTER IX 39
.
Plurisubharmonic functions on pseudoconvex domains . . . . . . . . . . . . . .
.
THE
265 273
. .
279
. .
287
IN PSEUDOCONVEX DOMAINS
Densely defined operators in Hilbert spaces . . . . . . . . . . . . . . . The 5 operator f o r L 2 differential forms .
. 41 . L 2 42 . C m 40
5 EQUATION
255
solutions of the solutions of the
. . . . . . . . . .
5 equation 7 equation . . . . . . . .
291 300 307
xi
CONTENTS
.
CHAPTER X
THE LEVI PROBLEM
43 . The Levi problem in Cn . . . . 44 . Holomorphic approximation in C n
. . . . . . . . . . . . . . . . 45 . The Levi problem in Banach spaces . . . . . . . 46 . Holornorphic approximation in Banach spaces . .
311 313 320 325
CHAPTER XI . RIEMANN DOMAINS 47 . Riemann domains .
. . . . . . . . . . . . . . .
48 . Distributions on Riemann domains 49 . Pseudoconvex Riemann domains . .
. . . . . . . . . . . . . .
50 . Plurisubharmonic functions on Riemann domains . . . . . . . . . . . . 5 1 . The equation in Riemann domains
331 339 346
. . . .
353 359
. . . .
361
. . . . . . . . . . . .
372
a
. . . . . . .
CHAPTER XI1 . THE LEVI PROBLEM IN RIEMANN DOMAINS 52
.
53 .
The Cartan-Thullen theorem in Riemann domains . . . . . . . . . . . . The Levi problem in finite dimensional Riemann domains
54 . The Levi problem in infinite dimensional Riemann domains . . . . . . . . . 55 . Holomorphic approximation in infinite dimensional Riemann domains . . .
. . .
380
. . .
391
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
397 400
. . .
408
CHAPTER XI11 . ENVELOPES OF HOLOMORPHY 56 . Envelopes of holomorphy 57 . The spectrum 58
.
Envelopes of holomorphy and the spectrum
BIBLIOGRAPHY. INDEX
. . . . . . . . . . . . . . . . . . . . . .
421
. . . . . . . . . . . . . . . . . . . . . . . . . .
431
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CHAPTER I
POLYNOMIALS
1. MULTILINEAR MAPPINGS T h i s s e c t i o n i s devoted t o t h e s t u d y o f multilinearmappings i n Banach s p a c e s . B e s i d e s t-.heir i n t r i n s i c i n t e r e s t , m u l t i l i n e a r mappings w i l l s e r v e a t w o f o l d p u r p o s e . Whereas s y m m e t r i c
mul-
t i l i n e a r mappings w i l l b e h e l p f u l i n t h e s t u d y o f p o l y n o m i a l s , a l t e r n a t i n g m u l t i l i n e a r mappings w i l l b e u s e d t o i n t r o d u c e d i f f e r e n t i a l forms. To begir. w i t h , w e e s t a b l i s h some n o t a t i o n . Throughout whole book t h e l e t t e r
iK
the
w i l l s t a n d e i t h e r f o r t h e f i e l d LPor
a l l r e a l numbers o r f o r t h e f i e l d
C of
all
complex numbers.
The s e t o f a l l s t r i c t l y p o s i t i v e i n t e g e r s w i l l b e d e n o t e d
by
fl u {O} w i l l b e d e n o t e d by no. U n l e s s s t a t e d o t h e r w i s e , t h e l e t t e r s E and F w i l l a l w a y s r e p r e s e n t Banach s p a c e s o v e r t h e same f i e l d M . W,
whereas t h e set
For each
1.1. DEFINITION.
m E W w e s h a l l d e n o t e by
t h e v e c t o r s p a c e o f a l l m - l i n e a r mappings
E,("E;F)
A : E m +. F,
whereas
w e s h a l l d e n o t e by S: ( m ~ ; ~ t) h e s u b s p a c e o f a l l c o n t i n u o u s members o f E a i m E ; F ) . F o r e a c h A E EalmE;F1 w e d e f i n e
When and
rn = 1
L('E;Fl
then a s usual we s h a l l w r i t e
= S:(E;F). When
write
EaimE;IK)
m = 1
and
= LaimE)
F = M
F = M and
E flE;F) = E a a ( E ; F )
a
then f o r short
we
shall
E f m E ; X ) = E f m E ) . F i n a l l y when
t h e n a s u s u a l w e s h a l l w r i t e E a ( E I = E*
E ( E l = E'.
1
and
MUJ I CA
2
1.2.
For each
PROPOSITION.
t h e foZZowing c o n d i -
A E Xa("E;FI
tions a r e e q u i v a Z e n t :
(a)
A
i s continuous.
(b)
A
is c o n t i n u o u s at t h e o r i g i n .
(c)
I I A II
1
and
Axl
1 . 8 . THEOREM. L e t
1
... ,x n )
we define
a
n
"1
if
an !
T h e n f o r each (x1,
A E d:,frnE;FI.
a = (al,
and e a c h
E En
Let
+ an,
a
... x n n A
= A
E d:zfmE;F).
if
rn = 0. xl
Then f o r a l l
,...,
xn
E
E
we h a v e t h e L e i b n i z F o r m u l a Afxl +
...
+ xnIm = 2
5
@l
Axl
.. .
a n xn
w h e r e t h e summation is taken o v e r a l l m u l t i - i n d i c e s E
such t h a t
WE
/ a / = m.
rn. The r e s u l t i s o b v i o u s f o r rn = 0 m = 1 . Assuming t h e formula v a l i d f o r a c e r t a i n rnzl one
PROOF.
and
..., an'
a = (al,
By i n d u c t i o n on
6
MUJ I CA
rn + 1. I n d e e d , i f
can r e a d i l y e s t a b l i s h it f o r
A
E
Ez(rn+lE;F)
t h e n one can w r i t e
...
A(xl +
. ..
+ x n ) rn+l = A ( x l +
...
+ xn)(xl +
+ xnIm
.. + xnI ,
and apply t h e i n d u c t i o n h y p o t h e s i s t o t h e mapping Alxl + . which b e l o n g s t o
Ez(rnE;FI.
1.9. COROLLARY.
Let
A
The d e t a i l s a r e l e f t t o t h e reader. Then f o r a l l
fz(rnE;F!.
E
x, y
f
E
we
h a v e t h e Newton BinomiaZ Formula
1.10. THEOREM.
Let
A
T h e n f o r aZT
f:frnE;F).
E
xo,
... ,xrn E E
we have t h e P o l a r i z a t i o n Formula A(xl,
..., x m )
1 -
=
m!Zrn
. . . €,A(xO
I:
A(xo +
. . .~ +ErnXrnlrn .
+X
~
E.;=?l
"
E
~
+X.
..
~ + E
x )rn =
rnrn
z
rn!
a ! 0
c1
... urn !
... C
+ c1rn =
1
...
a O J . ..,a
where t h e summation i s t a k e n e v e r a l l
+
E
By t h e L e i b n i z Formula 1 . 8 w e have t h a t
PROOF.
a
+
rn
ar n uo
E~
E EVo
m . Hence
. . . €,A(xO
+
E1Xl
+
...
+
E
E $1
X
rnrn
)
m
Clearly
whenever ai = 0
f o r some i w i t h
1
with Gk k = 0, , m, such that E Q(kM;F) for *
*
A
,
...
m
and therefore rn (3.6)
f(Xx) =
z Xk
Qk(x)
for all
k=O
From (3.5) and ( 3 . 6 ) we conclude that
x
E
M
and
h E M.
MUJ ICA
22
Thus
5
P k ( x ) = Q (x) k
for a l l
x E M
and
k
P*(x) = 0
for a l l
x
and
k > m.
k Pk(M E P ( M;F)
for every k
A mapping
3 . 6 . THEOREM.
flo
M
and the proof i s conplete.
is a p o l y n o m i a l of P ( a + Ab) is a p o l y n o m i a l i n a, b E E .
P : E
at m o s t m if and o n l y if d e g r e e at m o s t m f o r a l l
E
E
m
+
F
degree A of
The proof of t h e "only i f " p a r t was l e f t t o t h e r e a d e r a s E x e r c i s e 2.L. The " i f " p a r t f o l l o w s a t once from t h e proof of Lemma 3 . 5 . Indeed, it s u f f i c e s t o observe t h a t i n ( 3 . 5 ) w e have t h a t P k ( x ) = 0 f o r every k > m .
PROOF.
L e t f : E + F be a mapping such t h a t f ( a + hb) i s a p l y nomial i n A f o r a l l a , b € E . Then f need n o t be a polynomial i n g e n e r a l . However, w e have t h e f o l l o w i n g theorem.
A c o n t i n u o u s mapping
3 . 7 . THEOREM. if a n d o n l y if
P : E
+
F
i s a polynomial
P(a + Xb) i s a p o l y n o m i a l in X for a l l
a, b
E
E.
To prove t h i s theorem w e need t h e followinq lemma. 3 . 0 . LEMMA.
Let
f
: E
-+
F
be a mapping s u c h t h a t
f ( a + Xbl
i s a p o l y n o m i a l i n A f o r a l l a, b E E . If f i s i d e n t i c a l l y z e r o on a n o n v o i d o p e n s e t , t h e n f is i d e n t i c a l l y z e r o o n E .
Assume f i s i d e n t i c a l l y z e r o on a nonvoid open s e t F i x a E U, x E E and J, E F ' . Then J, o f [a + h ( x - a ) ] i s a M-valued polynomial i n A which i s i d e n t i c a l l y z e r o on a
PROOF.
U
C
E.
neighborhood of A = 0 . Hence Ji o f [ a + X ( x - a ) 1 = 0 for every X E IK, and i n p a r t i c u l a r J, o f(x) = 0. Since F ' separ a t e s t h e p o i n t s of F w e conclude t h a t f ( x ) = 0 . PROOF OF THEOREM 3 . 7 . T o prove t h e n o n t r i v i a l i m p l i c a t i o n asThen sume P(a + Xb) i s a polynomial i n A f o r a l l a, b € E. by Lemma 3 . 5 t h e r e i s a sequence of homogeneous polynomials k P k E P a ( E ; F I such t h a t
POLYNOMIALS
23
W
x
P(x) =
Pkfx)
f o r every
x E E,
k=O where f o r each x
E
E w e have t h a t
Pkfxl = 0
for a l l but f i -
n i t e l y many i n d i c e s . W e want t o show t h e e x i s t e n c e o f an rn El@ such t h a t P k = 0 f o r e v e r y k > m. F i r s t we show t h a t each P k i s c o n t i n u o u s . W e proceed by i n d u c t i o n on k , t h e a s s e r t i o n b e i n g obvious f o r k = 0. Let k > 0 and assume P i s c o n t i n u o u s f o r e v e r y j < k . Then f o r j a l l x E E and X E IK, X # 0, w e have t h a t
L e t ( 1 I be a sequence of nonzero scalars which converges n z e r o . For each n E nV l e t Qn : E + F be d e f i n e d by
to
By t h e i n d u c t i o n h y p o t h e s i s each Q n i s c o n t i n u o u s . By (3.7) converges p o i n t w i s e t o P k , and i s i n part h e sequence (Q n t i c u l a r p o i n t w i s e bounded on E . Then by Lemma 2 . 7 t h e sequence (Qn) i s uniformly bounded on a b a l l B ( a , r ) . Hence Pk is bounded on B f a , r ) as w e l l , and by P r o p o s i t i o n 2 . 4 w e may conclude t h a t P i s c o n t i n u o u s , as asserted. k To complete t h e proof o f t h e theorem w e c o n s i d e r t h e sets Am
= {x E E : P k ( x I =
0
for all
k > m).
a,
E =
U A and each A m i s c l o s e d i n E . S i n c e E i s a m= 1 m B a i r e s p a c e , some A m has nonempty i n t e r i o r . Then i t f o l l o w s from Lemma 3.8 t h a t P k i s i d e n t i c a l l y z e r o on E f o r e v e r y k > m . The proof i s now complete.
Then
24
MUJ ICA
Theorems 3 . 6 and 3 . 7 reduce t h e s t u d y of p o l y n o m i a l s t o the
case where t h e domain s p a c e i s one d i m e n s i o n a l . Next w e p r e s e n t
r e s u l t s t h a t r e d u c e t h e s t u d y o f p o l y n o m i a l s t o t h e case where t h e r a n g e s p a c e i s one d i m e n s i o n a l . A mapping
3.9. THEOREM.
P : E
is a p o l y n o m i a l of d e g r e e X i s a polynornial of degree
F
+
a t m o s t m if a n d o n l y if JI o P : E a t m o s t m f o r e v e r y JI E F’.
+
The n o n t r i v i a l i m p l i c a t i o n i n Theorem 3.9 f o l l o w s a t
once
from Theorem 3 . 6 and t h e f o l l o w i n g lemma.
a t m o s t m if rn
J l o P : 1K
+
for every
..,m
k = 0,.
each
m + 1
be
let
d i s t i n c t points i n
Lk E PIXI
be t h e unique
B!. F o r
polynomial
f o r j L O , . . .,m. j ki t h e n by C o r o l l a r y 3.2 t h e oolynomial Jl o P c a n b e
o f d e g r e e a t most m If
M
+
is a p o l y n o m i a l of degree is a p o l y n o m i c z l of d e g r e e at most F
--*
E F’.
lo,..., Am
Let
PROOF.
P : IK
A mapping
3.10. LEMMA.
Jl E F‘
L (h I = 6
such t h a t
k
r e p r e s e n t e d i n t h e form
m
m
f o r every
h E 1K.
Since
F ’ s e p a r a t e s t h e p o i n t s of F w e con-
clude that
P(XI
=
m Z
Plhk)LklhI
f o r every
E
M.
k=O
Thus P i s a polynomial of d e g r e e a t most
m , as a s s e r t e d .
W e complete t h i s s e c t i o n w i t h t h e f o l l o w i n g c o n t i n u o u s v e r -
s i o n of Theorem 3 . 9 . 3.11.
THEORFM.
i f and o n l y $ E
F’.
if
A mapping $ o P : E
P : E +
M
+
F
i s a continuous polynomial
i s a c o n t i n u o u s polynoiniaZ for every
POLYNOMIALS
25
T o prove t h i s theorem w e need t w o a u x i l i a r y lemmas.
3 . 1 2 . LEMMA. L e t (P,) C PflK; F) b e a s e q u e n c e of p o Z y n o m i a Z s o f d e g r e e at m o s t rn w h i c h c o n v e r g e s p o i n t w i s e t o a m a p p i n g
P :M
-+
Let
PROOF. each
F. Then P i s a l s o a p o l y n o m i a l o f d e g r e e at m o s t Ao,...,hrn
k = 0,.
. .,m
m + 1 L k E PlIKl
be
let rn s u c h t h a t
degree a t m o s t
Then by C o r o l l a r y 3 . 2 e a c h
L (A
k
Pn
i
d i s t i n c t points i n
rn.
X.
For
be t h e unique polynomial o f
= 6
)
ki
for
=
j
0,
.. . , m.
can b e r e p r e s e n t e d i n t h e form
rn
Pn ( A ) = n
By l e t t i n g
Hence
f o r every
m Z p(hk)Lk(h) k=O
f o r every
i s a l s o a polynomial of d e g r e e a t most
P
A
E
M.
we get t h a t
+. m
P(X) =
Z: p n l X k ) L k ( A I k=O
h E l??.
m , asasserted.
Now w e can improve Lemma 3 . 1 0 as f o l l o w s . 3 . 1 3 . LEMMA.
a
.+
a
PROOF.
A mapping
P :M
-+
F
i s a polynomia2 for every
J,
E
$ oP
P :
$ 0
F'.
m E W l e t Bm d e n o t e t h e s e t o f a l l Ji i s a polynomial of d e g r e e a t most m . By
For each
such t h a t
is a p o l y n o m i a l if
E
F'
hy-
m
pothesis
F' =
U
Brn.
m=l
i s c l o s e d i n F'. Indeed l e t (+n) be a sequence i n Bm which c o n v e r g e s t o some Ji E F'. men the sequence I $ , o P) c o n v e r g e s p o i n t w i s e t o $ o P, and it f o l l o w s from Lemma 3.12 t h a t $ E Bm too. Thus e a c h Bm i s c l o s e d i n W e claim t h a t e a c h
Bm
F t , as asserted.
Since
F'
i s a Baire s p a c e , some
But s i n c e c l e a r l y
Bm
h a s nonvoid interior.
we conclude t h a t Brn = F t . Thus $ o P i s a polynomial of d e g r e e a t most m f o r every $ E F ' and t h e n a n a p p l i c a t i o n of U r n 3.10 catpletes Bm
i s a vector s u b s p a c e of
F'
MUJ I CA
26 the proof.
PROOF OF THEOREM 3.11. To prove t h e n o n t r i v i a l i m p l i c a t i o n l e t P : E -* F b e a mapping such t h a t o P E P I E ) for every Jl E F ’ . Then $ o Pla + Abl i s a polynomial i n h f o r a l l Jl E F ’ and a , b E E. Then, by Lemma 3 . 1 3 , P(a + Ab) i s a polynomial i n A f o r a l l a , b E E . Then, by Lemma 3.5, t h e r e i s a sequence of k homogeneous polynomials Pk E P a ( E ; F ) such t h a t $J
W
P(x) = where f o r e a c h
x
2 Pk(x) k=O
E
E
f o r every
x E E,
Pk(x) = 0
w e have t h a t
f o r a l l but f i -
n i t e l y many i n d i c e s . Hence w e g e t t h a t m
(3.8)
Jlo P(x)
= E
Pk(x)
$ 0
for all
x E E
and
$ E F‘.
k=O Jl E F ’ . Then i t f o l l o w s from ( 3 . 8 ) t h a t $ o P k E P ( E l f o r e v e r y @ E F ’ . B17 E x e r c i s e k Mter 2 . 0 w e may conclude t h a t P k E P ( E ; F ) f o r e v e r y k E no. knowing t h a t e a c h P k i s c o n t i n u o u s , w e may p r o c e e d as i n t h e proof of Theorem 3 . 7 , and show t h e e x i s t e n c e o f an m E B such t h a t Pk = 0 f o r e v e r y k m . T h i s completes t h e p r o o f . By h y p o t h e s i s
J l o P E P ( E ) f o r each
k
EXERCISES
3.A. L e t bo,. ..,b,
lo,.
. .,Am
m + 1
be
d i s t i n c t p o i n t s i n M and
be a r b i t r a r y p o i n t s i n
F.
unique polynomial of d e g r e e a t most m j
=
0,.
. .,m.
such t h a t
i s g i v e n by t h e formula
Show t h a t L
n
m
L(hl =
L E PIlU;F)
Let
z
bk
k=O
ih
- A
(Ak
-
j#k
n jfk
be
L l A .I = l i . 3 3
let the for
.)
3
X.) 3
T h i s i s t h e Lagrange i n t e r p o l a t i o n formula. 3.B. Given any i n f i n i t e d i m e n s i o n a l Banach s p a c e function
f
: E
-+
M
such t h a t
f(a
6 , find
+ Ab) i s a polynomial i n
a h
POLYNOMIALS
for a l l
27
f i s n o t a polynomial. O f c o u r s e
a, b E E, but
the
f u n c t i o n f m o s t be d i s c o n t i n u o u s . L e t iP i C
P a I E ; F I b e a sequence of p o l y n o m i a l s of degree a t most m which c o n v e r g e s p o i n t w i s e t o a mapping P : E F. 3.C.
n
+
P i s a polynomial of d e g r e e a t most
(a)
Show t h a t
(b)
Show t h a t i f e a c h
Pn
i s continuous t hen P
m.
i s con-
t i n u o u s as w e l l .
4 . POWER SERIES
I n t h i s s e c t i o n w e i n t r o d u c e power series i n Banach s p a c e s i n t e r m s of homogeneous polynomials. W e e s t a b l i s h t h e CauchyHadamard Formula f o r t h e r a d i u s o f convergence. DEFINITION.
4.1.
A p o u e r s e r i e s from
E
into
around
F
the
m
point
a E E
i s a series o f mappings of t h e form
2 Pmlx-a), m=O
where
f o r every
Pm E P a ( m E ; F I
in f co
Note t h a t t h e power series m
m=O
AVO.
Pm(x - a )
I: A m ( x - a J m ,
w r i t t e n i n t h e form
where
also
can Am
be
=aC ~ Y " " ; F I ,
E
m=O
Am = Pm. 4.2.
The r a d i u s of c o n v e r g e n c e , o r more precisely,
DEFINITION.
m
t h e r a d i u s of uniform convergence of the
i s t h e supremum o f a l l u n i f o r m l y on t h e b a l l 4 . 3 . THEOREM.
Let
R
r
-
2
0
power series
B(a,r).
b e t h e r a d i u s o f c o n v e r g e n c e o f t h e power
a)
series
P
m=O
(a)
R
m (x
- a ) . Then:
i s g i v e n b y t h e Cauchy-Hadamard F o r m u l a 1 / R
Pm(x-a),
m=O such t h a t t h e series c o n v e r g e s
= l i m s u p IIPmII 1 / m m+
.
MUJ I CA
20
-
The s e r i e s c o n v e r g e s a b s o Z u t e Z y and unifomnZy on B(a;r)
(b) whenever PROOF.
0
5 r
< R.
First we show the inequality
Indeed, suppose R 0 and let 0 < r < R . Then the series converges uniformly on B ( a ; r ) . Set
Choose
mo
E
liV
such that m
for all
m
mo
and t and therefore
m > mo
E
x E B ( a ; r ) . Hence
and
B(0,r).
Whence
IIPm(t)ll
IIPmll 5 2 r - m
5 2
for all
for all m > m
0 '
l i m s u p IIPmll' I m - 1 / r .
Letting
r
+
R
we obtain (4.1).
Next we show the opposite inequality:
r Indeed, set L = l i m s u p IIPmIll/m, assume L < m , and take with 0 5 r < l / L . Choose s with r < s < 7 / L and choose m E W such that llPmlll/m < l / s for all m 2 mo Hence 0
.
Thus the series
I: P m ( x - a ) converges absolutely and m=O
formly on the ball
-
B ( a ; r ) . Hence
R
2 r , and letting
P
uni+
l/L
POLYNOMIALS
29
we obtain ( 4 . 2 ) . Thus w e have shown ( a ) . But i n t h e course of proving (a) we have a l s o shown ( b ) . m
4.4.
Let
PROPOSITION.
Z
Pm(x
-
a)
b e a power s e r i e s
m=O
E
i n t o F . If t h e r e i s
aZ2
then
B(a;rl,
3: E
m
r > 0
such t h a t
=
for every
Pm
0
X
m
-
Pm(x
n=O E
from
a) =O
for
No.
T h i s p r o p o s i t i o n i s an immediate consequence of
the
fol-
lowing lemma.
4.5. LEMMA.
m
( c ~ ) ~be= a~ s e q u e n c e i n
Let
there
If
F.
co
then
Z cmAm = 0 f o r a22 m=O for e v e r y m E W o .
cm = 0
By i n d u c t i o n on
PROOF. Assuming
co
=
...
m
deed, s i n c e
= cm m
X
cmr
1x1
A E M with
such t h a t
r > 0
m . Letting 1 = 0 = 0 w e show t h a t
is
5
P
w e g e t t h a t co= 0.
