FOUNDATIONS OF COMPLEX ANALYSIS IN NON LOCALLY CONVEX SPACES FUNCTION THEORY WITHOUT CONVEXITY CONDITION
NORTH-HOLLAND MATHEMATICS STUDIES 193 (Continuation of the Notas de Matem&tica)
S a u l LUBKIN University of Rochester New York, U.S.A. Editor:
ELSEVIER 2003 Amsterdam
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FOUNDATIONS OF COMPLEX ANALYSIS IN NON LOCALLY CONVEX SPACES FUNCTION THEORY WITHOUT CONVEXITY CONDITION
Aboubakr BAYOUMI
King Saud University Riyadh, Saudi Arabia
ELSEVIER 2003 Amsterdam
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To EGYPT To Cairo University To Iman, Mohammad and Youmna
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Preface Infinite Dimensional Holomorphic has a long history as a field of mathematics and yet its strong foundation for non locally convex spaces in the realm of contemporary mathematics is relatively new. In fact, the linear and topological properties of non locally convex topological spaces have received much attention in recent years. However the holomorphic properties of such spaces have been neglected. In 1978 we started our study and research towards the foundations of some of the holomorphic properties of certain non locally convex spaces at Uppsala University. During the last 25 years we have devoted our efforts to different fundamental holomorphic problems in several classes of topological spaces in the absence of convexity condition, and succeeded to solve most of them, like Levi problem and bounding sets problems and many other problems which may represent the backbone of Complex and Functional Analysis. I am now glad to present a collection of most of the published results in this field in this book " F o u n d a t i o n s o f C o m p l e x A n a l y s i s in N o n L o c a l l y C o n v e x S p a c e s " which may be the first book written in this new field. The book may also be entitled: F u n c t i o n s t h e o r y w i t h o u t c o n v e x i t y c o n d i t i o n . In fact, several of the results obtained show the s h a r p c o n t r a s t between it and the field of Holomorphy of locally convex spaces. In addition, new concepts and terminologies have been established for the first time by which we have been able to overcome the absence of non convexity and nonlinearity in our cases. Furthermore, these foundations verify, unlike what was previously believed by some mathematicians, t h a t there is no theory of holomorphy we can deal with it without convexity condition, see[175]. As it was expected we have encountered some obstacles and difficulties in developing this "field at the beginning. This normally arises from the absence of convexity of the fundamental base of neighbourhoods of non locally convex spaces. Therefore, it was essential first for us to generalize several of the important theorems in functional analysis to be suitable and helpful to tackle and solve the problems which had already appeared. Those theorems have been generalized for non linearity and non convexity. As for Functional Analysis we have been able to obtain some generalized
viii forms, as for example, of: H a h n B a n a c h t h e o r e m s , B r o u w e r ' s F i x e d Point T h e o r e m , L a g r a n g e M e a n - V a l u e T h e o r e m s , K a k u t a n i ' F i x e d Point T h e o r e m , B a n a c h S t e i n h a u s t h e o r e m and K r e i n - M i l m a n t h e orem. Regarding Infinite Dimensional C o m p l e x A n a l y s i s , w i t h o u t conv e x i t y c o n d i t i o n , we have introduced and studied the new concept of Quasi-Differentiability and its relations to the classical Fr~chet and G~teaux Differentiability. As a consequence we have obtained new and general versions, as for example, of the following Theorems: F u n d a m e n t a l T h e o r e m of C a l c u l u s , M e a n - V a l u e T h e o r e m for I n t e g r a l s , and L a g r a n g e
Mean-Value Inequalities. Also, the Levi p r o b l e m s o l u t i o n s in different classes of non locally convex spaces have been obtained which means the proof of c e r t a i n r i c h n e s s of t h e class of h o l o m o r p h i c f u n c t i o n s d e f i n e d on t h e s e s classes. During the study of Bounding and w e a k l y - b o u n d i n g (or L i m i t e d ) sets, it is showed that they are different, in several classes of non locally convex spaces. The role of the new terminology of what we have c a l l e d " p - e x t r e m e p o i n t s " or quasi extreme points of non convex sets, which arises from the geometry of the unit balls of our spaces, is rather important. It has applications in several branches of pure and applied mathematics such as: Complex Analysis, Functional Analysis, Numerical Analysis, Non Convex Analysis, Nonlinear Programming, and Differential Equations. Also the role of Quasi -Differentiability is very important and fundamental to build and establish this field of Mathematical science which I consider it sometimes a new field. We have given some applications to illustrate the significance of some of the above mentioned results in this book.Therefore, the book a i m s also to present some of the recent contributions to the theory of Functional analysis and Complex analysis without convexity condition.We hope that this purpose has been successfully achieved according to the given arrangement of our text. This text is mainly addressed to Complex analysis for topological spaces which are not necessarily locally convex with emphasis to locally bounded and locally p-convex spaces (0 < p < 1). We believe, however, that it will also attract some functional analysis and topology oriented readers because a great part of the material presented here has a functional analysis and topological flavour. Each chapter contains o p e n u n s o l v e d p r o b l e m s , so
ix that we hope that the present text will prove interesting for researchers, M.Sc. and Ph.D. students, and be successful enough to e m p h a s i z e t h e i m p o r t a n c e of this field to Pure~ A p p l i e d a n d I n d u s t r i a l m a t h e m a t i c s in particular, and to the other branches of science in general. Aboubakr Bayoumi K i n g S a u d University, R i y a d h
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xi
Acknowledgments The author would like to thank prof.C.O.Kisleman at Uppsala University for his encouraging support and his never failing interest in this field, to whom I am gratefully indebted for his friendship since I got my Ph.D. at Uppsala University in 1979. I am thankfull to him for valuable discussions. It is with pleasure that I express my gratitude to all my teachers at Mathematics Department of Uppsala University for their kindness and cooperation; Special thanks to my friends Mats Essen, Benget Josefson, Urban Cegrel, Gunner Berg, and Life Abrahmson. Also to my teachers and colleagues at Mathematics Department of Cairo University I want to express my deeply and sincerely felt thanks for their kindness and friendship, from where I got my B.Sc.degree. I want to express my appreciation to my colleagues at King Saud University for fruitful discussions and kind interest during the preparation of this book.
Ch.1 Fundamental Theorems (Functional Analysis) i Ch.2 Theory of Polynomials in F-Spaces
L I
......
/I
inm.v.s, ii
Levi Problem
........
L_J I
....................................................
I t
I ...............
Ch.9 Bounding and Weakly-Bounding Sets in T.V.S. i............................c,:i0
Ch.3 Fixed points and p-Extreme points -
chi4
...........................................
Quasi (Bayoumi) Differential ............................Calculus
' ............
Ch.6 Higher Quasi-Differential in F-spaces Ch.7
QuasiHolomorphic Maps
---I l t 1
.
Ch.5
Generalized MeanGener Valu, lue Theorems RealI &Complex Spaces
.
.
.
.
.
.
.
.
_
.....
i .................................................
Ch.8 New Versions of main Theorems
Contents 1
2
FUNDAMENTAL THEOREMS IN F-SPACES 1.1 L I N E A R MAPPINGS . . . . . . . . . . . . . . . . . . . . 1.1.1 Linearity and Boundedness ............. 1.1.2 The Space L(E,F) . . . . . . . . . . . . . . . . . . 1.2 HAHN-BANACH THEOREMS ............. 1.2.1 The Main Results . . . . . . . . . . . . . . . . . . 1.2.2 Hahn-Banach Theorem in Locally Bounded Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Failure of Polynomial and Holomorphic Extensions in F-Spaces ............
1 1 2 4
7 8 11 13
1.2.4
Hahn-Banach Theorem in Locally Pseudoconvex Spaces . . . . . . . . . . . . . . . . . . . . . . .
14
1.2.5 1.2.6
Hahn-Banach Theorem in t.v.s ........... Examples of F-spaces and Non F-spaces
17 18
1.3
OPEN
1.4
UNIFORM
THEORY
MAPPING
THEOREM
BOUNDEDNESS
OF POLYNOMIALS
.....
.............. PRINCIPLE
20 ......
25
IN F-SPACES
29
2.1
MULTILINEAR MAPS . . . . . . . . . . . . . . . . . . . 2.1.1 Continuous Multilinear Maps ........... 2.1.2 Locally Bounded Spaces L(E1,...,Em;F) ...... 2.1.3 Examples of Bilinear Maps ............. 2.1.4 Natural Isometry . . . . . . . . . . . . . . . . . . .
29 29 32 33 34
2.2
POLYNOMIALS 2.2.1 Symmetric
35 35
2.2.2
Multilinear
2.2.3 2.2.4
Polynomials Continuous
2.2.5
The
OF P-NORMED SPACES ...... Multilinear Maps ........... Formula
. . . . . . . . . . . . . . . . .
36
. . . . . . . . . . . . . . . . . . . . . . . Homogeneuous Polynomials ....
37 40
Generalized
Universal xiii
Constant
Trl, m q / p
m!
. . .
42
xiv
CONTENTS 2.2.6 2.2.7
The Space P(E,F) Banach-Steinhaus
. . . . . . . . . . . . . . . . . . .
Theorem
for P o l y n o m i a l s . .
45 46
3
FIXED-POINT AND P-EXTREME POINT 49 3.1 p - E X T R E M E P O I N T I N N O N L O C A L L Y C O N V E X SPACES ............................. 50 3.1.1 P r o p e r t i e s o f p - E x t r e m e P o i n t s ......... 54 3.1.2 G e n e r a l i z e d M i l m a n ' s T h e o r e m .......... 56 3.1.3 A p p l i c a t i o n s ...................... 58 3.2 GENERALIZED FIXED POINT THEOREM .... 62 3.2.1 G e n e r a l i z e d B r o u w e r s ' s F i x e d P o i n t T h e o r e m 63 3.2.2 Generalized Kakutani~s Fixed Point Theorem 66 3.3 GENERALIZED KREIN-MILMAN THEOREM . . 68 3.3.1 G e n e r a l i z e d K r e i n - M i l m a n T h e o r e m . . . . . . . 69 3.3.2 S e p a r a t i o n T h e o r e m s in S o m e S e q u e n c e FSpaces .......................... 72
4
QUASI-DIFFERENTIAL CALCULUS 4.1 Q U A S I - D I F F E R E N T I A B L E MAPS ........... 4.1.1 Q u a s i - D i f f e r e n t i a b l e M a p s in F - S p a c e s . . . . . 4.1.2 P r o p e r t i e s o f Q u a s i - D i f f e r e n t i a l s 4.1.3 Quasi-Differentials of Multilinear Maps ..... 4.1.4 Q u a s i - D i f f e r e n t i a l s o f P o l y n o m i a l s . . . . . . . . . 4.1.5 I n v e r s e M a p p i n g T h e o r e m ............. 4.1.6 R e a l a n d C o m p l e x C a s e s . . . . . . . . . . . . . .
77 77 78 83 84 85 87
5
GENERALIZED MEAN-VALUE THEOREM 89 5.1 M E A N - V A L U E T H E O R E M IN REAL SPACES . . 89 5.2 MEAN-VALUE THEOREM I N C O M P L E X S P A C E S 94 5.2.1 M e a n - V a l u e I n e q u a l i t y . . . . . . . . . . . . . . . . 94 5.2.2 Applications ...................... 96 5.2.3 Examples of Sequence Spaces ........... 99
6
HIGHER QUASI-DIFFERENTIAL IN F-SPACES 101 6.1 S C H W A R T Z S Y M M E T R I C THEOREM ....... 101 6.2 H I G H E R Q U A S I - D I F F E R E N T I A L S .......... 106 6.2.1 T h e Q u a s i - D i f f e r e n t i a l s Dmf a n d dmf . . . . . . . 107 6.3 G E N E R A L S C H W A R T Z S Y M M E T R I C THEOREM 109 6.4 D I R E C T I O N A L DERIVATIVES ............. 111 6.5 Q U A S I A N D F R I ~ C H E T D I F F E R E N T I A L S ..... 115
CONTENTS
7
6.5.1
Finite Dimensional
7.3
6.5.2
Infinite Dimensional
POWER
Case
116
..............
118
SERIES
IN F-SPACES
Series
.............
136
7.2.1
Power
7.2.2 7.2.3
Uniform and Normal Convergence ........ Generalized Cauchy-Hadamard Formula .....
138 139
7.2.4 7.2.5 7.2.6 7.2.7
R a d i u s o f N o r m a l C o n v e r g e n c e Pn . . . . . . . . R a d i u s o f A b s o l u t e C o n v e r g e n c e Pa . . . . . Uniqueness of Power Series ............. Quasi-Differentials of Power Series ........
144 144 146 147
. . . . . . . . . . . . . . . . . . . . . .
QUASI-ANALYTIC MAPS ................. 7.3.1 Quasi-Analytic and Quasi-Holomorphic 7.3.2
Principle
7.3.3
Integral
of Quasi-Analytic Domain
QA(U)
. . .
Maps ....
153
Continuation
...............
154
BOLZANO'S INTERMEDIATE THEOREM 8.2.1 Finite Dimensional Spaces ............. 8.2.2 Degree Theory .....................
8.3
INTEGRAL
8.2.3
BOUNDING 9.1
157 . . . 157
R i e m a n n I n t e g r a t i o n o n [~,/~] . . . . . . . . . . . Curvilinear Integrals ................. Fundamental Theorem of Calculus ........
8.2
Infinite Dimensional MEAN-VALUE
AND
WEAKLY-BOUNDING
BOUNDING SETS 9.1.1 Bounding Sets 9.1.2 Bounding Sets 9.1.3 Bounding Sets
. . in in in
.....
Spaces ............. THEOREM
136
151 . 151
NEW VERSIONS OF MAIN THEOREMS 8.1 F U N D A M E N T A L THEOREM OF CALCULUS 8.1.1 8.1.2 8.1.3
9
Case ...............
QUASI-HOLOMORPHIC MAPS 123 7.1 F I N I T E E X P A N S I O N S AND TAYLOR'S FORMULA124 7.1.1 Finite Expansion .................... 125 7.1.2 Taylor's Formula .................... 128 7.1.3 Quasi-Differential of Taylor Polynomials .... 129 7.1.4 General Mean-Value Theorem ........... 130 7.1.5 Taylor's Formula with Lagrange Remainder . . 133 7.2
8
xv
158 160 161 163 163 164 171
....... SETS
176 179
. . . . . . . . . . . . . . . . . . . . 180 L o c a l l y B o u n d e d F - S p a c e s . . 181 S e p a r a b l e M e t r i c S p a c e s . . . 186 Locally Pseudoconvex S p a c e s 189
xvi
CONTENTS 9.1.4
9.2
B o u n d i n g S e t s in N o n L o c a l l y P s e u d o c o n v e x Spaces .......................... WEAKLY-BOUNDING (LIMITED) SETS ......
191 ' 192
9.2.1 9.2.2
9.3
9.4
W e a k l y - B o u n d i n g S e t in L o c a l l y B o u n d e d Spaces192 T h r e e D i f f e r e n t C l a s s e s of H o l o m o r p h i c F u n c tions ........................... 196 9.2.3 E x a m p l e s o f H o l o m o r p h i c F u n c t i o n s . . . . . . . 199 P R O P E R T I E S OF B O U N D I N G A N D L I M I T E D SETS206 9.3.1 B o u n d i n g S e t s a n d C o m p a c t L i n e a r M a p s . . . 206 9.3.2 P r o p e r t i e s of T h e D i f f e r e n t W e a k l y - B o u n d i n g Sets ........................... 209 9.3.3 W e a k * C o n v e r g e n c e in t h e D u a l o f a p - B a n a c h S p a c e A s D i f f e r e n t f r o m N o r m C o n v e r g e n c e . . 211 HOLOMORPHIC COMPLETION ............ 217 9.4.1 H o l o m o r p h i c C o m p l e t i o n in F - S p a c e s . . . . . . 217 9.4.2 H o l o m o r p h i c E x t e n s i o n P r o b l e m ......... 221
10 L E V I P R O B L E M I N T O P L O G I C A L S P A C E S 227 10.1 L E V I P R O B L E M A N D R A D I U S O F C O N V E R G E N C E 2 2 9 10.1.1 P B - S p a c e s ....................... 231 10.1.2 P r o p e r t i e s o f t h e R a d i u s of C o n v e r g e n c e . . . . 234 10.1.3 T h e L e v i P r o b l e m in P B - S p a c e s . . . . . . . . . . 241 10.2 L E V I P R O B L E M ( G R U M A N - K I S E L M A N APPROACH) 243 10.3 L E V I P R O B L E M ( S U R J E C T I V E LIMIT APPROACH)253 10.4 L E V I P R O B L E M ( Q U O T I E N T MAP APPROACH) 258 Bibliography Notations Index
Chapter I
FUNDAMENTAL THEOREMS IN F-SPACES This chapter is devoted to the introduction of some of the fundamental theorems in functional analysis in their new versions. Those versions may be suitable to tackle several types of problems in Infinite Dimensional Holomorphy, in the absence of convexity condition. In fact this chapter consists of ~ parts deal with: I - L i n e a r m a p pings, I I - H a h n - B a n a c h t h e o r e m s , I I I - O p e n m a p p i n g theorem, and I V - U n i f o r m boundedness principles. The well known results of the open mapping theorems have been introduced with different proofs, which may be similar to the classical ones. The results on Hahn-Banach theorems and of the uniform boundedness principles are totally new and the reader may not find them in any other books on functional or complex analysis. They have several applications as we we shall see here in our book.
