LINEAR TOPOLOGICAL SPACES OF CONTINUOUS VECTOR-VALUED FUNCTIONS Liaqat Ali

350 Pages·2013·1.82 MB·English

LINEAR TOPOLOGICAL SPACES OF
CONTINUOUS VECTOR-VALUED FUNCTIONS
Liaqat Ali Khan
AP
Academic Publications
Linear Topological Spaces of
Continuous Vector-Valued Functions
Authors: Liaqat Ali Khan
Address: Department of Mathematics
Faculty of Science
King Abdulaziz University, Jeddah
DOI: 10.12732/acadpubl.201301
Pages: 350
Typesetting system: LATEX
(cid:13)cAcademic Publications, Ltd., 2013
Allrightsreserved. Thisworkmaynotbetranslatedorcopiedinwholeorin
part without the written permission of the publisher (Academic Publications,
Ltd., http://www.acadpubl.eu), except for scientific or scholarly analysis.
(cid:13)c Academic Publications, Ltd., 2013
http://www.acadpubl.eu
Contents
Introduction vi
Preface vii
Acknowledgements xiii
Chapter 1. The Strict and Weighted Topologies 1
1. Strict and related Topologies on C (X,E) 2
b
2. Weighted Topology on CV (X,E) 20
b
3. Notes and Comments 24
Chapter 2. Completeness in Function Spaces 27
1. Completeness in the uniform topology 28
2. Completeness in the Strict Topology 33
3. Completion of (C (X,E),β) 38
b
4. Completeness in the Weighted Topology 41
5. Notes and Comments 43
Chapter 3. The Arzela-Ascoli type Theorems 45
1. Arzela-Ascoli Theorem for k and β Topologies 46
2. Arzela-Ascoli Theorem for Weighted Topology 53
3. Notes and Comments 61
Chapter 4. The Stone-Weierstrass type Theorems 63
1. Approximation in the Strict Topology 64
2. Approximation in the Weighted Topology 72
3. Weierstrass Polynomial Approximation 77
4. Maximal Submodules in (Cb(X,E),β) 79
5. The Approximation Property 82
6. Notes and Comments 87
Chapter 5. Maximal Ideal Spaces 89
1. Maximal Ideals in (C (X,A),β) 90
b
2. Maximal Ideal Space of (C (X,A),β) 94
b
3. Notes and Comments 99
iii
iv CONTENTS
Chapter 6. Separability and Trans-separability 101
1. Separability of Function Spaces 102
2. Trans-separability of Function Spaces 107
3. Notes and Comments 114
Chapter 7. Weak Approximation in Function Spaces 115
1. Weak Approximation in (C (X,E),β) 116
b
2. Weak Approximation in CV (X,E) 120
o
3. Notes and Comments 126
Chapter 8. The Riesz Representation type Theorems 127
1. Vector-valued Measures and Integration 128
2. Integral Representation Theorems 135
3. Notes and Comments 140
Chapter 9. Weighted Composition Operators 141
1. Multiplication Operators on CV (X,E) 142
o
2. Weighted Composition Operators on CV (X,E) 152
o
3. Compact Weighted Composition Operators 159
4. Notes and Comments 166
Chapter 10. The General Strict Topology 167
1. Strict Topology on Topological Modules 168
2. Essentiality of Modules of Vector-valued Functions 177
3. Topological Modules of Homomorphisms 180
4. Notes and Comments 187
Chapter 11. Mean Value Theorem and Almost Periodicity 189
1. The Mean Value Theorem in TVSs 190
2. Almost Periodic Functions with Values in a TVS 194
3. Notes and Comments 203
Chapter 12. Non-Archimedean Function Spaces 205
1. Compact-open Topology on C(X,E) 206
2. Strict Topology on C (X,E) 214
b
3. Maximal Ideals in NA Function Algebras 222
4. Notes and Comments 227
Appendix A. Topology and Functional Analysis 229
1. Topological Spaces 230
2. Topological Vector Spaces 243
3. Spaces of Continuous Linear Mappings 261
4. Shrinkable Neighborhoods in a TVS 268
CONTENTS v
5. Non-Archimedean Functional Analysis 272
6. Topological Algebras and Modules 275
7. Measure Theory 296
8. Uniform Spaces and Topological Groups 300
9. Notes and Comments 310
Appendix. Bibliography 313
Appendix B. List of Symbols 329
Appendix C. Index 333
vi CONTENTS
Introduction
This monograph is devoted to the study of linear topological spaces