= 0
too.
In-
C
such
converges t h e r e i s a c o n s t a n t
rn=O
that
IIcmllrmI C
f o r every
m . Since
c0
=
...
= cm = 0
it fol-
lows that
Then f o r 0 < l h
Letting
X
-+
0
I 5
r / 2
we get t h a t
we get t h a t
= 0
and t h e lemma has
been
proved.
Let
E and F be Banach spaces over X , w i t h
s i o n a l . W e have seen i n E x e r c i s e 2 . 5 t h a t each admits a unique r e p r e s e n t a t i o n of t h e f o r m
E P
n dimenE
P(mE;F)
30
MUJ I CA
...
where c a E F and where C,, , 6 , denote the coordinate functionals associated with a given basis. Thus each power m
Z Pm ( x - a ) from E into F can be written, at
series
least
m=O
formally, as a multiple power series
where the summation is taken over all multi-indices a = (a1 , , a n ) E IN:. If the multiple series converges absolutely and uniformly on a certain set X then it is clear that
...
W
the
I: P m f x
power series
m=O . -
-
a ) also converges absolutely and
uniformly on X and they coincide there. Conversely, we the following proposition. m
Let
4.6. PROPOSITION.
m
Z €‘,(+I m=O
from el,
E
i n t o F with radius
..., e n
for e a c h that
=
I: Amxm be a power se ri e s m=O
of
convergence
...
= IIe n I1 = 1 , s e t
a = (a1 ,. . ., a n ) E A$
with ( a ( = m.
E
B
with
Ilelll
=
m
a1 Z ~ , i ~+ .~. . e+ ~5 n e n I = catl a m=O
lcll
have
R
...
>
0.
T h e n we
a
5,
Given
have
n
...
w hen ever + + l C n l < R / e . Both s e r i e s converge abso0 whenever ZuteZy and u n i f o r m l y f o r 15, I + . . . + 15, I - r - r < R/e. PROOF.
By the Leibniz Formula 1.8 we have that
~ , ( < , e ~+
...
+ 5n e n ) = z Ial=rn
al m! a! E l
a
. . . c n ‘ n Arnei 1 ... e an n .
On the other hand, using Exercise 2.G and applying the sical Leibniz formula, we get that
clas-
POLYNOMIALS
If
0
5 r
t h e n t h e l a s t w r i t t e n series c o n v e r g e s
< R/e
I 0 such t h a t B f b ; r ) C U and choose n s u c h t h a t an E B f b ; r l . Then i f f o l lows from ( a ) t h a t f i s i d e n t i c a l l y z e r o on Bfb;r). Hence b E A and t h e proof i s complete. t o show t h a t A
i s closed i n
38
MUJ I CA
Next w e e x t e n d t h e Open Mapping P r i n c i p l e . PROPOSITION. L e t U b e a c o n n e c t e d o p e n s u b s e t of E, and l e t f E X ( U l . If f is n o t c o n s t a n t o n U t h e n f l v l i s an open s u b s e t o f 6 f o r e a c h o p e n s u b s e t V of U. 5.8.
PROOF. s e t of
C l e a r l y i f s u f f i c e s t o show t h a t
convex
open s u b s e t of
for e a c h convex open s u b s e t V U and l e t x E V . c i p l e 5.7 t h e f u n c t i o n f i s n o t c o n s t a n t is a p o i n t y E V such t h a t f l x l # f ( y l . t h e open s e t @,
f l V ) i s an open subo f U. L e t V be a By t h e I d e n t i t y Prinon V and hence there S i n c e V i s convex,
i s convex as w e l l . The f u n c t i o n
gfXI = f [ i s holomorphic on
A and
3c
+ Afy
-
3c)l
gfOl = f f x ) # f ( y l = g i l ) .
By
the
Open Mapping P r i n c i p l e f o r holomorphic f u n c t i o n s o f one complex v a r i a b l e t h e set
g l A ) is open i n
w e conclude t h a t
fiVl
6. S i n c e
i s a l s o open i n
6.
A s an immediate consequence w e o b t a i n t h e M a x i m Principle.
Let
U be a c o n n e c t e d o p e n s u b s e t o f E , and a E U s u c h t h u t lfirll 5 I f f d l t h e n f i s c o n s t a n t on U.
5.9.
PROPOSITION.
Ze t
f E X ( U I . I f there e x i s t s
f o r every
x E U
i s n o t c o n s t a n t o n U. Then by t h e Open Mapp i n g P r i n c i p l e 5.8 t h e set f ( U ) i s open i n @, and hence cont a i n s an open d i s c A ( f ( a ) ; r l . But t h i s i s i m p o s s i b l e , s i n c e by PROOF.
Assume f
hypothesis
lf(xl
I 5
Iffa) I
f o r every
x
E
U.
To end t h i s s e c t i o n we g e n e r a l i z e L i o u v i l l e ' s Theorem.
39
HOLOMORPHIC MAPPINGS 5.10.
PROPOSITION. I f a m a p p i n g t h e n i t is c o n s t a n t o n E .
f E J C ( E ; F ) is b o u n d e d
E
on
PROOF. L e t x E E and J, E F t . Then the f u n c t i o n g ( x ) = $ o f ( X x ) i s holomorphic on C and bounded t h e r e . By t h e classical L i o u v i l l e ' s theorem, g i s c o n s t a n t , and i n p a r t i c u l a r J, o f ( x ) = J, o f ( 0 ) . S i n c e F t s e p a r a t e s t h e p o i n t s of F w e conclude t h a t f ( x ) = f ( 0 ) and t h e proof i s complete. EXERCISES 5.A.
'
be Banach s p a c e s , and l e t V S E L f E ; F I , f E JC(V;G) and J C ( S - l ( V ) ; G ) and T o f E K ( V ; H i ) .
Given
F.
f oS
E
U be a n open subset o f E , and l e t f E J f ( U ; F ) l e t f a : U - a -t F be d e f i n e d by f o r every t E U - a . 5.B.
Let
(a) Show t h a t f a E J C ( U - a ; F ) and f o r e v e r y t E U - a and m E n o . Show t h a t t h e mapping f morphism between J C f U ; F I and X f U (b)
5.C.
Let
b e a n open
E , F, G , H
Let
subset of show t h a t
U be a n open s u b s e t of show t h a t
f , g E JCfU)
-+
-
T
E
L(G;HI,
a E E . For each faft) = f f a + t)
Pmfa(t) = Pmf(a + t )
i s a v e c t o r space iso-
fa
a;F). E.
two
Given
functions
J€(u) and
fg E
rn P r n ( f g )( x )
x
=
pm-j f ( x ) P j g ( x )
j=O
for a l l
m
E
liVo
and
x
-
a)
E
U.
m
5.D.
Let
m=O
P,(x
b e a power series from
w i t h r a d i u s o f convergence R > 0 and w i t h each Let f : B(a;R) F be d e f i n e d by
E
co
x m=O
Pm(x
- a)
f o r each
x
E
F,
Pm cmtinuous.
-+
f(x) =
into
B(a;R).
MUJ I CA
40
Show t h a t f i s holomorphic on t h e b a l l B ( a ; R / e I . t h a t f i s holomorphic on t h e b a l l B ( a ; R ) ? 5.E.
Let
(a)
Can you shcw
X be a t o p o l o g i c a l s p a c e . Show t h a t each c o n t i n u o u s mapping
f : X
+
F
i s lo-
c a l l y bounded. I f X i s m e t r i z a b l e show t h a t a mapping f : X F is l o c a l l y bounded i f and o n l y i f f i s bounded on each compact s u b s e t of X. (b)
5.F.
+
U be a connected open s u b s e t of E , and l e t f Suppose t h e r e are a nonvoid open s u b s e t V of U and a c l o s e d subspace N of F such t h a t f(V) C N. Show that f(U) C N.
E
Let
JC(U;F).
5.G.
Let
E JC(U;F).
for all 5.H.
Let
U be a connected open subset o f E , and let f I f t h e r e i s a p o i n t a E U such t h a t IIffz)ll 5 IIf(a)ll 2 E U, show t h a t I l f l l i s c o n s t a n t on U. F = B2
be d e f i n e d by
w i t h t h e norm o f t h e supremum. L e t
f ( z ) = ( 1 , ~ )f o r e v e r y
(a)
Show t h a t
f E Pf5;F).
(b)
Show t h a t
II f II
(c)
Show t h a t
f
(d)
Show t h a t
II f II
6 . VECTOR-VALUED
f: 5 + F
z E 5.
i s c o n s t a n t on
A(0; 1 I .
i s n o t c o n s t a n t on
AfO; I ) .
i s n o t c o n s t a n t on
5.
INTEGRATION
W e assume t h a t t h e r e a d e r i s f a m i l i a r w i t h
theory of Lebesgue measure and i n t e g r a t i o n , and by t h i s w e mean i n t e g r a t i o n of s c a l a r - v a l u e d f u n c t i o n s . However, throughout t h i s book w e s h a l l o f t e n f i n d d e s i r a b l e t o i n t e g r a t e f u n c t i o n s with values i n a Banach space. With t h i s i n mind w e p r e s e n t a few e1errenta-y f a c t s r e g a r d i n g t h e Bochner i n t e g r a l . These few f a c t s w i l l be the
HOLOMORPHIC MAPPINGS
41
s u f f i c i e n t f o r o u r needs. 6.1.
L e t (X, 8 , 11) be a f i n i t e measure
DEFINITION.
mapping
sets
f : X
-+
Al,...,Ak
E
z
and v e c t o r s
k
Then f o r e a c h
A
is s a i d t o b e s i m p l e i f t h e r e are d i s j o i n t
F
flxl =
space.
xA
Z j=1
j
(x)b j
bl,...,bk
E
F
for all
x
E
such t h a t X.
we d e f i n e
A E Z
The v e r i f i c a t i o n of t h e f o l l o w i n g lemma i s s t r a i g h t f o r w a r d , and i s l e f t as an exercise t o t h e r e a d e r . 6.2.
LEMMA.
L e t (X,
x , ~ b) e
a f i n i t e measure s p a c e ,
f : X F be a s i m p l e mapping. Then f o r each we h a v e t h a t : +
A
E Z
and and
let + €
F'
I
6 . 3 . DEFINITION.
(a)
L e t (X,
A mapping
x,
f : X
p) b e a f i n i t e measure s p a c e .
-+
F
i s s a i d t o be measurable i f there
e x i s t s a sequence of s i m p l e mappings v e r g e s t o f almost everywhere. (b)
A measurable mapping
f : X
:
f,
+
X
-+
F
which
i s said t o be Bochner
F
i n t e g r a b l e i f t h e r e e x i s t s a sequence of s i m p l e mappings
X
-+
F
such t h a t
lim n-+m I n t h i s case w e d e f i n e
1,
/Ifn
- flldu =
con-
0.
f,
:
42
MUJ I CA
€or each
A
f
2.
Lemma 6.2 g u a r a n t e e s t h a t t h e Bochner i n t e g r a l
is
IAfdll
w e l l d e f i n e d . I n d e e d , on o n e hand Lemma 6.2 inplies that
f.f,fndlJl
i s a Cauchy s e q u e n c e , a n d on t h e o t h e r hand Lemma6.2 g u a r a n t e e s t h a t the d e f i n i t i m of sequence I f , ) .
i s i n d e p e n d e n t o f t h e c h o i c e of t h e
J,fdp
F i n a l l y , from Lemma 6 . 2 and
the d e f i n i t i o n
of
t h e Bochner i n t e g r a l w e can e a s i l y o b t a i n t h e f o l l o w i n g propos i t i o n . The d e t a i l s are l e f t t o t h e r e a d e r . 6.4. let
PROPOSITION. L e t IX, Z, P I b e a f i n i t e m e a s u r e s p a c e , and f : X + F b e a Bochner i n t e g r a b l e m a p p i n g . T h e n :
(a)
f o r each
(b)
f o r each
6.5.
The f u n c t i o n
$I E
F’
and
The f u n c t i o n
IJJ
o f : X
+
6
i s i n t e g r a b Z e und
A E Z.
II f I l
: X
+
lR i s i n t e g r a b Z e and
A E 2.
PROPOSITION.
Hausdorff space
X.
Let
u
be a f i n i t e Bore2 m e a s u r e on a compact
T h e n e a c h c o n t i n u o u s mapping
f : X
-+
F
i s
Bochner i n t e g r a b l e .
PROOF. C l e a r l y i t s u f f i c e s t o show t h a t f i s t h e uniform lim it of a s e q u e n c e o f s i m p l e € u n c t i o n s . L e t n E liV be given. S i n c e f i s c o n t i n u o u s a n d X i s compact w e c a n f i n d p o i n t s al,
..., a k E
X
such t h a t
HOLOMORPHIC MAPPINGS
For e a c h
. . .,k
j = I,
U
Then
43
set
j
= f-'
[ B ( f ( a j ) ; l/n)l
are d i s j o i n t B o r e l sets which cover
AI,...,Ak
X. I f w e
define
k
fn(x)
then
COROLLARY.
Hausdorff space from
X
Then f
fx)f(a.) 3
f o r every
x E X,
i s a s i m p l e f u n c t i o n and
fn
II f n ( x )
6.6.
z x Aj j=l
=
into
-
flx)Il < l / n
Let X.
p
f o r every
x E X.
be a f i n i t e B o r e l m e a s u r e on a c o m p a c t
L e t f f n ) b e a s e q u e n c e o f continuous mappings
F w h i c h c o n v e r g e s u n i f o r m l y on X
t o a mapping f.
i s c o n t i n u o u s and
For each
A E z.
So f a r w e have only c o n s i d e r e d Bochner i n t e g r a t i o n w i t h re-
s p e c t t o f i n i t e p o s i t i v e measures, b u t
the extension t o real
measures o r t o complex measures may p r o c e e d e x a c t l y as
i n the
s c a l a r case. EXERCISES 6.A. fX,Z,
T h i s i s E g o r o f f ' s Theorem f o r v e c t o r - v a l u e d mappings.Let u ) be a f i n i t e measure s p a c e . L e t I f , ) be a sequence o f
measurable mappings from everywhere t o a mapping
X
into
F
which
converges
almost
f . By r e p l a c i n g a b s o l u t e values by norm
a t t h e a p p r o p i a t e p l a c e s i n t h e s t a n d a r d proof of t h e scalar E g o r o f f ' s theorem, show t h a t f o r e a c h E > 0 t h e r e e x i s t s a
44
MU J ICA
set A E Z with p(X \ A ) f uniformly on A .
5
E
and such t h a t
(f,)
converges t o
6.B. L e t (X, 2 , p l b e a f i n i t e measure space. L e t (f,) be a sequence o f measurable mappings from X i n t o F which c o n v e r g e s a l m o s t everywhere t o a mapping f .
(a)
Using E g o r o f f ‘ s Theorem 6.A f i n d a sequence o f sets A , E Z and a sequence of s i m p l e mappings g n .- X F such t h a t p ( X \ A n ) 5 2-n and Ilgn(x) - f ( x l l l 5 2-n f o r every x E A n . +
m
(b)
Let
Bj
k2j Ak
=
f o r every j formly on e a c h B
< 2 -
f o r each
j
i7V.
E
and show t h a t ( g
Show t h a t p ( X \ B . ) 3
converges t o
n
f
uni-
i’
m
(c)
B =
B Show t h a t jZl j ’ ( g ( x ) ) converges t o f ( x ) f o r e v e r y n shows t h a t f i s measurable.
Let
= 0
p ( X \ BI
U
x
E
B.
and s h m t h a t
In p a r t i c u l a r this
6.C. L e t (X, 2 , v) be a f i n i t e measure s p a c e . Using Exercise 6.B show t h a t a measurable mapping f : X + F i s Bochner i n t e g r a b l e i f and o n l y i f Jx I(f 11dl.1 < m. T h i s i s Bockncr’s nharac-
t e r i z a t i o n of Bochner i n t e g r a b l e mappings. 6.D.
T h i s i s t h e Dominated C o n v e r g e n c e Theorem f o r Bochner in-
t e g r a b l e mappings. L e t (X, Z, P I b e a f i n i t e measure s p a c e . L e t If,) be a sequence of Bochner i n t e g r a b l e mappings from X i n t o F which converges a l m o s t everywhere t o a mapping f . Suppose there e x i s t s an integrable function
X
lR such t h a t IIfn(xJII 5 g l x ) f o r e v e r y II E IN and a l m o s t e v e r y ~ € 1Using . Bochner’s c h a r a c t e r i z a t i o n 6.C and t h e s c a l a r Dominated Convergence Theorem show t h a t
-f
o
and
g :
+
f i s Bochner integrable, $* II f, - fll du
SA fndu -,SAffdu f o r e a c h
A
E
X.
(X, 2 , be a f i n i t e measure s p a c e . L e t ( f n I be a sequence of Bochner i n t e g r a b l e mappings from X i n t o F which converges u n i f o r m l y on X t o a mapping f . Show t h a t f is Bcchner 6.E.
Let
45
HOLOMORPHIC MAPPINGS
Jxllfn
integrable, A E Z. 6.F.
flldu
and
0
-C
JAf n d u
+
IA fdu
for
each
1-1 b e a B o r e 1 p r o b a b i l i t y measure on a compact Haus-
Let
d o r f f space (a)
X, and l e t
Given
$l,.
nj = and l e t
-
T
E
i,
e(Fm;
f
. .,$, $0
: X E
+
b e a c o n t i n u o u s mapping.
F
let
(FBI
j = 1,
for
fdp
..., n,
B n ) b e d e f i n e d by
~y = ($zIy),...,$nIy))
f o r every
y
E F.
Using t h e Hahn-Banach s e p a r a t i o n theorem show t h a t
where (b) point
c o ( B ) d e n o t e s t h e convex h u l l o f t h e set
B.
Using a compactness argument show t h e e x i s t e n c e of a
-
y E co(f(XI)
such t h a t
I n p a r t i c u l a r t h i s shows t h a t t h e Bochner i n t e g r a l S X f d p l i e s i n t h e c l o s e d , convex h u l l
c o (f (X))
of
f (X)
.
7. THE CAUCHY INTEGRAL FORMULAS A f t e r t h e i n t e r m i s s i o n on v e c t o r - v a l u e d i n t e g r a t i o n i n t h e mapp r o c e d i n g s e c t i o n , w e c o n t i n u e o u r s t u d y of holomorphic pings. I n t h i s s e c t i o n w e e s t a b l i s h t h e Cauchy i n t e g r a l f o m l a s
we study more c l o s e l y t h e q u e s t i o n of convergence of t h e T a y l o r series. and d e r i v e some o f t h e i r consequences. I n p a r t i c u l a r
7.1. THEOREM.
Let
U b e an o p e n
subset o f
E , and l e t
f
E
JC(U;F).
46
MUJ I CA
r > 0 b e s u c h t h a t a + r;t E U f o r a l l 5 E a(0;r). T h e n f o r e a c h A E A ( O ; r l we h a v e t h e Cauchy I n t e g r a l Formu l a Let
a E U, t E E
and
i
-
f ( a + A t ) = 2 In z
-+
:('
dr;.
Act)
lLl=P PROOF.
$ E F'
If
glr;) = $ o f ( a
then t h e function
holomorphic on a neighborhood of t h e c l o s e d d i s c
-
+
is
Ct)
By
A(0;r).
t h e Cauchy i n t e g r a l formula €or holomorphic f u n c t i o n s
of
one
complex - 7 a r i a b l e w e have t h a t
f o r each
A E A(O;r).
Since
F'
s e p a r a t e s t h e p o i n t s of
F the
d e s i r e d c o n c l u s i o n follows.
7.2.
U b e a n o p e n s u b s e t of E , and Z e t fE a E U, t E E and r > 0 b e s u c h t h a t a + Ct E U Z ( 0 ; r ) . The f o r e a c h A E h ( O ; r ) we h a v e a s e r i e s
COROLLARY.
JC(U;F).
Let
for aZZ
5
E
Let
expansion o f t h e form m
f(a
+
At) =
z m=O
where c
m
=-
c
m lm
f f a + Xtl d
0 Pe .such t h a t
,..., t n E
...
E U f o r a12 < E i n f O ; r ) . T h e n for e a c h 'ntn we have t h e Cauchy I n t e g r a Z Fo rm u l a +
49
HOLOMORPHIC MAPPINGS f(a + Altl
PROOF. Rl
Since
> rl
all
...
+
5
+ A ntn )
-n A (0;r)
the polydisc
,..., Rn
>
E An(O;R).
rn
a + Cltl
such t h a t $
If
E
F'
w e can f i n d E U for 'ntn
i s compact,
. ..
+
+
then t h e func t ion
i n s e p a r a t e l y holomorphic i n e a c h of t h e v a r i a b l e s C l , ..., ' n when t h e o t h e r v a r i a b l e s are h e l d f i x e d . T h e n r e p e a t e d a p p l i c a t i o n s o f t h e Cauchy i n t e g r a l formula f o r holomorphic f u n c t i o n s
of one complex v a r i a b l e l e a d t o t h e f o r m u l a
+
$ of(a
X
f o r every
...
+
Altl
+ Antn)
An(O;r).
E
Since t h e func ti on
i s c o n t i n u o u s on t h e compact s e t
a o A n (0;
P),
Fubini's
Theorem
allows u s t o r e p l a c e t h e i t e r a t e d i n t e g r a l by a m u l t i p l e i n t e g r a l . And s i n c e
F'
separates points, the desired
Let
conclusion
follows.
7.8.
E
U,
U be a n o p e n s u b s e t of t l , ..., t , E E and r l
+
...
+ cntn
COROLLARY.
X(U;F).
Let
a
cltl
that
a +
each
A E An(O;r)
E,
,...,
E
U
for a l l
and l e t
f E
n > 0 be s u c h
5 E zn(O;r).
Then f o r
we h a v e a s e r i e s e x p a n s i o n of t h e f o r m
MUJ I CA
50
where
T h i s m u l t i p l e s e r i e s c o n v e r g e s a b s o Z u t e l y and u n i f o r m l y f o r E
iZn(o;si
o
. 0 be such
..
+
‘ntn
E
U
for a l l
m E N o and e a c h m u l t i - i n d e x h a v e t h e Cauchy I n t e g r a l F o r m u l a
each
a
E
5 E EVE
zn(O;p).
with
la
Then f o r
= m we
.. dc, . PROOF. form
By P r o p o s i t i o n 4 . 6 w e have a series e x p a n s i o n
of
the
51
HOLOMORPHIC MAPPINGS
where
c
f o r each
u
ci E
“I
... t n
= +m’A
m
with
I c t I = m . T h i s m u l t i p l e series converges
f(a)tl
-
a b s o l u t e l y and u n i f o r m l y on a s u i t a b l e p o l y d i s c
An(O;c). A f t e r
comparing t h i s series e x p a n s i o n w i t h t h e series e x p a n s i o n g i m by C o r o l l a r y 7 . 8 ,
Lemma 4 . 7
an a p p l i c a t i o n of
completes
the
proof. 7.10.
Let
COROLLARY.