1,1
LINEAR MAPPINGS
Linear maps between p-normed spaces and linear functional on a p-normed space play a central role in functional analysis and the study of calculus and holomorphy. In this chapter we study the interrelationship between continuity and boundedness. As a consequence we prove the three basic principles of p-Banach spaces, namely the Hahn-Banach theorem, the Open Mapping Theorem and the Uniform Boundedness Theorem, see parts II,III &IV. All the spaces considered in this book unless otherwise stated are corn-
2
CHAPTER
1 FUNDAMENTAL
THEOREMS
IN F-SPACES
plex locally bounded spaces (Lbs), so they are p-normed spaces for some 0 < p _ 1, and consequently they are quasi-normed spaces, Rolewicz [186]. In particular E and F will denote the p-normed and the q-normed spaces respectively, where (0 < p, q _ 1). 1.1.1
Linearity and Boundedness
D e f i n i t i o n 1 A p - n o r m on a vector space E over K is a mapping from E to ~+ satisfying (i) Ilxll = 0 iS and only if x=O (ii) II~xll- I~lPllxll, So~ every A E K, x E E
I1.11
(iii) IIx + Yll -< Ilxll + IlYlI, for every x, y ~ E. D e f i n i t i o n 2 A q u a s i - n o r m on a vector space E function II" II on E satisfying: (a) Ilxl[ - 0 if and only if x -- 0
is a real non-negative
(b) Iltxll- Itl Ilxll
(~) IIx + Yll < o- (llxll + IlYll) for all x , y E E, t E K , and some constant ~ > 1, which is independents of x, y. The smallest a for which (c) holds is called the q u a s i - n o r m c o n s t a n t
oS (E, I1 II). The sets {x" Ilxll < E}, E > 0, form a base of neighbourhoods of 0 for a metrizable vector topology on E. If (E, I1" [[) is quasi-normed, then there exists 0 < p _ 1 and a p- norm [1" lip which determine the topology of E. The function ]]. [[1/p is then a quasi-norm on E which determines the topology of E. D e f i n i t i o n 3 Let E and F be p-normed, and q-normed spaces respectively over the same field K . A map A " E --~ F is said to be l i n e a r if A ( t x + y) = t A ( x ) + A(y) for all x, y in E
and for all t i n K .
A linear map may not be continuous in general as the following example shows.
1.1 L I N E A R
MAPPINGS
3
E x a m p l e 1 Let E = C ( I , K ) be the p- normed space of all continuous functions on the unit interval I - [0,1] under the sup p-norm: Ilfllp = supxe/If(x)lp. Let F - C~(I,tr) be the p-norrned subspace of E consisting of those functions f which have continuous derivative dr. Then the linear map f ---, d f of F into E is not continuous. In fact, if we set f ~ ( x ) = (sin n x ) / n , then the sequence (f~) converges to O, whereas the sequence (df~) does not converge to O.A D e f i n i t i o n 4 Let U C E be a nonempty set of a p-normed space E and let F be a q-normed space, (0 < p , q 0 where a >_ 1 here and throughout this book. q is called a b s o l u t e l y s u b l i n e a r o f t y p e a if (iO is replaced by q(tx) = Itlq(x) for scalars t. It is obvious that a quasi-seminorm is an example of such q, see [207] and [186]. The following theorem provides us with a new version of the HahnBanach Theorem in vector spaces. We extend a linear functional f which is majorized on a subspace M C E by a sublinear functional q of type a defined on E. The majorization will not remain fixed under the extension if a > 1. T h e o r e m 4 [18](1990) Let E be a real vector space, M a subspace of E with c o d i m M - n, and q a sublinear functional of type ~r. Suppose f is a linear functional on M such that, f ( x ) p > 0), does not have the HBEP. He showed that a subspace M of lp, with an/1-precompact unit ball, cannot be extended to the whole of lp. Shapiro [190] generalized this result to every non locally convex F-spaces E with basis. Kalton [131] dropped the assumption that E has basis in Shapiro's result. We use Kalton's result to study the extension and separation properties for polynomials and holomorphic functions. D e f i n i t i o n 8 A topological vector space E is said to have the p o l y n o m i a l H a h n - B a n a c h e x t e n s i o n p r o p e r t y ( P - H B E P ) if every continuous polynomial on a closed subspace can be extended continuously to the whole space. Similarly, E is said to have the H o l o m o r p h i c H a h n - B a n a c h ext e n s i o n p r o p e r t y ( H - H B E P ) if every holomorphic function (continuous and Gdteaux analytic) on a closed subspace can be extended holomorphically to the whole space. The following interesting theorem is a consequence of Kalton'result [131]. T h e o r e m 8 [18](1990)
Assume E is a non locally convex F-space. Then E polynomial Hahn-Banach extension property.
does not have the
14
CHAPTER
1 FUNDAMENTAL
THEOREMS
IN F-SPACES
P r o o f . By Kalton's result [131] mentioned above, there exists a continuous linear functional f on a closed subspace M of E which does not have the continuous extension to E. Since continuous linear functionals are continuous polynomials, f represents the required continuous polynomial on M which does not have a continuous extension to E. I Let us now consider the holomorphic extension in F-spaces. That is " W h e t h e r every holomorphic function on a subspace c a n b e e x t e n d e d t o a h o l o m o r p h i c f u n c t i o n o n E ".
M
of E
In fact, several authors studied holomorphic extension in locally convex spaces. For example Dineen [82] showed that, not every holomorphic function on the closed subspace in lcr of co can be extended holomorphically to l~. That is l ~ does not have H-HBEP. Aron and Berner [[8] ,propl.1] characterized those holomorphic functions f on co which can be extended holomorphically from co to l~. He proved that f should be bounded on bounded sets. In non locally convex spaces we studied the holomorphic extension when M is either dense or a closed subspace of certain metric spaces, (see section 9.4, Ch.9, on holomorphic extension problems). In what follows we extend some of their results to F-spaces. T h e o r e m 9 [18](1990) A s s u m e E is a non locally convex F-space. Then E holomorphic Hahn-Banach extension property.
does not have the
P r o o f . Since a continuous linear functional on a closed subspace M of E is holomorphic, the conclusion will be reached if we apply the above Kalton's result [131]. I 1.2.4
Hahn-Banach Pseudoconvex
Theorem Spaces
in L o c a l l y
Let E be a locally pseudoconvex topological vector space (Lps), that is, the topology of E is defined by a fundamental family of absolutely pseudoconvex neighbourhoods of the origin. D e f i n i t i o n 9 A subset A of a vector space E is called a b s o l u t e l y p s e u d o c o n v e x if it is absolutely p-convex for some i >_ p > O. Here A is absolutely p-convex if ax + by E A whenever x, y E A and lal p + Ib]p _p>_olpwith this q-topology is a complete separable non locally convex Lps and has the unit vectors (en) as its symmetric Schauder basis.
1.2
HAHN-BANACH
THEOREMS
19
A sequence converges in Olp if and only if it is contained and converges in some lp. A set is compact in Ulp if and only if it is contained and compact in some lp; no closed infinite dimensional subspace of Dlp is contained in lp; and no infinite dimensional subspace of Ulp is metrizable. Hence Ulp itself is not metrizable, Stiles [202]. Since E is Lps, Theorem 10 implies that every continuous linear functional on a closed subspace M of E has a linear extension to E which turns out to be continuous if codlin M < oo. A
E x a m p l e 4 Space o f m a p p i n g s with r a p i d l y decreasing a p p r o x i m a tion numbers Let E be a Banach space and L(E) be the space of all continuous linear mappings of E equipped with the linear mappings norm. For T 6 L(E) we define the rth-approximate number by a~(T) = inf { l I T - SI]; S 6 L(E), with dim S(E) A r } On the set O0
lp(E) = {T 6 L(E);
~-~[ar(T)] p < oo},
(0 < p _< 1),
r=l
we define a quasi-norm II" II given by IITI] = (lITIIp) l/p, where
O0
JJTlip r--1
This quasi-norm satisfies
lIT + sll l nB1 and A
F
:
is a surjection,
Un> 1nS1 -- Un> 1nS1.
By the Baire category theorem, there is an integer k such that kS1 has non empty interior. Thus for some point ~ E F and some a > 0, BF(~, a) C kS1, where BF(~, a) is the open ball in F centered at ~ with radius a. Since the mapping y ~ ky is a homeomorphism, $1 contains an open ball BE(q, r) for some r > 0 and r/E $1. Then
BF(0, ~) c S] - V c S-~ - S-~ c 2 ~ / ~
- &
1.3 O P E N
MAPPING
where So -'- 21/qs1.
THEOREM
21
Hence BE(O, r) C r
and
2-1/qBF(O, r/2) C ~,~1 because
BE(O, r) = r l / q B F ( O , 1).
(1.25)
So inductively, we get the following inclusion 9
BF(0, Now let y E F Xl E B1 with
r
all n.
c
such t h a t Ilyll < r / 2 . Since y E
~1
(1.26)
by
(1.26), there exists
r
Ily - A(z )II
0, we write
BE( ) = {x E; Ilxll
O, there exists 5 > 0 such that Ilxll < 5 =:> IIA,,:(x)ll < ,~ Vk e I.
D) {Ak}k~,
is bounded pointwise, i.e. for each x E E,
Mx > O such that
IIA~(x)ll 0
is uniformly bounded, i.e.
that
IIAkll (b). Assuming the statement (a), we can find a 5 > 0 such that Ilxll < ~ ~ IIAk(x)ll < 1 for all k E I. If x =/= 0, then Proof.
51/Px
lIAr( ilxil l/p) II < 1 , i.e. for all k E I.
IIAk(*)ll
l Dn. Then the Baire category theorem ensures that s o m e Dn contains a closed ball BE(~, r). Consequently we have
ilAk(x)l I k for all k" If ( f o A ) is continuous for all f E F'~ then the sequence (f[A(ak)]) is bounded for every f in F ' , so the sequence (A(ak)) must be bounded by Theorem 16 of uniform boundedness. This contradicts the fact that IIA(ak)ll > k; thus A is continuous. II
2.1.2
Locally Bounded
S p a c e s L ( E 1 . . . , E m ; F)
Let us denote by L ( E 1 , . . . , Era;F) the set of all continuous multilinear maps of E1 x ... x Em into F. Then L ( E I , . . . , Era;F) becomes a vector space under the pointwise addition and scalar multiplication For any A E L ( E 1 , . . . , E,~; F), we define
IIAII -
sup{ IlA(x)II; Ilxll~ _< 1}
Then by Theorem 18 (c) IIA(x) II -< IIAll Ilxlll q/pl 9 9 9IIx~ll q/pro
and IIAII is the smallest M q > 0 satisfying the inequality in Theorem 18 (c) and IIAII is a q-norm on L(E1, " " , E m ; F ) . In case that we are concerned with continuity of multilinear mappings we write the vector space of all (algebraic) multilinear mapping, from
2.1 M U L T I L I N E A R M A P S E1 x . . . x Em we write
to F
33
by L a ( E 1 , ' " , E m ,
F). When E - E1 = "" = Era,
L ( m E , F) - L(E1, " " ,Era; F), La( E , F ) - La(E1, " ' , E m ; F ) m
o
T h e o r e m 20 [36] If F is p-Banach space, then L ( E 1 , . . . , Em; F) is a p-Banach space for any pi-normed spaces Ei (1 < i _< m). P r o o f . This is modeled on after the proof of Theorem 1 (subsection 1.1.1). m
2.1.3
Examples
Example 6 . (a) Let E , F
of Bilinear Maps
and G
r
be p-no~vned spaces. Then the map
L(E; F) • L(F; G) --~ L(E; G)
defined by
r
g) - g o f
is bilinear, and furthermore, it is continuous since IIg o fll p > 0), helped us to introduce the concept of the arc segment A b, and the new terminology of p-extreme points. The following lemma characterizes the new concept of the p-extreme points of some non-convex sets in a vector space X . Lemma4 Let X be a vector space and l > p > O. Let K be a n o n e m p t y p-convex subset of X and z E K . T h e n the following s t a t e m e n t s are equivalent : (i) z is a p - e x t r e m e point of K . (ii) Let x, y E K be such that x # y, and t, s C [0,1]. I f z=sx+ty,sP+tP=l,
then
s=0
or t = 0 .
Equivalently, if z - (1 - t) 1/px + tl/Py, then we have either t - 0 o r t = (iii) I f x, y E K are such that 1 z -
Ov)
t -
+ y),
then
x -
y -
(2)I/p-Iz;
1.
(3.8)
( 89 1/p
{z }
P r o o f . This follows from the definition of the p-extreme points, noticing t h a t z could not be an internal point of any arc segment A y joining x, y inK. I P r o p o s i t i o n 2 Let X be a locally p-convex space and K be a p - c o n v e x subset of X . Then K ~ Ep(K) = r That is, any interior point of K , when it exists, is not a p- extreme point of K . P r o o f . For K ~ = r the result is trivial. Assume now t h a t K ~ ~ r and t h a t x E K ~ So we can assume t h a t K contains 0. Let x C K ~ T h e n there exists an absolutely p-convex 0-neighborhood V in X such t h a t VCK
with x + V C K .
The m a p ~ ~ /~x f r o m t ~ i n t o X is continuous, say, at ~0 T h e n for this x-neighborhood x + V, there is an r > 0 such t h a t :
(3.9) 1_ 1 2~ .
56
CHAPTER
3 FIXED-POINT
1 --1
]A-2~
[_ _ 0 a n d ~ t ~ P - 1 . Let K b e c o m p a c t , K C R n+l But x E C p ( K ) if and only if X -- t l X l + ... + tn+lXn+l for some t C S and xj c K , ( 1 < j < n + l ) . In other words, Cp(K) the image of S x K n+l under the continuous mapping : ( t l , . . . , t n + l , X l , .., X n + l ) --+ t l X l + ... + tn+lXn+l. Thus Cp(K) is compact, m T h e o r e m 31
is
(Generalized Milman Theorem)[2~(2001)
If K is compact in a locally p-convex space X and if Cp(K) compact, then every p-extreme point of Cp(g) lies in K, that is C_ K.
is also
(3.15)
58
CHAPTER
3 FIXED-POINT
AND
P-EXTREME
POINT
P r o o f . Suppose t h a t some p-extreme point a r Cp(K) is not in K. Then there is an absolutely p-convex 0-neighborhood V in X such that
(a+V) AK-r Choose x l , . . . , Xn in K
(3.16)
so that K C Ul(Xj + v ) . Each set
Aj -- Cp(K A (xj + V))
(1 _< i < n)
(3.17)
is absolutely p-convex and also compact since Aj c Cp(K). Also note t h a t K C A1 U . . . U An. But L e m m a 5 shows t h a t m
Cp(K) C Cp(A1 U ... U An) -- Cp(A1 U ... U An)
(3.18)
N
The opposite inclusion holds also, because Aj C Cp(K) fore,
for each j. There-
Cp(K) - Cp(A1 U ... U An).
(3.19)
In particular, a = tlYl + . . . + tnyn for each yj in some Aj, where each tj > 0 and E1n tjp _ 1. Now the grouping
t2y2 + . . . + tnYn a -- flY1 + (1 - tPl)1/p ~2 ~-~ ~--t-tP) llp
(3.20) !
exhibits a as a p-convex combination of two points of Cp(K) by (3.18). Note t h a t the denominator equals (1 - t~) 1/v. Since a is a p-extreme point of Cv(K), we conclude yl - (t2y2 + ... + t~y~)/(t~ + ... + t{) 1/p and hence a -- (tl + (1 - t~)l/p)y I
e
A1.
Thus for some j, m
a E Aj C xj + V C K + V
(3.21)
which contradicts (3.16). Note that Aj C xj + V by (3.17). I 3.1.3
Applications
Although the notion of the p-extreme point seems to be a purely algebraic one, we have the several interesting applications in non convex analysis and m a t h e m a t i c a l programming. Below we introduce some of them.
3.1 P - E X T R E M E
POINT IN NON LOCALLYCONVEX
SPACES59
A p p l i c a t i o n s to N o n C o n v e x A n a l y s i s
It may be useful to give examples of p-extreme points of the closed unit ball in some concrete p-Banach spaces. The following theorem studies the set of p-extreme points in the F-space l P ( 1 > p > 0). The result for 11 is similar and well known. T h e o r e m 32 [27](2000) In
lp, the set of p-extreme points of the closed unit ball Blp
is given
by Ep(Blp) = {Ae("); I)~1- 1, n -
(3.22)
1,2,...}.
Consequently, the p-convex hall Cp(A) of the following set A represents a p-extreme set of Bl~,
1)p !
A-
x(n);x(n)-((n
_l ) nth place ~ 0 , " " . ) , n E N }
"
P r o o f . Suppose IAI- 1 and )~e(n) = Then
1
21/p (x + y), 1
)~ -
2--~/p ( Xn nu Yn )
where x - (Xn), y - (Yn) e Blp.
and 0 -
1
2-~(xj+YJ) jCn.
As A, with IAI - 1, is a p-extreme point of Bz~ - Bzp NK, K is the scalar field, it follows that x~ - Yn and hence 1 - I A I< }--~ I xk IP< 1 and 1 - I A I< ~ l Yk IP~ 1: that is, xj - - y j - - 0 (for j ~ n). Consequently, 21/p/~e (n) -- x -- y.