of continuous bounded vector-valued functions endowed with the uni-
form, strict, compact-open and weighted topologies. In the past four
decades, several major results onvector-valued function spaces have been
extended to the non-locally setting. These include generalized versions of
some classical results such as the Stone-Weierstrass theorem, the Arzela-
Ascoli theorem, and the Riesz representation theorem. These also in-
clude maximal ideal spaces of function algebras, separability and trans-
separability, weak approximation, composition operators, general strict
topology on topological modules, the mean value theorem and almost
periodicity for vector-valued functions, and non-Archimedean function
spaces. Our main objective is to present recent developments in these
areas. Several examples and counter-examples are included in the text.
Background material on Topology and Functional Analysis is included
in the Appendix. Also, an up-to-date bibliography is included to assist
research in further studies.
AMS Subject Classification. [2000] Primary 46E10, 46E40; Sec-
ondary 46A10, 28A25, 46H25, 47B33
Preface
The purpose of this monograph is to present an up-to-date study of
linear topological spaces of continuous bounded vector-valued functions
endowed withtheuniform, strict, compact-openandweighted topologies.
In the past four decades, several new results have had been obtained
in this direction. The main topics include generalized versions of some
classical results such as the Stone-Weierstrass theorem, the Arzela-Ascoli
theorem, and the Riesz representation theorem. These also include the
studyofmaximalidealspacesoffunctionalgebras,separabilityandtrans-
separability, weak approximation, composition operators, general strict
topology on topological modules, the mean value theorem and almost
periodicity for vector-valued functions, and non-Archimedean function
spaces. Our main objective is to present recent developments in the non-
locallyconvexsetting, thatis,intopologicalvectorspaces, notnecessarily
locally convex. So it covers both the cases: (a) locally convex spaces, (b)
topological vector spaces which are not locally convex. We mention that
”non-convexity” has also been the main theme in the monographs by
L. Waelbroeck (1971), N. Adasch, B. Ernst and D. Keim (1978), N.J.
Kalton, N.T. Peck and J.W. Roberts (1984), S. Rolewicz (1985) and A.
Bayoumi (2003); see also the papers by W. Robertson (1958), V. Klee
(1960a, 1960b) and J. Kakol (1985, 1987, 1990, 1992).
Our exposition relies mostly on the original papers (published during
1972-2012) with some additional clarifications. We have tried to make
the text easily readable and as self-contained as possible. The only pre-
requisites for reading the bookare topology, functional analysis andmea-
sure theory of the undergraduate level (e.g. the material given in H.L.
Royden’s book: Real Analysis). For this purpose, we have included back-
ground material on Topology and Functional Analysis (such as topolog-
ical spaces, topological vector spaces, non-Archimedean functional anal-
ysis, topological algebras and measure theory) in Appendix A. Also, an
up-to-date bibliography is included to assist research in further studies.
The monograph is intended basically a monograph for researchers work-
ing in vector-valued function spaces. However, those working in this and
vii
viii PREFACE
related fields may conveniently choose some topics for a one or two se-
mester course work. In fact, the author has been using portions of the
text in his lectures to M.Sc./M.Phil. students as a one semester course
before initiating their research in a relevant field of function spaces.