A E L s f r n E ; F ) and l e t
T h e n for a l l a , t l J . . . t n E E and a l l we have t h e p o l a r i z a t i o n f o r m u l a
PROOF.
Apply C o r o l l a r y 7 . 9 w i t h
f
ci E
P = iDE
A^
P(mE;F).
E
la1 = m
with
= P.
W e r e c a l l t h a t a s e t A i n E c o n t a i n i n g t h e o r i g i n is said t o be b a l a n c e d i f closed u n i t d i s c i f t h e set
-
A
<x E A f o r e a c h x A . I f a E A then A
-
a
E
and e a c h
A
1 and a neighborhood V o f K i n U such t h a t t h e s e t
U , and f i s bounded on B as w e l l . Hence
i s a l s o contained i n w e can w r i t e
f
Ia +
C(X
- a)]
5 - 1
= m=O z
f [ a + ~ ( -x a ) ] f1+1
and t h i s series c o n v e r g e s a b s o l u t e l y and u n i f o r m l y f o r and
1 5 ) = P. A f t e r i n t e g r a t i n g o v e r t h e c i r c l e
x
151 = r
a p p l y i n g t h e Cauchy I n t e g r a l Formulas 7 . 1 and 7 . 3
E
V
and
we c o n c l u d e
that
and t h i s series c o n v e r g e s a b s o l u t e l y and u n i f o r m l y f o r
7.12.
DEFINITION. J C ( U ; F ) and l e t a (a)
Let E
U be an open s u b s e t of
E, l e t
3:
E
f
V. E
U.
The r a d i u s of
boundedness o f
f
at
a i s t h e supremum
-
of a l l
U and f i s bounded on B ( a ; r ) . The r a d i u s of boundedness of f a t a w i l l b e d e n o t e d by r b f ( a ) . r > 0
such t h a t
Bla;r)
C
(b) The r a d i u s of convergence o f t h e T a y l o r series of f a t a w i l l b e d e n o t e d by r e f l a ) . For s h o r t r c f ( a ) w i l l be r e f e r r e d t o as t h e r a d i u s of c o n v e r g e n c e o f f a t a .
(c) The d i s t a n c e from ci t o t h e boundaryof U w i l l b e den o t e d by d u ( a ) . When U = E t h e n f o r convenience w e d e f i n e d , ( a ) = m. 7 . 1 3 THEOREM.
Let
U be a n o p e n s u b s e t o f
E,
let
f
E JC(U;F)
HOLOMORPHIC MAPPINGS
and l e t
a
53
Then
E U.
rbf ( a l = m i n { r c f ( a ) , d , l a ) } .
PROOF.
W e observe a t t h e o u t s e t t h a t
and hence
rbfia)
2
d u ( a ) . Thus t o show t h e i n e q u a l i t y
rbf (a)
(7.1)
5
min { r e f ( a ) , d , f a ) 1
i t s u f f i c e s t o show t h a t r b f ( a ) < refla). Let 0 2 r < rbf(a). Then B ( a ; r ) C U and f i s bounded, by c s a y , on B ( a ; r ) . I t f o l l o w s from t h e Cauchy I n e q u a l i t i e s 7 . 4 t h a t I I P m f ( a ) l l 5 e r - m f o r e v e r y rn E Ih7 and an a p p l i c a t i o n o f t h e Cauchy-Hadamard Formula 4 . 3 shows t h a t r e f l a ) 5 r . L e t t i n g r r b f ( a ) w e get that rbf(a) < r c f ( a ) and ( 7 . 1 ) f o l l o w s . +
Next w e show t h a t o p p o s i t e i n e q u a l i t y :
Let
that
0 - r < s < min { r e f ( a ) , d,la)).
B(a;si C U
f o r every
x
Since
s < d U ( a ) it f o l l o w s
and t h e n Theorem 7 . 1 1 i m p l i e s t h a t
E B(a;s).
On t h e o t h e r hand, it f o l l o w s from
the
Cauchy-Hadamard Formula t h a t
and hence t h e r e e x i s t s
c > 1
such t h a t
(7.4)
for e v e r y
m
E
W o . Then from ( 7 . 3 ) and ( 7 . 4 )
it f o l l o w s t h a t
54
MUJ I CA
f o r every
7.14.
x
B i a ; r l . Hence
E
and ( 7 . 2 )
r
U be an open s u b s e t of
Let
REMARK.
rbf(a)
E,
follows.
let
and
f
E
i s f i n i t e dimensional then each c l o s e d b a l l w i t h f i n i t e r a d i u s i s compact, and whence i t f o l l o w s t h a t r h f ( a ) = d, (a and r e f l a ) 1. d u ( a ) f o r e v e r y a E U. I n s h a r p c o n t r a s t w i t h t h e f i n i t e d i m e n s i o n a l s i t u a t i o n w e have t h e f o l l o w i n g reJC(U;F).
If
E
sult
7.15 PROPOSITION. S u p p o s e t h e r e e x i s t s a s e q u e n c e (q,) i n E ’ l i m q ix) = 0 f o r every s u c h t h a t llqmll = I f o r e v e r y m a n d m x E E . Then t h e r e e x i s t s a f u n c t i o n f E J C ( B ) whose r a d i u s of boundedness a t t h e origin e q u a l s one. By Example 5 . 5 t h e f u n c t i o n
PROOF.
i s holomorphic on a l l of Formula t h a t
rcf(0)
E.
I t f o l l o w s from t h e Cauchqr-Hadamrd
= 1 . An a p p l i c a t i o n o f Theorem 7 . 1 3
com-
pletes the proof.
7.16. C
E‘
EXAMPLE.
E = c
0
or
Lp
(1
5
p
such t h a t
0
By Theorem 7.13 the
f a t t h e o r i g i n c o n v e r g e s t o f u n i f o r m l y on E > 0 w e c a n f i n d k o E I N such t h a t
T a y l o r series of
B(0;Zr).
r
Then g i v e n
5
and
2r
.
k > ko
By a p p l y i n g t h e Cauchy Inqualities
7.4 we get t h a t
for
/ I t / (5 r
k
and
2 k0
. Thus m
Prnflti =
P r n [ PjfIOll ( t l j=O
and t h i s series c o n v e r g e s u n i f o r m l y f o r 1 I t II 5 r . Since Prn[$f(0)]
= 0
whenever
rn > j
w e c a n even w r i t e m
Prnflti =
2 P"[ P"+JfIOll It). j=O
(b)
I n t h e g e n e r a l c a s e t a k e an a r b i t r a r y p o i n t
If we define
t
E
U - a
fa : U - a
+
by
F
fa(t)
= fla + t l
t h e n by E x e r c i s e 5.B w e have t h a t
and P m f a ( t l = P m f ' ( a
+ tl
f o r every
-
t E U
Moreover, i f f i s bounded on B ( a ; 3 r ) B I O ; 3 r ) . Hence u s i n g ( a ) w e g e t t h a t
=
z: j=O
then
P"1P"fjfa(0)l
-
f a E
a fa
I t )
and
a
E
U.
f o r every JcfU - a;F)
rn
E
INo.
i s bounded on
56
MUJ I CA
Iltll 5 r.
and t h e l a s t w r i t t e n series c o n v e r g e s u n i f o r m l y f o r S i n c e Pm [ P m + j f l a ) ] E P ( j E ; P ( m E ; F ) )
t h e proof i s complete.
COROLLARY. L e t U b e an open s u b s e t of E, and l e t If f o r e a c h rn E I N o a n d t E E we d e f i n e PTf : U 7.18.
T o end t h i s s e c t i o n w e g e n e r a l i z e t h e
f E JCIU;FI. -+
classical
by
F
Schwarz'
Lemma as f o l l o w s .
7.19.
THEOREM.
pose t h a t exists
m
U = B(a;r)
Let
Ilf~xIII < c E
Ilf(x)ll
W
5
f o r every
such t h a t 113:
c(
-
P'fla)
a l l rn )
C
and l e t f E X ( U ; F ) . S u p x € B ( a ; r l and s u p p o s e t h e r e = 0 for every j m. T h e n E
for every
x E B(a;rl.
PROOF. F i x x E B ( a ; r ) w i t h x # a , and f i x 11 II = 1 . L e t g be t h e f u n c t i o n of one complex
+
$
F'
E
with
variable
de-
f i n e d by
By Theorem 7 . 1 1 t h e T a y l o r series of
wise t o f on t h e b a l l
f
at
a
converges p o i n t -
B ( a ; r ) . Whence t h e f u n c t i o n
g
can
be
w r i t t e n a s t h e sum of t h e power series m
g(h) =
z
o p j f i a ) (x - a )
j =m
/ Ilx - a l l ) . I n p a r t i c u l a r g i s holomorphic on t h a t d i s c . Take s w i t h llz - a II < s < r . Since IIfIl 5 c on t h e d i s c on
A(0;r
B ( a ; r l i t follows t h a t
HOLOMORPHIC MAPPINGS
for
IA I
= s / l l z - a II,
and t h e r e f o r e f o r
57
IX I
t h e c l a s s i c a l Maximum P r i n c i p l e . By a p p l y i n g with
X =
< s /Il x -
this
- a II
by
inequality
we get t h a t
1
s
After l e t t i n g
+
r, an a p p l i c a t i o n of t h e Hahn-Banach Theorem
completes t h e proof.
EXERCISES 7.A. most
E P(E;FI
b e a c o n t i n u o u s polynomial
a, t
Let
E
E,
r > 0
f E 7CIE;F).
a constant
c > 0
and
k > m.
Suppose t h e r e i s an i n t e g e r m E I N o
such t h a t
II f(xlII
f
7.C.
U be an open s u b s e t of
Let
pose t h e r e i s a c l o s e d s u b s p a c e
7.0.
x
E
5 c
112 Ilm
for all
i s a polynomial of d e g r e e a t most
Show t h a t
every
at
of d e g r e e
m. Show t h a t
for a l l 7.B.
P
Let
U. Show t h a t
Show t h a t i f
N
E , and l e t
of
F
x
and E
E.
m.
f
Sup-
E K(U;F).
such t h a t f f x )
€
N
for
f E Jc(U;N).
U i s a p r o p e r open s u b s e t of
E
then
U be an open s u b s e t of E, and l e t f E X f U ; F ) . Show t h a t e i t h e r r f l x ) = 00 € o r e v e r y x E U ( i n t h i s case U = E l , 7.E.
Let
b
5e
MUJ 1 CA
or else
rbffxl
for a l l x, y
E
m
for every
x
E 11,
and in the latter case
U.
7.F. (a) Let E be a separable Banach space. Using Cantor's diagonal process show that each bounded sequence in El has a U(E',E)-convergent subsequence. (b) Show that each infinite dimensional, separable Banach space satisfies the hypothesis in Proposition 7.15. 7.G. (a) Let E be a reflexive Banach space. By considering a suitable separable, closed subspace of E show that each bounded sequence in E has a ofE,E')-convergent subsequence. (b) Show that each infinite dimensional, reflexive Banach space satisfies the hypothesis in Proposition 7.15
8.
G-HOLOMORPHIC MAPPINGS
In this section we show that a mapping is holomorphic if it is continuous and its restriction to each complex line is holomorphic. This is a useful characterization, and in m y situations this is the easiest way to check that a given mapping is holomorphic. 8.1. DEFINITION. Let U be an open subset of E . A mapping f : U F is said to be G - h o l o m o r p h i c or G - a n a l y t i c (G for Goursat) if for all a E U and b E E the mapping X f ( a + Ab) is holomorphic on the open set {A E @ : a + Xb E U}. We shall denote by J C G f U ; F ) the vector space of all G-holomorphic mapJEG(U;@i pings from U into F. If F = @ then we shall write = JC,(U). +
+
8.2. EXAMPLE.
PROOF.
P(a
PafE;F)
C
JCG(E;F).
+ Ab) i s a polynomial in X for
all
a, 0
E
E.
59
HOLOMORPHIC MAPPINGS
REMARKS. (a) The Identity Principle, the Open Mapping Principle, the Maximum Principle and Liouville's Theorem, all of them established in Section 5 for holomorphic mappings,= actually true for G-holomorphic mappings. A glance at the corresponding proofs shows this at cnce. 8.3.
(b) An examination of the corresponding proofs shws that Theorem 7.1 and Corollary 7.2 are still valid for G-holomorphic mappings.
(c) Finally, an examination of the corresponding proofs shows that Theorem 7.7 and Corollary 7.8 are still valid for those G-holomorphic mappings whose restrictions to finite dimensional subspaces are continuous. PROPOSITION. L e t XG(U;F). F o r e a c h a
8.4. E
U b e a n o p e n s u b s e t of E , a n d l e t E U and m E W o L e t P m f ( a ) : E --t
f F
be d e f i n e d by
where
r
is c h o s e n s o t h a t
0
a + < tE U fop a l l
5 E
a(O;r).
Then:
(a)
The d e f i n i t i o n o f
(b)
f o r all
(c)
P m f ( a ) ( t ) i s i n d e p e n d e n t from t h e
r.
choice o f
The m a p p i n g
t
E
E
and
m
P f l u ) i s m-homogeneous,
i.1 E C.
If U i s a - b a l a n c e d
s e r i e s expansion
t h a t is
t h e n f o r each
3:
E
U
we have t h e
60
MUJ I CA (a)
PROOF.
Let
a + r;t
that
E
U
t
E
b e g i v e n and l e t
E
for a l l
r;
E
r
0 < s
be
such
X ( 0 ; r ) . Then by Remark 8.3(b) we
have t h a t
f o r every
X
E
A(0;sl.
f o r every
m
E
Do.
t
Let
(b)
and
p E 6
be g i v e n .
If
i s suf-
r > 0
t h e n a g a i n by Remark 8 . 3 ( b ) w e have t h a t
ficiently s m a l l
for every
E E
By Lemma 4.5 w e c o n c l u d e t h a t
Then a n o t h e r a p p l i c a t i o n
XEA(0:r).
of
Lemma
4.5
y i e l d s t h e d e s i r e d conclusion.
(c)
r;
E
a.
Given
LC E
a + c ( x - a)
w e have t h a t
U
By a compactness argument w e c a n f i n d
a + 1
U
for all
such t h a t
Then by Remark 8 . 3 ( b ) we
E X(0;r).
have t h a t
f o r each
X
E A(0;r).
Letting
X = 1
we get the desired
con-
clusion. 8.5. that
Under t h e s e t t i n g o f P r o p o s i t i o n 8.4 it i s
REMARK.
Pmf ( a )
E
PalmE;F)
for a l l
a
E
U
and
m E Do,
proof of t h i s f a c t , w i t h o u t a d d i t i o n a l h y p o t h e s e s on have t o w a i t u n t i l S e c t i o n 3 6 , f o r i t rests on a deep of Hartogs on s e p a r a t e
analyticity.
true but
a
f, w i l l theorem
61
HOLOMORPHIC MAPPINGS
8.6. E
Let
PROPOSITION.
JCG(U;F). T h e n
f
be an o p e n s u b s e t of
U
i s c o n t i n u o u s if and o n l y i f
and l e t f i s locally
E,
f
bounded.
To b e g i n w i t h w e remark t h a t Schwarz'
PROOF.
v a l i d f o r G-holomorphic mappings.
Lemma
is
7.19
I n d e e d , the sane proof
amlies
p r o v i d e d w e use P r o p o s i t i o n 8 . 4 1 ~ )i n s t e a d o f Theorem 7 . 1 1 . Now, l e t Given - c
a
f : U
for all
mapping
F
be G-holomorphic and l o c a l l y bounded.
w e choose r > 0 and c > 0 s u c h t h a t 1 I f(x)ll x E B ( a ; r ) . By a p p l y i n g Schwarz' Lemma t o t h e
U
E
+
-
f(x)
f ( a ) we get t h a t
f o r a l l x E B f a ; r l , p r o v i n g t h a t f i s c o n t i n u o u s a t the p o i n t a . S i n c e t h e r e v e r s e i m p l i c a t i o n i s c l e a r , t h e proof i s complete.
Now w e can e s t a b l i s h t h e
characterization
of
holomorphic
mappings announced a t t h e b e g i n n i n g of t h i s s e c t i o n . 8.7.
Let
THEOREM.
mapping
f : U
--t
F
U be an o p e n s u b s e t of
E.
(a)
f
i s holomorphic
(b)
f
i s c o n t i n u o u s and G - h o t o m o r p h i c .
(c)
f
i s c o n t i n u o u s and
f
f i n i t e dimensional subspace M
of
PROOF.
The i m p l i c a t i o n ( a )
( b ) * ( c ): and l e t
Let
Then
f
:
U
-+
I
each
U n M
i s hoZomorphic for each
E.
* (b) i s clear. P
be G-holomorphic and continuous,
M be a f i n i t e d i m e n s i o n a l s u b s p a c e of
and l e t ( e l , ..., e n ) be a b a s i s f o r
M.
E. Lst
+ Alel +
...
+ A e I = n n
"1
z1 c a X l c1
aE U nM
Then by Remark 8 . 3 ( c ) *
have a series e x p a n s i o n of t h e form f(a
for
the following conditions are equivalent:
"n ... in
67
MUJ I CA
where t h i s m u l t i p l e series converges a b s o l u t e l y and on a s u i t a b l e p o l y d i s c Pm E P l m M ; F )
An(O;rl.
I f f o r each
uniformly
rn E W o we define
by
t h e n w e have a power series e x p a n s i o n W
f l a + Alel
+
... +
Xn e n ) =
w i t h uniform convergence on
(c) * ( a ) : subspace of
E
Let
U. I f
C
... +
M i s a f i n i t e dimensional
c o n t a i n i n g a t h e n by h y p o t h e s i s
-
M
-
a) from M i n t o
U
power
M and N a r e two f i n i t e
dimen-
such t h a t
F W
x
f(x) =
I
M is series
f
holomorphic and t h e n by Theorem 7 . 1 1 t h e r e i s a .Z P ( x m=O m
Anen)
T h i s shows ( c ) .
An(O;r).
B(a;r)
+
Z Prn(Alel m=O
M
- a)
Pmlx
m=O x E M n E(a;r). If
f o r every
a t h e n i t f o l l o w s from t h e M N = P,(t) uniqueness of t h e T a y l o r series e x p a n s i o n t h a t P , ( t l for a l l t E M N and a l l rn E W o .L e t Pm : E' F b e deM = P rn ( t ) i f M i s any f i n i t e dimensional subspace f i n e d by P , ( t l of E c o n t a i n i n g a and t . Then Pm E ? ( m E ; F ) by E x e r c i s e a
s i o n a l s u b s p a c e s of
E
containing
+
2.B,
and W
f(x) =
f o r every
a ball Ilf(x)ll
-
x E B(a,r).
E(a;s)
5
L'
C
Now s i n c e f and
E(a;r)
f o r every
Z Pm(x m=O
x
E
- a)
i s continuous w e can
a constant
B ( a ; s ) . Given
t
l e t M b e any 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 and
0
c E
E E
such with
find that
Iltll < 1
containing
t. Then by t h e Cauchy I n t e g r a l Formula 7 . 3 w e g e t t h a t
a
63
HOLOMORPHIC MAPPINGS and i t f o l l o w s t h a t
IIPmII
5 cs-m . Hence e a c h P
m
i s continuous
W
and t h e power series
2
Pm(x
-
a ) has a r a d i u s ofconvergence
m=O
g r e a t e r than o r equal t o
s . T h i s show ( a ) and t h e theorem.
U be a n open s u b s e t of
Let
Then e a c h
Cn.
mapping f : U F is separately f ( i , , . . ., r n ) i s holomorphic i n e a c h
G-holomorphic
hoZomorphic,
+
that
is,
when t h e o t h e r v a r i ‘j a b l e s a r e h e l d f i x e d . The f o l l o w i n g r e s u l t on s e p a r a t e l y h o l o -
morphic mappings p a r a l l e l s P r o p o s i t i o n 8 . 6 .
U be a n o p e n s u b s e t of C n , and l e t f : U - + F be s e p a r a t e l y holomorphic. T h e n f is c o n t i n u o u s if and only if f is locally b o u n d e d . 8.8.
LEMMA.
PROOF.
Let
f
Let
U F b e s e p a r a t e l y holomorphic and l o c a l l y a E U choose r > 0 and c 0 such that f o r e v e r y 5 E A n ( a ; p ) . Then f o r e a c h 5 E A n ( a ; r ) :
+
bounded. Given
llfl 0 . S i n c e F i s e q u i c o n t i n u o u s e a c h p o i n t a E K h a s aneighborhood Va such t h a t IIfIz.) - f i a l l l 5 E f o r a l l x E Va and f E F . S i n c e K i s compact t h e r e i s a f i n i t e s e t A C K s u c h t h a t K C V { V a : a E A ? . Whence i t follows t h a t PROOF.
Let
s u p II f izc)II XE K
2
s u p 11 f I z ) l l x€ A
+
E
F. B y a p p l y i n g t h i s argument t o t h e s e t F - F (which i s also e q u i c o n t i n u o u s ) w e can f i n d a f i n i t e s e t B C K
f o r all
f E
such t h a t
for a l l
f, g E
F.
I t follows t h a t
MUJ I CA
for each
fo E
F and the proof is complete.
Now it is easy to prove A s c o l i ' s Theorem. 9.12. THEOREM.
Let
X
b e a t o p o Z o g i c a l s p a c e . Then e a c h e q u i -
c o n t i n u o u s , p o i n t w i s e bounded s u b s e t of compact i n
ClX)
is
re2ativeZy
C l X ) f o r t h e compact-open t o p o Z o g y .
PROOF. Let F be an equicontinuous, pointwise bounded subset of C ( X i , and let 'i denote the closure of F in ex. Then F is clearly pointwise bounded, and therefore compact in C A by Tychonoff's product theorem. Now, the set 'i is equicontinuous by Lemma 9.10, and hence the product topology and the compactopen topology coincide on F by Proposition 9.11. Thus F is a compact subset of ( C ( X ) , T ~ ? ) and the proof is complete. After establishing some topological properties of the spaces of continuous mappings, we devote our attention to the spaces of holomorphic mappings. 9.13. PROPOSITION. I f U is an o p e n s u b s e t of E t h e n K(U;FI is a c l o s e d v e c t o r s u b s p a c e of i C ( U ; F . J , T,). I n particular (JC(U;FI, -re) i s complete.
PROOF. The proposition is essentially a restatement of Exercise 8.A. Let If . I be a net in J C ( U ; F I which converges to a 2 mapping f E C i U ; F ) for the compact-open topology. Given a E U, b E E and $ E F ' set gilXl = $ o f i ( a + Ab) and g(h) = $ o f ( a i Ab) for every X E A = { A E g : a + hb E U}. Then each g i is holomorphic on A and the net ( g i ) converges to g uniformly on each compact subset of A . By the well known theorem of Weierstrass for holomorphic functions of one complex variable, the function g is holomorphic on A. Then it follows from Theorems 8.7 and 8.12 that f E K I U ; F ) . The last assertion in the proposition follows from Proposition 9.5.
73
HOLOMORPHIC MAPPINGS
COROLLARY.
9.14.
U
If
a n o p e n s u b s e t of @
7 : s
n
t h e n (JC(U;F),T~)
i s a Frechet s p a c e . 9.15.
PROPOSITION.