Thus /~e (n) e E p ( B l p ) , if I A I-- 1. Conversely, if r / - (r/j) e Ep(Blp), then Ilrlll- 1; To see this we claim that there is some n such that r/j -- 0, for all j 7~ n. Hence, r/ - (~j) -- ~n e(n) with I r/n I-- 1, for we have II~ll- 1. Assum, on the contrary that there are j and k with j < k such that
rjj r 0 and r~k r O. Then j
0 p > O) of Lp [0, 1] points.
has no p-extreme
Proof. It suffices to consider any f E BL, with llfllp - 1 since the interior points will n a t u r a l l y be non p-extreme points. In fact one can choose c E [ 0 , 1 ] such t h a t f o f f l p d t - 1 / 2 , f~lffPdt-1/2. Now let
g-
21/p f.x[O,c], h -
21/Pf.x[c,11
where x[0,c], X[c,l] are respectively the characteristic functions of [0, c] and [c, 1]. T h e n Ilgllp = Ilhl[p- 1 are such t h a t 1
1
f - 2-i~/pg + 2-g~h, g ~r h Hence f
is not a p - e x t r e m e point of BLp. m
3.1 P - E X T R E M E
POINT IN NON LOCALLY CONVEX
SPACES61
Applications to Operations Research In the following we give another application in m a t h e m a t i c a l p r o g r a m m i n g of the p-extreme points. We solve a system of nonlinear inequalities and find the maximum or the minimum of a given nonlinear function restricted to this system.
Example 9 Consider the following example of a nonlinear system of inequalities y < ( x - 1) 2 y < ( x + 1) 2 (3.23) y > - ( x + 1) 2 Y >_ -(~
m
1) 2
Figure 1: The d o m a i n of i n t e r s e c t i o n of t h e four inequalities (3.23) In fact this can be solved if we first consider the four following arcs drawn in the four different quadrant anticlockwise.
y-x2+2x-1
=0
y- x2 - 2x- 1 = 0
(3.24)
y+x2+2x+l=O y+x2-2x+l
=0
These four arcs in R 2 represent the graphs of the four intersecting parabolas whose vertices are (• 0), (0, i l ) , see the figure. The shaded area represents the solution of the system of inequalities (3.23). Since the p-extreme points of the shaded area are (0, i l ) , (=t=l,0), the maximum of the given function will be taken at some of these points. For example, the maximum of the nonlinear function :
62
CHAPTER
3 FIXED-POINT
AND
P-EXTREME
POINT
f ( x , y) -- x 2 - 3y 3
(3.25)
restricted to the system (3.23) will be equal to 3 and is taken at the p-extreme point (0, - 1). The above example is simple but an interesting one to explain the benefit of our new concept, the p-extreme points. However the concept of p-extreme points can be used widely to solve a general problem in mathematical programing. For example the nonlinear programming problem (NLP) can be expressed as follows :
E x a m p l e 10 Find the values of the decision variables that
X l ~X2 , ...Xn
such
max (or rain) z -- f ( X l , X 2 , . . . x n ) s.t. gl ( Xl, X2, ...Xn) ( )bl
g2( Xl, X2, ...Xn)
( ~ ,--, or >_)b2
gin( Xl, X2, ...Xn)
( ~ , - - , or ~ ) b m .
(3.26)
As in the linear programming, f(Xl,X2,...x,~) is the NLP's object function, and gl( Xl, x2,...Xn) (_)bl., ..., gm( Xl, x2, ...xn) (_)bin are the constraints. NLP with no constraints is an u n c o n s t r a i n e d NLP. The solution will then be obtained whenever the p-extreme points of D, the intersection domain of the constraints, are obtained.
3.2
GENERALIZED
FIXED POINT THEOREM
Fixed point theorems play an important role in many parts of analysis and topology. They have many applications, for example, in the fields of linear algebraic equations, ordinary and partial differential equations, and integral equations. In fact, one of these basic theorems is due to Brouwer which is considered as one of the most celebrated theorems in mathematical analysis. Fixed points of continuous and holomorphic functions on compact convex sets in locally convex spaces have been studied by several authors, see for example ,[108] and [188].[206]
3.2
GENERALIZED
FIXED POINT THEOREM
63
We devote this section to extend Schauder's and Tychnoff's Fixed Point versions, of Brouwer's Fixed point theorem, to p-convex sets K in locally p-convex spaces X, (1 > p > 0), by proving that K has the fixed point property. We also generalize Kakutani's Fixed Point Theorem to locally p-convex F-spaces whose dual separates points. We prove that an equicontinuous group of affine maps of K has a fixed point.These depend on making use of the new terminology, the p-extreme points of non-convex sets, see the preceding part. In [24], we studied the set of fixed points of holomorphic mappings of some non convex bounded domains U in r n and proved that it represented an analytic submanifold of U. Moreover, in [37], we studied the fixed points theorem of holomorphic mappings f of a bounded Riemannian manifold M and proved that the set of all fixed points of f is a complex submanifold of M. D e f i n i t i o n 23 Let X and Y be vector spaces and T " K --. Y satisfies T ( s x + ty) = s T ( x ) + t T ( y ) whenever x, y e K , 0 < s, t. Then T is said to be a j ~ n e . Let us recall that a point z E A is said to be a p-extreme point of A if z is not an internal point of the arc segment A y for x, y E A. That is, if z - s x + ty, x , y E A, 0 < s , t , s p + t p - l t h e n x - y [ = ( s + t ) - l z ] . Ep(A) denotes the set of p-extreme points of A, see the preceding part. D e f i n i t i o n 24 Let X be a vector space and K C X . A set G is said to be a n e q u i c o n t i n u o u s g r o u p of maps of a set K if it takes K into K , and G is a group. That is, every T C G is a 1-1 map of K into K whose inverse T -1 also belongs to G (so T maps K onto K ) and that T1T2 E G whenever T1, T2 E G. Here (T1T2)x = T1 (T2x). We note that the composition of two affine maps is affine. Also, by equicontinuity we mean that, to every neighborhood W of 0 in X corresponds a neighborhood V of O in X such that T x - T y E W whenever x E K , y C K , x-yCV andTCG.
3.2.1
Generalized
Brouwers's
Fixed Point Theorem
The infinite-dimensional version of Brouwer's theorem, concerning the fixed point property of closed balls in ~ , of convex sets of normed spaces has been given by Schauder [188] and of locally convex spaces by Tychnoff [205].
64
CHAPTER
3 FIXED-POINT
AND
P-EXTREME
POINT
In what follows, we generalize their fixed point theorems to p - c o n v e x s e t s in l o c a l l y p - c o n v e x F - s p a c e s , need not be locally convex, (1 > p >
0). The problem is nonlinear and involves Minkowsky p-functionals. T h e o r e m 34 ( G e n e r a l i z e d B r o u w e r ' s theorem)[28](2000). If K is a nonempty compact p-convex set in a locally p-convex F-space X , and f " K ~ K is continuous, then f ( a ) - a for some a E K. P r o o f . C o n t r a r y to the conclusion, we assume f fixes no point of K. T h e n its graph G - {(x, f ( x ) ) E X • X; x E K } (3.27) is disjoint from the diagonal A of X x X and is compact. Hence there is an absolutely p-convex neighborhood V in X such t h a t G + (V x V) does not intersect A. In particular we get
f(x) ~ x + V
(x E K ) .
(3.28)
Let qv be p-Minkowsky functional of V. Now qv is continuous on X and q v ( x ) < 1 if and only if x E V, see [207]. Define a ( x ) -- max{0, 1 - q v ( x ) } Choose X l , . . . ,
Xn E
K
(x E x).
(3.29)
so t h a t the sets xi + V (1 _ i _< n) cover K, i.e. K C U~xi + V.
P u t a i ( z ) - c ~ ( x - xi)
and define
I OZi(X) I lip ~i(X) -- O~1(X) ~- : : :--nc tin(X)
(X E K, 1 < i
0 such t h a t B(a, r) C U and IIAxll 1/q 0 such t h a t B(a,r) C U, and if x e B(a,r), IIAxlI1/q< e i l x - all 1/p. Hence (4.20) becomes ]Ix - a - A - l ( f ( x ) - f(a))]] lip < c llx - all lip for x E B(a,r). Now choose a > 0 such t h a t B ( b , a ) C U and g(B(b, 5 ) ) C T h e n if y E B(b, 5) and x = g(y), using (4.19) we obtain
IIg(Y) - g(b) - A - l ( y - b)ll 1/p p>0,
(5.1)
f o r some A E (0, 1). I f p - 1, we have A b - [ a , b] and the classical m e a n value theorem for n o r m e d spaces is obtained.
P r o o f . Let p : ~ +
--~ E
be defined by 1
=
1
+
(1 -
(5.2)
so that ~(0) - a, ~(1) - b, and ~([0, 1]) - A b. Since U is open and ~p is continuous, there is a number ~ > 0 such that p ( - 5 , 1 + 5) C U. By the chain rule it follows that
5.1 M E A N - V A L U E
THEOREM
1
( f o ~)'()~) - D f ( b ) ~
IN REAL
i
1
)~-
1
+ (1 - A)~a)(
SPACES
91
i
(1 -- A); -1 b-
a).
P
(5.3)
P
If we apply the classical m e a n value t h e o r e m to ( f o ~), we deduce t h a t there exists A0 E (0, 1) such t h a t f(~;(1)) - f ( ~ ( 0 ) ) - ( f o ~;)'(A0)(1 - 0), and by
(5.3) we have,
f(b) - f(a)
-
i
i
b
i--1
D f ( A ~ b + (I - A 0 ) ~ a ) ( ~ / ~
- - ( 1 - )~0) 1-1)
(5.4)
_. . . . Df(c)()~o/P-1 .. b _ (I - ,k0) i - 1 a)
P 1
(5.5)
P
1
where c - ) ~ b + (1 - A0)~a E A b C U. If we p u t p - 1, we obtain A b - [a, b], and the classical mean-value t h e o r e m in n o r m e d spaces is o b t a i n e d . T h i s completes the p r o o f , i We shall now see t h a t the locally b o u n d e d spaces E which are not locally convex have balls BE satisfying the hypotheses of the preceding T h e o r e m . Let us recall t h a t , a subset U of E is said to be a b s o l u t e l y p - c o n v e x if for e v e r y x , y E U and for every #1, #2 E ~ with [#11p + I # 2 1 p O. Since (Df~) converges uniformly on B(r ,a), there exists no > 0 such that for rn, n > no, we have IIDfn(Y) - Df~(Y)[I < eq for every y E B(a, r), moreover, [ I g ( a ) - D fn(a)l ] < eq (5.14) If we let m ---+ cx) in (5.11), we obtain
I l l ( x ) - f(a) - [ f n ( x ) - A(a)]ll
_ p > O. A
THEOREM
Chapter 6
HIGHER Q U A S I - D I F F E R E N T I A L IN F-SPACES In this part we introduce the concept of higher order quasi-differentials for maps between locally bounded F-spaces, and prove an analogy to the theorem, which states that 9 "All h i g h e r quasi-differentials can be c o n s i d e r e d as s y m m e t r i c m u l t i l i n e a r m a p s ". We shall also introduce the Ggteaux differential and study its relationship with the Quasi-differential.
6.1
SCHWARTZ
SYMMETRIC
THEOREM
S e c o n d Differential Let E and F be p-normed and q-normed spaces respectively (0 < p, q _< 1), U an open subset of E and f " U --~ F quasi-differentiable. Then
D f " U ---+L ( E , F). Since L ( E , F) is a q-normed space, let us define the second differential of f.
D e f i n i t i o n 31 Assume that f 9 U ~ F is quasi- differentiable on U. The function f is said to be twice quasi- d i f f e r e n t i a b l e (or quasid i f f e r e n t i a b l e o f order two) at a C U ( or on U ) when the mapping I01
102CHAPTER 6 HIGHER QUASI-DIFFERENTIAL
IN F-SPACES
D f : U --, L(E; F) is quasi-differentiable at a (or on U ), respectively. In this case D(DF)(a) e L(E; L(E; F)) is called the second quasi-differential of f
on U and is denoted by
DZ f 9U ~ L(E; L(E; F)).
This concept of second quasi -differential, obtained by a simple repetition of differentiation of order one, has the apparent disadvantage that the values of the second quasi differentials belongs to the q-normed space L ( E ; L ( E ; F ) ) . In this form this space appears complicated, since it is a q-normed space of continuous linear maps. However, because of its natural isomorphism to L(2E; F) which follows from Theorem 21,(Ch.2, subsection 2.1.4, p.34). So we are able to treat the concept of the second quasidifferentials as follows : Consider D2 f 9U ~ L(E; L(E; F)) and denote by
d2 f (a) e L(2E; F) the element of L(2E; F) that corresponds to D2f by the natural isometric isomorphism given in Theorem 21, we note that the relation between D2f(a) and d2f(a)is characterized by
d2 f(a)(s,t) -
D2 f(a)(s)(t)
for any s, t C E. Therefore, the second quasi-differential
D2 f(a) 9A --, L(E; L(E; F))" corresponds to the second quasi-differential
d2 f 9A --~ L(2E; F).
Thus, quasi-differential of order two now takes values in the q-normed space L(2E; F). We recall that this is done by identifying L(E; L(E; F)) with L(2E; F), the space of all c o n t i n u o u s b i l i n e a r m a p p i n g s of E x E into F, which appears less complicated.
6.1 S C H W A R T Z S Y M M E T R I C
THEOREM
103
R e m a r k 12 In the case of quasi-differentials of order one, we also use the notation df (a) - D f (a), df - D f . We point out here that: without assuming that f is quasi-differentiable on the entire subset, we can say more generally that f is twice quasid i f f e r e n t i a b l e at the point a C V if : 1 - f is quasi-differentiable on a neighborhood of V of a. 2- The mapping D f : V --~ L(E; F) is quasi-differentiable at the point a.
Schwartz S y m m e t r i c T h e o r e m of T w i c e Quasi-Differentiable M a p s In elementary calculus, it is known that if U C ~2, f . U ~ ~, a C U, 02f 6yO~ 02f are continuous on U, then and the partial derivatives O~Oy,
02f
02f
OxOy
OyOx
We will derive in this section a generalization of this symmetric theorem for twice quasi-differentiable functions on locally bounded F-spaces. This can be obtained from the following generalized theorem for normed spaces, (see Chae [69]).
T h e o r e m 53 ( G e n e r a l i z e d S c h w a r t ' s s y m m e t r i c theorem)J29](2003) Let f " U --~ ~ be twice quasi differentiable at a E U. Then d2 f(a) is a symmetric bilinear map. That is, for all k, h E E, d2 f ( a ) ( h , k) -
d2 f ( a ) ( k , h).
(6.2)
P r o o f . Since f is twice quasi-differentiable at a, there is some r > 0 such that B(a, r) C U and f is quasi-differentiable on B(a, r). Consider the function
A (h, k) - f ( a + h + k) - f ( a + h) - f ( a + k) + f ( a ) whereh, kcE are such that a + h + k , a + h , a + k belong t o U . precisely, we may assume [[ h [[_< r / 2 [[ k I[_< r/2. We note that
More
(h, k) - a (k, h) We wish by a suitable process to approximate d2f(a)(h, k) with A (h, k) to show that d2f(a) is symmetrtic and to obtain the desired equality
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2 and f o r any fixed x2, ...Xm E E, if g" U ~ F is defined by g(x) -- d m - l f ( x ) ( x 2 , . . . , x r a ) ,
110CHAPTER 6 HIGHER QUASI-DIFFERENTIAL
IN F-SPACES
then g is quasi-differentiable at a and dg(a)(x) -
d m f ( a ) ( x , x2, ..., x,~)
for all x E E. P r o o f . The m a p g:U --~ F satisfies the composition
g - )~or 1oD m - 1f. where
r
L ( E m - 1 ; F) --* L m-1 (E; F )
is the natural isometric isomorphism and A" L m - I ( E ; F) --* F is given by A(A) - A(x2, ..., Xm)
for all
A E L m - l ( E ; E).
is clearly continuous and linear. Now
Dg-
n(s
- s162
- ~or
Hence, for all x E E, we have
dg(a)(x)
-
D g ( a ) ( x ) - ()~or
f(a))(x) - A(r
f(a)(x)))
= A ( d m f ( a ) ( x ) ) - d m f ( a ) ( x , x2, ...,xm). which is what set out to prove, m T h e o r e m 56 ( G e n e r a l i z e d S c h w a r z T h e o r e m ) (2002) Let f " U --~ F be m-quasi-differentiable at a E U. Then d m f (a) is a symmetric m-linear map; i.e.
dm f (a) E Ls(mE; F). P r o o f . For m - 2, this was proved in Theorem 53 (section 6.1,p.103). We now proceed by induction on m for m >_ 3. Assume t h a t the theorem has been proved for m - 1. Then d m f ( a ) is the quasi-differential of the m a p p i n g d i n - i f , which by our assumption exists in a neighborhood V of a. By the induction hypothesis, we can assume d i n - i f 9V --~ Ls(mE; F).
F
By the l e m m a 11, p.109, for fixed x3, ..., xm C E, the mapping defined by g(x) -- dm-2 f ( x ) ( x 3 , ..., Xm)
g" V
6.4 D I R E C T I O N A L
DERIVATIVES
111
is quasi-differentiable at a and its quasi-differential at a is given by
dg(x)(y) - dm-l f(x)(y, x3,...,Xm) for all x E V and for all y E E. Similarly, applying the lemma to the map X --+ d m - l f ( x ) ( x 2 ,
..., xm)
where x2,..., Xm are fixed, we obtain
d2g(a)(x2,xl) - dm f ( x ) ( x l , x 2 , ...,Xm) for all X 1 E E .