F
family
JC(U;FI
C
U b e a n o p e n s u b s e t of
Let
E . The? for each
the f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t .
(a)
F
is b o u n d e d i n
(b)
F
is l o c a l l y b o u n d e d .
(c)
F
i s e q u i c o n t i n u o u s and p o i n t w i s e b o u n d e d .
(JC(U;FI,
T ~ ) .
( a ) * ( b ) : I f F i s n o t l o c a l l y bounded t h e n w e c a n a E U , a s e q u e n c e (f,) C F and a s e q u e n c e ( a n ) s u c h t h a t / l a n - all < l / n and Ifnlan) 1 > n for every
PROOF.
find a point C
U
n. I f w e set
then
K
i s a compact s u b s e t of
unbounded on
K. Hence
U
and
the
sequence
(f,)
is
F i s n o t bounded i n (JCfU; F1, . c c l .
F is l o c a l l y bounded, t h a t i s uniformly bounded on a s u i t a b l e n e i g h b o r h o o d of e a c h p o i n t o f U. Then c l e a r l y F i s u n i f o r m l y bounded o n e a c h compact s u b s e t of U, t h a t i s F i s bounded i n I J C I U ; F I , T ~ ) (b)
4
(a):
Assume
(b)
3
(c):
If
F i s l o c a l l y bounded t h e n
and
f o r every
c
x
E
;0
be such t h a t
B ( a ; r l and
f
E
F.
Then
it follows
Cauchy i n e q u a l i t i e s t h a t m
~ ~ f i z- i f i a i l l
5
Z: I I p r n f ( a ) ( x m=l
c IIx < -
is obviously
F i s equicontinuous let a E U , B(a;rl C U and I1 f(x)ll < c
p o i n t w i s e bounded. T o show t h a t
r > 0
F
r-
- all
112-
all
U ~ I I
from
the
74
MUJ I CA
for every (c) is
c
x E Blu;ri
-
f E F.
and
0
c + I
-
i s equicontinuous.
a E U. S i n c e F i s p o i n t w i s e bounded there such t h a t I1 flail1 5 c f o r e v e r y f E F . S i n c e F i s
(b):
Let
e q u i c o n t i n u o u s t h e r e i s a neighborhood
IIf(x)
F
Hence
5
ffa)ll
for a l l
1
x
E
f
a in
F . Then
U such t h a t Ilf~'xiII 5
LC
E
V
and
f
E
F , completing t h e p r o o f .
V
and
V of
for all
By combining A s c o l i ' s Theorem 9 . 1 2
E
and P r o p o s i t i o n s
9.13
and 9.15 w e o b t a i n a t once t h e f o l l o w i n g r e s u l t , which e x t e n d s t h e c l a s s i c a l M o n t e 1 's T h e o r e m .
9.16. PROPOSITION. b o u n d e d s u b s e t of
U b e a n o p e n s u b s e t of E . l X ( U I , T ~ i) s r e Z a t i v e 2 y c o m p a c t . Let
Then
each
EXERCISES
Show t h a t each c l o s e d s u b s p a c e of a k-space i s a k-space.
9.A.
Show t h a t e a c h
a Hausdorff
apen s u b s p a c e of
k-space
is
a
k-space. 9.B.
UI
Let
and
K
b e a compact s u b s e t of a Hausdorff s p a c e b e open s u b s e t s of
U2
compact sets
KI
C
and
Ul
K2
such t h a t
X C
U
2
K
such t h a t
X.
Let
u U2 ' Find K = K1 u K2 . CUI
U 2 b e two open s u b s e t s of E . Show t h a t t h e s p a c e ( X ( U l U U 2 ) , r c l can be c a n o n i c a l l y i d e n t i f i e d w i t h a c l o s e d v e c t o r subspace of t h e p r o d u c t ( M I U a ), T ~ x) (X(UzI,r e ) . 9.C.
Let
Ul
and
G e n e r a l i z e t h i s t o an a r b i t r a r y f a m i l y ( U i l i E I of
9.D.
of open subsets
E. L e t (xi)
be a n e t
in a
X tvith
to p o logi ca l space
the
p r o p e r t y t h a t e v e r y s u b n e t of (xi) h a s a s u b n e t whizh c o n v e r g e s t o a fixed point 9.E.
Let
x. Show t h a t
(J:?:)
converges t o x.
U be a connected open s u b s e t of
bounded sequence i n / X ( U ) , T ~ ) and suppose ( f n l l t ' ) l converges i n
6
f o r every p o i n t
E.
that x
Let
the
(f,.' !>e a sequence
i n a nonvoid
open
HOLOMORPHIC MAPPINGS
set
V
C
(a)
U. U s i n g M o n t e l ' s Theorem 9.16 a n d E x e r c i s e f ( x ) = lim f (xi e x i s t s €or e v e r y n
that the l i m i t (b)
75
Show t h a t
compact s u b s e t of
converges to
(f,)
f
show
9.D.
x
uniformly
E
U.
on
each
U.
T h i s r e s u l t e x t e n d s t h e classical V i t a l i ' s Theorem. 9.F.
Let
U be a n o p e n s u b s e t of a s e p a r a b l e B a n a c h s p a c e
E.
U s i n g C a n t o r ' s d i a g o n a l p r o c e s s show t h a t e a c h b o u n d e d sequence i n ( X ( U ) , T ~ )h a s a c o n v e r g e n t s u b s e q u e n c e .
This sharpens
the
c o n c l u s i o n i n M o n t e l ' s Theorem 9 . 1 6 . 9.G.
Show t h a t i f
i s a n o p e n s u b s e t of
I/
i s a complemented s u b s p a c e o f t h a t for each
a
E
p r o j e c t i o n from ( K ( u ; F ) , 9.H.
Show t h a t i f
Show t h a t i f
Banach s p a c e
~
-+
.
T,)
m
and
hVo
E
t
More p r e c i s e l y , show
P"f(a)
o) n t o I P ( " E ; F / ,
t h e n (P(mK;F), T ~ ) is a
continuous
T ~ ) .
E
t h e n t h e mapping
E B.
i s an open s u b s e t o f a f i n i t e dimensional
U
i:' t h e n t h e mapping
i s c o n t i n u o u s €or e a c h
9.J.
T
f
U i s a n o p e n s u b s e t of
i s continuous f o r each 9.1.
(K(U;F),
t h e mapping
U
E
Show t h a t i f
s i o n a l Banach s p a c e
I/
DJ0.
rri F
i s an open s u b s e t of an i n f i n i t e
K , and i f
i s n o t c o n t i n u o u s for a n y
rrt
E
F # {O},
dimen-
t h e n t h e mapping
M. F u r t h e r m o r e , show t h a t i f
i:'
76
MUJ I CA
s a t i s f i e s t h e hypothesis i n Proposition rn E lN
7.15
o n e c a n even f i n d a sequence ( f y L ) i n
v e r g e t o zero i n
( J C ~ U P; ? E ; F I
d o e s n o t con-
)
), T ~ ) .
A l o c a l l y c o n v e x s p a c e i s s a i d t o be b a r r e l l e d
9.K.
each
such t h a t
JCIU;FI
c o n v e r g e s t o zero i n ( J C ( U ; F ) , T ~ )b u t ( P m f
(f,)
for
then
c l o s e d , convex, b a l a n c e d , a b s o r b i n g set i s
if
each
a neighborhood
of
zero. (a)
Using t h e
Category
Baire
Frgchet space i s b a r r e l l e d . relled i f space
Theorem
Conclude t h a t
show
that
each
is
bar-
(JE(U;FI,T~)
U is an open s u b s e t of a f i n i t e dimensional
Banach
E.
(b)
Show t h a t i f
m e n s i o n a l Banach s p a c e
U i s a n o p e n s u b s e t of a n i n f i n i t e d i E l and i f
F # 101,
then f o r each n E U
t h e set
i s a closed, convex, b a l a n c e d , a b s o r b i n g s u b s e t o f b u t i s n o t a neighborhood of zero.
( 3 C ( U ; F ) , T~:),
is
Hence I J C ( U ; F I , -ri,l
not
barrelled.
NOTES AND COMMENTS Most o f t h e r e s u l t s i n C h a p t e r 11 h a v e b e e n
known
l o n g t i m e a n d can a l r e a d y b e f o u n d i n t h e book o f E . R.
Phillips I 11
.
Among t h e r e s u l t s t h a t a p p e a r e d
for
H i l l e and
within
l a s t t w e n t y y e a r s w e m e n t i o n Theorem 7 . 1 3 , d i , e t o
a the
Nachbin
L.
I , P r o p o s i t i o n 7.15, d u e t o S. Dineen I 4 I , and P r o p o s i t i o n 9 . 1 6 , n o t i c e d by H . A l e x a n d e r 11 1 . I t w a s a l s o 13. Alcxnnder 1 1 I who showed t h a t (ZftUi, I(,) i s n e v e r b a r r e l l e d when LI i s a n 12
open s u b s e t o f a n i n f i n i t e d i m e n s i o n a l Banach s p a c e ,
a result
t h a t w a s l e f t t o t h e r e a d e r a s E x e r c i s e 9.K. P r o p o s i t i o n 7.15 h a s an i n t e r e s t i n g s e q u e l , f o r 7.F
and
7.G
that
each
raised
Indeed,
we
separLib1.e,
or
a n a t u r a l q u e s t i o n i n t h e t h e o r y of Banach s p a c e s .
know from E x e r c i s e s
it
77
HOLOMORPHIC MAPPINGS
r e f l e x i v e , i n f i n i t e d i m e n s i o n a l Banach s p a c e s a t i s f i e s t h e hypothesis i n Proposition 7.15.
I t i s t h e n n a t u r a l t o a s k whether
e v e r y i n f i n i t e d i m e n s i o n a l Banach s p a c e s a t i s f i e s t h e hypothesis i n Proposition 7.15.
T h i s q u e s t i o n w a s answered i n t h e
affir-
mative by B. J o s e f s o n [ 2 1 , and i n d e p e n d e n t l y b y A . N i s s e n z w e i g [ 1]
.
T h i s i s a deep r e s u l t and t h e i n t e r e s t e d r e a d e r i s refer-
r e d t o t h e o r i g i n a l p a p e r s of B. J o s e f s o n [ 2 I and A. Nissenzweig
[ l ] , o r t o t h e r e c e n t book of J. D i e s t e l [ l ] , f o r a proof o f t h i s theorem. Many of t h e r e s u l t s i n S e c t i o n s 5,7 and 8 can b e found t h e books of L. Nachbin [ 1 ]
,
[ 2
J
and
T.
Franzoni
and
in E.
V e s e n t i n i 111. Our b r i e f p r e s e n t a t i o n of t h e Bochner i n t e g r a l i n S e c t i o n 6 f o l l o w s e s s e n t i a l l y t h e book of J. D i e s t e l
[
Our p r e s e n t a t i o n o f t h e compact-open t o p o l o g y i n S e c t i o n 9
11. is
q u i t e s t a n d a r d and can be found f o r i n s t a n c e i n t h e book of S. Willard [ 11. For t h e p r o p e r t i e s o f holomorphic mappings between l o c a l l y convex s p a c e s t h e r e a d e r i s r e f e r r e d t o t h e books of M. [
1 1 , P . Noverraz [ 3 I ,
Colombeau
I
1]
.
G.
H e r d
Coeurs [ 1 1 , S . Dineen [ 5 ] and J. F.
CHAPTER I11
DOMAINS OF HOLOMORPHY
1 0 . DOMAINS O F HOLOMORPHY
I n t h i s s e c t i o n w e i n t r o d u c e t h e n o t i o n s o f domain lomorphy a n d domain
o f e x i s t e n c e , and s t u d y t h e i r
of ho-
elementary
properties. W e b e g i n by p r e s e n t i n g some e x a m p l e s t o m o t i v a t e t h e d e f i n i t i o n s . A s i n t h e p r e c e d i n g c h a p t e r a l l Banach spces considered
w i l l b e complex. EXAMPLE.
10.1.
2
and
V
U and
If
V
a r e t w o o p e n sets i n
connected, then t h e r e i s a function
V
h a s no h o l o m o r p h i c e x t e n s i o n t o Since
PROOF.
the function
flzl = ( z
f E J C l U ) d e f i n e d by
holomorphic e x t e n s i o n t o
with
U
V.
i s connected t h e r e is a p o i n t
V
C
f E J C ( U ) vhich
-
a
V n aU. Then
E
has
a)-’
no
V.
For holomorphic f u n c t i o n s of
n
2
2
variables the
situa-
t i o n i s e n t i r e l y d i f f e r e n t , as t h e f o l l o w i n g e x a m p l e shows. 10.2.
where
EXAMPLE.
c’
-q
r . 3
R .
a unique e x t e n s i o n
figure i n PROOF.
2 D = A (0;R)
Let
5
m
-
for
f E JC(D).
and l e t
j = 1,2.
Then e a c h
f
E
X(H)
has
The p a i r (H,DI i s c a l l e d a Hartogs
c2.
Choose
pI
with
rl < p 1 < RI .Given 79
f
E
at(f1) define
80
MUJ 1 C A
f o r every
i n the polydisc
z
(cl -
A f t e r expanding
z,l
i n t e g r a t i o n shows t h a t each
g
-1
= A 2 ( D ; R ' ) where R ' = ( p l , R 2 ! .
D'
. i n powers o f
f i x e d . On t h e o t h e r h a n d ,
z2
t h e i n t e g r a l s i g n w e see t h a t
zI
z 2 f o r each
zI, a term
z1
by d i f f e r e n t i a t i o n
for
under
i s a holomorphic f u n c t i o n
g
fixed. Since g
term
by
is a holomorphic f u n c t i o n of
of
i s c l e a r l y l o c a l l y bounded, a n
a p p l i c a t i o n o f Lemmas 8.8 and 8.3 shows t h a t
g
by t h e Cauchy I n t e g r a l Formula for h o l o m o r p h i c
E
Now,
7C(D').
functions
of
o n e v a r i a b l e , w e have t h a t g l z l = f ( z l f o r e v e r y z E C 2 w i t h I z 2 I < r 2 , and t h e r e f o r e f o r e v e r y z E D' n R, I z l I < PI and since
D ' n 11
i s c o n n e c t e d . Then t h e f u n c t i o n
= f on H
f i n e d by
and
7
= g
-
f E ?C(D)
de-
on D ' i s t h e r e q u i r e d exten-
s i o n . The u n i q u e n e s s o f t h e e x t e n s i o n i s c l e a r . T h i s example m o t i v a t e s t h e f o l l o w i n g d e f i n i t i o n . 10.3.
DEFINITION.
set V o f
E
Let
containing
U be an open s u b s e t of
U i s s a i d t o be a h o l o m o r p h i c extension
o r hoZornorphic c o n t i n u a t i o n of
7
extension
E
E . An opensub-
U i f each
f
E
M ( U ) has a unique
K(v).
W e want t o s t u d y t h o s e open sets
U i n E which a r e i n saw
s e n s e t h e l a r g e s t common domains o f d e f i n i t i o n f o r a l l the func-
tions
f
E
XtUl.
These
open
sets w i l l be
holomorphy. How s h o u l d w e d e f i n e domains o f
called
domains o f
holomornhy?
We
might be i n c l i n e d t o d e f i n e a domain of holomorphy a s a n
open
s e t i n F: which h a s no p r o p e r h o l o m o r p h i c c o n t i n u a t i o n , but such
a d e f i n i t i o n would t u r n o u t t o b e i n a d e q u a t e . A c t u a l l y , s u c h a d e f i n i t i o n would be a d e q u a t e i f w e e n l a r g e d t h e c l a s s
of
ob-
j e c t s u n d e r d i s c u s s i o n by r e p l a c i n g open s e t s i n Banach s p a c e s
b y Riemann domains o v e r Banach s p a c e s . W e s h a l l i n d e e d do t h i s i n Section 52, but f o r the t i m e being
we
s h a l l restrict
s t u d y t o domains o f holomorphy i n Banach s p a c e s , and
case t h e d e f i n i t i o n i s t h e f o l l o w i n g .
in
our this
81
DOMAl NS OF HOLOMORPHY
U i n E i s s a i d t o b e adomain of h o l o m o r p h y i f t h e r e are no open sets V and W i n E w i t h t h e DEFINITION. An open set
10.4.
following p r o p e r t i e s :
i s connected and n o t co n ta in e d i n
(a)
V
(c)
For e a c h
U.
I
f
unique) such t h a t If
f E JclUl
= f
on
there exists W.
i s a domain o f holomorphy
U
(necessarily
f € Jc(V)
then
U
clearly
has
110
proper holomorpic c o n t i n u a t i o n , b u t t h e converse i s n o t t r u e i n general.
1 0 . 5 . PROPOSITION.
Let
U
b e a n o p e n s u b s e t of E . Assume t h a t
for e a c h s e q u e n c e ( a . ) i n U w h i c h c o n v e r g e s t o a p o i n t a 3
there e x i s t s a function Then
f E J c t U l w h i c h i s u n b o u n d e d on
E
aU
la .1. 3
U i s a d o m a i n of hoZomorphy.
PROOF.
Suppose
i s n o t a domain o f holomorphy, and let V and
U
W b e t w o open s e t s s a t i s f y i n g t h e c o n d i t i o n s i n D e f i n i t i o n 10.4.
By t h e I d e n t i t y P r i n c i p l e w e may assume t h a t in
W
i s a connected
W
V . By E x e r c i s e 1 0 . F t h e r e i s a s e q u e n c e ( a .) 3 which c o n v e r g e s t o a p o i n t a E V n aU n aW. By h y p o t h e -
component o f
U
fi
sis t h e r e is a f u n c t i o n hand
f ( a .) = f l a
10.6.
COROLLARY.
3
f
E
KCfUl which i s unbounded
F ( a . ) converges t o
Then on one hand
3
.)
3
i s unbounded.
E v e r y o p e n s e t in
?(a),
and
on
fa.). 3
on t h e 2 t h e r
This is impossible. &
i s a domain of holomorphy.
L e t U be a n open s e t i n @ a n d l e t f a . ) b e a sequence 3 U which c o n v e r g e s t o a p o i n t a E aV. Then t h e f u n c t i o n f ( z ) = ( z - a )- 1 i s h o l o m o r p h i c on U and unbounded o n ( a . ) .
PROOF. in
3
10.7.
COROLLARY.
ho Zomorphy PROOF.
E'uery c o n v e x o p e n s e t i n
E
i s a domain
of
.
Let
U be a convex o p e n s e t i n E
and l e t
(a.) 3
be
a
82
MUJ I CA
a E aU.By t h e HahnReq(a) Banach Theorem t h e r e e x i s t s P E E ? such t h a t R e q ( x l f o r e v e r y x E U. Then t h e f u n c t i o n f ( x ) = [ q(x - a ) ]- I i s holomorphic on U and unbounded on ( a . 1 . sequence i n U which converges t o a p o i n t
3
If
U i s a domain of holomorphy t h e n f o r e a c h p a i r of open s e t s V and W s a t i s f y i n g t h e c o n d i t i o n s ( a ) and (b) i n Eefinibe t i o n 1 0 . 4 t h e r e e x i s t s a f u n c t i o n f E s C ( U ) , which c a n n o t 3: E K ( V ) such extended t o V i n t h e s e n s e t h a t t h e r e i s no t h a t 7 = f on W. I n g e n e r a l t h e f u n c t i o n f depends on t h e f for open s e t s V and W. I f w e can t a k e t h e same f u n c t i o n a l l V and W t h e n w e s h a l l s a y t h a t U i s a domain of e x i s t e n c e . More p r e c i s e l y , we have t h e f o l l o w i n g d e f i n i t i o n . 10.8. DEFINITION. An open s e t U i n E i s s a i d t o be t h e domain of e x i s t e n c e of a f u n c t i o n f € J C ( U ) i f t h e r e are no open sets
I/
and
i n E and no f u n c t i o n
W
f
E X(V)
w i t h t h e follai-
ing properties: (a)
V
(c)
7
i s connected and n o t c o n t a i n e d i n
= f
on
U.
W.
C l e a r l y e v e r y domain o f e x i s t e n c e i s a domain of En t h e two phy. The n e x t theorem shows t h a t i n
holomorconcepts
coincide. 10.9.
THEOREM.
E v e r y d o m a i n of h o l o m o r p h y i n
En
i s a domain
of existence. W e s h a l l p r e s e n t l y g i v e an e x i s t e n c i a l proof of Theorem 10.9
we s h a l l g i v e a c o n s t r u c t i v e proof of a theorem of H . C a r t a n and P . T h u l l e n , which i m p l i e s Theorem 1 0 . 9 . The key t o t h e first proof of Theorem 1 0 . 9 i s t h e f o l l o w i n g l e m m a .
based on t h e B a i r e Category Theorem.
Let
10.10. LEMMA. Banach s p a c e
E.
In
t h e next s e c t i o n
U be a domain of h o l o m o r p h y i n a s e p a r a b l e
Let
F d e n o t e t h e s e t of
a l l f u n c t i o n f E JC(UI
83
DOMAINS OF HOLOMORPHY
f. T h e n F i s a
U i s n o t t h e domain o f e x i s t e n c e o f
such that s e t of
the f i r s t category i n (JCIUl,
PROOF.
For e a c h p a i r of open sets
T ~ ) .
V
and
satisfying
W
c o n d i t i o n s ( a ) and ( b ) i n D e f i n i t i o n 1 0 . 4 l e t t h e v e c t o r s u b s p a c e of a l l
f
E
the
X ( U , V, W) d e n o t e
7
x(Ul f o r which t h e r e e x i s t s
x(V/ ( n e c e s s a r i l y u n i q u e ) such t h a t 7 = f on W. S i n c e U i s a domain o f holomorphy, J c t U , V , W ) i s a p r o p e r v e c t o r subspace
E
of
x(Ul. For e a c h
m E mT
let
K m ( U , V, WI d e n o t e t h e set of a l l
such t h a t 171 5 m on V. W e claim t h a t JC,(U,V,Wl be a n e t i n i s a c l o s e d s u b s e t of ( X ( U l , r c l . I n d e e d , l e t If,) X m ( U , V, Wl which converges t o some f i n ( K ( U ) , Tc). S i n c e I fi I < m on V f o r e v e r y i an a p p l i c a t i o n o f M o n t e l ' s Theorem f E Jc(U,V,Wl
9 . 1 6 y i e l d s a s u b n e t of
(Fi)
which c o n v e r g e s t o a f u n c t i o n g Whence it f o l l o w s t h a t f E 3Cm (U,V, Wl and f = g , I
i n (JctVl,- r e ) .
and o u r claim h a s been proved. S i n c e v e c t o r subspace of
3C(U/
Jc(U,V,Wl
NOW,
a proper
and h a s t h e r e f o r e empty i n t e r i o r
( J C ( U I , T ~ ) ,w e c o n c l u d e t h a t t h e smaller s e t
c l o s e d , nowhere
is
d e n s e s u b s e t of
xm(U,V,W)
in
is
a
Let
V
( 3 C ( U ) , -re*).
l e t D d e n o t e a c o u n t a b l e d e n s e s u b s e t of
d e n o t e t h e c o l l e c t i o n of a l l open b a l l s
all.
V whose c e n t e r s belong
are r a t i o n a l . L e t P d e n o t e t h e c o l l e c such t h a t V E V and W i s a c o n n e c t e d component of U n V. C l e a r l y P i s c o u n t a b l e , and t o complete t h e p r o o f o f t h e lemma w e s h a l l show t h a t F i s t h e union of t h e s e t s JcmIu,v,wl w i t h I V , W l E P and m E IN. L e t f E F. Then w e can f i n d open sets V and W i n E and a f u n c t i o n f E X l V ) s a t i s f y i n g t h e c o n d i t i o n s (a), ( b ) , ( c ) i n D e f i n i t i o n 10.8. Without l o s s of g e n e r a l i t y w e may assume t h a t W i s a connecV . By E x e r c i s e 10.F t h e r e i s a p o i n t a t e d component of U E v n a u n aw. Choose V ' E V such t h a t a E V ' C V and 171 i s bounded, by rn s a y , on V ' . S i n c e a E a W t h e r e i s a p o i n t b E w n v'. L e t W ' d e n o t e t h e c o n n e c t e d component o f U n V' which c o n t a i n s b . Then I V ' , W ' l E P and f E 3Cm ( U , V ' , W ' ) , c o m t o D and whose r a d i i
t i o n of all p a i r s ( V , W )
I
pleting t h e proof. PROOF O F THEOREM 10.9.
Let
U be a domain o f holomorphy i n
61".
84
MUJ I CA
Then ( J c t U l , ~ ~ i) s a F r g c h e t s p a c e a n d , by Lemma 1 0 . 1 0 , t h e s e t
f 6 J c t U ) s u c h t h a t U i s t h e domain o f e x i s t e n c e of f , i s o f t h e s e c o n d c a t e g o r y i n ( J c ( U ) , T,) , and i s i n p a r t i c u l a r
of a l l
nonempty
.
Theorem 1 0 . 9 d o e s n o t g e n e r a l i z e t o a r b i t r a r y Banch spaces. I n d e e d , A . H i r s c h o w i t z [ 1 ] h a s g i v e n a n example of a nonsep a r a b l e Banach s p a c e whose open u n i t b a l l i s n o t a domain of e x i s t e n c e . But t h e f o l l o w i n g p r o b l e m r e m a i n s open. 10.11. PROBLEM.
Let
E be a separable Banach s p a c e . Is
domain o f holomorphy i n
E
every
a domain of e x i s t e n c e ?
In Sectjon 45we s h a l l present a p a r t i a l positive
solution
t o Problem 10.11.
EXERC ISES
10.A.
Let
where
n
2
D = An(O;R)
and
and l e t
0 < r
< R < w j j D i s a holomorphic c o n t i n u a t i o n of 2
a Hartogs f i g u r e i n
j = I,.
for H.
z" ( a ; r ) .
10.C.
Let
Showthat
The p a i r (H,DI i s called
Cn.
10.B. L e t V be a c o n n e c t e d open s e t i n -n A ( a ; r ) be a compact p o l y d i s c c o n t a i n e d i n V \
. .,n.
Show t h a t
V
8
with
n
and
V,
2
2. Let
U = U.
let
i s a holomorphic c o n t i n u a t i o n of
U and V b e t w o open s u b s e t s o f E w i t h
U
C V.
Show
t h a t V i s a h o l o m o r p h i c c o n t i n u a t i o n of U i f and o n l y i f e a c h f E X ( U ) h a s a n e x t e n s i o n f E K ( V ) , and e a c h c o n n e c t e d compon e n t of
V
c o n t a i n s p o i n t s of
U.
be a h o l o m o r p h i c c o n t i n u a t i o n o f an open s e t
10.D.
Let
in
Using E x e r c i s e 8 . 1 show t h a t i f
E.
then each
V
f
E X(U;F)
F
i s any Banach
has a unique extension
-
f
E
JCIV;FI.
U
space
DOMAINS Show t h a t an open s e t
10.E.
85
OF HOLOMORPHY
U i n E i s a domainofholomorphy
( r e s p . a domain o f e x i s t e n c e ) i f and o n l y i f e a c h connected cam-
U i s a domain of holomorphy ( r e s p . a domain o f e x i s -
ponent o f tence)
.
U and V be open s u b s e t s o f E , w i t h V c o n n e c t e d U. L e t W be a c o n n e c t e d component o f U n V . Show t h e e x i s t e n c e o f a p o i n t a E V n aU n aW. 10.F.
Let
and n o t c o n t a i n e d i n
L e t (H,D) be a Hartogs f i g u r e i n
10.G.
E).
(D \
Show t h a t
U h a s no p r o p e r
8
2
,
and l e t
holomorphic
U = H u
continuation,
U i s n o t a domain o f holomorphy.
but
L e t Ui be a domain of holomorphy i n E f o r e a c h i €I. 10.H. Show t h a t t h e s e t U = i n t n Ui i s a domain of holomorphy a s -LEI
well. 10.1.
q E X i E ) and l e t
Let
U = cp-'(A)
the set
A
b e an open s e t i n
i s a domain o f holomorphy i n
8 . Show that E.
10.J. Given open sets A ] , . . . , A n i n 8 show t h a t t h e p r o d u c t U = A l x . . . x A n i s a domain of holomorphy i n tn. 10.K.
Show t h a t an open s u b s e t U o f
tence of a fun cti on f o r every 10.L.
Let
x
E
f
E
JC(UI
i s t h e domain of exis-
5 duix)
rcf(xl
U.
U b e an open subset o f
pose t h a t f o r e a c h b a l l connected component of
E , and l e t
B ( a ; r ) with center
f i c i e n t l y small r a d i u s , t h e function of e x i s t e n c e of
E
i f and o n l y i f
a
f €
E
X(U).
Sup-
and
suf-
aU
i s unbounded on e a c h U n B ( a ; r l . Show t h a t U i s t h e domain f
f.
11. HOLOMORPHICALLY CONVEX DOMAINS I n t h i s s e c t i o n w e i n t r o d u c e t h e n o t i o n of holomorphicconv e x i t y and e s t a b l i s h a c l a s s i c a l theorem o f H.
Cartan
and
P.
86
MUJ I CA
T h u l l e n , which c h a r a c t e r i z e s domains of holomorphy i n terms of holomorphic c o n v e x i t y . To m o t i v a t e t h e d e f i n i t i o n of h o l o m o r p h i c a l l y convex domains
w e b e g i n by p r e s e n t i n g some p r o p e r t i e s of convex sets. 11.1. PROPOSITION.
Let
c o n t i n u o u s a f f i n e forms o n
A^ceEr
d e n o t e t h e v e c t o r s p a c e of a l l
C @ E'
A b e a s u b s e t of E and l e t
let
E,
denote t h e s e t
Then: The s e t
(a)
i s a l w a y s c o n v e x and c l o s e d , and i n
ACeE,
-
p a r t i c u l a r contains the closed, convex h u l l
-
A
(b)
I f
A
i s bounded t h e n
AC
c o ( A ) of A .
@
E , = co(A).
(c) If A i s bounded ( r e s p . c o m p a c t ) t h e n ed ( r e s p . c o m p a c t ) a s w e l 2 .
-
W e s h a l l prove t h a t
PROOF.
C
=(A)
A l l t h e o t h e r a s s e r t i o n s are clear. L e t
Banach Theorem t h e r e e x i s t
p
E
E'
$
and
A^c3Er
is bound-
when A i s bounded.
9 a
Z I A ) . By the HahnE
1R
such
that
R e V ( y ) f o r all x E = ( A ) . S i n c e Q ( ZA () ) is bounded t h e r e i s a d i s c A ( < ; r ) such t h a t P ( Z ( A ) I C z ( < ; r )
ReV(x) < a
f E c CB E' be d e f i n e d by f ( x ) - p ( x ) - 5 s u p l f l 5 sup 5 P < f f y ) , proving
and q ( y ) 9 7 i ( < ; r ) .L e t f o r every
3:
E
If(
Then
E.
A
B A^C
that
y
11.2.
PROPOSITION.
ZG(A)
@El.
F o r an o p e n s u b s e t
U of E
the
foZ:owing
conditions are equivalent: (a)
U
i s convex.
(b)
c U
(c)
z C C B En, U
f o r e a c h compact s e t
K C U.
i s compact f o r e a c h c o m p a c t s e t
K
C
U.
a7
DOMA 1 NS OF HOLOMORPHY
( a ) * ( b ) : I f K i s a compact s u b s e t of U t h e n t h e r e i s a b a l l V = B ( 0 ; r ) such t h a t K + V C U. Hence K c a E I = e o ( K I C c o ( K ) + V = c o ( K + V ) C U. PROOF.
A
(b) * (c):
zgeEl
This i s obvious s i n c e
each compact s u b s e t K
( c ) * ( a ):
of
segment
K = Ix,y).
and set
Let x , y E U
t i o n 11.1 t h e l i n e
1
[z,y
KcaEl.
=
equals
can w r i t e [ z , y ] = A u B , where A K6 a E l are two d i s j o i n t compact sets. S i n c e [ x , y ] conclude t h a t
must be empty. Thus
B
i s compact
for
E.
[x,y]
(-I
C
By
Proposi-
we B=KtBEl\U Hence
U and i s connected w e U and U i s con-
vex. With t h i s m o t i v a t i o n i n mind w e i n t r o d u c e t h e f o l l o w i n g & finition. 11.3.
DEFINITION.
Let
U b e a n open s u b s e t o f
J C ( U I - h u l l of a s e t
(a)
The
(b)
The open s e t
A
C
U
E.
i s d e f i n e d by
U i s s a i d t o b e holomorphically c o n v e x i s compact f o r e a c h compact s e t K C U.
if & C i U l
U b e an open s u b s e t o f
Let
E . W e s h a l l set
d ( A ) = inf d u ( x f
x€A € o r each set clear that Since
-
KJC(u,
-
i s a compact s u b s e t i s c o n t a i n e d i n t h e compact
A C U. I f
K
A
is c l e a r l y closed i n
K~(,,,,
i s compact i f and o n l y i f
du(isccu,’
p h i c a l l y convex i f and o n l y i f
set 11.4.
U w e conclude t h a t > 0 . Thus
U
KJC(UI
i s holomor-
du(K^JCiui) > 0 f o r e a c h compact
K C U. THEOREM.
F o r a n open s u b s e t
U of
Zoving c o n d i t i o n s : (a)
U
i s a domain of e x i s t e n c e .
E
consider the
fol-
MUJ I CA
88 U
(b) A
such t h a t
j
For e a c h s e q u e n c e ( a
(c)
a
E
is t h e u n i o n of a n i n c r e a s i n g s e q u e n c e of open s e t s d u ( ( b j ) x ( uI) > 0 f o r e v e r y j . .)
3
there e x i s t s a function
aU
U w h i c h crnvergss t o a point x(UI w h i c h i s unbounded on
in f
E
(a. ) . d
i s a domain of h o l o m o r p h y .
(d)
U
(el
du(~Xtu)I = d U (KI
(f)
U
f o r e a c h compact s e t
K C U.
i s hoZomorphicaZZy c o n v e x .
Then t h e i m p l i c a t i o n s ( a )
always t r u e . I f
(b) * ( c ) * ( d ) * ( e ) * ( f ) are
=*
i s separable then (a)
E
(b).
( a ) * ( b ) : Suppose U i s t h e domain of e x i s t e n c e of
PROOF.
a
f E X I U ) . C o n s i d e r t h e f o l l o w i n g open sets:
function
B
j
= Ez
E
u
= {x
E
B
: If(z)l
l / j } .
j
m
Then
U =
U
j=1
A
j
and
A
C Aj+l
j
f o r every
j. F u r t h e r m o r e , the
+ E(O;l/j), and j l P m f ( x ) (t)I 5 j f o r e v e r y .c E A j and whence i t f o l l o w s t h a t > 1 / j for every y E t E Z ( O ; l / j l . W e claim t h a t r c f ( y ) function f
i s bounded by
T o show t h i s l e t
x(U)
E
< jm+l
-
y
j on t h e s e t
(zi)K(u).
.
E
it follows t h a t
A
and
(Aj)x(u)
I P l f (yl I
t
E
% ( O ; l / j ) . Since
< s u p I P:f
A.
I
< j . Thus II Pmf (Y
3
f o r e v e r y m and i t f o l l o w s from t h e Cauchy
Formula t h a t
rcf(y) 2 I/j,
as
PTf
asserted.
Hence
the
Hadamard
series
m
Z Pm f l y ) ( t ) m=O
defines a function
f
Y'
holomorphic on t h e
B(y;Z/jl, and which c o i n c i d e s w i t h f on a neighborhood we o f t h e p o i n t y . S i n c e U i s t h e domain of e x i s t e n c e of f
ball
conclude t h a t
B ( y ; l / j ) C U. T h i s shows t h a t
and ( b ) i s s a t i s f i e d .
du((ii),iul) 2 l / j
DOMAl NS OF HOLOMORPHY
89
( b ) * ( c ) : By h y p o t h e s i s U i s t h e union of an sequence of open sets
= (Zj)x(u,
every
j. S e t
= B
L e t ( a .I b e a sequence i n 3
i'
B~
du((Aj)x(uii
such t h a t
Aj
f o r every
increasing >
for
0
and note that (fi.)x(u)
j
3
U which c o n v e r g e s t o a p o i n t
aU. A f t e r r e p l a c i n g ( a . ) and ( B , ) by s u i t a b l e subseqwnces, 3 i f n e c e s s a r y , w e may assume t h a t a 9 B j and a j E B j + c l f o r in
j
every
in
j. S i n c e
a
j
such t h a t
JcfU)
9 B~~ = ( E j ) J C t u ) we can f i n d a sequence (9 .I 3
s u p 19 . I 3
Bi
[9.(a.)l
1
3
f o r every
3
j .
By
t a k i n g s u f f i c i e n t l y h i g h powers of e a c h p j w e c a n i n d u c t i v e l y and f i n d a sequence If.) i n J C ( U ) such t h a t supIfjl 5 2 - j 3
Bj
m
j. whence it f o l l o w s t h a t t h e series
f o r every
v e r g e s u n i f o r m l y on e a c h Bi t o a f u n c t i o n f >
j
f o r every
(d) * (el: L e t
Y
XtU) and
E
fines a function f w i l l coincide with
Proposition 10.5.
U and s e t r = ( a ) =. ( b ) w e s h a l l prove t h a t t h e series P m f ( y ) ( t ) de-
K b e a compact s u b s e t o f
d l i ( K i . By modifying t h e proof of
f
Y
zJCiu)
E
f
on a neighborhood of
and
ixiu,. Given
y E
E B(0;~).
find
E
and t h e r e f o r e
U
C
Then
> 0
m=O B(y;r).
holomorphic on t h e b a l l
K + apt
t
dU(Kx(Ui) = r .
E B(0;r)
choose
B = K
+
fix
NOW,
'iZpt
+
B
= ep
-m
p t
U and w e can is con-
B(0;pc.l
U and f i s bounded, by e s a y , on B . I f t h e n i t follows from t h e Cauchy I n e q u a l i t y 7 . 4 t h a t
sup
f E JC(V)
such t h a t
p > I
tained i n
such
0
t h a t the
m
series
2 Pmf(y)(t m=O
+ h ) converges u n i f o r m l y f o r
h
BIO;El.
E
m
T h i s shows t h a t t h e series
2 Pm f ( y ) I t ) d e f i n e s a holomorphic m=O
function f
Y
on t h e b a l l
B ( y ; r ) and t h e proof o f
(d)
=$
(e) i s
complete. S i n c e t h e i m p l i c a t i o n (el t o show t h a t ( b )
=$
( f ) i s o b v i o u s , i t o n l y remains
D b e a count a b l e dense s u b s e t of U. For e a c h x E D l e t B(x) denote t h e l a r g e s t open b a l l c e n t e r e d a t x and c o n t a i n e d i n U , t h a t i s B ( x ) = B ( x ; d U ( x ) / . By modifying t h e p r o o f of ( b ) * (c) w e s h a l l c o n s t r u c t a f u n c t i o n f E X ( U ) which i s unbounded on B ( x ) f o r e v e r y x E D. NOW, l e t (x .I be a sequence i n D w i t h t h e 3 p r o p e r t y t h a t e a c h p o i n t of D a p p e a r s i n t h e sequence (2.) 3 i n f i n i t e l y many t i m e s . By h y p o t h e s i s U i s t h e union of a n i n c r e a s i n g sequence of open s e t s Aj s u c h t h a t d u ( ( b j ) l r i u i )> 0 f o r every
=*
j. set
( a ) when B i s s e p a r a b l e . L e t
B~ = ( 2 j ) x ( u , f o r e v e r y
j . Note t h a t
Blx)
B . f o r e a c h x E D and j E ilv. Hence, a f t e r r e p l a c i n g (B<j) by a subsequence, i f n e c e s s a r y , w e c a n f i n d a sequence ( y3- ) i n f o r every U such t h a t y j E B ( x3. ! , y j 9 B j and y j B i + l j. Then, p r o c e e d i n g as i n t h e proof o f ( b ) * ( c ), w e can cons t r u c t a f u n c t i o n f € K i l l ) such t h a t l f ( y j ) l 2 j f o r e v e r y j. W e claim t h a t f i s unbounded on t h e b a l l Biz) f o r e v e r y x E D. I n d e e d , g i v e n x E D w e can f i n d a s t r i c t l y i n c r e a s i n g sequence (j,) i n JI? such t h a t x = x f o r every k . Hence jk y E B ( x ) f o r e v e r y k and f i s unbounded o n B ( x 1 , as as-
$2
jk s e r t e d . To complete t h e proof w e s h a l l
prove
that
U is
the
domain of e x i s t e n c e of f. I n d e e d , suppose t h e r e e x i s t o p e n sets V and W and a f u n c t i o n E X ( V ) satisfying the conditions i n Definition 10.8. any
r > 0
B ( a ; r ) . Then
Take a p o i n t
E V
n aU
aW
and
consider
B ( a ; 2 r ) C V . Choose a p o i n t x E D n W n
such t h a t du(x)
a
r
and
B l x ) C B ( a ; 2 r ) C V . Since
B(x)
U n V , w e conclude t h a t B i z ) C W. Thus = f i s unbounded on B ( x i , and t h e r e f o r e on B ( a ; B r ) . S i n c e r > 0 can be t a k e n a r b i t r a r i l y s m a l l w e c o n c l u d e t h a t 3 i s n o t l o c a l l y bounded a t a , a c o n t r a d i c t i o n . Hence U i s i s connected and c o n t a i n e d i n
91
DOMAi NS OF HOLOMORPHY
t h e domain of e x i s t e n c e of
f , and t h e proof of t h e theorem i s
complete. Now i t i s e a s y t o prove t h e Cartan-ThuZZen T h e o r e m :
11.5. THEOREM.
For a n open s u b s e t
U of
t h e foIZowing oon-
6'"
d i t i o n s are equivalent:
(a)
U
i s a domain of e x i s t e n c e .
(b)
U
i s a domain of holornorphy.
(c)
U
i s holomorphically convex.
PROOF. hold.
* ( b ) * ( c ) always * ( a ) c o n s i d e r t h e compact s e t s
By Theorem 1 1 . 4 t h e i m p l i c a t i o n s ( a ) To show t h a t ( c )
Kj = { x
E
U
:
llxll 5 j
co
Then
U =
U
and
du(xl
2
l/j}.
0
and Kj C Kj+l
K
f o r every
j . Since
U i s ho-
j=l
Lomorphically convex t h e set 0
I f we set
> 0
A
i
= K
f o r every
i
(ij)xcu i s )compact
f o r every j .
m
then
U =
U
A j,
Aj
C
Aj+l and d U ( ( Aj I J C ( U I )
j=l
j. By Theorem 1 1 . 4 ,
U i s a domain of existence.
Theorem 1 1 . 5 does n o t g e n e r a l i z e t o a r b i t r a r y Banach spaces. I n d e e d , B. J o s e f s o n
[ 1]
h a s g i v e n a n example
of
a holomor-
p h i c a l l y convex open s e t i n a n o n s e p a r a b l e Banach s p a c e
i s n o t a domain of holomorphy. But t h e f o l l o w i n g problem
which
re-
mains open.
11.6.
PROBLEM.
Let
E
b e a s e p a r a b l e Banach s p a c e .
h o l o m o r p h i c a l l y convex open s e t i n
E
Is
every
a domain of e x i s t e n c e , o r
a t l e a s t a domain of holomorphy? I n Section 4 5 w e s h a l l p r e s e n t a p a r t i a l p o s i t i v e t o Problem 1 1 . 6 .
solution
To complete t h i s E e c t i o n w e g i v e two a p p l i c a -
t i o n s of Theorem 1 1 . 4 .
MUJ I CA
92
11.7.
PROPOSITION.
Let
and Z.et
U =
T E L(E;FI
V be a domain of e x i s t e n c e i n -1 T ( V ) . Then:
i s a domain of hoZomorphy.
(a)
U
(b)
I f
E
i s separabZe t h e n
U i s a domain of e x i s t e n c e .
One can r e a d i l y check t h a t
PROOF.
f o r each set
A C U. NOW,
s i n c e V i s a domain
Theorem 1 1 . 4 y i e l d s an i n c r e a s i n g sequence of and a sequence of 0-neighborhoods m
u B
and f B j ) x ( V ,
j=1
j
and
U
u =
let
F,
j
= T
-1
+
V
C
Vj
in
f o r every
V
F
of
j . Set
u
A
j
B V
j
3
j . Then u s i n g (11.1) w e g e t
and
€or every
n
j =
= T-1 ( B . 1
A
m
j=l
sets
open
such t h a t
j
( V . ) f o r every 3
existence,
. By
that
Theorem
1 1 . 4 w e may conclude t h a t a r b i t r a r y , and t h a t
U i s a domain of holomorphy if E i s U i s a domain of e x i s t e n c e i f E i s sepa-
rable. Next w e show t h a t i n t h e case of s e p a r a b l e Banach
spaces
t h e c o n c l u s i o n of C o r o l l a r y 1 0 . 7 c a n bc improved as f o l l o w s .
11.8. PROPOSITION.
E v e r y c o n v e x o p e n s e t i n a s e p a r a b l e Banach
s p a c e i s a domain of e x i s t e n c e .
U be a convex open s e t in a s e p a r a b l e Banach space Then U i s t h e union of t h e i n c r e a s i n g sequence of open sets
PROOF. E.
Let
A . d e f i n e d by 3
Using t h e i d e n t i t y
93
DOMAINS OF HOLOMORPHY
f o r a l l a , B 2 0 w i t h a + B = I , w e can see t h a t e a c h A j convex. Then i t f o l l o w s from P r o p o s i t i o n 11.1 t h a t (2.)
JC(UI
J
is C
j . Thus an a p p l i c a t i o n of Theorem 1 1 . 4 c o m p l e t e s t h e p r o o f .