(6.13)
From (section 6.6, p.103 ) we have
d2g(a) (Xl, X2) -- d2g(a)(x2, Xl) and hence
dmf(a)(x2,xl,x3, ...,Xm) = dmf(a)(xl, ...,Xm). But din-if(a) is symmetric and from (6.13), for Xl E E, the map
(6.14)
xl,...,Xm are fixed but arbitrary, so
dmf(a)(xl) : (x2,...,xm)--+ dmf(a)(xl,...,x,~)
(6.15)
is symmetric. This shows that the multilinear map dmf(a) : E m --+ F is symmetric by(6.14) and (6.15), so the proof is complete. I
6.4
DIRECTIONAL
Definition 34
DERIVATIVES
(Gdteaux Differential)Let U
space E. If for a given a E U and h E E, Of
be open in a p-normed the limit
[ f (a + Ah) - f (a) ]
(616)
exists, where A E K, then f is said to be "Gateaux differentiable at a in the direction of h "; Sometimes we write Of(h,a) instead of ~ (a) " Also ~h(a) is called the G a t e a u x derivative o f f at a in the Oh direction h.
112CHAPTER 6 HIGHER QUASI-DIFFERENTIAL
IN F-SPACES
D e f i n i t i o n 35 If f is Gdteaux differentiable at a in any direction, we say that f is " G d t e a u x d i f f e r e n t i a b l e at a " and the mapping of o f ( ~ ) . h e E -~ ~ ( a )
(6.~7)
e F
is called the " G d t e a u x d i f f e r e n t i a b l e " o r G d t e a u x d e r i v a t i v e .
D e f i n i t i o n 36 I f f is Gdteaux differentiable at each point of direction h, the mapping of
~ u ~
Oh
is called the" d e r i v a t i v e o f f o n
U
of ~(x)
U
in the
(6.18)
~ F
i n the d i r e c t i o n h ".
E x a m p l e 19 We give examples which show that G~teaux derivative is not in general a linear map, and that f may have G~teaux derivative without being continuous. Let us first start with this example of a directional derivative in its simple form. (1)
Let f : K n ~ K
and ej = ( 0 , . . . , 0 , 1 , 0 , . . . 0 ) , j = l , . . . n .
Then
n X--(Xl,...,Xn)--~-'~xje.j 1 and Of
Of
o%-7(x)- ~ (x) which is the partial derivative of f with respect to i th coordinate. the partial derivative is the derivative in the direction ej. (2) Let
Thus
f "K 2 ---, K be defined by
I XlX~ f (x) -
(~+;~) , x # o,
0
,
x-O
Then ~ (0) - f ( h ) , which shows that the G~teaux derivative may not be a linear map. (3) Let f "K 2 --* K
be defined by
d ~r f(x) -
~'
0,
.
x-O
6.4 D I R E C T I O N A L
DERIVATIVES
113
Then f is not continuous. However O0-~h(0) - 0 for all h E K 2. That is, f is Gdteaux differentiable at 0 but f is not continuous at 0. G a t e a u x Differentials. We have shown that G~teaux differentials may not be linear and G~teaux differentiability does not necessarily imply the continuity of the function. We now show that quasi-differentiability is stronger than G~teaux differentiability (or directional differentiability ).
T h e o r e m 57 [29](2002) Let E and F be p-normed and q-normed spaces respectively. If
f :E---~F is quasi-differentiable at a E E, and
then
of
Oh (a) - D f ( a ) h ,
f
is G~teaux differentiable at a
i.e. O f ( a ) - D f ( a ) .
(6.19)
P r o o f . We have
lim II f ( a + x) - f(a)
-
x ---*O
D f ( a ) x II1/q /II x II~/p= o
Replace x by Ah to get the desired result. In fact lim~_~0 II f ( a + A h ) - f ( a ) -
1 II h II1/~
~%II
Df(a)(Ah)111/q /II Ah
f ( a + Ah) - f(a) ~
ADf(a)(h) ill~q_ 0; -
~
i.e., lira [ f ( a + Ah) - f(a)] _ D f ( a ) ( h ) ,~---.0 ,X Hence
m
II~/p
Of Oh (a) - D f ( a ) h .
-
114CHAPTER 6 HIGHER QUASI-DIFFERENTIAL
IN F-SPACES
Corollary 7 Let E and F be p-normed and q-normed spaces respectively, (0 < p , q _ < 1). If f " U ~ F is quasi-differentiable at the point a C U, then df(a + Ah)
d~
Of
I~,--o=~
(a)
(6.20)
=Of(h,a)
That is, the usual derivative of f ( a + A h ) with respect to A at A - O is equal to the G~teaux derivative at a in the direction h. I Proof. Define g" V--, F by g ( A ) - f ( a + A h ) where V is a neighborhood of 0 in K. So by the chain rule we obtain the derivative g' (0) - D f (a)h,
since g ' ( 0 ) - lim~__,0 [g(~)-g(0)]x . Hence, g'(0)- li~ [f(a+ Ah)-f(a)l
_ -~ Of ( a ) - D f ( a ) ( h )
I
T h e o r e m 58 ( C o m m u t a t i v i t y
of directional derivatives)(2002) Let E and F be p-normed and q-normed spaces respectively, and U C E open, (0 < p,q < 1). Let f : U ~ F be twice quasi-differentiable on U. Then for any h, k E E,
of " U ~ F Ok
is quasi-differentiable on U such that for any x C U 02 f d2 OhOk (X) f(x)(h,k). Hence 02 f
02
OhOk (~) - OkOh(x) f Proof. The mapping
of Ok
9x E U ~ D f ( x ) ( k ) ,
(6.21)
S
6.5 Q U A S I A N D F R E C H E T D I F F E R E N T I A L S
115
is the composition of the mappings
x e U ~ D f ( x ) e L(E; F), and
A e L ( E ; F ) ~ A(x) e F Thus
is quasi-differentiable, and its quasi-differential at x is given
by
of
o2f
D(--~-~(x))(h) - OhOk (x). We have then
d2 f ( x ) ( h , k ) _
02 f OhOk (x).
Since d2f(x) is symmetric by Theorem 53 (p.103), this establishes the commutativity of the directional derivatives, m R e m a r k 13 The above Theorem (and hence Schwartz theorem ) generalizes the rule of the classical differential calculus, where for some w - f ( x , y ) , we have
i)2 f OyOx
02 f OxOy
d
6.5
QUASI AND FRECHET DIFFERENTIALS
This part is devoted to studying the relationship between the concepts of Quasi-Differentiability and Fr~chet-Differentiability. These will be called Qdifferentiability and F-Differentiability, and their classes will be denoted by QD and FD, respectively. We have seen, in the preceding part, that quasi-differentiability is a continuous linear mapping on spaces of maps between locally bounded Fspaces. This is similar to the situation for the Fr~chet differentiabilty and normed spaces. That is, both are continuous and satisfy
D ~ ( f + g) - D ~ f + Dag,
Da(,X f) - ,X(D~f)
whenever f, g are simultaenously differentiable in any sense.
116CHAPTER
6 HIGHER QUASI-DIFFERENTIAL
IN F-SPACES
However, our goal here is to prove that: " T h e t w o c o n c e p t s of diff e r e n t i a b i l i t y are t o t a l l y different ". But the new one, t h a t is the Q-differentiability, is more suitable for all F-spaces which are locally convex or not. It may play a role in analysis and applied mathematics, where the classical Fr6chet differentiability can not be used.
6.5.1
Finite Dimensional
Case
The following lemma shows the equivalence between the Q-differentiability and G-differentiability in some finite dimensional spaces.
L e m m a 12 Let E and F be any two finite dimensional p-normed spaces and f be a map between them. Then f is Q-differentiable if and only if f is F-differentiable. P r o o f . Note t h a t lim IIf(x) - f ( a ) - T~(x - a)lll/p _ 0 ~-~ IIx - all lip
(6.22)
if and only if lim ilf(x) - f ( a ) - T~(x - a)l [ = 0
IIx--alf for a continuous linear mapping Ta E L ( E , F), where a , x C E. We know that finite dimensional topologies are equivalent, m The following theorem shows that we still have the linear transformation, defined by the Jacobian matrix, to find out quasi-differentials when we deal with finite dimensionM spaces. T h e o r e m 59 [29](2002) Let E and F be any two finite dimensional p-normed and q-normed spaces with dimensions n and m respectively, (1 > p,q > 0). Let U C E be open and f 9U ---+~ m a function defined by
f (a) - (fl (a), ..., fro(a)), where If of~ ox~ (a) D f(a) matrix
fi " U - - + ~ , ( 1 < i < m ) . f is quasi-differentiable at aC U, each of the partial derivatives exists, (1 < i < m, 1 < j < n). Furthermore, the quasi-differential
9~n ___+~ m is the linear transformation defined by the Jacobian of f at a. II
6.5 QUASI A N D F R ~ C H E T DIFFERENTIALS P r o o f . Let L - D f (a) 9j ~ n ~ s t a n d a r d basis for ~ n and ~m. f at a, so t h a t
Let Assume
117
el,...,en and e l , . . . , e m b e the (aij) is the Jacobian m a t r i x of
~m.
m
Lej - ~
aijei,
l<j O. Notice that the dual space of /if(I) equals {0}. Therefore, the function G is not quasi-differentiable at f - 0. However, we notice that LI(0, 1) C Lll2(0, 1).
(6.27)
For if f E L1 (0, 1), then
j/O1 [1 f [1/2 . l i d
x < ( ~I [ f l dx)l/2.(
/i
1
ldx)l/2;
120CHAPTER 6 HIGHER QUASI-DIFFERENTIAL
IN F-SPACES
t h a t is, f E LlZ2(0, 1). Therefore w e consider our function f in L I ( 0 , 1). We claim t h a t G(f ) = Ilfll is a Fr6chet differentiable function at 0. Notice t h a t To(x) C (L1) ', the dual space which is isomorphic to Lo~. Now lim
Ila(f)
- G(o) -
f~o
To(f -
0)[ I =
lim I a ( f ) To(f)[ X-~0 IlfllL~ -
[If - OIIL~
lim I Ilfll- T o ( f ) f ~O IlfllL1
=
l _ 0
0 in L 1, where To(f) E (L1) ' -~ L c~ may be taken as equal to I l f l l - f0~ I f dx, since f is considered in L 1 9 This completes the proof of the theorem, m as f ~
[1/2
In what follows we prove t h a t the class of Quasi-differentiable maps may be not contained in the class of Fr~chet-differentiable maps.
T h e o r e m 61 [33](2002) Not every Quasi-differentiable map f from a p-normed space E into a q-normed space F is Fr~chet-differentiable, (1 > p, q > 0), that is
QD(E; F) ~ FD(E; F).
(6.28)
P r o o f . As in the proof of Theorem 60, we take E = LP(I), the space of integrable functions on I = [0, 1], F = K , with the q-norm, and f = 0, the zero function in LP(I). Consider the function G : LP(I) ---,K given by
G(f) - I l f l l -
I f Ip dx.
(6.29)
'O
We have seen t h a t lim
f---,o
I I G ( f ) - G ( 0 ) - 0)ll 1/q
Ilf - oil 1/p
Ilfll ~/q
= lim
f~o Ilflll/p
1 f~o ]lflll/p-1/q
= lim
Now if q1 p1 > 0, the limit will approach zero and G(f) will be a quasi-differentiable function. But if p = q we have non-zero limit since 1
lim f--,0 So G(f)
1
Ilfll~/Itfll~
- 1.
will never be a Fr6chet-differentiable function, m
(6.30)
6.5 Q U A S I A N D F R E C H E T
121
DIFFERENTIALS
R e m a r k 15 The relation between the calsses of Q-differentiability and F-
differentiablity, proved so far, may be summarized by the following diagram Gateaux differentiable ~ .
Qu
-
.
.
.
9
9
ii1
C~F d,
Figure 2: R e l a t i o n between the different calsses of differentiability's.
This Page Intentionally Left Blank
Chapter 7
QUASI-HOLOMORPHIC MAPS The subject of this chapter is the holomorphic mappings of metrizable spaces, in particular locally bounded F-spaces which are not necessarily locally convex. We first review the basic concepts of complex variables with values in Let f 9t2 --~ (~ be a function defined in an open subset ~ of the field (Y of a complex variable with values in (~. We say f is holomorphic if for every point a in ~ there is a convergent series such that oo
f ( a + t) -- ~
ck(a) t k o
holds for all complex numbers t of sufficiently small modulus. This is one of several possible definitions and it is suitable for generalization to vector spaces. D e f i n i t i o n 37 Let open set in E. Then
E
be a topological vector space over (~ and t2
is said to be G d t e a u x a n a l y t i c convergent series such that
if for any a C ~
Oo
f (a + tb) - ~
ck(a, b) t k 0
123
and b E E
an
there is a
124
CHAPTER
7 QUASI-HOLOMORPHIC
MAPS
for It I small. If E is of finite dimension, it is a classical result that such a function is already continuous, in fact it is infinitely differentiable. If E is of infinite dimension it is desirable to impose some kind of regularity: It is convenient to require f to be continuous. See example1, p. 3, of a non continuous linear mapping. Thus we have D e f i n i t i o n 38 We say that f : ~ -.(~, C E open in an infinite dimensiononal space, is h o l o m o r p h i c if f is Gdteaux analytic and continuous. The theory of holomorphic functions on topological spaces has developed rapidly in locally convex spaces in general during the last 50 years. However it is totally neglected for non locally convex spaces till our thesis [11] was written in 1979 containing the solution of the Levi problem in some of such spaces to prove that : T h e r e is s o m e t y p e o f r i c h n e s s to the class o f h o l o m o r p h i c f u n c t i o n s o n s u c h spaces. Therefore we proceed to solve and describe some of the basic problems and the new concepts in the field to which we have consecrated our book.
7.1
FINITE EXPANSIONS FORMULA
AND TAYLOR'S
This chapter is devoted to extend the Taylor~s T h e o r e m to maps between locally bounded F-spaces. It says that : " I f a f u n c t i o n f is m - q u a s i - d i f f e r e n t i a b l e ~ t h e n f m a y b e a p p r o x i m a t e d locally by a p o l y n o m i a l of d e g r e e m". Let us first introduce the following concept. D e f i n i t i o n 39 ( m - q u a s i t a n g e n c y ) L e t E and F be p-normed and qnormed spaces (0 < p, q ~_ 1) respectively and U a nonempty open subset of E. Two functions f,g:U-.F are said to be m - q u a s i - t a n g e n t (or m - ( p q ) - t a n g e n t ) aEU if lira I] f ( x ) - g(x) ]Ip = O. x---~a ]] X - a ]]mq
at a f i x e d p o i n t
7.1 F I N I T E E X P A N S I O N S That is, then
f o r every e > 0
AND TAYLOR~S FORMULA
there e x i s t s 5 > 0
]1 f ( x ) - g ( x )
such that if, II x - a
]]P< e ] i x - a l l
125 II< 5,
mq .
In particular, for m - 1, we regain the definition of quasi-tangency at a point a which was introduced in chapter 4. As in part X we can easily show the following properties 9 If f and g are m - q u a s i - t a n g e n t at a, then, (i) ( f - g) is continuous at a and f (a) = g (a). (ii) f and g are ( m - 1 ) quasi-tangent at a. (iii) ( f - g) is m-quasi-tangent to 0 at a. (iv) m - q u a s i - t a n g e n c y is an e q u i v a l e n c e r e l a t i o n on the vector space of all m a p p i n g s from U to F which are continuous at a. (v) The notion of m-quasi-tangency depends only on the topologies of E and F, not on the F - n o r m s used to induce the topologies.
7.1.1
Finite Expansion
If f 9U --~ F is quasi-differentiable at a point a E U, f is 1-quasi-tangent to an atone linear m a p (i.e. a polynomial of degree 0
such t h a t
II x - a II< 5 ~11 f (x) - f (a) - f (a) (x - a) lip O, there exists 5 > 0 with B (0, 5) C U such that for any m elements x l , . . . , Xm in U with II xl II1/p + . . . + II Xm Ill~P< 5, the polarization Cm ( f ) of f with respect to Xl , . . . xm satisfies the inequality
II r
~ (11 Xl
II lip §
§
II x~ II~/P) ~/p
7.1 F I N I T E
EXPANSIONS
AND
TAYLOR'S
FORMULA
127
P r o o f . It suffices to show, by the definition of Cm (f) (see p.44), t h a t + emXm)IIlZq< e (il X~ II + ' ' " + II Xr~ li) "~zp
II f (ClXl + . . .
for all ek - +1, k - 1, ..., m. to 0 at the origin. In fact, lim il/(~)-~ ~--,o II~llr~/p
But this is obvious, since f is m - q u a s i - t a n g e n t
= 0
implies t h a t given e > 0
we get for small
v~]u~ of ~, II f (x)lll/~< ~ II x IIm/~. Thus It f(s
+--.
nc s
1/q~_ s
( II Xl II1/p + . . .
)
m
II Xm II lip
~ .
I is a polynomial of degree < m m-quasi-tangent to 0 at a point a C E, then P - O.
Lemma
Proof.
14 If P E P ( E , F )
If P
and if P
is
is m - q u a s i - t a n g e n t to 0 at a point a, i.e. limll P (x) HI]/q /II x -
a lim/P= O,
x---->a
then the polynomial Pa defined by P~ (x) - P (x + is m - q u a s i - t a n g e n t to 0 IIP(~+~)ll alzllm/p~/~. Therefore,
~)
at the origin. Notice t h a t lim
IIP(x)ll~/q - l i m
x---,a iix--all mlp
it is suffices to consider the case where a
--x~O
is the
origin. Let P - Po + P1 + . . . + P,~. T h e n for Am C Ls (mE; F ) , P,~ - -Amwe have from L e m m a 13 and relation (7.3) above t h a t for any e > 0, there exists > 0 such t h a t
II Am (Xl...,Xm) II1/q< _ e
if II
(11 x~
II1/p + . . . +
II xm II~/P)"~/q
(7.4)
]11/p
Zl -~-''" -~- II Xm Ill/p< 6. T h e preceding inequality (7.4) holds for any x, ..,x,~ m-linear map. In fact, we have
II Am (Axl...,Axm)II
~/q- I~Xl"~/q II A ~
since Am is an
(xl...,x~)II 1/q .
Since the inequality (7.4) is valid for e > 0 and for any Xl ~ " ' ~ Xm conclude t h a t A m (Xl,..., X m ) -- 0
we
CHAPTER
128
7 QUASI-HOLOMORPHIC
MAPS
for all Xl..., Xm. Thus Pm = 0. Consequently, the degree of P is less than m. Since P is m-quasi-tangent to 0 at the origin, P is also ( m - 1)quasi-tangent to 0 at the origin. Repeating the above argument, we have P,~-I = 0. Thus by induction we obtain Po = P1... = Pm = O. m
T h e o r e m 62
Let f : U---, F and a E U. If P1 and P2 f at a, then =
are two "m-expansions "of
P2.
P r o o f . Let P = P1 - P 2 . We claim that P = 0. Since both P1 and P2 are m-expansions of f, it follows that P is m-quasi-tangent to 0 at the point a. By Lemma 14, we conclude that P = 0. li 7.1.2
Taylor's Formula
Let E and F be p-normed and q-normed spaces respectively (0 < p, q _< 1) and f : U ~ F, where U is an open subset o r E . If f is m-quasidifferentiable at a C U, then it was shown that
dk f (a) E Ls (kE; F ) for k - 0 , 1 . . . , m . Hence we can associate dkf(a) the u n i q u e k hom o g e n e o u s p o l y n o m i a l which we will denote by dkf (a) (See subsection
2.2.3). Thus
dk f (a) E P (kE; F) and
dkf (a) (x) - dk f (a)x k. As dkf (a) is a k-homogeneous polynomial, the mapping xEE~dkf(a)(x-a) is a p o l y n o m i a l of d e g r e e k.
cF
7.1 F I N I T E E X P A N S I O N S A N D TAYLOR'S F O R M U L A
Definition 41 By a "Taylor p o l y n o m i a l " Tm,l,a a, we mean the polynomial defined by
of order m
129 of f
at
m
Tm,l,a (x) -- ~
~.d f (a) (x - a)
k--o m
= ~
1
~ d k f ( a ) ( x - a) k
k--o
Example 22 If P C P (E, F) is of degree < m, then for every a C E P-
T~,p,~.
To see this we may assume without loss of generality that P is a homogeneous polynomial, since the general case follows from the homogeneous case by adding a finite number of homogeneous polynomials. Let P - ft, where A C Lsm (E; F). Then we have by the binomial formula and Theorem 5~, p. 108. m
P (x) -
ix-ol
A x m - A (a + (x - a) )m k--o m
1
= ~
k
~ d f ( a ) ( x - a) k
-
Tin,p,
a
k=o
which shows that P -
7.1.3
Tm,P,a for any a E E.
Q u a s i - D i f f e r e n t i a l of Taylor P o l y n o m i a l s
If f - U --+ F is m-quasi-differentiable at a point a C U, then df (a) e L (E,F)
and d m - 1 (dr) (a) E L
(m-lE; L (E; F ) ) .
On the other hand, dm f (a) e n (mE; F) .
Therefore we do not have d m f - d m-1 (df) unlike D m f - D m-1 ( O F ) . However, the map r n (mE; F) --+ n (m-lE; L (E; F)) defined by r (A) (Xl,..., Xm-1) (Xm)
-
-
A (xl,..., Xm)
CHAPTER
130
7 QUASI-HOLOMORPHIC
MAPS
is an isometric isomorphism. Under this correspondence, we can identify d m f (a) with d m - 1 (df) (a) with respect to r Then
d m-1 (dr)(a)(xl,..., Xm-1) (Xm)
dmf (a)(Xl,... , Xm)
--
and in particular,
d m-~ ( d r ) ( a ) ( x - a) m-~ (x - a) - d m f ( a ) ( x - a) m .
We have also from Theorem 45 (section 4.1.4, p.84)
d ( d k f (a) ( x - a ) k) = k d k f (a) ( x - a ) k-1 The following theorem gives the q u a s i - d i f f e r e n t i a t i o n of t h e T a y l o r p o l y n o m i a l of o r d e r m of f at a and the f o r m of t h a t o r d e r (m-l) of df a t a.
T h e o r e m 63
Iff
is quasi-differentiable at a, then we have
U--,F
dTm,f,a - Tm-:,df,a. Proof. m 1
m 1
dTm,/,a - d(~-'~ ~.dk f (a) (x - a) k) -- E k=o
m
1 k--1
m-1
= ~
-
a)
k--o
m =
(a))(x
~.d (dk f
d k f ( a ) ( x - a) k-1 -- ~ "
ld k -~. ( d f ) ( a ) ( x -
1
~(k -d 1)w k-1 ( d r ) ( a ) ( x - a) k-1
k = l
a) k
__ T m - l , d f , a
"
(X).
k--o
7.1.4
General Mean-Value
Theorem
For a given function f : U --* F, it is not certain that f admits an mexpansion at a point a in U. However, if f is m - q u a s i - d i f f e r e n t i a b l e , we will see t h a t f a d m i t s a n m - e x p a n s i o n . This result is the so called Taylor~s t h e o r e m . But we first need the following generalization of the Mean-Value theorem (Theorem 50, p.94) for Taylor's theorem.
7.1 F I N I T E
EXPANSIONS
AND
TAYLOR'S
FORMULA
131
T h e o r e m 6 4 (Generalized Mean- Value Theorem)J29](2002) Let U be an open subset of a p - n o r m e d space E such that U contains the arc segment A b and F is q-normed. I f
f "U--,F is quasi-differentiable and p ' [ 0 , 1 ] - - ~
is differentiable such that
II D f [)~l/pb + ( 1 - ) k ) 1/p a] I1 0, let X following condition-
II f (t) - f (a)/ll/q 0 such that
II b - c I1> ~, B (c, ~) c U and if t C t3 (c, 6), II f (t) - f (c) - D f (c) (t - c) IIlZq< E lit
-
c II1/p/2
or
II f (t) - f
(c) Ill~q
0 with such t h a t i f l A t - ) ~ c l < r l ,
rl 0 bea given real number. Since L > K, there exists an infinite subset NK of the natural numbers such t h a t 1
IlPmll~
> K
Vm r NK.
Now let r ~1 . We want to show t h a t the series does not converge uniformly on B - (a, r). This implies t h a t p - 0. Using of the q-norm IIP,~II, it is easy to find Ym e E , I l y m l l - 1 such that [IPm(ym)ll > K mq for each m E NK. Let X m - a + r y m . It follows that
IIPm(Xm-
> rmq(r1)mq
a)]] - rmqllPm(ym)l I
m
for m E N K
_ l
This proves t h a t the power series does not converge uniformly on B - ( a , r).
(2)Assumethat Let
0 0 ande-
1 Then there exists N > 0 such t h a t 2rq/v" IIP.~II _< e "~
If x E B - ( a , r ) ,
,for m >_ N.
we have
IIP,~(x- a)ll N.
Notice t h a t IlP,~(x)ll _ IIP~ll.llxll ~q/p for p-homogenous polynomial (see section 2.2.4.). This shows t h a t the power series converges uniformly on B - ( a , r). Since r > 0 is arbitrary, we conclude t h a t p - oe. / C o r o l l a r y 10 I f p is the radius of uniform convergence for a) and O < r < p, then
E~%0Pro(x-
oo
IIP~ll ~ / ~
< o~
(7.14)
rn--0
P r o o f . As in the case (2) of the preceding proof, apply the root test with am -I[Pmll r mq/p t o obtain the result, m
CHAPTER
142
7 QUASI-HOLOMORPHIC
MAPS
F is a q-Banach space, the following statements are equivalent 9 (a) E m ~ = 0 P m ( x - a) converges uniformly on B - ( a , r ) , r > O.
C o r o l l a r y 11 If
1
(b) lira sup IIPmll~ < oo. (c) The sequence ( IlPmll -~ ) is bounded, (d) There exists co > O and c > O such that lIPmll ~ coc mq/p
liP011 ~
(In (d), to avoid
V m - o, 1, 2, ..
(7.15)
1, w~ u ~ the constant co)
P r o o f . We note that the implications (a)=~ (b) =~ (c) =~ (d) are clear. For ( d ) ~ ( a ) ' l e t r > 0 be such that r c o < l . Then OO
oo
I I P m ( x - a)ll _< ~
co(cr) mq/p < c~
(7.16)
m--O
m--O
for x e B(a, r), which shows that the series converges normally on B - ( a , r). We used here the fact, IIPm(x)ll O. The series, if it exists, is unique by Theorem 72 (subsection 7.2.~, p.1~6).
152
CHAPTER 7 QUASI-HOLOMORPHIC MAPS
It deserves to point out that if f has a Taylor Series expansion (or representation) at a, then the function f is infinitely quasi-differentiable in a neighborhood of a as shown in Theorem 74(subsection7.2.7, p.149). Moreover we have the following relations:
Pm = --m!l~ f (a)
,
Am - -~dm
(a)
for m = 0, 1, 2 ..... Consequently, if f has a Taylor Series expansion at a, then 1
f (x) - ~
~
1
-~..dmf (a)(x - a) - ~
lrt-- O
m--
--~.dmf (a)(x - a) m O
in some neighborhood of a in U. Since the open ball of uniform convergence of the Taylor Series at a and the domain of this function f could be different, we usually write OO
f (x) _ ~
~1. dAmf (a) (x -- a ) - ~
1 --~..dmf (a)(x - a) m
?Tt"-- O
to indicate formally, the relationship between f and its Taylor Series expansion of a. D e f i n i t i o n 49 A function f : U --~ F
is said to be quasi-analytic(or Bayoumi analytic) on U if f has a Taylor Series expansion at each point a E U. If K = (~, a quasi-analytic mapping will be specially called quasiholomorphic(or Bayoumi holomorphic). It is clear that any linear combination of quasi-analytic mappings is quasi-analytic, and hence we have a v e c t o r s p a c e of all q u a s i - a n a l y t i c m a p p i n g s f r o m U t o F . This space will be denoted by QA (U; F ) . In caseK =(Y, we write QH (U; F) instead of QA (U, F ) . For simplicity, we denote QA (U) = QA (U;K) and QH(U) = QH(U,(T). Since QA ( U ; F ) is a vector subspace of Qc$, ( U , F ) , the space of infinitely q u a s i - d i f f e r e n t i a b l e m a p p i n g s from U to F, we have for f C QA (U; F) the following quasi-differentiable mappings.
din f : x e U ~ d ~ f (x) e L~ (mE; F)
(7.43)
d m f " x E U ~ dmf (x) e P (mE, F)
(7.44)
We also have from Ch.6(sec.6.2 ), the following quasi-differentiable operators
7.3 Q U A S I - A N A L Y T I C M A P S
153
rim: f E QA (U; F) ~ d ~ f E QA (U; L~ (mE; F))
~"
f E QA (U; F ) -~ ~ f
E QA (U; P (mE; F))
R e m a r k 18 (Local Property)Quasi-Analyticity is a local property in the following sense: If f E QA ( U ; F ) and V is a nonempty open subset of U; then f i y E QA (V, F) conversely if U - - U V~, where V is an open nonempty subset of aEI
oc
U and f : U --, F is such that for each a E I, fiv~ E QA ( V ~ , F ) , then f E QA(U; F). The following theorem gives a condition on a power series to be a Taylor series to its sum.
T h e o r e m 76 Let ~ Pm ( x - a) be a power series between locally bounded F-spaces whose radius of uniform convergence r is positive and f (x) be its sum for II x I1< r. Then f is quasi-analytic in B (a,r). Proof. This is a restatement of Theorem 74 (subsection 7.2.7, p.149). I 7.3.2
Principle of Quasi-Analytic Continuation
To solve the problem of q u a s i - a n a l y t i c c o n t i n u a t i o n means: " G i v e n a q u a s i - a n a l y t i c m a p f in a connected open set U a n d a connected open set V c o n t a i n i n g U, find a n a n a l y t i c m a p g o n V s u c h t h a t giu = f ". The following theorem guarantees that such a mapping g is u n i q u e if it exists. This is the principle of analytic continuation in its s t r o n g form.
T h e o r e m 77 ( s t r o n g form)[34] (2002) Let E and F be a p-normed and a q-normed spaces respectively, (0 p, q _ 2 does not hold. For example, let g -- ~ 2 , f (Xl, x2) -- Xl and g (Xl, X2) -- 0 Then f , g E QA (E) and f - g on { ( x l , x 2 ) ' X l 0}, but f # g. In general, if M is a vector subspace of E which is not dense in E, then we can find a bounded linear functional f satisfying I] f ]I- 1 and f - 0 on M , (see Ch.1, p.12).
7.3.3
Integral D o m a i n QA (U)
If f, g ~ QA (U) then the multiplication
f.g ~ QA (U) as it can be shown from multiplication of power series. Therefore, QA (U) is a ring.
7.3 Q U A S I - A N A L Y T I C
MAPS
155
T h e o r e m 79
QA (U) is an integral domain if and only if U is connected. P r o o f . It is clear that, if QA (U) is an integral domain then U is connected. Assume that U is connected and f , g C QA (U) are such t h a t f.g = O. If f ~ O. Then there exists a nonempty open subset V of U such t h a t f 5r 0 on V. Thus g must be zero on V. Since g is analytic, g must be zero identically on U by Theorem 77 on the strong form of analytic continuation. In what follows we give an organogram (schedule) which shows the generalized theorems in the book obtained by the author between 1979 (the year he has obtained his Ph.D.) and 2003 (the year of writing the book).
GenralizedTheoremsinourbook (1979.2003) ,,
CompleAnal x ysisi ,I
I
,
Schwatz Levip~oblem 'e~= i Symmetric Steinhaust Theorem Theorem I _L_
!
~vex ~es [/I | Separable T.V.S
,,
I
Boundingi Sets t ! ! I
t
!
I
I
,-
I
,
,
.os i1..msI .anva.ue,,
Lagrange IFund, an oe~,llF,~o,,s a, I ,n,eg~,i' i o-n~ MeanValue nterrTled~I Theorems ofCaculus Theorems[ V~ Thee
ied I We dy'
_
hn' lach
.
.
.
.
1
Realani l Cmplex ! =t~!
bdedSpaces~
....
....
............ 1.
.
.
I .
!
....
Krien . Milman Milman 1 Theorem
J
.
!
l e,s II; , |s, ' B~
,
FunctionalAnalysis
!
,,
]
.
1 ........
F-Spaces i S:~:s~ ! .actor ! Spaces I
Chapter 8
NEW VERSIONS THEOREMS
OF MAIN
This chapter is devoted to v e c t o r - v a l u e d continuous or quasi h o l o m o r phic m a p p i n g s . The reader should be familiar with the two equivalent ways of finding a holomorphic mapping, the one based on complex q u a s i d i f f e r e n t i a b i l i t y , and the other based on convergent p o w e r series. We point out here that the concept of vector-valued holomorphic mappings arises naturally, as in the study of Banach algebras as well as in some other contexts. We have already enlarged the classical definition of holomorphic mappings from complex to vector-valued ones in locally bounded F-spaces. We use some of the new concepts introduced here, as the concept of quasi-differentiability, to present more generalized fundamental theorems in complex and functional analysis such as : F u n d a m e n t a l Theorem of Calculus, Integral M e a n - V a l u e Theorem, and B o l z a n o ' s I n t e r m e d i a t e - V a l u e Theorem.
8.1
FUNDAMENTAL CALCULUS
THEOREM
OF
In this part we extend one of the most celebrated theorems of mathematical analysis, the Fundamental Theorem of Calculus~ to vector-valued functions of p-Banach spaces using the concept of quasi-differentiability. Hence, we have established a new general form of the fundamental theorem 157
158
CHAPTER
8 NEW
VERSIONS
OF MAIN
THEOREMS
of calculus suitable for applying to the integral forms of continuous vector -valued maps of locally bounded F-spaces, not necessarily locally convex. In fact, using the curvilinear integrals, the theory of vector-valued holomorphic mappings, like that of complex-valued holomorphic mappings can be developed most efficiently. We first need to generalize the Riemann Integral to vector- valued functions. 8.1.1
Riemann
Integration
on [a,~]
D e f i n i t i o n 50 ( S t e p M a p s ) . Let I = [a, ~] be a closed interval in ~, and E be a p-Banach space (0 < p < 1) over K . A s t e p m a p
f.
E
is a map for which there exists a partition P. aand elements Y l , . . . , Y n E E
ao
CA is the characteristic function on the set [a,/3]. Let S(I, E) be the v e c t o r s p a c e of all s t e p m a p s from I to E. Then S ( I , E ) is a v e c t o r s u b s p a c e of the vector space B ( I , E ) of all bounded maps from I to E. Having the p-sup norm, [[fl[ - sup
{[f(x)[[
xEI
on the space B ( I , E ) , B ( I , E) becomes a p - B a n a c h s p a c e . D e f i n i t i o n 51 ( R e g u l a r F u n c t i o n s ) L e t R(I, E) - S(I, E) be the closure of S ( I , E ) in B ( I , E ) . A map in R ( I , E ) is said to be regular. Since a continuous map f " I --~ E is uniformly continuous, we can conclude that
C(I, E) C R(I, E)
(s.2)
8.1 F U N D A M E N T A L
THEOREM
OF CALCULUS
159
Definition 52 (Integra 0 We define the integral of a step map f " [~, ~] --, E,
n
f - ~
yi)~.(C~i__l,(~i ) 1
by
(8.3)
f - ~ ( o ~ i - oLi-1)Yi. 1
It is easy to see that the integral
J Z.S(I,E)-~E is linear and
II
E
fll 1/p O,
for x E OU
(8.12)
164
CHAPTER
8 NEW
VERSIONS
OF MAIN
THEOREMS
where OU denotes the b o u n d a r y of U - (a, b). T h e n f has at least one zero in U. He then proved t h a t " I f U is a b o u n d e d d o m a i n o f (T c o n t a i n i n g t h e o r i g i n , a n d
(s.13) is a n a l y t i c in U a n d c o n t i n u o u s o n Re z - f ( z ) then result. If closed
> 0,
U- and such that for z e / ) U
(8.14)
f h a s o n e z e r o in U. He has used Rouche's theorem to prove that It says that : f and g are each functions which are analytic inside and on a simple contour C and if the strict inequality
If(z)-
g(z)] 0 we have
det(A + eI) - d e t ( p - l ( A + e I ) P ) - d e t ( p - 1 A p
+ eI) 7s O.