EXERC ISES Show t h a t an open s u b s e t U of E i s h o l o m o r p h i c a l l y con-
ll.A.
vex i f and o n l y i f e a c h c o n n e c t e d component of
U
i s holomor-
p h i c a l l y convex.
ll.B. each
Ui
Let
i = I,
2.
b e a h o l o m o r p h i c a l l y convex open s e t i n E Show
t h a t t h e open s e t
U = U 1 n U,
for
i s holo-
m o r p h i c a l l y convex as w e l l . Given a h o l o m o r p h i c a l l y convex open s e t U i n E
ll.C.
function
f
E
J C ( U l show t h a t t h e open s e t
a V = {x E U: If(z)l< l l and
i s h o l o m o r p h i c a l l y convex as w e l l .
11.D. L e t V b e a n open subset of F , l e t T E 8 l E ; F I and l e t U = T - 1 ( V ) . Show t h a t i f V i s h o l o m o r p h i c a l l y convex t h e n U i s h o l o m o r p h i c a l l y convex as w e l l .
U i b e a h o l o m o r p h i c a l l y convex open s u b s e t of a Banach s p a c e Ei f o r i = 1,2. Show t h a t U l X U, i s a h o l o m o r p h i c a l l y convex open s u b s e t of E l x E 2 . ll.E.
ll.F. for
El
x
Let
Ui b e a domain of e x i s t e n c e i n a Banach s p a c e Ei i = 1,2. Show t h a t U l x U 2 i s a domain of holomorphy i n E2. Let
If
ll.G.
U i s a convex open s e t i n E show t h a t
= d U f K i f o r e a c h compact set
dU(zceE,)
K C U.
11.H. L e t U b e a h o l o m o r p h i c a l l y convex open s e t i n C n , and l e t l a .) b e a sequence i n U such t h a t e a c h f E J C ( U I is bounded 3
MUJ I CA
94
on (a.). 3
Show t h a t t h e s e t
(a)
8 =
If
E
XfUI
: sup1
f (aj)1 5 1 ) i s
a c l o s e d , convex, b a l a n c e d , a b s o r b i n g s u b s e t o f ( J C ( U ) , T ~ ) . Using Exercise 9 . K
(b)
set K i n U s u c h t h a t (c)
aj
show t.he e x i s t e n c e of
E ,(,?i
f o r every
a
compact
j.
Using P r o p o s i t i o n 1 0 . 5 conclude t h a t
U is a
domain
g i v e an
alteron t h e
of holomorphy. T h i s exercise, t o g e t h e r w i t h Theorem 1 0 . 9 ,
n a t i v e proof of t h e C a r t a n - T h u l l e n Theorem 1 1 . 5 , based F r g c h e t s p a c e p r o p e r t i e s of ( J C ( U ) , T ~ ) .
1 2 . BOUNDING SETS
I n t h i s s e c t i o n w e i n t r o d u c e t h e n o t i o n of b o u n d i n g s e t a n d s t u d y i t s c o n n e c t i o n w i t h domains o f holomorphy and domains of existence.
DEFINITION. L e t U be an open s u b s e t o f E . A s e t i s s a i d t o be a b o u n d i n g s u b s e t of U , o r . JCIUI-bounding, each f E J C ( U ) i s bounded on B . 12.1.
B C
U if
U b e a n open s u b s e t o f E . Then e a c h rel a t i v e l y compact s u b s e t of U i s JCfU)-bounding. Moreover, for e a c h compact s u b s e t K of U t h e s e t K x ( U I i s JCIUI-bounding.
12.2. EXAMPLES.
Let
&
U b e a n o p e n s u b s e t of E , and L e t B U. T h e n , for e a c h i n c r e a s i n g s e q u e n c e ( A . ) of o p e n s u b s e t s o f U w h i c h c o v e r U, t h e r e e x i s t s j s u c h
12.3.
PROPOSITION.
Let
b e a b o u n d i n g s u b s e t of 3
that
B
PROOF.
C
(A^j)JC(u,.
Set
B~
=
(Alj)xtu,
f o r every
3’.
If
j t h e n , a f t e r r e p l a c i n g ( B . ) by a s u i t a b l e 3
n e c e s s a r y , w e c a n f i n d a sequence Iz . ) i n 3
and
xj
E
Bj+]
f o r every
B
R
Eli
f o r each
subsequence,
if
such t h a t x
3’ 9 Bj
j. Then t h e proof of t h e b r p l i c a t i o n
95
DOMAINS OF HOLOMORPHY
( b ) * ( c ) i n Theorem 1 1 . 4 y i e l d s a f u n c t i o n
f
E X t U ) which
unbounded on ( x . ) , and t h e r e f o r e unbounded on
B , a contradic-
3
tion.
12.4.
COROLLARY.
For an open s u b s e t
is
U of E c o n s i d e r t h e f o l -
lowing c o n d i t i o n s :
(a)
U
(b)
dU(B) > 0
(c)
U
Then
(a)
i s a domain of e x i s t e n c e . f o r each bounding s u b s e t
B
of
U.
i s a domain of holornorphy.
* (b)
(c).
Apply Theorem 1 1 . 4 a n d P r o p o s i t i o n 1 2 . 3 .
PROOF.
1 2 . 5 . THEOREM. E v e r y b o u n d i n g s u b s e t of a s e p a r a b l e Banach space
i s r e Z a t i v e Zy c o m p a c t . PROOF. L e t B b e a bounding s u b s e t o f a s e p a r ' a b l e Banach E . L e t (a.) be a d e n s e s e q u e n c e i n E . 3
Given
E
> 0,
space
l e t ( A . ) be 3
.i
t h e i n c r e a s i n g s e q u e n c e of open s e t s d e f i n e d by m
Then
E =
u
and by
A
A . = u B(ai;El. 3 i=I
Proposition 12.3 there e x i s t s j
such
j
j=1
Then, by P r o p o s i t i o n 11.1, B C ( A ^ j ) 6 e E , that B C (A^jIJc(E). c o ( A . ) . S e t K = c o f a l , . . . , a . } . Then c o ( A . ) C K . + B ( O ; E ) and
s
j
=(A
B C
that
B
Thus
B
.) C
3
K
j
3
+ B ( 0 ; 2 ~ ) . Since
3
Kj
3
i s compact w e
c a n b e c o v e r e d b y f i n i t e l y many b a l l s
conclude
of r a d i u s
i s p r e c o m p a c t and t h e r e f o r e r e l a t i v e l y compact i n
3E.
E.
U b e a domain o f e x i s t e n c e i n a separable Banach s p a c e . T h e n e a c h b o u n d i n g s u b s e t of U i s r e l a t i v e l y com12.6.
COROLLARY.
pact i n
Let
U.
T h e o r e m 1 2 . 5 d o e s n o t g e n e r a l i z e t o a r b i t r a r y Banach spaces. I n d e e d , S . Dineen [ 2 ]
. ., O ,
(0,.
has
shown t h a t t h e u n i t v e c t o r s
I, 0,. . . 1 form a b o u n d i n g s e t i n
Em.
un =
96
MUJ I CA
E XE RC ISES 12.A.
U b e an open s u b s e t of E , and l e t B b e a bounding U. Show t h a t e a c h f f X ( U ; F ) i s bounded on B .
Let
s u b s e t of
12.B. E
U be an open s u b s e t of E , l e t A C U and l e t y 1 I ffu)II 5 s u p IIf1x)II f o r e v e r y f E JC(U;FI.
Let
A^x(uI.
Show t h a t
XE
12.C.
U b e an open s u b s e t of
Let
f JCfU;FI,
s u b s e t of
(a)
E , and l e t
F be a bounded
T ~ ) .
U i s t h e union of an i n c r e a s i n g sequence such t h a t t h e f u n c t i o n s f E F a r e u n i f o d y
Show t h a t
of open s e t s
Aj
bounded on e a c h (b) tions
A
If
B
f E F
(c)
Aj. i s a bounding s u b s e t of
U , show t h a t t h e fun-
are u n i f o r m l y bounded on
B.
B i s a bounding s u b s e t of U w i t h d U f B ) > 0, f E F are u n i f o r m l y bounded on t h e B + B f O ; € ) , for a suitable E > 0. If
show t h a t t h e f u n c t i o n s
set
U b e a b a l a n c e d open s u b s e t of E . Show t h a t t h e b a l a n c e d h u l l o f e a c h bounding s u b s e t of U i s a l s o a bounding s u b s e t of U. 12.D.
Let
12.E.
Let
subset of
U be an open s u b s e t of
s u b s e t B of x
E , and l e t
F . Given a bounding s u b s e t A V , show t h a t
A x B
V
be an
open
of
U , and a bounding i s a bounding s u b s e t o f U
v.
U and V be open s u b s e t s of E. s u b s e t A of U and a bounding s u b s e t B o f B i s a bounding s u b s e t of U + V . 12.F. L e t
12.G.
Let
U be a convex open s e t i n U.
f o r e a c h bounding s u b s e t B of
Given a boundi.ng V , show t h a t
E . Show t h a t
du(BI
A +
> 0
DOMAINS OF HOLOMORPHY
97
NOTES AND COMMENTS The proof
of
Theorem 1 0 . 9
b a s e d on t h e B a i r e
Theorem i s t a k e n from t h e book of L. Nachbin [ 3
Category
1 . The c h a r a c -
t e r i z a t i o n of domains of holomorphy i n Theorem 11.5 i s due H.
C a r t a n and P. T h u l l e n [ 1 1 .
[ 1]
to
Theorem 1 1 . 4 i s due t o S. Dineen
and A. H i r s c h o w i t z [ 3 1 , and r e p r e s e n t s an a t t e m p t t o ex-
t e n d t h e C a r t a n - T h u l l e n Theorem t o i n f i n i t e d i m e n s i o n a l Banach s p a c e s . Bounding sets were i n t r o d u c e d by H. Alexander [ l ] , who o b t a i n e d Theorem 12.5 f o r s e p a r a b l e H i l b e r t
spaces.
Theorem
12.5 i s a s p e c i a l case of more g e n e r a l r e s u l t s o b t a i n e d by
S.
Dineen [ 4 ] and A . H i r s c h o w i t z [ 2 1 . The proof of Theorem 1 2 . 5 g i v e n h e r e i s due t o M.
Schottenloher [ 2
I.
This Page Intentionally Left Blank
CHAPTER IV
DIFFERENT IABLE MAPPINGS
1 3 . DIFFERENTIABLE MAPPINGS T h i s s e c t i o n i s d e v o t e d t o t h e s t u d y of d i f f e r e n t i a b l e m a p p i n g s between Banach s p a c e s . U n l e s s s t a t e d o t h e r w i s e , t h e letters F w i l l r e p r e s e n t Banach s p a c e s over t h e same f i e l d M .
E and
1 3 . 1 . DEFINITION.
f
:
U
+
F
U b e a n open s u b s e t o f
Let
mapping
A
i s s a i d t o be d i f f e r e n t i a b l e i f f o r each p o i n t a
t h e r e e x i s t s a mapping
A E
I1 f ( x )
lim
Let
REMARKS.
-
f
fE;F)
such t h a t
-
- all1
f(a) II x
x+a 13.2.
E.
Alx
=
- a II
11 be an open s u b s e t o f
(a)
Eacn d i f f e r e n t i a b l e mapping
(b)
The mapping
-+
U
0.
Then:
E.
f : U
E
F
is
continu-
ous. A E E(E;FI
t h a t appears
in
Definition
f and a . I t i s c a l l e d t h e d i f a t a a n d w i l l be d e n o t e d by Dffal. Thus a d i f f e r e n t i a b l e mapping f : U + F i n d u c e s a mapping Df : U
1 3 . 1 i s u n i q u e l y d e t e r m i n e d by
f e r e n t i a l of
f
+
L((E;F).
(c)
If
E
and
F
are comDlex Banach s p a c e s t h e n w e
t o d i s t i n g u i s h between t h e complex d i f f e r e n t i a b i l i t y of C
E
-+
F
arid t h e r e a l d i f f e r e n t i a b i l i t y of
f
:
U
I t i s c l e a r t h a t complex d i f f e r e n t i a h i l i t y i m p l i e s
C
EB
real
have f : U .+
FIR.
dif-
f e r e n t i a b i l i t y , b u t t h e converse i s n o t t r u e . Indeed, t h e function
e
E
C
+
z
E
6:
Is
07-differentiable
99
without
being
MUJ I CA
100
@-differentiable. We shall soon study the connection between lR- differentiable mappings and C-differentiable mappings. 13.3. EXAMPLE. If f : E F differentiable and Df(x) = 0 then A is differentiable and
is a constant mapping then f is for every a E E. If A E f ( E ; F 1 DA(a) = A for every a E E.
+
1 3 . 4 . EXAMPLE.
where PROOF.
Every P then
A E LS(mE;F)
P(mE;F)
E
is differentiable. If P = i m-1 for every a E E .
D P f a l = mAa
By the Newton Binomial Formula
for all a , x E E . If we s e t
then it is clear that
p ( x ) / II x
- a II
+
0
when
x
a.
--t
1 3 . 5 . EXAMPLE.
Let E and F be complex Banach spaces and let U be an open subset of E . Then every f E K ( l J ; F l is C-differentiable and D f ( a ) = P 1 f ( a ) for every a E U.
PROOF.
Let
a E U
and l e t
1 f(x) = f ( a ) + P f ( a ) ( x
for every
x E B(a;r).
0 < P
0 be given. By a p p l y i n g (13.1) t o t h e mapping + ( x , t l = DLC+(x,tJ w e can f i n d 6 > 0 such t h a t A E C(U;C(E;FIl.
whenever ll h I1 2 S
and
t
E T.
Then by C o r o l l a r y 1 3 . 9 w e
that
whenever
I1 /z II
5 6 and
t
E
T.
Hence
have
DIFFERENTIABLE MAPPINGS
whenever
II h II < 6.
107
The d e s i r e d c o n c l u s i o n f o l l o w s .
As w e have a l r e a d y remarked, e v e r y @ - d i f f e r e n t i a b l e mapping The n e x t p r o p o s i t i o n t e l l s u s when an B- d i f f e r e n t i a b l e mapping i s @ - d i f f e r e n t i a b l e . is R-differentiable.
13.15 PROPOSITION.
Let
and
E
F
be c o m p l e x Banach s p a c e s , l e t
U be a n o p e n s u b s e t of E , and l e t f : U --t F b e a n I ? - d i f f e r e n t i a b l e m a p p i n g . L e t D f ( a ) d e n o t e t h e r e a l d i f f e r e n t i a 2 of f a t a , and l e t D ’ f f a ) and D r r f ( a l be d e f i n e d b y
t
f o r every D”f(a)(t)
=
f
0
E.
i s @ - d i f f e r e n t i a b l e if and onZy if a E U and t E E . In t h i s case Dffal
Then f
f o r every
= D ’ f ( a ) i s also t h e eoinplex d i f f e r e n t i a l o f PROOF.
By P r o p o s i t i o n 1 . 1 2
f
a t a.
D f ( a ) i s C - l i n e a r i f and o n l y
if
D ” f f a l = 0 . Thus i t s u f f i c e s t o a p p l y t h e d e f i n i t i o n o f 6 - d i f f e r e n t i a b i l i t y and t h e u n i q u e n e s s of t h e d i f f e r e n t i a l a t a given point. The d i s t i n c t i o n between
B-differentiability
and
C-dif-
f e r e n t i a b i l i t y i s emphasized by t h e f o l l o w i n g theorem.
13.16. THEOmM.
U
Let
E
be a n o p e n s u b s e t of
and E.
it
f e r e n t i a b l e if and o n l y i f D f ( a ) = P f f ( a ) f o r every
PROOF.
F
a
be c o m p l e x Banach spaces, and l e t
Then a mapping
f : U
+
In
i s hoZornorphic.
F
is 6-dif-
this
case
follows
from
E (1.
Suppose f i s C - d i f f e r e n t i a b l e .
Then
it
108
MUJ I CA
t h e Chain Rule 13.6 t h a t t h e f u n c t i o n
g o t ) = $ a f ( a + Xb)
is
t - d i f f e r e n t i a b l e on t h e open set A = (1 E U : a + Xb E U ) f o r e v e r y a E U , b E E and I/J E F’. Hence g i s holomorphic, by the c o r r e s p o n d i n g r e s u l t f o r f u n c t i o n s of one complex variable. Thus
f
i s G-holomorphic, by Theorem 8.12,
and t h e r e f o r e holanorphic, s i n c e f i s c l e a r l y c o n t i n u o u s . S i n c e t h e re-
by Theorem 8.7,
v e r s e i m p l i c a t i o n was e s t a b l i s h e d i n Example 1 3 . 5 , of t h e theorem i s complete. 13.17.
Let
COROLLARY.
E
and
U be a n o p e n s u b s e t o f
let
F
E.
hoZomorphic i f and o n l y if f
the
proof
be c o m p l e x Banach s p a c e s , and
Then a m a p p i n g f : U is i R - d i f f e r e n t i a b l e a n d
+
F
is
D”f
is
F
and
i d e n t i c a l l y zero.
EmRCISES 13.A.
U be a n open s u b s e t of
Let
g : U
f
:
U
+
b e two d i f f e r e n t i a b l e mappings.
F
+
E , and l e t
af + Bg
i s d i f f e r e n t i a b l e for all a , D ( a f + B g l l a l = c l D f ( a ) + B D g ( a l f o r e v e r y a E U.
(a)
Show t h a t
(b)
Show t h a t i f
and
13.B.
L e t E and F
F = Fl
X
...
X
F = D(
t h en t h e product
D f f g ) ( a ) = g(a)Dffa)
f e r e n t i a b l e and a E U.
j
(j = 1 , .
Fm. Let
.., m )
b e Banach s p a c e s ,
U b e a n open s u b s e t of
.
fg
+ f(a)Dg(a)
.
BE
dif-
is
for
every
and
B , let
iK
let
fj
:
( j= 1 , . . , m ) and l e t f = (f,,. ., fm) : U F. Shm t h a t j f i s d i f f e r e n t i a b l e i f and o n l y i f e a c h f,. i s d i f f e r e n t i a b l e .
U
-+
F
+
J
Show t h a t i n t h i s case a
E
D f ( a ) = (Dfl(a),
..., D f m ( a ) )
for e v e r y
U.
13.C.
Let
...
E
j
( j
= I,.
-+ F m‘ i s d i f f e r e n t i a b l e and
A : El
X
. ., m )
and F b e Banach s p a c e s , and l e t b e m - l i n e a r and c o n t i n u o u s . Show t h a t A
DIFFERENTIABLE MAPPINGS
,..., am) and
a = lal
for a l l
,...,
tm) i n
E1 x
... x E m .
F . ( j = l ,..., rn) and G b e Banach spaces. L e t
Let E,
13.D.
t = Itl
109
3
be an open s u b s e t of
E , let
f e r e n t i a b l e , and l e t
A
: Fl
fj : U x
.. .
continuous. Show t h a t t h e mapping
F
-+
Fm g = A x
i -+
o
U
( j = l ,..., m) b e d i f G
be
(f,,
m-linear
. . . ,f m )
:
U
and --t
G
i s d i f f e r e n t i a b l e and
f o r every
a
Let E
13.E.
product (x
I
and
U
E
Let
-+
F
E
E.
be a r e a l o r complex H i l b e r t s p a c e , and
with
function
U b e a connected open s u b s e t o f
E , and l e t
b e a d i f f e r e n t i a b l e mapping whose d i f f e r e n t i a l
L IE;F)
i s a c o n s t a n t mapping. Show t h e e x i s t e n c e o f
and
E
b
F
such t h a t
inner
f(z) = IIcc1I2 is B-difDf(a) (tl = ZReltlal f o r a l l a, t E E.
Show t h a t t h e
y).
f e r e n t i a b l e on E 13.F.
t
f(x) = Ax + b
f o r every
x
f : U
Df : U
+
A E LfE;F) E
U.
n E = X m and F = X , and l e t G be any Banach space o v e r X. L e t U C E and V C F b e t w o open sets, and l e t f : U + F and g : V G be two d i f f e r e n t i a b l e mappings witlrl f(U) C V. Let E l , . . .,Ern d e n o t e t h e c o o r d i n a t e f u n c t i o n a l s of El and l e t q l , . . . , n , d e n o t e t.he c o o r d i n a t e f u n c t i o n a l s of F. Show t h a t i f we w r i t e f = i f , , . . ,fn) t h e n 13.G.
Let
-+
.
f o r every
a E U
..
and
j = 1,.
. . ,m.
Let z I , . ,z n d e n o t e t h e complex c o o r d i n a t e functionals denote t h e corresponding real C n and l e t x l,yl, . . ,xn, y, c o o r d i n a t e f u n c t . i o n a l s . L e t U be an open s u b s e t of C n l l e t F
13.H.
of
.
110
MUJ ICA
be a complex Banach s p a c e , and l e t
af/axj
t i a b l e mapping. L e t
and
f : U
af/ayj
+
F bean W - d i f f e r e n -
d e n o t e t h e real partial
d e r i v a t i v e s i n t h e s e n s e of P r o p o s i t i o n 1 3 . 1 1 a n d l e t and be t h e f u n c t i o n s d e f i n e d by
af/azj
af/azj
a
f o r every
(a)
U.
Show t h a t
a
f o r every
on
E
t
and
U
E
(b)
Show t h a t
U for
j = I,.
f
. ., n .
6
8.
i s holomorphic i f and o n l y i f
af/ai. = 3
0
These are t h e Cauchy-Riernann equations.
Note t h a t t h e o p e r a t o r
j u s t defined coincides,when j a p p l i e d t o holomorphic f u n c t i o n s , w i t h t h e complex p a r t i a l d e i n t h e s e n s e of P r o p o s i t i o n 1 3 . 1 1 . r i v a t i v e a/az 3/32
i
13.1.
f
:
U
Let +
5.
function 13.J.
Let
If
be an open s u b s e t of
E.
Show t n a t
i s B - d i f f e r e n t i a b l e i f and o n l y -
f : U
-+
5.
is
if
a
function
the c o n j u g a t e
B-differentiable.
U be a n open s u b s e t of
Cn
a n B - d i f f e r e n t i a b l e € u n c t i o n . Show t h a t
and l e t
f : U
+
6
be
DIFFERENTIABLE MAPPINGS
f o r every
j = 1,
111
..., n .
1 4 . DIFFERENTIABLE MAPPINGS OF HIGHER ORDER T h i s s e c t i o n i s d e v o t e d t o t h e s t u d y of d i f f e r e n t i a b l e m a p p i n g s of h i g h e r o r d e r . U n l e s s s t a t e d o t h e r w i s e , t h e l e t t e r s
w i l l r e p r e s e n t Banach s p a c e s o v e r t h e same f i e l d
and F
14.1.
DEFINITION.
f : U
+
Let
U b e a n open s u b s e t o f
t i a b l e and t h e d i f f e r e n t i a l aF
Df
:
U
+
d:(E;FI
A mapping
E.
i s s a i d t o be t w i c e d i f f e r e n t i a b z e i f
F
E
aC.
f
is differen-
is differentiable
well.
1 4 . 2 . REMARKS.
Let
U be a n open s u b s e t o f
E
and l e t f : U + F
be a t w i c e d i f f e r e n t i a b l e mapping. Then t h e d i f f e r e n t i a l of the mapping
t i a l of
a t a point
Df
f
at
a
a E U
i s c a l l e d t h e second d i f f e r e n -
and w i l l b e d e n o t e d by
DZf(a).