(8.18)
m
Lemma
21 Let U be an open bounded set in(F ~ and let f, g : U - --, (~ n
(8.19)
be two continuous maps. Let w C ~T n and
(8.20)
w ~ f ( O U ) u g(OU). A s s u m e further that e satisfies
0 < e < min
{llf(z)- ~11;
c ou}.
(8.21)
all z C OU,
(8.22)
z
If I l f ( z ) - g(z)ll < e for then
deg(f, U, w) = deg(g, U, w).
(8.23)
P r o o f . Define the homotopy H : U - x [0, 1] ---, Cn
(8.24)
by H(z,t) :=(1-t)f(z)+tg(z),
forzEU-
andtE
[0,1].
(8.25)
170
CHAPTER
8 NEW
VERSIONS
OF MAIN
THEOREMS
By assumption, it is easy to see that H ( z , t ) ~ w for z C OU and t E [0, 1]. By homotopy invariance theorem the result follows, m Proof
of T h e o r e m 82, Define the homotopy H " ~ - • [0, 1] --~ 07 n
by H(z,t)'-(1-t)z+tf(z),
forze~-
andte[0,1].
By assumption, H ( z , 0) # 0 and H ( z , 1) ~= 0, for z e / ) ~ and Re z - . H ( z , t ) > Re z-.(1 - t)z > O,
(s.26)
for z E Ogt and t E (0, 1). By the homotopy invariance theorem, deg(f, ~, 0) - deg(H(z, t), fl, 0) - deg(I, ~, 0) - 1.
(8.27)
Notice that f-~(0) is a compact subvariety of ~t. We claim that f - l ( 0 ) is finite. For, let M be a component subset of f - l ( 0 ) . Then M is a compact subvariety of ~. Since each projection ~ j ( z l , .., .Zn) - zj is analytic on M, ~j is constant by applying the result of Gunning&Rossi[[ll7],pl06]. So M is a singleton; thus f - l ( 0 ) is discrete and hence finite. Now, let ~1,---, ~k denote the zeros of f. Let Aj be a neighborhood of ~j such that the closed sets A~- are pairwise disjoint and Aj C ~. Let
So K is a closed subset of ~t- which does not contain a zero of f. By the excision and additivity properties of the degree, Schwartz [[188],p.86] we have deg(f, ~, 0) - deg(f, ~ - K, 0) - ~
deg(f, Aj,0).
(s.29)
J We claim that deg(f, Aj, 0) _> 1,
for each j.
(8.30)
If deg(J G (f)) =fi 0, where J~j (f) denotes the complex Jacobian matrix of f at ~j, then deg(f, Aj,0) - 1 by Lemma 19. Suppose now that d e t ( J ~ (f)) - 0.
(8.31)
8.2 B O L Z A N O ' S I N T E R M E D I A T E
THEOREM
171
Then by Lemma 20, det(JCj (g)) # 0 when
g(z)-e(z-~j)+f(z),
and e > 0
is small enough.
Consequently it follows from Lemmas 19 and 21 that deg(f, Aj, 0) = deg(g, Aj, 0) _> 1.
(8.32)
By (8.27) and(8.29), the theorem is established, m 8.2.3
Infinite Dimensional
Spaces
In infinite dimensional spaces Wlodraczyk[208], obtained the following extension of Bolzano's intermediate-value theorem for h o l o m o r p h i c m a p s in certain c o m p l e x B a n a c h spaces. B a n a c h Spaces of C o n t i n u o u s Linear M a p s
Let E , F be complex Hilbert spaces; let L(E; F), as before, denotes the Banach space of all continuous linear maps A : E ~ F with the mapping 12orm
IIAII-- sup IIA(x)ll.
(8.33)
L1 (E, F) C L(E; F)
(8.34)
Ilxll_ O, for A EOFt
(8.36)
be continuous in ~ - . (1) /f
172
CHAPTER
8 NEW
VERSIONS
OF MAIN
THEOREMS
and ( I - f)(f~) is contained in a compact subset of L(E; F), where A'is the conjugate of A, then f has at least one zero in ft, i.e. f (A) - 0 for at least one A E ft.
(2) If additionally f is holomorphic in f~ and D r ( A ) inverse for A E ft, then f has exactly one zero in f~. Proof.
has a bounded
To prove (1), let
ht(A) - A -
f(t,A),
where f (t, A) - t [A - f(A)] ;
i.e. ht(A) - ( 1 - t)A + t f ( A ) ,
AEft
and0_ (1 - t) Re {A'A} > 0. Since 0 E gt, we apply the homotopy property to [[200],p.811, to have
ht (A), see Smart
deg(I - f(0, .), ~, 0) = d e g ( I - f(1, .), a , 0) = 1
(8.51)
or equivalently deg(f, a , 0) = deg(I, a , 0) = 1. Hence it follows that
f-l(0)
(8.52)
is nonempty.
(2) Since
A ~ Df(A) is a quasi-holomorphic map of ~ into L(lp;K), by the inverse map theorem for complex p-Banach spaces, the map f is locally quasi-biholomorphic in ~t, that is, f and f - 1 are quasi-holomorphic, see Theorem 46(Ch.4, subsection 4.1.5, p.85). i.e. for each A E ~, there exists a neighborhood UA of A in ~ such that
f(Un) = VA is open in Ll(lp;K), f - 1 exists and is quasi-holomorphic in
(8.53)
VA.
We now prove that f - l ( 0 ) contains one mapping : Let us first show that f - l ( 0 ) is finite. Towards a contradiction, let Ak E f - ~ ( 0 ) , for all k - 1,2,...We have h(Ak) - Ak, k - 1,2, ...,where
h=I-f. Since h is compact and continuous in ~ - , there exists a subsequence of (Ak), say (Ak), and A e ~t- such t h a t
ilA-Akrl 0
as k ~ ~
and
h(A)-A
(8.54)
This yields
f(A) = 0
and
R e { A * f ( A ) } = 0.
(8.55)
176
CHAPTER
8 NEW
VERSIONS
OF MAIN
THEOREMS
Consequently, A E ft. But f is biholomorphic in UA and Ak E UA for sufficiently large k. This yields a contradiction. Thus f - l ( 0 ) is finite. If f - 1 (0) -- {A1, ..., A n } ,
(8.56)
then by Smart[J200], propertiesl0.3.h&10.3.6, p.80)], we have n
deg(f, t2, 0) - deg(I, t2/U, 0) - ~
deg(f, Uk, 0)
(8.57)
k=l
where Uk is a small neighborhood of Ak such that the sets Uk are pairwise disjoint, U~- C f~ and U - t2-/(U~=lUk ). Further we have f - I - h , h has isolated fixed points in t2; h is compact and L I ( E , F ) is complex. Thus the multiplicity of each eigenvalue of
D(h)(Ak) - D ( I - f ) ( A k ) ,
k - 1, 2, ..., n
(8.58)
is even by Kransnoseliski[[164]Lemma 4.1 and Theorem 4.7], this yields
deg(f, Uk, 0) - deg(I - h, Uk, 0) - 1 By (8.51), (8.57) and the proof. I
(8.59)
k - 1, 2, ..., n.
we deduce that
(s.59)
n - 1, which completes
R e m a r k 22 Every separable locally bounded space Ep is isomorphic to a quotient space of lp, (0 < p 0 by ignoring the coordinates for which bi - 0. Consider the mapping r __,/~ defined by ...,
where t -
(tl,...,tn) E R n. Let
-
...,
to)
(9.3)
182CHAPTER
9 BOUNDING
AND WEAKLY-BOUNDING
/
B * -- d/)-l(Trn(Blp)) --
t e ~n; ~
n
eptj < 1
SETS
/
(9.4)
j=l
and b; - 05-l(b)
The set
- (log bl, ..., log
bn).
(9.5)
B* is closed and convex in/~n. This is because n
t --+ ~
e ptj
(9.6)
j=l
is a continuous convex function in JRn. Since b* ~ B*, it follows by the classical Hahn- Banach Theorem that there exists a continuous linear functional T o n ~]~n such that sup T ( t ) < T(b*) tCB*
where T takes the form n
T(t) - ~ mjtj, j=l
t E ~n
where necessarily m j >_ O,j - 1,...n, since point s with 8j < tj. But the hyperplane
/
n
t c ~n; ~ . ~ j t j
B*
- T(b*)
(9.7)
contains with
t
every
/
j--1
can be moved so that all the m j become rational numbers, j - 1, ...n. Thus there exists a positive integer m such that m m j - kj are all non-negative integers. The monomial P(x)
kl xk2 .. kn -- x 1 .x n
(9.8)
satisfies the following" n
P ( b ) -- rIjn=l.bknj -- exp(~--~, kjb;) - e x p ( m T ( b * ) ) ,
j=l
(9.9)
9.1 B O U N D I N G
SETS
183
and sup [ P ( x ) 1 -
sup e x p ( m T ( t ) )
l , ( b ) is replaced by IIx+yllp- 5A > o,
when j 7~ k
for some suitable number A > 0 and some c~ E I. From now on the proof can be given along the same lines as in the above theorem. This completes the proof of the theorem. IH R e m a r k 26 For the quasi-complete locally convex spaces the result is due to Schottenloher [196] and for separable or reflexive Banach spaces it is due to Dineen[86]. Theorem
92 [16](1990)
A subset A of the pseudoconvex F-space
Zo - n~>o/~ -
x - (x~); II x Iio= sup~-~.lXn IP< oo p>O
1
,
(9.23)
9.1 B O U N D I N G
SETS
191
(0 < p < 1), of the intersection of all spaces lp of complex sequences x = (xn) with the F-norm II x Iio= supp>0 ~ - ~ I x~ IP< oc is bounding if and only if A is relatively compact. P r o o f . First the space lo is a complete metrizable pseudoconvex space with respect to the F-norm, oo
if x i]o= s u p ~-~lXn ]P. p>O
(9.24)
1
Hence by Theorem 90, the bounding subset of lo are relatively compact. In fact Io has polynomially convex balls, for the function
f(xl, ...x~) =ll ~,---, ~
JIo
is convex for x = (x], ...x~) C ( ~ n n E N, see Corollary 15. This completes the proof, i 9.1.4
Bounding
Sets in Non Locally Pseudoconvex
Spaces
In infinite dimensional locally convex spaces E the size of the bounding subsets has been investigated by Dineen [88] and Josefson[122], who answered the following questions negatively for l ~ : "Does there exist an infinite dimensional normed space E such t h a t t h e u n i t ball BE is b o u n d i n g . ? " In some non locally convex spaces the situation is different as it will be asserted by the following theorem.
T h e o r e m 93 / / 6 ] ( 1 9 9 0 ) A subset A of the non locally pseudoconvex F-space
z(~/-
9
(x~); li x ii- ~ l ~
I~n< ~ , P~ -~ 0
(9.25)
1
(0 < p~ < 1), Pn ~ O, of all complex sequences x = (xn) with the Fnorm [I x I[- ~ ] xn ]pn< oo, is bounding if and only if A is bounded. Consequently, the unit ball B l(pn)(0, 1) is bounding. P r o o f . The s p a c e l(pn) , (O < Pn < 1),p~ --~ 0, is Schwartz and hence it is Montel space. It is metrizable but not locally pseudoconvex since there is no countable family of pa-semi norms defining its metric topology, see
1 9 2 C H A P T E R 9 B O U N D I N G A N D W E A K L Y - B O U N D I N G SETS nolewicz [[186],p.153]. Now as l(pn), Pn ~ O, is a Montel space then every closed bounded subset A of E is compact and consequently A is bounding. On the other hand the bounding subsets of this space are relatively compact since its balls are polynomially convex, see the theorem above. Since B z(p~)(0, 1) is topologically bounded with respect to the only topology define by the given F-norm, it is bounding, and the proof of the theorem is complete, m C o n j e c t u r e 1 Are the weakly bounded subsets of l(p~), (0 < p~ < 1),p~ -~ 0, weakly-bounding? We have a negative conjecture.
9.2
WEAKLY-BOUNDING
9.2.1
(LIMITED)
SETS
W e a k l y - B o u n d i n g Set in Locally B o u n d e d Spaces
Definition 62 A subset A of a topological vector space E, whose dual E ~ separates its points is called. " w e a k l y - b o u n d i n g " ( o r limited) if every sequence (On) C E', which converges pointwise to zero in E, converges uniformly to zero on A. That is,
lim Cn (x) -- 0, for every x E E
lim
n---~oo
II
n - - ~ oK)
II Cn II-o,
(9.26)
IIA= sup A Ir
Proposition 3 A subset A of a locally bounded p-normed space E is weaklybounding if and only if II f ]]=sup If ( x ) ] < oo, for all A
xcA
f-
~
q5~ E H (E)
(9.27)
1
where ( cpn) C E' . P r o o f . =~)" Assume A is weakly-bounding. Then ]] On IIA--' 0 whenever r (x) --* 0 for every x in E. Let
1
9.2 W E A K L Y - B O U N D I N G
(LIMITED)
SETS
193
Since f - ~ r is analytic, the boundedness of f on A is given from the analyticity of the polynomials c n n C N and the assumption that II Cn IIA--~ 0 to have ~ ] 1 II On ]]~< c~. Notice that II ~-~.~ Cn n IIA~ EF
II
IIA< oo.
r If ~ r n G H (E) then ~-~ (On (x)) n tn/p converges for every x E E and t E (/'. By Generalized Cauchy-Hadamard formula in one variable (see subsection 7.2.3, p.139), limsup 1On (x) n 88 Ip -- lim 1r (x)I p -- 0. n--~oe
Now if oO
II
r
11,4
lp/M
in
196CHAPTER 9 BOUNDING
AND WEAKLY-BOUNDING
SETS
be the quotient mapping which maps the Fr6chet space lp onto the Fr6chet space lp/M with respect to the F-norms IJ 7r (x)I[= inf IJ x - m lip, where meM
II x lip= E F Ixjl p, for x -
(xj) C lp.
For the weakly-bounding subset B of lp given above, we claim that the subset 7r (B) of lp/M will satisfy the desired property, that is, it is non bounding but weakly-bounding in lp/M" Let (On)C (lp/M)' with (~n (X)~ 0 for every x in lp/M. Then oo
II ~ r 1
(x)
II ~ ~'~ II r 7r(B)
~-
(9.35)
1
This is because d/)nOTr is a sequence of continuous linear functionals on E which vanishes on M and B is a weakly-bounding subset of E. On the other hand 7r (B) is not bounding in lp/M. Notice that 7r(B) is not bounded in lp/M. This completes the proof. II
9.2.2
T h r e e Different Classes of H o l o m o r p h i c
Functions
The above theorem can be formulated using holomorphic functions as follows.
T h e o r e m 96 [16] Let E be a complex p-Banach space with a Schauder basis, (0 < p < 1). Then there exists a holomorphic function f in H (E) and a non-relatively compact set A in E such that
II f IIA= ~ pointwise in E where (r bounded set in E.
a~d
II On IIA~ 0
if On(X) ~ 0
(9.36)
C_ E'. Moreover A could be an unbounded weakly
P r o o f . Every bounding subset A of E is weakly-bounding but the converse is not always true; Remark the sets A , B of the preceding Theorem 95. Hence, not every set A in E satisfy II r ]IA~ 0, whenever Cn(x) ~ 0 pointwise for every (On) C_ E', is relatively compact. Therefore a function f E H (E) may exist and satisfies II / IIA= c~. Of course f will not of the form ~ r with II r IIA~ 0 wheneverr n ~ 0 pointwise since A it is not weakly- bounding, in accordance with Proposition 3, given at the begining of this part. Such a set A may be a proper weakly bounded set in E. This completes the proof. II
9.2 W E A K L Y - B O U N D I N G
( L I M I T E D ) SETS
197
R e m a r k 28 This last situation is almost similar to the finite dimensional
case(T n in which every f r H ((Tn) is unbounded on unbounded sets o f t ~. Theorem 95 shows the remarkable relevance of the concepts of bounding and weakly-bounding sets. We may consider them as basic concepts in holomorphy. To show this, let E be a locally bounded F-space and let By (E) and WBg (E) denote the classes of bounding subsets and weakly-bounding subsets of E respectively. According toTheorem 95 if E is not locally convex space with basis, one may obtain
Bg (E) ~ WBg (E) .
(9.37)
Moreover the class WBg (E) consists of two subclasses: (1) The bounded weakly-bounding subsets of E, (WbBg (E)), and (2) The unbounded weakly-bounding subsets of E, (W~Bg (E)). Therefore we have precisely the following relations
Bg(E) WbBg (E
(E)
Bg (E) C WbBg (E) C WWBg (E) . Figure 5: R e l a t i o n b e t w e e n t h e d i f f e r e n t classes of b o u n d i n g sets Consequently we will have the following result.
198CHAPTER
9 BOUNDING
AND WEAKLY-BOUNDING
SETS
Corollary 17 [16](1990).Let E be a locally bounded space with a Schauder basis. Let us denote by H b (E) the class of holomorphic functions of type Cn n, (r C E' which are bounded on all bounded sets in E. i.e.
H(E) = I
f=Er
nell(E);
(r
V bounded sets B in E
J C_
and by H b (E) the class of holomorphic functions of type E r E', which are bounded on all weakly bounded sets in E, i.e.
H (E) = I
f - ~ Cn n e H (E) ; (r
C E', II f lie< ~ ,
V weakly bounded set C in E
J
Then
H b (E) C H b (E) C H (E).
(9.38)
That is, the class of holomorphic functions H (E) on E contains properly two other different classes.
Proof.
See the proof of Theorem 95. m
(E)
H(E)
H b(E) C H D(E) C H ( E ) Figure6" Relation between the different classes H b ( E ) , H b (E)
and H (E) of holomorphic functions
199
9.2 W E A K L Y - B O U N D I N G (LIMITED) SETS
Remark 29 We notice that these above three classes of holomorphic functions coincide for many locally convex spaces, e.g. if E shown by Josefson [122] and Dineen [82].
lp (c~ >~p >~ 1) as
Remark 30 As we have seen,
the theorems of the two preceding parts explore a new class of bounding sets which appear in the study of holomorphy in the non locally convex spaces E. It is the class of bounded weakly-bounding sets ( B - W B g ( E ) or bWBg (E) for short).So recall that we have so far the following relation between the three classes of bounding sets in some non locally convex spaces : Bg(E) ~ B-
9.2.3
WBg(E) ~ WBg(E).
(9.39)
E x a m p l e s of H o l o m o r p h i c Functions
We introduce some of the remarkable topological vector spaces which satisfy the hypotheses of the theorems of the preceding subsections 9.2.1 and 9.2.2. We will also construct certain holomorphic functions and non bounding weakly- bounding sets in some spaces to explain the performance and effects of these concepts in Complex and Functional analysis.
Construction of Certain Holomorphic Functions ( O n L o c a l l y B o u n d e d Sequence Spaces) Example 31 Let E - lp, (0 < p < 1). According to the above comment the relation
means that not every A C_ E satisfying lim II r
IIA-- 0 i f
lim
Cn (X) -- 0 , e v e r y
x e E
(9.40)
is relatively compact. Equivalently, there is A C_ E which may be unbounded satisfying II r IIA--~ 0 i r e s ( x ) --~ 0, x E E, (r C_ E', and a function f = E r f C H (E) exists with II f IIA< c~. This is because weakly- bounding subsets are not necessarily relatively compact by Theorem 96.
200CHAPTER
9 BOUNDING
AND WEAKLY-BOUNDING
SETS
To show this, consider the following set given in the proof of Theorem 95.
B -- { x(n);
x (n) - -
1 ( 1 ) :/p (1, (~)l/p,..., ,0,...),
n:g
}
.
We have proved that B e WBg (E) /Bg (E) and B is unbounded in E. Now the following function (:x:)
1
:<x>
+niX>
~
~
o
n
I
will be the required one, i.e. II f liB< oo. Obviously (r
(1)
1)1//) __
1{~n {{B-- ( 1 ) 1( 1-1 )
"(n a s Tt---~ cx).
1
(9.41)
X n
c E' and hence
1
~(1-;)
1
--,0
Now oo
1 )l_~.Xnn e H (E)
(9.42)
1 and it is bounded on B"
m(1)
II fllB=sup ~ xmgB
m
1
!n P
n
=
1+~
-E
o
is not Gelfand-Phillips spaces with respect to the sup F - n o r m oo
II x IIo:Supll x lip=sup ~
p>o
where
p>o j=l
Ixjl p
x -- ( xj ) E lp.
Proof. The bounding subsets of lo are relatively compact since lo has polynomially convex balls. Notice that the function f ( X l , . . . ,xn) =11 ( e ~ , --- , e ~ ) I1~ois convex for x - ( X l , . . . ,xn) E ~ " , n E N, see the author[[11],
p. 17]. The limited (weakly-bounding) subsets of lo could be unbounded. example the set A-
{
x(~); x ( ~ ) - ( 1 ,
(1) 1/p ~
(1) lip ,...,
For
} ,0,...), nEN
is weakly-bounding in lo for it is weakly-bounding in each lp. Notice that lo is dense in each lp(O < p < 1). Moreover A is not bounded in lo :
n(_.~)l/p.p
II x(n) Illo--sup ~
m-1
n 1
= ~
- - + oo
rn-1 m
as n ~ cx~. Consequently a weakly-bounding set in lo may not be compact.
214CHAPTER
9 BOUNDING
AND WEAKLY-BOUNDING
SETS
35 The importance of this F-space lo is that every separable Banach space E is isomorphic to a quotient space of lo; i.e. for a subspace M of lo, we have E ~_ lo/M. (9.68)
Remark
In what follows we prove that the F-spaces lp~ which may be locally bounded or not (this depends on (Pn)) are not Gelfand-Phillips spaces. Theorem
103 (Sequence F - S p a c e s ) T h e spaces oo
< ~ } , (1 > pn > 0)
lpn - - { X - - (Xn); X n e (~, ~
1
are not Gelfand-Phillips spaces Proof. F i r s t : I f pn /---~0, then each of lpn is metrized by a p-norm and lp~ become sequence locally bounded F-spaces. Consequently limited (weakly-bounding) sets in lp~ are not necessarily relatively compact, see [[12], Theorem3. 2] where we have proved that limited set could be unbounded S e c o n d : If pn ~ 0, the spaces lp~ are non locally pseudoconvex F-spaces. We claim that they are not Gelfand-Phillips spaces neither. We have to choose an unbounded set which is bounding or weaklybounding. In fact, the unit ball as we have seen [12] are bounding. Remember that these spaces are Schwartz and hence Montel spaces. Therefore every bounded closed subset of them is bounding. Let us consider the following set
A -
1
x(~); x ('~) - (1, (~)
1/pl
_1)1/pn, 0,. . .) , n o N } . '""(n
(9.69)
This set is unbounded; notice that
f;x
n
1 ----+ (:~
II1
m
aS Tt----+ 00.
However, A is weakly-bounding, i.e.
lim On (x) -- 0, for every x e lp,~ :=~ lim II r n----~
n----~(x)
IIA-- o,
(9.70)
9.3 P R O P E R T I E S
OF BOUNDING
AND
LIMITED
SETS
215
This is because the dual of lp~ is loo and A is weakly b o u n d e d in /1; we note t h a t O(3
lim sup ~ Ixjl-- O. n x C A .J - - n This completes the proof. II R e m a r k 36 Referring to the preceding theorems 102 and 103, we have seen a limited set may not be bounded( see [[12], Theorern3.2] ). Recall that the intersection between the classes of limited sets and of bounded sets is bigger than the class of bounding sets; consider for example the set of
B-
{
x(~) ; x ( n ) - (
/1) 1/p (1) 1/p -n
' ... '
-n
' 0,...),
}
of lp . Hence our space E - lp will have b o u n d e d and u n b o u n d e d limited sets which are not relatively c o m p a c t with the priority t h a t all its b o u n d i n g sets are relatively compact(see[16]). T h e following example explains the preceding theorems. E x a m p l e 3 7 A limited set may be unbounded by Theorem 103 and [[12],Th.3.2]. That is an unbounded set D may exist such that all holomorphic functions of type 1
will satisfy II r liD< C E*. {Sm}) To show this, let E - l(p~), inf Pn --~ O, be a locally bounded space. We now give an example of a function g (x) -- ~ r
C H (E)
such that g lD is bounded.
and a non-bounded Limited subset D of E Set (x)
-
and let 1 D - { x (~) ;x (n) - (0, (21og22)I/p
~---~
(
1 )liP 0 , . . . ) , n o N } n log2n '
216CHAPTER
with P=n p (1 -
9 BOUNDING 1 ~ log log n ]
AND
andO 0 such that (x)
E
en log d(Tcny, y) > - c o .
n=l
The function (:x:)
n(~) - ~ ~ log d(..x,
x).
xCE
(9.75)
n=l
is plurisubharmonic and hence
- {x e E; R(x) < R(y)} is a pseudoconvex domain of E. We notice that F C_ ~, since R ( x ) ~ - - o o , for x ~ U ~ ( E ) - F.
9.4 H O L O M O R P H I C C O M P L E T I O N
219
Now apply the Levi problem solution given by the author[12], [13],and [14], (see also Ch.10 ,subsection 10.1.1, p.231). We note ft is a domain of existence of a holomorphic function f E H(E). Hence the holomorphic completion F5 is contained in ft to have in particular y ~ Fs. Since y E E / F is arbitrary we conclude that
F-F~. This completes the proof of the theorem. I
Examples of Holomorphically Complete F-spaces Example 38
The complex spaces
l(p~)- { x--(Xn);~--~.lxn IP~< cx~ ,
0 < Pn olp-
x-
(Xn); sup p>0
I xn IP< c~
,
(9.77)
1
lp -- Uq_plq -- { x e l ;suP lXj
,0 < p < _ 1
(9.80)
q>-P 1
are non locally convex spaces which satisfy the hypotheses of the above theorem. Notice that lp C l+ C lp+e.
(9.81)
220CHAPTER
9 BOUNDING
AND WEAKLY-BOUNDING
SETS
Relation Between Bounding Sets and Holomorphic Completion In what follows we give the relation between the union of the bounding subsets of a locally convex pseudoconvex F-space E and its holomorphic completion E.
T h e o r e m 105 [17](1989) If E is a locally pseudoconvex F-space with the bounded approximation property, then E D -- U A E B g ( E ) A -
where A orE.
is the closure of A, and Bg(E)
is the class of bounding subsets
For the proof of the theorem we need the following lemma.
L e m m a 24 Let E be a locally pseudoconvex F-space, with the bounded approximation property, f E H ( E ) , and A is a bounding subset of E. Then there exists a zero neighborhood of V in E such that
flfllA§ P r o o f . For f E H(E), and x E E
fz(Y) :
(9.82)
< we
define f~ : ~ - ~
by
f (x + y).
(9.83)
It is clear that f~ E H(E), for every x in E. Similar to the method of locally convex spaces, see Dineen [[82], proposition 4.22], we can show that the family (fx)~eA is t o - b o u n d e d subset of H(E), where ro is the compact open topology on H(E). In fact, if K is an arbitrary compact in E, then sup IlfyllA =
yEK
sup
yEK, xEA
IIf(x + Y)II = sup I]LIIK < xEA
Since E is metrizable, (fz)zEA is a locally bounded subset of H(E) and hence there exists a neighborhood V of zero, for example a ball B(0,e), such that sup IIfzIiv < oo.
xEA
Hence sup xEA,yEV,
] f ( x + y)I--]lfiiA+y < o o . I
9.4 H O L O M O R P H I C
COMPLETION
221
P r o o f o f T h e o r e m . If x C E5 then there exists (xn)~ C E such that x as n ~ oc. Since x C E5, Xn lim f(Xn) exists for every f E H(E), n----~ o o
and hence sup I f ( x ~ ) I < oo. n
Thus {xn} is a bounding subset of E
and x E UAEBg(E)A, that is
E5 C UACBg(E ) A.
On the other hand if A is a bounding subset of E and f E H(E), then by the above lemma a pseudovonvex balanced neighborhood V of zero exists in E such that LI/IIA+v < o o . By using the Taylor series expansion about points of A we find that there exists a holomorphic function f ~ on A + W such t h a t f A+W -- f A+V
where W is the interior of the closure of V in E. As A C A + W, we will have A C_ E5 and hence E D - UACBg(E)A-.
This completes the proof of the theorem. I
9.4.2
Holomorphic Extension Problem
In this section we study the holomorphic extension problem which is purely an infinite dimensional problem in some F-spaces : Let F be a subspace of a locally pseudoconvex metrizable space E, whose dual E' separates the points of E. " W h e n can every h o l o m o r p h i c function on F be e x t e n d e d to a h o l o m o r p h i c f u n c t i o n o n E ?"
222CHAPTER
9 BOUNDING
AND WEAKLY-BOUNDING
SETS
Holomorphic Extensions of Dense Subspaces The following theorem gives a counterexample to holomorphic extension problem. More precisely, it supplies us with a dense subspace which cannot be extended to the whole space all holomorphic functions.
Theorem 106 [19](1989) Not every holomorphic function on lo
-
Np>olp
with the sup-topology, that is, the one defined by the F-norm OG
jixll- sup p>0
,xo ,p 1
can be extended to a holomorphic function
on
lp
(1 > p > 0).
P r o o f . lo is a dense subspace of each lp (1 > p > 0), by Stile[[201],p.117]. Hence every bounding subsets of lo is bounding as a subset of lp. Since bounding subsets of lo and of lp are relatively compact, by Theorem 90, p.186, it suffices to pick up a non compact subset of lo which is compact as a subset of lp. For example the set
D-
~0,x(1),x(2),...~,
where
(0,. ,0,( n1 )I-l/p, 0,...), n E N
(9.84) (9.85)
is compact in lp if (1 > p > 0), but it is not compact in lo. Notice that oo
]lx(n)llo
-
-
s u p ~ - ~ . l x n Ip p>O 1
= sup(-1 )(1-1lp)p
----
n 1-p --+ oo
p>0 ?%
as n ~ c~. This completes the proof of the theorem, m As another counter example where the holomorphic extension fails to occur from a non locally convex subspace to a locally convex one we obtain the following result.
9.4 H O L O M O R P H I C
COMPLETION
223
T h e o r e m 107 If f is a holomorphic function on lp (1 > p > 0), then f may not be extended to a holomorphic function on 11. That is, not every holomorphic function on lp can be extended holomorphically to its Banach-envelope 11. P r o o f . The set A -
1 )i/p
x(~); x (~) - ( ( n
is not bounding as a subset of lp, for it is not relatively compact. However it is bounding as a subset of 11. See also proof of T h e o r e m 95. This completes the proof of the theorem, m
R e m a r k 37 It follows from the preceding theorem and the fact that the linear completion l~ - 1 1 that
lp C lpt . , also that ( lp)5 = lp. Holomorphic Hahn-Banach Extension Theorem We consider the second type of holomorphic extension problem : " W h e t h e r eve r y h o l o m o r p h i c f u n c t i o n d e f i n e d o n a c l o s e d s u b s p a c e F of a c e r t a i n F - s p a c e E can be e x t e n d e d analytically to E". Since there exists a continuous linear functional on a subspace of lp (1 > p > O) which cannot be linearly extended to lp, by Stiles [201], we will deduce the following theorem.
T h e o r e m 108 Let M be a closed subspace of lp (1 > p > 0) whose unit ball is ll-precompact. Then not every holomorphic function f in H ( M ) can be extended analytically to lp. P r o o f . By the result of Stile[[201], p . l l l ] , where under this assumption one can find a continuous linear functional on a subspace of lp (1 > p > 0) which cannot be linearly extended to lp. m Consider now the following normalized sequence (u~) which has disjoint supports:
224CHAPTER
9 BOUNDING
=
u2 -
AND WEAKLY-BOUNDING
SETS
(1,0,..),
1 1 (0, (~)I/P, (~)l/p, 0, ...), ..
Un--(O'""(n
1 )l/p
'
ooo~
(1)l/p, 0,. ) n ~
oo
(9.86)
nth-place Then we obtain the following theorem. T h e o r e m 109 Let M be a closed subspace of ~ which is spanned by the above sequence {u~; n E N } . Then not every holomorphic function on M can be extended analytically to 11. P r o o f . The set {Un;nEN}
is not bounding in the subspace M, for it is not relatively compact. On the other hand it is compact in 11. Hence it is bounding in 11. m R e m a r k 38 The above theorem implies that M~ = M r lp. The locally convex space l~ has the property given by this theorem. This is because the set D = {ej; j C N } o f unit vectors in l ~ is bounding in l~ but not as a subset of Co. That is the holomorphic completion satisfies
(Co)5 5r lc~
(9.87)
Discussion We have studied some problems which arise in infinite dimensional complex analysis. In fact, the study of bounding and weakly-bounding subset of non locally convex spaces provides information which is of fundamental interest in Holomorphy. There are essential relations between this study of this chapter and the radius of convergence problem suggested by Kisleman [134]. That is,
9.4 H O L O M O R P H I C
COMPLETION
225
"To c o n s t r u c t a h o l o m o r p h i c f u n c t i o n w i t h p r e s c r i b e d r a d i u s of c o n v e r g e n c e " and the Levi problem, which will be discussed in the next chapter, that is, to answer " W h e t h e r e v e r y p s e u d o c o n v e x d o m a i n is a d o m a i n of h o l o m o r phy". For example we can easily deduce that there are entire functions of H(E) with finite radius of convergence after we have characterized the bounding subsets to be relatively compact, see the examples of this chapter. In addition we can check : " W h e t h e r t h e Levi p r o b l e m will n o t h a v e a s o l u t i o n " if we can find a certain type of bounding subsets domain of E as Josefson [121] has done for the space loo. This chapter indeed explores an important point that : " N o t e v e r y h o l o m o r p h i c p r o p e r t y in locally c o n v e x s p a c e s c a n b e i n h e r i t e d b y n o n locally c o n v e x s p a c e s " . One of the reasons behind that is, for example, that the boundedness with respect to the original and the weak topologies are not the same. This has been explained via the study which has been held for the weaklybounding sets. It is hoped that the present study in this chapter will be helpful in claiming some achievments in pure and applied mathematics. The next chapter discusses the different approaches which are used to solve the Levi problem in seprable topological vector spaces. The following figure explains the Levi spaces among this class of the separable toplogical vector spaces.