Thus
D2f(a)
2
d: ( E ; C f E ; F ) ) or L ( E; F ) , in view of t h e c a n o n i c a l isomorphism g i v e n i n P r o p o s i t i o n 1.4. I f 2 t h e i n d u c e d mapping D f : U L( 2 E ; F ) i s c o n t i n u o u s , t h e n f i s s a i d t o be t w i c e c o n t i n u o u s Zy d i f f e r e n t i a b l e .
may b e r e g a r d e d a s a member o f
-+
The n e x t r e s u l t g e n e r a l i z e s t h e c l a s s i c a l Schwarz 14.3.
THEOREM.
Let
U be an o p e n s u b s e t of
E
’
Theorem.
and k t f : U + F
be a t w i c e d i f f e r e n t i a b l e m a p p i n g . T h e n t h e b i l i n e a r mapping 2 2 D f l a ) E 5 1 E;F) i s s y m m e t r i c f o r e a c h a E U. I n o t h e r words,
D 2 f ( a l ( s , t l = D 2f ( a l ( t , s ) f o r a l l PROOF. For
Let
a E U
s , t E B(O;rl
and c h o o s e
s, t
r > 0
E
E.
such t h a t
B ( a ; 2 r ) C U.
w e d e f i n e t h e mapping
g ( s , ~ )= f ( a
+
s + tl
- f(a +
s)
- f(a +
t) + f(a).
The i d e a o f t h e p r o o f i s t o e s t i m a t e t h e d i f f e r e n c e g(s,tl 1 D ’ f ( a l ( s , t l f o r s and t n e a r z e r o . F i r s t n o t e t h a t i f w e s e t
MUJ I CA
112
for
IIIp/x)II /
then
E U
3:
113:
-
a It
-+
0
when
x
-P
a.
Hence
w e can w r i t e
Now, f i x
each
x
and d e f i n e
s E B(O;r)
Dhlx) = D f f x + s )
B ( a ; r , J . Noting t h a t
E
g f s , t ) = h(a
h(xl = f ( x + sl
+ t l - h l a l , and u s i n g C o r o l l a r y
= l l h l a + t)
-
< II t II
IIDhfa + A t )
sup
h(aj - Dhla)ftill
- Dh(a)ll.
Now,
- Dh(al
= [ D f ( a + At
+ sl
[ Df(a + sl 2
= [ D fla)fAt +
- D f l a l ] - [ D f l a + At) - Dflall -
- Dfla)] s ) + cp(a + A t + s ) ]
2
[ D f ( a l i s ) + iola + s)
1
= cpfa + At + s l - cp(a + A t ) Thus w e g e t t h e estimate IIg(s. t )
-
2
D f l u ) is, t l l l
for and
13.9 we g e t t h a t
O_ < A_ 0
E
IIlp(x)II
/ I I x - all 5
s, t
BI0;61
E
for a l l BI0;rl
the
r o l e s of
s
t , s E B I O ; 6 ) . And s i n c e we conclude t h a t 2
-
II D f l a ) I s , t l f o r all
s,
t
0 < 6 < r
z E B(a;261.
such t h a t for
all
for all
s, t
Then
we g e t t h a t
By i n t e r c h a n g i n g
E
6 with
f o r every
E
and
t we get t h a t
gfs,t)
2
= g(t,sl
( 3 s (11s II + II t II j
D f f a ) (t,s)ll
2
s, t E E . D f l a ) Is, t l =
but then, obviously f o r a l l
E BIO;6/,
Since E > 0 w a s a r b i t r a r y , we conclude t h a t 2 D f ( a ) f t , s l f o r a l l s, t E E , a s w e w a n t e d .
2
U be a n open s u b s e t o f E . B y i n d u c t i o n f : U .+ F t o b e k times d i f f e r e n t i a b l e i f f i s k - 1 times d i f f e r e n t i a b l e and i t s I k - 1 ) t h k-1 d i f f e r e n t i a l Dk-'jf : U I f E ; F ) i s d i f f e r e n t i a b l e . A mapp i n g f : U + F i s s a i d t o be i n f i n i t e Z y d i f f e r e n t i a b z e i f i t i s k t i m e s d i f f e r e n t i a b l e f o r e a c h k E lN. 14.4.
on
k
DEFINITION.
Let
w e d e f i n e a mapping
+
14.5.
REMARKS.
be a
k
U be a n open s u b s e t o f
Let
E , and l e t f : U - + F
times d i f f e r e n t i a b l e mapping. Then t h e d i f f e r e n t i a l o f
t h e mapping
f e r e n t i a l of
Dk-lf f
at
at a point a
a
E
U
i s called t h e
k t h difk D fIal k -C( E ; F ) ,
and w i l l b e d e n o t e d by D k f ( a l . Thus
may be r e g a r d e d a s a member o f
L(E;l(k-lE;F))
or
114
MUJ I CA
under t h e i d e n t i f i c a t i o n g i v e n , b y P r o p o s i t i o n 1 . 4 . I f t h e i n k k d u c e d mapping D f : U + C ( E ; F I i s c o n t i n u o u s t h e n f i s s a i d
t i m e s continuousZy d i f f e r e n t i a b z e . For convenience w e
k
t o be
Dof = f .
also d e f i n e
Now w e c a n e a s i l y g e n e r a l i z e Theorem 1 4 . 3 . 14.6.
THEOREM.
Let
U b e an o p e n s u b s e t o f
and l e t
E
f : U
+
F
be a k t i m e s d i f f e r e n t i a b Z e m a p p i n g . T h e n t h e k - Z i n e a r mapping D k f ( a ) E J ( k E ; F ) i s s y m m e t r i c f o r e a c h a E U. k = 2
k . For
By i n d u c t i o n o n
PROOF.
this
is
just
Theorem
k > 3 a n d assume t h e t h e o r e m t r u e f o r k - 1. If k t h e n D f l u ) i s t h e d i f f e r e n t i a l a t a o f t h e mapping g
14.3. L e t
a E U
--
D k - l f , w h i c h , by t h e i n d u c t i o n h y p o t h e s i s , t a k e s i t s
in
k
D f ( a ) ( t l ,t,,
and hence
t2,.
. .,t k .
values
Thus
ls(k-lE;F).
. . . ,t k ) i s
symmetric i n t h e
variables
k On t h e o t h e r h a n d , D f (a) i s t h e s e c o n d d i f f e r e n t i a l
h = Dk-2f,
a t a of t h e mapping
and i f fol lo ws
from
Theorem
14.3 t h a t
k
D f(a)(tl,tz,
Hence
..., t k ) i s
symmetric i n t h e v a r i a b l e s
tl
t 2 , and t h e d e s i r e d c o n c l u s i o n f o l l o w s .
and
The n e x t t h e o r e m s t r e n g t h e n s
the
conclusion
in
Theorem
13.16. 14.7.
U
THEOREM.
Let
E
b e an o p e n s u b s e t o f
and E.
F
b e c o m p l e x B a n a c h s p a c e s , and Let
Then f o r e a c h mapping
f o Z Z o w i n g conditions are equiuaZent: (a)
f
i s holomorphic.
(b)
f
is C - d i f f e r e n t i a b l e .
f : U +F
the
DIFFERENTIABLE MAPPINGS
(c)
i s infinitely 6-differentiable.
f
Dk f ( a l = k ! A k f f a l
t h e s e conditions are s a t i s f i e d t h e n
If
a E U
f o r every
k
and
E
lN.
I n view o f Theorem 1 3 . 1 6 it i s s u f f i c i e n t t o show t h a t
PROOF.
f : U
if
115
i s holomorphic, then
F
-+
t i m e s C-differen-
is k
f
Dk f ( a ) = k ! A k f ( a ) f o r e v e r y
t i a b l e and
a
E
U
and
k
E W.
W e p r o c e e d by i n d u c t i o n on
k , t h e statement being t r u e f o r
= 1
k
by Theorem 1 3 . 1 6 . L e t
Pk-'f
E
every
E
U. Hence
Dk-l f = ( k
-
I)!
Ak-lf
and by Theorem 1 3 . 1 6 w e may c o n c l u d e t h a t
D k f l u ) = P1(Dk-'f)(al
f e r e n t i a b l e and f o r every
a
E
U. Thus f
f
is
k-1
tims
By Theorem 7 . 1 7 ,
P 1 ( P k - ' f l ( a ) = Pk-l(Pkffa))
~ C I U ; P ( ~ - ' E ; F ) ) and
a
and assume
2
Dk-' f = ( k - l)! A k - l f .
8 - d i f f e r e n t i a b l e and
is
= k !
k
k
k
E
for
x(U;d:s ( k - 1 E ; F ) ) , is
Dk-'f
6-dif-
= (k - I ) ! P1(Ak-'f)(a)
t i m e s @ - d i f f e r e n t i a b l e and
k
A f(a)t
.
k k S i n c e D f i a ) and A f ( a l a r e b o t h s y m m e t r i c , w e c o n c l u d e t h a t k k D f i a ) = k! A f ( a ) , a s asserted. 14.8.
k
E
W
pings
b e an open s u b s e t o f E . For each k w e s h a l l d e n o t e by c ( U ; F I t h e v e c t o r s p a c e o f a l l mapLet
DEFINITION. f : U
+
U
which a r e
F
+
F
t i m e s continuously
B-differen-
t h e v e c t o r space of all k which b e l o n g t o C (U;FI f o r e v e r y k E T?.
t i a b l e . W e shall d e n o t e by mappings f : U
k
Cm(U;F)
For c o n v e n i e n c e w e a l s o d e f i n e C o ( U ; F ) k of C ( U ; F ) , where k E W o U { m } , are
= C(U;FI. often
The members
c a l l e d mappings
116
MUJ I CA
of class k
ck .
F = M
When
thenweshall w r i t e for short
Ck(U)
= c IU;M). THEOmM.
14.9.
and
Let
E , F , G be Banach s p a c e s o v e r
M . Let
U cE
b e two o p e n s e t s , a n d l e t f : U F and g : V + G k be two mappings of c l a s s c , w i t h f ( U ) C V . Then t h e composite V
+
F
-+
g of
mapping
:
u
+.
G
i s a l s o of c l a s s
c ~ .
we proceed by i n d u c t i o n on k , t h e theorem b e i n g obk = 0. Let k > 1 and assume t h e theorem t r u e f o r k - 1. By t h e Chain Rule 1 3 . 6 t h e rmpping g o f : U G is B - d i f f e r e n t i a b l e and D ( g o f ) ( x ) = D g l f l x ) ) o D f ( x ) f o r e v e r y x E U. Thus t h e mapping Dlg o f ) : U +. E ( E ; G ) i s o b t a i n e d by
PROOF.
vious f o r
-+
composition o f t h e mappings
The mapping Cm,
Df : U
+.
and
IT
d e f i n e d by
T i s b i l i n e a r and c o n t i n u o u s , a n d h e n c e i s o f class
by E x e r c i s e 1 4 . B .
On t h e o t h e r hand, t h e mapping
i n a product
i t s values
S
d: I E ; F )
and
p o t h e s i s t h e mappings
and i t s components
Dg o f : U Df
and
-+
Dg o f
a g a i n by Hence
takes
are t h e mappings
By t h e i n d u c t i o n hyk- I are b o t h or c l a s s c ,
6 IF;G).
and then it f o l l o w s from E x e r c i s e 1 4 . A
of c l a s s Ck-' t o o . Then, D ( g o f ) i s of class Ck-'.
S
t h a t t h e mapping
the induction
g o f
i s o f class
S
is
hypothesis, C k and t h e
proof i s complete.
EXERCISES
E' and F (j = 1 , . . . , m ) b e Banach s p a c e s , and l e t j x F . Let. U be an open s u b s e t of E, l e t f : U F = Fl x ... m j P ( j = 1 ,..., m ) and l e t f = (f,,..., f,) : U + F . Show t h a t Jf i s k t i m e s ( c o n t i n u o u s l y ) d i f f e r e n t i a b l e i f and o n l y i f each f i s k t i m e s (continuously) differentiable. j 14.A.
Let
+
117
DIFFERENTIABLE MAPPINGS 14.B.
Let
A : El
E
...
x
j
x
(j = 1 , Em
F
+
..., m )
and
F
be Banach s p a c e s l a n d l e t
be m - l i n e a r and c o n t i n u o u s . Show t h a t A D kA = 0 f o r every k > m + 1.
i s i n f i n i t e l y d i f f e r e n t i a b l e and
E i s a r e a l o r complex H i l b e r t s p a c e t h e n 1 1 ~ 1 1 i ~s o f class Cm on E .
Show t h a t i f
14.C.
the function
3:
+
14.D. L e t U be an open s u b s e t of E , let f : U times d i f f e r e n t i a b l e mapping, and l e t [ a , a + t ] segment which i s e n t i r e l y c o n t a i n e d i n U.
+
be a
F
a
be
(a)
Using t h e c l a s s i c a l T a y Z o r ' s f o r m u l a show t h a t
(b)
Applying ( a ) t o t h e mapping g(x)=f(x)-
k
line
1 k k D flal(x-a) k.'
show t h a t
U be an open s u b s e t of
14.E.
Let
space
C"(U; F ) ,
endowed
with
E , and c o n s i d e r t h e v e c t o r
the
locally
convex
g e n e r a t e d by a l l t h e seminorms o f t h e form where
j
E
liVo
f
and K i s a compact s u b s e t of
s u p II D 3 f ( d l l , xE K U. Using Propo+
s i t i o n 1 3 . 1 3 show t h a t t h e l o c a l l y convex s p a c e always complete. Show t h a t i f
14.F.
( e l , . . . , e n ) d e n o t e t h e c a n o n i c a l b a s i s of
Let
5 , , . . . ,en
(a)
f : U
Show t h a t t h e p a r t i a l d e r i v a t i v e
-+
e x i s t s and e q u a l s j ,
E
(1
and
;Xn
j l a
a
k
... a S j
akf ( a ) / a S
D K f ( a ) ( e ,..., e ) f o r every jl jk
,..., n l .
be
F
1.
,...,
then
d e n o t e t h e c o r r e s p o n d i n g c o o r d i n a t e functionals.
L e t U be an open s u b s e t of M n and l e t t i m e s d i f f e r e n t i a b l e mapping.
j,
is
Cm(U;F)
i s f i n i t e dimensional
i s a Frgchet space.
Cm(U;F)
let
E
topology
E
U
k and
118
MUJ ICA
(b)
akf/aCj
show t h a t
1
... 3 5
i s a symmetric f u n c t i o n j k
of t h e i n d i c e s Using t h e L e i b n i z
(c)
Formula 1 . 8
show t h a t
Dkf(u
f o r every
use t h e n o t a t i o n index
a with
14.G.
Let
t = (tl,.
and
u E U
. ., t,)
k
"1
= a fiat,
a"f/ag"
an.For s h o r t w e shall 01 . . . atnn f o r e v e r y m u l t i -
la1 = k .
U be an open s u b s e t of
: U
+
F
rivatives on
Ck
i s of class
of o r d e r
aaf/ax"
.
x l , . . , 3: n iRn. Show t h a t a mapping
iRnl
denote t h e coordinate f u n c t i o n a l s of
f
E
and l e t
i f and o n l y i f a l l t h e p a r t i a l dela1 2 k e x i s t and are c o n t i n u o u s
U.
1 5 . PARTITIONS O F U N I T Y I n t h i s s e c t i o n w e i n t r o d u c e p a r t i t i o n s of u n i t y , a fundamental t o o l which i n many d i f f e r e n t s i t u a t ' i o n s i s used t o cons t r u c t o b j e c t s w i t h c e r t a i n g l o b a l p r o p e r t i e s by patching together o b j e c t s which have l o c a l l y t h e same p r o p e r t i e s . Before s t a t i n g t h e main r e s u l t w e g i v e two p r e p a r a t o r y lemmas.
15.1. LEMMA. that
~ ( 3 : )
then there i s a f u n c t i o n
b
a
If
= 0
x 5 a,
i f
~ ( x = ) 1 a 2 3: 5 b .
strictZy increasing i f
PROOF.
x > 0 15.Al
S t a r t with t h e f u n c t i o n f and
ffz) = 0
if
3:
f i s s t r i c t l y p o s i t i v e for x 5 0 . Then t h e f u n c t i o n
a l s o of class
Cm
on
p e CmiEJsuch
and
p(z)
is
f ( z l = e -l/z i f C m i l R i by E x e r c i s e
d e f i n e d by
5 0 . Then
zero f o r
x > b
i f
x
f E
0 and f i s i d e n t i c a l l y gfx) = f [(x - a ) ( b - 2 1 1 i s
R l i s s t r i c t l y p o s i t i v e on t h e i n t e r v a l
a < z < b , and i s i d e n t i c a l l y z e r o o u t s i d e t h a t i n t e r v a l . Then the function
b
pP(3:) = J z y ( t ) d t / J a g ( t l d t
is
identically
zero
DIFFERENTIABLE MAPPINGS
x L b.
x < a , and identically one for
for
/
g(x)
that
Iab g ( t l d t ~p
Since
we conclude that IP is of class Cm on
is strictly increasing for
15.2. LEMMA.
119
Let
cp'(x) =
lR and
a 5 x 5 b.
E b e a H i l b e r t s p a c e , a n d l e t 0 < r < R. Then p E C m ( E ) s u c h t h a t p(x) = 1 i f IIxII
zr,
there i s a function p(x) = 0
i f IIxII 2 R
r < IIxII < R.
0 < p ( x 1 < 1 if
and
PROOF. By Exercise 14.C the function f (x) = - II x I1 2 isof class Cm on E . By Lemma 15.1 there is a function g E C m ( i R ) such 2 that g ( t ) = 0 if t 5 - R , g l t ) = 1 if t 2 - r 2 and 0 < g ( t ) < 1 if - R 2 i t < - r Z . Then the function P(x) = g o f i x : ) 2 = g i- II x II l has the required properties. 15.3. DEFINITION. a Banach space.
Let X be a topological space and let F be
(a) The s u p p o r t of a mapping f : X F is the closure of the set {x E X : f ( x l # 0). The support of f will be denoted by s u p p f. +
(b)
A collection ( f i j i E I
of mappings from X into F is
said to be Z o c a l l y f i n i t e if each point of X has a neighborhood which meets only finitely many of the sets s u p p f i . (c) A p a r t i t i o n o f u n i t y on X is a locally finite collection ( p i ) i E I of continuous functions from X into [ 0, 1 ]
c
such that
pi(")
= 1
for every
x
E
X.
i E I
(d)
A partition of unity ( p i ) i E I
s u b o r d i n a t e d to an open cover i U i ) i E I
for every
i
E
on
X
is said
of X if
to be
s u p p pi C U i
I.
If X is an open subset of a Banach space then it makes sense to talk about Cm p a : q t i t C o n s of u n i t y on X : thismeans of course that each member of the partition of unity is a Cm function. Actually this is the situation we shall be primarily interested in. Indeed, we have the following theorem.
MUJ i CA
120
1 5 . 4 . THEOREM. L e t U be an o p e n s u b s e t of a s e p a r a b l e Hilbert E . Then f o r e a c h o p e n c o v e r ( U i ) i E I o f U there i s a Cm
space
on
partition of unity
which
U
to
is s u b u r d i n u t e d
.
(UiliE1
i s a sequence
of open b a l l s B ( a n ; r n ) whose u n i o n i s U and s u c h t h a t each U i . By t h e axiom of c h o i c e there B ( a n ; 2 r n l i s c o n t a i n e d i n some Since U i s a Lindelof
PROOF.
is a function n
every
0 < f,(x)
and
< 1
in
2
s u p p g,
r n < IIz
- anll
< 2rn.
by
g1 = f ,
and
CmiE/
...
Then i t i s clear t h a t
2.
=
if
- f,!
gn = ( 1
n
for
By Lemma 1 5 . 2 t h e r e i s a s e q u e n c e (f ! i n ?(El n fn(xl = 1 i f 1 1 2 - anll 5 r n , f n ( z ! = 0 i f I I z - a n II
another sequence ( g n !
if
B(an;Brn) C UTin)
I such t h a t
-+.
liV.
E
such t h a t > 2rn
: liV
T
space t h e r e
S”PP
f,
=
=
Bn(an;2rn)
more, one c a n r e a d i l y p r o v e by
f o r every
n. Since
fn = 1
on
-
(1
0
5
Define
fn-l’fn
gn 5 1
UT&
on
f o r every
and t h a t Further-
E
n.
induction t h a t
-
B ( a n ; r n ) i t follows from(l5.1)
that (15.2)
g,
...
+
-
ig
n = 1
on
B(an;rnf
and
(15.3)
g j =
on
B(an;rn)
f o r every
j > n.
@
Thus ( 1 5 . 3 ) g u a r a n t e e s t h a t t h e s e q u e n c e (9,) i s l o c a l l y f i n i t e in
U , whereas
every
x E
(15.2)
guarantees t h a t
U. F i n a l l y we d e f i n e
vi(x!
z
nexi =
g (xi
T(n)=i
=
g,(x)
1
for
f o r each
DIFFERENTIABLE
and
x
each
MAPPINGS
121
U. S i n c e t h e s e q u e n c e ( g ) i s l o c a l l y f i n i t e i n U, n pi i s w e l l defined and belongs t o Cm(U). Furthermore, E
t h e set
x
every
suppgrL is closed i n
U T
U and t h e r e f o r e
supp
(pi
( n )=i
U,
E
(viiiE1
i s t h e required p a r t i t i o n of unity.
U b e an o p e n s u b s e t of a separable HiZbert b e two d i s j o i n t c Z o s e d s u b s e t s o f U. Then t h e r e i s a f u n c t i o n p E C m ( U / s u c h t h a t 0 _.< P z 1 o n U, IP = 1 o n a n e i g h b o r h o o d of A i n U, and (p = 0 on a neighborhood of B i n U. 1 5 . 5 . COROLLARY.
space
Let
E.
Let
and
B
By Theorem 1 5 . 4 t h e r e are t w o n o n n e g a t i v e f u n c t i o n s V ,
PROOF. $ in
A
$ = 1
U. Then
on
(p
contains
A , whereas
contains
B.
s u p p J, C U \ A and cp + on t h e o p e n s e t U \ s u p p JI, which
supp 9 C U \ B,
C m l U l such t h a t
= 1
= 0
(p
on t h e o p e n s e t U \ s u p p
which
(p,
EXERCISES 15.A.
Consider t h e function
x
and
0
(a) and
k
E
f(x) = 0
(b)
P Z k ( t l i s a polynomial i n
where
Using L ' H o s p i t a l r u l e show t h a t
15.C.
k
E
x > 0
t of d e g r e e 2 k .
f ( k ) ( 0 ) exists
I?. Conclude t h a t
n
(pn
E W,
Let
L e t (U,)
: B
and
-+
B such t h a t
Zim n+
(pn(zl =
(pn(xl = 0 m
for e v e r y
and
f E CmllR7R).
Find an i n c r e a s i n g sequence o f convex, i n c r e a s i n g
functions and
for
f ( k ) ( x ) = e - 1 / x P 2 k ( l / x ) for e v e r y
equals zero f o r every 15.B.
f ( x J = e -7/x
d e f i n e d by
x 5 0.
for
Show t h a t 1Iv,
f
€or every z
Cm
x 5 0
0.