Levi Problem in Separable t.v.s. Separable t.v.s.
F-Spaces with plurisubharmonic
tp(1 > p > 0)
Locally pseudoconvex spaces(Lps) with b.a.p.
~logarithmic metric with f.d.Schauder decomposition
/.C.8.
Banach spaces with b.a.p.
9
t~
" I p n ~ p r t "-"~ 0~
Ii x II=
Inductive spaces Up>tip ,
II 9 II= ~up~>0E~
~,~%~ Ix,, ITM Lps N F-spaces
loo 9is not a Levi space. Lp :(1 > p > 0) : is not aLevi soace. Figure 8 : S o m e L e v i s p a c e s a m o n g s e p a r a b l e t . v . s .
I~
I~
Chapter 10
LEVI PROBLEM IN TOPLOGICAL
SPACES
One of the important and interesting problems in the field of holomorphy is the Levi problem. To solve the Levi problem means to prove that the class of holomorphic functions has a certain type of richness. In one complex variable, given a domain ~ there exists a holomorphic function f in ~ which cannot be continued beyond the boundary of ~ as a holomorphic function. For instance we may prescribe the values f(zk) arbitrarily if (zk) is a discrete sequence in ~. This means that ~ is the natural domain of existence of f . In two complex variables, we have already domains like
no - {z ~r
1 _O
~D
= inf inf inf Irish(w, z) - inf inf ~(~-, z)Sa(x z) ~ z
T>0
t
Z
T>0
for 5a(x, y) = inf(ltl; x + ty e Oft). Hence - log dfl(x) = sup s u p [ - log ~(~-, z) - 7 log 5a(x, z)] z
(10.10)
T>0
and since this function is either - - c o or else is continuous, it is plurisubharmonic as the s u p r e m u m of a family of plurisubharmonic functions. I R e m a r k 39 In the above theorem we have used the following property of a pseudoconvex domain f~ C E : if ~ C E and E is a topological vector space, then f~ is pseudoconvex if and only if the function
(x,y) H - l o g S a ( x , y )
is plurisubharmonic on ~ x E
w h e ?~e
5a(x, y) = sup(r; x + r D y C ~ ) , x E t~, y E E,
(lo.11)
D = {t C ~ ; Itl _< 1}. In fact, this is equivalent to our definition of pseudoconvexity, i.e. ft N F is pseudoconvex in F for every finite-dimensional subspace F of E, see Noverraz [ [181], Lemma 2.1.5]. I As an application of the above theorem we get Example
41
A complex vector space E
d(x, y)
with the metric given by
llx-yll - 1 + IIx-
YlI'
(10.12)
234CHAPTER
10 L E V I P R O B L E M
IN TOPLOGICAL
SPACES
where
I1 II i~ p-homogeneous norm, (0 < p < 1) is a PB-space. that the function
-
log d(etx, O) - log(1 +
Iletxll)
- log
is convex in t ~T for every fixed homogenous.A
Iletxll
- log(1 + oze-ltlp),
x C E. Notice that
d
We note 1 OZ-'-
IIxll
is not p-
40 According to the definition of a PB-space E: for every pseudoconvex domain f~ in E, - l o g dn is plurisubharmonic in ft. The converse is also true: Indeed, assuming that - l o g d~ is plurisubharmonic in ft and F is an arbitrary finite-dimensional subspace of E, f~ N F is pseudoconvex in view of the classical properties of plurisubharmonic functions of finitely many variables. Hence f~ is pseudoconvex. Consequently if E is a PBspace, then f~ is pseudo convex if and only if - log da is plurisubharmonic in f~.
Remark
10.1.2
Properties
of the Radius
of Convergence
Let us recall that the r a d i u s o f c o n v e r g e n c e R s ( x ) of a function f E H(f~) at a point x E f~ is the least upper bound of all numbers r > 0 such that the Taylor series of f at x converges uniformly in B ( x , r ) , the closed ball of centre x and radius r in E. Also the r a d i u s o f b o u n d e d n e s s Rb(x) of a numerical function
at a point x C f~ is the least upper of all numbers r > 0 bounded above in B ( x , r ) with B ( x , r ) C f~.
such that
u is
For a function f 6 H(ft),
-
sup(r > O; IlflIB( , )
is finite, B ( x , r ) C ~),
x E ~.
(10.14)
By these definitions of the radius of convergence R I and the radius of boundedness R} for a function f C H(ft), it is easy to prove t h a t
R} - inf(Rs, da)
(10.15)
where da(x) - infyeE/a d(x,y), x C f~, is the distance function on f~ defined by the metric d of E.
10.1 L E V I P R O B L E M
AND
RADIUS
OF CONVERGENCE
235
It may happen that f can be continued analytically beyond the boundary of Ft and hence R} will be less than R I (i.e. RbI n~(y) > 0. Then u is bounded above in B ( x , r - d ( x , y ) ) C B ( x , r ) for all r < Rb(x). Hence Rb(y) >_ r - d(x, y) and letting r tends to Rb(x) we get
R b (x) > R b (y) > R b (x) - d(x, y). This proves (10.16) in this case, and by s y m m e t r y the estimate holds everywhere, m L e m m a 26 If f
vector space E
is a holomorphic function on a subset f~ of a metric with a translation invariant metric d satisfying
d(tx, O) then
R ,b( x ) - d(x, y) - R z ( x ) - d(x, y). In conclusion we have
> R f ( y ) - Rg(y) > R gb ( y ) - d(x,y) = Rf(x) - d(x, y)
Rf(x)
t h a t is, (9.16) is proved. This completes the prove of the lemma, m The following result shows t h a t R~ for f E H(f~) admits a certain geometric property which is a consequence of the PB-property. Theorem
111
Let ~ be a pseudoconvex domain in a PB-space. Then - l o g R b is plurisubharmonic in
(10.18)
if u is plurisubharmonic in Consequently -logRbf is plurisubharmonic in
(10.19)
for every f E H(gt). P r o o f . Let u be a plurisubharmonic function in f~. Let
~
- {x e ~; ~(x) < k}, k e N.
(~0.20)
For every k E N, the set f~k is pseudoconvex. This follows from the definition of pseudoconvexity and the fact u is plurisubharmonic in f~. Let
d~k(x ) -- inf d(x,y),
x e f2k, k C N.
YEO~ k
Hence
R b - supd~ k k
lira d~k, k-~
(10.21)
10.1 L E V I P R O B L E M
AND
RADIUS
OF CONVERGENCE
237
that is - log R b - ir~f( - log d~ k).
(10.22)
The functions - l o g d~ k are plurisubharmonic in f~k for every k C N, since ftk is pseudoconvex and E is PB-space. Thus - l o g R b is plurisubharmonic as a decreasing limit of a sequence of plurisubharmonic functions. Since log l f ] is plurisubharmonic in ft for every function f E H(ft), we also get b ] ---- - log R} - log/i~loglf
(10.23)
is plurisubharmonic in Ft. I Let
Ep
be a complex vector space with the p-homogeneous norm
II.llp d(x,y) -II x - y l l p ,
such that
IItxllp -I t lp d(x, 0),
xEE, tE~. The following Lemma gives a formula for the radius of convergence R I for I E H(ft), ft C Ep open, which is well known for normed spaces, i.e. the 1-homogeneous spaces (see chapter 7). The given formula is called the pCauchy-Hadamard formula(or the Generalized Cauchy-Hadamard formula).
(p-Cauchy-Hadamard formula)The radius of convergence R S for f c H ( f ~ ) , ft is open inEp (O_n,
nEN,
and R S < R
Here a covering (Vn) on ~ will be defined such t h a t
I[flly~ - sup [ f ( X n ) I < xC Vn
for
n e N.
(10.29)
10.1 L E V I P R O B L E M
AND RADIUS
OF CONVERGENCE
239
The proof can be completed as in the forthcoming Theorem 117, p.247, (see also the author [12land Schottenloher [198]). T h e o r e m 113 [12](1979) Let ~ be a domain in an infinite-dimensional metric vector space E with a monotone Schauder decomposition (~n) and with a translation invariant metric d such that log d(x,O) is plurisubharmonic function in E. Let R: ~ - - ~ + be such that
- l o g R plurisubharmonic in E
and R _n (2) For all x E V and r n. (3) For a l l x E U , t h e r e e x i s t h > 0 andnEN w i t h B ( x , 5) CVn. (4) The sequence (x,,) has the properties t h a t x , C Vn+l/Vn and hence Xn ~ 71"n+l (Yn). The set,
7rn+l(Yn)- {x e LnA+I nTrnl(LA);dv(vr,x) >_ s --]--d(TrnX,X)}
(10.52)
is not necessarily holomorphically convex since -logdu(x) and logd(x, O) m a y not be plurisubharmonic functions in general. Thus we may have xn in 7rn+l(Vn) A although Xn ~ 7rn+l(Vn) as in (4). Fortunately we have the following lemma. L e m m a 30 The previous cover (Vn) of U has the property that, for every fixed n E N, there is a natural number mn olP(E)
a metrizable non locally convex vector topology is generated by the sets
Bp(To, e)
-
-
T e S(E); ~
[ar(T
-
To)]p < e
,p > 0,~ > 0
r--1
S ( E ) is complete and it has the bounded approximation property if E is complete, as it is shown by Pietsch[[184],p.139]. It is worthy to point out that the function
log dp(T, 0), T e S(E), is a plurisubharmonic function where oo
0) :
(10.61)
r:l
This is also the case for the function Oo
- log dn (T) -
sup - log ~ ToCO~
[a~ (T - To)]P, T e ~t
r=l
It is plurisubharmonic if ~ is any pseudoconvex domain of S(E). That is, S(E) is a PB-spaces. So by Theorems 113, 115, p.,239, 242, it is a Levi space. Note that L(E) is not a Fr~chet space if E is not Banach. A
252CHAPTER
10 L E V I P R O B L E M
IN TOPLOGICAL
SPACES
E x a m p l e 46 ( T h e I n f i n i t e P r o d u c t o f Fr~chet Spaces) We show that the Levi problem can be solved for the product H ~ E j of Fr~chet spaces (complete metric spaces) Ej, when the topology of Ej is given by the metric d(x, y)E~ -- IIx --
yll~j,
II.ll~j
where is a homogeneous pseudonorm, j E N, and each Ej has the bounded approximation property (b.a.p.), j E N. Let us first show that: the product
E-
II~~
(10.62)
has the b.a.p, if each Ej, j C N, has the b.a.p. By Theorem 114, the Fr~chet space Ej, j E N, is a complemented subspace of a space Fj with basis; hence it can be written as Ej - Fj |
(10.63)
Gj.
and there exists a continuous projection IIj
nj.Ej--,Fj with H~-1 (0) - Gj. Then E - nFEj
- nFFj
@ nF
Gi
Let r 9 ~ N ~ [0,--~oo] be convex and homogeneous of degree 1, F-normllxl] E can be defined for x - (Xl, x2, ...) e E by
]l.llE - r
(10.64) t h e n an
IIx211E=,---),
For example we can take oo
r
IIx211E~., ...) -
(~-~ IlxjllqEj)1/q j=l
where 1 l p , t h a t is,
E ~_ lo/M for a closed subspace M of E, see Stiles [203], and since E is a Levi space, see Corollary 20, p.240, we can obtain the following interesting partial result for the Levi problem. 120 (Levi p r o b l e m in certain separable B a n a c h space)[l~] Let U be a pseudoconvex domain in a Banach space E. Suppose that E is isomorphic to a quotient space of lo, that is,
Theorem
E ~_ lo/M,
(10.79)
and lo/M has the bounded approximation property. Then U is a domain of holomorphy. P r o o f . Since l0 is a Fr~chet space, its quotient space lo/M is a Fr6chet space. Now the space lo/M is assumed to have the b.a.p. Hence a direct application of Corollary 20, p.242, will imply the required result.This completes the proof of the theorem, m Let Ep be a p-Banach space. The fact t h a t every separable locally bounded space is isomorphic to a quotient space of lp(O < p p > 0).
The Hardy space Hp of all analytic functions f on the unit disc of (~ is separable, locally bounded, non locally convex space with respect to the pnorm
Ilfll-
~-~llim/02~ I f @ d ~ Ip dO,
f
H,(r
The Banach envelope of Hp is isomorphic to 11, that is H p ,.o ll
(10.81)
by Kalton [130]. This implies that H p has the b.a.p. Moreover, Hp contains a non locally convex closed subspace Mp of E isomorphic to lp (0 < p < 1), that is
Mp ~_ lp,
(10.82)
see Shapiro [189]. Now every pseudoconvex domain in this complemented subspaces M of
Hp, in H ~ is a domain of holomorphy, by corollary 20, p.242. That is, H p and Mp are Levi spaces. A
260CHAPTER
10 L E V I P R O B L E M
IN TOPLOGICAL
SPACES
R e m a r k 45 ( I m p o r t a n t ) T h e previous results of this chapter can be generalized to non-schlicht domains over a suitable space E. Let us first recall the following concepts which are analogues of those considered by Schottenloher [19~ for locally convex spaces : A Riemann
d o m a i n s p r e a d o v e r a metric vector space E
(ft, q) where ft
is a pair
(10.83)
is a connected Hausdorff space and
q" ~ --~ E
(10.84)
is a local homeomorphism. That is, for every x E f~ there exists a neighborhood w of x such that qw " W --~ E (10.85) is a homeomorphism of w onto q(w). If q is injective, the domain (ft, q) is called s c h l i c h t d o m a i n , and can be identified, via q, with a domain in E. The boundary distance function dn on a fixed domain (ft, q) over E is defined by: dn(x) -
sup(r; there exists a neighborhood U of x s.t.
q I u " U ~ B ( q ( x ) , r ) is a homeomorphism), x E f t
(lo.86) (lO.87)
The ball B ( x , r ) for x E ~, r > dn(x) is just the component of q - l ( B ( q ( x ) , r ) ) which contains x. The plurisubharmonic, the holomorphic or for that matter any locally defined class of functions, can now be defined on (ft, q) using restrictions q[~ of the projection q, see the author[12], [13], and [1~].
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lp(O
p > 0) : Hardy spaces. I I ~ E j " Infinite product of Ej. NLP: Nonlinear programming. mrnq /P m! : New universal constant. qu(x): Minkowsky functional. E - limieA Ei 9Surjective limit of Ei. d(x, y): Distance between x, y. H ( E ) : Entire functions(continuous and Gateaux analytic). Hbb(E) 9Entire functions of type ~ CX, bounded on bounded sets of E. /)" Linear completion. !
279 Ea : Holomorphic completion.
HBEP: Hahn Banach extension property. HBSP: Hahn Banach separation property. CP(E) : The set of all continuous pseudonorms. (f t, q) : Schlichet domain.
La(E, F): Space of linear mappings. L(E, F): Space of continuous linear mappings. La(mE, F): Algebraic m-linear maps of E into F. L~s(mE, F): Algebraic symmertric m-linear maps of E into F. L(E m, F): Space of continuous m-linear maps of E into F. L~(mE) = Ls(mE,K) : Continuous symmertric m-linear maps. L(E1, .., Em; F): Continuous multilinear maps of E1 x ... x Em into F. P~(mE; F): Algebraic m-homogenous polynomials of E into F. P(mE; F): Continuous m-homogenous polynomials of E into F. p(mE): Continuous m-homogeneous polynomials of E into K. A b - { (1 - t)l/pa + tl/pb}, 0 _< t _< 1 "Arc segment between a and b. BE(0, 1): Unit ball of E of center at the origin. Cp(A)" Closed p-convex hull Of a set A. Ep(K) : p - E x t r e m e points of K. d(x, O) =11 x lip: p-norm defined by a p-homogeneous metric d. L p ( I ) - {f; f : l f
Ip< oo}" Lebsgue integrable functions on [0, 1].
A(x) = f (x) - f (a) - T(x - a): Affine linear functional. QA (U,F): Quasi-analytic (Bayoumi analytic)maps from U to F. QH (U; F): Quasi-holomorphic(Bayoumi holomorphic)maps fromUto F. QA (U) = QA (U; IK): Quasi-analytic( Bayoumi analytic)maps. QH(U) = QH(U,~) : Quasi-holomorphic(Bayoumi holomorphic)maps. Qc~ (U, F) : Quasi-C~Analytic maps from U to F. ~(a~_l ~) 9Characteristic function on (c~i-1, ai). deg(f, M, x ) : Degree of a mapping f On M at x C f(M). det(A) : Determinant of A.
II I llA= sup A JI (x)IT0: Compact open topology. Bdd(E) : Bounded susets of E. WBdd(E) : Weakly bounded subsets of E. Bg (E) : Bounding sets in E. WBg (E) 9Weakly-bounding sets(Limited sets) in E. WbBg (E) :Bounded Weakly-bounding sets(Bayoumi limited sets) in E. P C X ( E ) : O p e n pseudoconvex sets of E. PSH(U) : Plurisubharmonic functions on U.
280
a.p." Approximation property. b.a.p: Bounded approximation property. [ei]~ 9Cosets of the basis ei. H~ 9Banach envelope of Hp, l~ - {x - ( ~ ) ; x~ e r E ~ I~nP < ~ } , ( 0 < p < 1). l o - n ; > 0 1 ; - { x - (x~); sup;>0 E ~ I ~ I;< o o } . l~-Uq_plq - {x E/q; supq>__p~--}7 I xj Iq< c~ }, (0 < p _< 1). f ~ ( s ) - sup~