U be a n open s u b s e t o f a s e p a r a b l e H i l b e r t
space.
be a n i n c r e a s i n q s e q u e n c e of open s e t s whose union i s
U , a n d l e t ( e n ) be a n i n c r e a s i n g s e q u e n c e o f r e a l numbers. Using
MUJ I CA
122
Theorem 15.4 f i n d a f u n c t i o n on
and
U3
cp
2
on
en
Un \
C”tU; iR) such t h a t
cp E
Un-l
f o r every
n
2 el
cp
2.
U be an open s u b s e t of a s e p a r a b l e H i l b e r t s p a c e . find a function g E Cm(U; R) Given a f u n c t i o n f E C ( Y ; i R ) s u c h t h a t g 2 f on U. 15.D.
Let
15.E.
Let
f : X?
+
R
be a f u n c t i o n which i s b o u n d e d a b o v e o n
e a c h i n t e r v a l o f t h e form f -
m,
b).
Find a f u n c t i o n g ECm(B;?R)
x
0
g ( x ) = constant f o r
such t h a t every
x
15.F.
Let
E
-
and
g ( x ) 2 f1x)
for
iR. K be a compact s u b s e t o f a H i l b e r t s p a c e
and
E,
l e t U be an open neighborhood of K . By a d a p t i n g t h e proof of Theorem 15.4 f i n d a n o n n e g a t i v e f u n c t i o n cp E C m ( E ) such t h a t s u p p cp C U, 15.G.
Let
cp
5
1
on
E
and
= 1
cp
on a neighborhood o f
K be a compact s u b s e t of
a H i l b e r t space
.
such t h a t
...
+
+ cpn =
1
s u p p pj
C
Uj,
cpl
+
on a neighborhodd of
...
p D n- 1
+
and
E,
l e t Ul,. . ’ un be open s u b s e t s of E which c o v e r K . E x e r c i s e s 9 . B and 15.F f i n d n o n n e g a t i v e f u n c t i o n s ( P I , E Cm(E)
K.
on
Using
...
3
‘n
E , and
K’.
K be a compact s u b s e t of a c o m p l e t e l y r e g u l a r Hausbe open s u b s e t s of X which d o r f f s p a c e X , and l e t U I J . . . , U n Find n o n n e g a t i v e f u n c t i o n s p I , . . .,pn E C t X i such cover K . 15.H.
Let
supp p j
that
C
U
pl +
j’ on a neighborhood o f
= 1
...
+
p,
5
7
on
X , and p I +
...
+
‘n
K.
1 6 . TEST FUNCTIONS
I n t h i s s e c t i o n w e i n t r o d u c e t h e space
of
test f u n c t i o n s
and e s t a b l i s h some p r o p e r t i e s t h a t w i l l be of f r e q u e n t u s e
in
t h i s book.
16.1.
of a l l
DEFINITION. f E Cm(lRn)
W e s h a l l d e n o t e by
D(3RM,
t h e vec t or space
which have compact s u p p o r t . Each f
E
D(Bnl
DIFFERENTIABLE MAPPINGS
is called a t e s t function.
123
U i s a n open s u b s e t o f
If
Rn t h e n
w e s h a l l d e n o t e by D I U l t h e v e c t o r s p a c e of a l l f E D(Wn! such t h a t s u p p f C U. L i k e w i s e , i f K i s a compact s u b s e t of Rn t h e n w e s h a l l d e n o t e by D ( K ) t h e vector s p a c e of a l l f E s u p p f c K.
such t h a t
DtlRn)
I t i s n o t a t a l l obvious t h a t t e s t f u n c t i o n s e x i s t ,
from t h e z e r o f u n c t i o n . But t h e r e s u l t s i n t h e p r e c e d i n g
apart
sec-
t i o n g u a r a n t e e t h e e x i s t e n c e of l a r g e c o l l e c t i o n s o f t e s t func-
U be a n open s u b s e t o f R n l l e t i U i ) b e an open c o v e r of U , and l e t ( 9 ; ) be a Cw p a r t i t i o n o f u n i t y on U , subordinated t o t h e cover I l l i ) . I f each Ui is relatively compact i n V t h e n t h e e n t i r e c o l l e c t i o n ( p i ) i s c o n t a i n e d i n D t V ) . Another example of a t e s t f u n c t i o n . which i s v e r y u s e f u l f o r it s e r v e s t o g e n e r a t e new t e s t f u n c t i o n s , i s t h e following. Indeed, l e t
tions.
16.2.
EXAMPLE.
II x II < 1
if
p : Rn
Let
and
plxi = 0
if
lR b e defined by P I X ) =ke-"iz-'lsrl'21 II x II 1 , where t h e c o n s t a n t k
p d h = 1 , and the l e t t e r
i s chosen so t h a t
0
-+
lRn
d i m e n s i o n a l Lebesgue measure. Then
n
1 5 . A , and
let
x
E
s u p p p = B ( 0 ; l ) . More g e n e r a l l y ,
p6 E D(Bnl
Bn.
for
p6dX = 1
Then
lRn
then we
standsfor
, by E x e r c i s e each
p g ( x ) = 6-np(x/6) and s u p p p g = B ( 0 ; 6 1 .
be d e f i n e d by
U i s an open s u b s e t of
If
p E C"(lRn)
A
3
for
s h a l l denote
6 > 0 every
by
1
I U i t h e Banach s p a c e of a l l e q u i v a l e n t classes of Lebesgue 1 U. We s h a l l d e n o t e by L ( U , l o c ) t h e v e c t o r s p a c e of a l l e q u i v a l e n t classes of Lebesgue m e a s u r a b l e f u n c t i o n s on U which are i n t e g r a b l e o v e r e a c h compact s u b s e t L
i n t e g r a b l e f u n c t i o n s on
of
u.
U b e an open s u b s e t of B n l l e t 6 3 0 and l e t U g = { x E U : d , l x ) > 6). Given f E L ' I ? J 3 l o c i and M 9 E D ( B ( 0 ; 6 ) 1 w e d e f i n e t h e i r ( ~ ( i n o o l u t i o n f * l p : U6 by 16.3.
DEFINITION.
Let
-+
if
* p i (xi =
I
R(0;61
f(x - y i p i y i d h i y i =
1
f(yl9Px
'Glx;61
- gidh(y)
MUJ I CA
124
x
for every
E
Ug.
16.4. PROPOSITION. L e t U b e a n o p e n s u b s e t o f 1 L (U, l o c l a n d l e t 9 6 P ( B ( 0 ; 6 ) ) . T h e n :
f *p E D(U6)
In particular
p a c t s u b s e t of
PROOF.
U2&
For each
-
Ug
x E
Bn, l e t
i f t h e s u p p o r t of
f
f E
i s a corn-
we can write
By differentiation under the integral sign we 1 E C ( U s ) and
get that
for every j . Then (a) and (b) follow by induction. Since is clear, the proof of the proposition is complete.
f * l p
(c)
16.5. PROPOSITION. L e t U b e an o p e n s u b s e t o f E n and f E CtU) b e a f u n c t i o n w i t h c o m p a c t s u p p o r t . T h e n f * p 6 c o n o e r g s s t o f u n i f o r m l y o n Li when 6 0. '(2t
+
PROOF. Since f is continuous and has compact support, it is uniformly continuous on U. Hence, given E > 0 we can find > 0 such that s u p p f c U Z 6 and IJ: - y i - f ( . c i [ 5 E for
If
0
x
every
E
Ug
and
y
j '
3: E
U
6
we have that r
E B(0;60).
If
0
6
then for every
DIFFERENTIABLE MAPPINGS
125
16.6. PROPOSITION. L e t U b e an o p e n s u b s e t o f Bn f E L 1 (U) be a f u n c t i o n w i t h compact s u p p o r t . T h e n :
PROOF. ( a ) Since f * p6 vanishes o u t s i d e of t h e F u b i n i Theorem shows t h a t
U6
and
let
an a p p l i c a t i o n
( b ) S i n c e t h e c o n t i n u o u s f u n c t i o n s w i t h compact s u p p o r t a r e dense i n
L 1 (U!, g i v e n
E
w e can f i n d a f u n c t i o n g E C ( U ) ,
> 0
w i t h compact s u p p o r t , such t h a t
< dUfsupp gl and l e t
K = sup
1 6 . 5 w e can f i n d
with
< E/X(K) 0 .< 6 < 6
JIu l f
*p6
whenever 0
PROOF.
of
u
-
fldh 5
Let
E.
g + B ( 0 ; r l . Then by
0
0 i s s u f f i c i e n t l y s m a l l , and T * p 6
E
D'(Ul
6
when
PROOF. and
supp 9
whenever
9 C
E
D(Ul
U2?.
choose
Then
r > 0
T ( p ) when
6
+
p6
in
supp T C U2r
such t h a t
T * p g E D(Url
and
cp * p 6
DIUrl
E
0 < 6 c r . U s i n g Lemma 1 7 . 1 5 a n d E x e r c i s e 16.F,
observing t h a t
*
0.
+
Given
+
let
9 T
p6
=
we g e t t h a t
(T
*
p 6 j (9)
= T(p
and
*
+
0.
From P r o p o s i t i o n s 1 7 . 1 2 a n d 1 7 . 1 6 we g e t a t o n c e t h e
fol-
lowing c o r o l l a r y .
If
1 7 . 1 7 . COROLLARY.
i s dense i n
U i s a n open subse't o f
IRn
then
D(Ul
D'lU).
W e end t h i s s e c t i o n w i t h t h e f o l l o w i n g r e s u l t , which
will
be needed later on.
1 7 . 1 8 . PROPOSITION. --t
Let
U b e a n o p e n s u b s e t of R
n
.
Let
f:U
b e a f u n c t i o n w h i c h is L i p s c h i t z c o n t i n u o u s , tizut is, t h e r e
1K
i s a constant
k > 0
all
x , y E U. T h e n
and
p
If l x l -
such t h a t
f(yl
I 5
a f / a z j E Lp(U;Zoc) f o r e v e r y
-
k IIz y II for j = 1 , ... , n
1.
T o prove t h i s p r o p o s i t i o n w e n e e d t h e f o l l o w i n g lemma.
17.19.
LEMMA.
Let
Cf.)3
be a sequence o f
f u n c t i o n s d e f i n e d on a measurable space
zE X PROOF.
for w h i r h t h e s e q u e n c e For each
m,n
E
X-valued
measurable
X. Thcn t h e s e t of p o i n t s
(f.(x)) c o n v e r g e s is m e u s , r a b Z e . 3
IN c o n s i d e r t h e m e a s u r a b l e s e t
136
MUJ I CA
S i n c e t h e s e t of p o i n t s m
m
n
U
m=l
n=l
f o r which ( f . ( x i l c o n v e r g e s i s
x E X
3
, t h e desired conclusion follows.
Amn
PROOF OF PROPOSITION 17.18.
n = 1 . W i t h o u t loss U i s a bounded open interval.
F i r s t suppose
of g e n e r a l i t y w e may assume t h a t
Then t h e L i p s c h i t z c o n d i t i o n i m p l i e s t h a t t i n u o u s / and i n p a r t i c u l a r o f bounded
i s a b s o l u t e l y con-
f
v a r i a t i o n . By a t h e o r e m
f ' ( a l e x i s t s f o r almost e v e r y
o f Lebesgue t h e d e r i v a t i v e
a E
If'lall 5 k w h e r e v e r i t e x i s t s . N O W / s i n c e f i s a b s o l u t e l y c o n t i n u o u s , t h e i n t e g r a t i o n by p a r t s formula
U, and c l e a r l y
cp
-
f 'dx =
(p'
fdx
is v a l i d f o r every
Dill),
(p E
and
hence
f i n t h e s e n s e of d i s t r i b u t i o n s c o i n c i d e s with
t h e d e r i v a t i v e of
t h e c l a s s i c a l d e r i v a t i v e . The d e s i r e d c o n c l u s i o n f o l l o w s .
n
Next s u p p o s e
W e s h a l l prove t h a t
2.
af/ax,
E LpIU,locl
p 1 . The p r o o f f o r af/ax i s analogous. Without j loss o f g e n e r a l i t y w e may assume t h a t U = A x B , where A i s ;? bounded open i n t e r v a l i n B and B i s a bounded open set i n f o r every
nn- 1 . L e t
C d e n o t e t h e s e t of a l l p o i n t s ( a , b l
that the partial derivative
E
A
( a , b l e x i s t s . Then
such
H
x
is
C
a
axl
measurable s u b s e t of
let
denote
Cb
the
A
x
set
B , by Lemma 1 7 . 1 9 .
of
a l l points
For e a c h
a
E
b
E B
such
A
that
aS
( a , b l e x i s t s . Then, a g a i n by Lemma 1 7 . 1 9 , e a c h C b i s a 3x1 m e a s u r a b l e s u b s e t of A , and c l e a r l y C U iCb x { b } ) . Let DEB
C' d e n o t e t h e complement of t h e complement of
u
(C6
x
Cb
in
C
A,
in
A x B , and l e t
f o r each
b
E
B.
Ci
denote (7 '
Then
I b F ) . NOW, s i n c e f i s L i p s c h i t z c o n t i n u o u s ,
it
= is
bkR
c l e a r t h a t f o r e a c h b E R t h e f u n c t i o n f i z l , b l i s absolutely c o n t i n u o u s , a n d i n p a r t i c u l a r of bounded v a r i a t i o n on A . Then i t f o l l o w s a g a i n from Lebescjue's Theorem t h a t t h e s e t C i has o n e d i m e n s i o n a l Lebesque measure z e r o €or e a c h h t B . Then a d i r e c t a p p l i c a t i o n o f t h e F u b i n i Theorem shows t h a t t h e s e t C has
71
d i m e n s i o n a l Lebesgue m e a s u r e z e r o . Thus t h e p a r t i a l de( a , b l exists for
rivative i)x1
almost e v e r y i a , b l
E
ll,
and
137
DIFFERENTIABLE MAPPINGS
1 aax f(a,b) I
clearly
I
5 k
wherever it e x i s t s .
f o l l o w s from t h e a b s o l u t e c o n t i n u i t y of f ( x l J b l
rivative
af/axl
the derivative
before,
As
that the
it de-
i n t h e sense of d i s t r i b u t i o n s coincides w i t h af/ax, i n t h e classical sense. This completes
t h e proof.
EXERCISES
U b e a n open s u b s e t o f B n , l e t a E U and l e t : P t U ) + IK be d e f i n e d by G a l c p l = c p ( a ) f o r e v e r y cpE V t U ) . Show t h a t G a i s a d i s t r i b u t i o n on U , c a l l e d t h e D i m e measure 17.A.
Let
a.
at
17.B.
Let
Y : LP YIxl =
b e t h e H e a v i s i d e function,
B
+
which
is
x < 0, and Y t x l = 1 i f x > 0. S b t h a t Y d e f i n e s a d i s t r i b u t i o n on IR. Show t h a t t h e d e r i v a t i v e d e f i n e d by of
if
(I
i s t h e Dirac measure
Y
17.C.
Let
U b e a n open s u b s e t o f
T E D'lUl
tion
60.
iR
n
Ti E
Let (UiliE1
D'(1l.l
E
U(Ui
E
PIUi)
i
E
T E o'(Ul
and e v e r y
i
E
Ui n U
Let
Ti(cpl
#
j such t h a t
I. Show t h a t
U be a n open s u b s e t o f Show t h a t i f a f / a x j = g f o r some o f d i s t r i b u t i o n s , t h e n af/axj = g 17.E.
U
of
there exists a distribution
I
with t h e property t h a t
n Uj), whenever
a distribution
distribuT extends
b e an o p e n c o v e r o f a n open s u b s e t
Suppose t n a t f o r e a c h
LPn.
Show t h a t a
c"(u).
as a c o n t i n u o u s l i n e a r f u n c t i o n a l t o 17.D.
.
h a s compact s u p p o r t i f and o n l y i f
@.
= T.(cp)
for
3
every
Using Theorem 1 5 . 4 f i n d
Ttcpl = Tilcp) T
i s unique.
lRn
and l e t
j (j = 1 , .
f o r every
cp
f, g E c ( U ) .
..,n)
i n the sense
i n t h e classical sense.
NOTES AND COMMENTS
The m a t e r i a l i n t h i s c h a p t e r c a n b e f o u n d i n many s t a n d a r d
138
MUJ I CA
books. We have included only that material which is required for the rest of the book. The material in Sections 13 and 14 can be found, for instance, in the texts of J. Dieudonn6 [ 1 1 , H. Cartan [ 2 ] and L. Nachbin [ 4 1 . The remaining three sections constitute a brief introduction to the theory of distributions. The standard reference f o r the theory of distributions is of course the book of L. Schwartz 1 1 1 . The book of L. Hormander [ 1 ] containsa concise introduction to the subject. The proof of Proposition 17.18 given here is taken from the book cf S. M. Nikol'skii [ 1 1 .
CHAPTER V
DIFFERENTIAL FORMS
1 8 . ALTERNATING MULTILINEAR FORMS I n t h i s s e c t i o n w e introduce t h e exterior product
of
al-
t e r n a t i n g m u l t i l i n e a r f o r m s . T h i s i s i n d i s p e n s a b l e f o r the study
of d i f f e r e n t i a l f o r m s , t o b e g i n i n t h e n e x t s e c t i o n .
18.1. DEFINITION.
Given
A
E
XaimE)
t e r i o r p r o d u c t o r wedge p r o d u c t
and
A I\ B
B E Ea(nE)
t h e i r ex-
E a ( m + n E ) i s d e f i n e d by
€
I n o t h e r words,
The e x t e r i o r p r o d u c t t h e mappings A
or
the mapping i A , H I 18.2. (-
PROPOSITION.
?irnnA
A
A
A B
c a n also b e d e f i n e d i f o n e o f
B t a k e s v a l u e s i n a Banach space. -+
A
If
A B
Clearly
i s b i l i n e a r and c o n t i n u o u s .
A E XaIm,Y)
arid
H
LainE)
then B A A =
B.
PROOF: I f c1 d e n o t e s t h e p e r m u t a t i o n t a k i n g 11, . . . , m + n ) i n t o i n + 1 , . . . , n. + m , 1 , . . . , n i t h e n i t i s c l e a r that (- ] = l a
i-
u " ~ Hence . 139
M U J I CA
140
18.3. PROPOSITION.
If
A E ea("El,
5
E
g"(ng)
and
C t Lff('EI
then
T h i s p r o p o s i t i o n i s a n immediate c o n s e q u e n c e o f t h e f o l l o w i n g lemma.
18.4. LEMMA.
Let
A E EemEl,
B E d : ( n E ) and
(a)
( A eP B l a = 0
(b)
(Aa 8 Bia
= (A 8 B a j a = ( A 8 B i a .
(C)
[ ( A 8 B)'
8 C l a = [ A 8 (B 8
whenever
A'
= 0
C E E ( I P E ) . Then:
or
Gala
B"
= 0.
= (A 8 B Q C ) ' .
( a ) W e show t h a t ( A 8 B l a = 0 whenever An = 0. The o t h e r statement i s p r o v e d s i m i l a r l y . L e t T d e n o t e t h e s u b g r o u p
PROOF.
which l e a v e m+n i s o m o r p h i c t o Sm and
of a l l
T E
S
m + 7,
..., m + n
f i x e d . Then T i s
DI FFERENTI AL j = 3,
..., rn + n .
Sm+n
a
( x ~ , . . . , x ~1 + ~
-
(b) Since ( A a and t h e r e f o r e
0
= (A
Q
i s t h e u n i o n of t h e cosets a T , and since
nT are e i t h e r d i s j o i n t o r i d e n t i c a l , w e get t h a t
(rn + n)! ( A 8 P )
=
141
Then
Since t h e group t h e cosets
FORMS
A)a
= 0,
(a) implies t h a t
( A a 8 B l a = (A 8 B l a .
[ (Aa
- A)
8 B1'
The e q u a l i t y (A Q Ballz
is proved s i m i l a r l y .
B)'
( c ) f o l l o w s a t o n c e from ( b ) . If A 8 B
A E GaimE)
and
R E
gains) then the tensor
i s a l t e r n a t i n g i n t h e f i r s t m v a r i a b l e s and
i n the last
product
alternating
n varia.bles. This motivates t h e following d e f i n i -
tion.
1 8 . 5 . DEFINITION. s p a c e of a l l
rn
A
€
W e s h a l l d e n o t e by firn+nE;F)
E a m n (rn+nE; F)
the
sub-
which are a l t e r n a t i n g i n t h e f i r s t
v a r i a b l e s and a l t e r n a t i n g i n t h e l a s t
n variables.
denote t h e set of a l l permutaticns
o
Snt+n such that n i ? ) < ... elm) and a i m + 1 ) ... olrn + n ) . Note t h a t S,, n a s (rn + n l ! / i n ! n ! e l e m e n t s . Then w e h a v e t h e foll o w i n g p r o p o s i t i o n , whose p r o o f i s s t r a i g h t f o r w a r d and i s left as a n exercise t o t h e r e a d e r . Let
Smn
18.6. P R O P O S I T I O N . be d e f i n e d by
For e a c h
A
E L(rn+nE;F)
let
E
E E(rn+nE;F!
MUJ I CA
142
2'hen
Aamn
mapping
A
= A
a
for every
the
amn m+n continuous p r o j e c t i o n from E I (E;F)
induces
-+
A E Lamn(m+nE.J. I n p a r t i c u l a r
t a("+"E; F ) .
onto As
an immediate c o n s e q u e n c e w e g e t a n o t h e r , somewhat
dif-
f e r e n t formula f o r t h e e x t e r i o r product.
1 8 . 7 . PROPOSITION.
If
and
A E La(n'E)
then
B E LafnE)
In o t h e r w o r d s , (A A B)(xl,
..., x m i n
W e end t h i s s e c t i o n w i t h t h e f o l l o w i n g r e s u l t . , which paral-
l e l s Theorem 1 . 1 5 .
18.8. THEOREM, If E a n d F a r c c o m p l e x fic?n,i(.h sp,; n.
each
A
E
ea(rnE;F)
c a n be u n i q u e l y r e p r e s e n t e d as a sum
E F and t h e summation is t a k e n o v e r a l l multij,. . . j r n i n d i c e s i j , , . . , , j m ) s u c h t h a t I 5 jl ... jm 5 p i .
where
c
(c) m
-.
Show t h a t
h a s dimension
Given
and
i
n
)
whenever
0 -
( j = i ,
..., n )
n.
n
(d)
AELajnR;F)
z.= J
xikekE E
j=1
MUJ I
144
CA
show that A ( x l ,..., x )
n
18.B. of
Let
zl,
...,
2
=
det(x
jk
I A ( e , ,..., e n ) .
be the complex coordinate
n
functionals
and let F be a complex Banach space.
Cn
(b) Show that if 0 p 5 n, 0 2 4 5 n and p + I 2 1 then each A E d : a ( p q C n ; F ) can be uniquely represented as a sum
where
and the summation is taken over a l l
multi-indices (j],. . .,jp,k , , n. and 1 5 k l i . . . i k
. . . ,k 4 )
such that
i<j,
,
..