To introduce the material in its historical perspective, consider a
topological space X and a topological vector space E over K (= R or
C). Let C(X,E) (resp. C (X,E)) be the vector space of all continuous
b
(resp. continuous and bounded) E-valued functions on X; if E = K,
these spaces are simply denoted by C(X) and C (X). If X is compact,
b
then C(X,E) = C (X,E) and C(X) = C (X); in this case the uni-
b b
form topology (i.e. the sup norm topology) u is the appropriate one to
study on C(X). After the appearance of M.H. Stone’s paper of 1937, the
space (C(X),u) in the case of compact X has been an intensively studied
mathematical object. Its interest arises in part from its rich structure:
under the uniform topology u, C(X) is a Banach space; under pointwise
multiplication, it is an algebra; under the natural ordering, C(X,R) is
a lattice (see, e.g., Kakutani (1941), Kaplansky (1947b), Hewitt (1948),
Myers (1950), M. and S. Krein (1940), etc.). If X is not compact, the
uniform topology u on C(X) is not well-defined since C(X) may contain
some unbounded functions. However, in this case, the compact-open
topology k is the most useful topology to be considered on C(X) and on
C(X,E) with E even a topological space. It has been studied by several
authors over the past sixty years and significant contributions have been
made in this field, among others, by Fox (1945), Arens (1946), Myers
(1946), Hewitt (1948), Warner (1958), Wheeler (1976), McCoy (1980);
see also the monographs by Semadeni (1971), Schmets (1983), McCoy
and Ntantu (1988), Arkhangel’skii (1992) and Tkachuk (2011). If X is
againnon-compact andwe consider thespaces C (X)andC (X,E), then
b b
both u and k topologies are well-defined and we have k 6 u.
In 1958, R.C. Buck introduced the notion of strict topology β on
C (X,E) in the case of X locally compact and E a locally convex space.
b
The problems discussed in the Buck’s paper (1958) are:
(1) Relationship between the β,k and u topologies on C (X,E) (e.g.
b
k 6 β 6 u with k = u iff X is compact; k = β on u-bounded sets);
(2) Completeness of C (X,E),β);
b
(3) Stone-Weierstrass theorems for (C (X),β) and C (X,E ),β);
b b n
(4) Characterization of maximal β-closed ideals in C (X);
b
(5) Identification of the β-dual of C (X) with the space M(Bo(X))
b
of regular Boreal measures on X, via the integral representations;
(6) The open problem whether or not (C (X),β) a Mackey space.
b
PREFACE ix
After the appearance of the Buck’s paper, a large number of pa-
pers have appeared in the literature concerned with extending Buck’s
results to more general cases or studying further properties of β, and
also with obtaining some variants of β. In particular Todd (1965) and
Wells (1965)], independently, established the Stone-Weierstrass theorem
for (C (X),β). Todd (1965) also characterized the maximal β-closed
b
C (X)-submodules of C (X,E) while Wells (1965) identified the β-dual
b b
of C (X,E) with a certain space M(X,E∗) of E∗-valued measures on X.
b
Conway (1967) and LeCam (1957), independently, proved that, if X is
locally compact and paracompact, (C (X,E),β) is a Mackey space but
b
it is not so in general. Collins and Dorroh (1968) proved that (C (X),β)
b
has the approximation property. Dorroh (1969) also showed that β is
the finest locally convex topology on C (X) which agrees with k on u-
b
bounded sets.
The next major development in the study of β topology on C (X)
b
has been its extension to the case of a completely regular space X, given,
independently, by Van Rooij (1967), Giles (1971), Fremlin, Garling and
Haydon (1972), Gulick (1972), Hoffman-Jφregensen (1972), and Sentilles
(1972). In fact, Fremlin, Garling and Haydon (1972) and, independently,
Sentilles (1972) introduced on C (X) three types of strict topologies: the
b
substrict topology β , the strict topology β and the superstrict topology
o
β with β being equivalent to the strict topology ‘β’ of Buck (1958) in
1
the case of X a locally compact space. They also identified the topo-
logical duals of (C (X),β ), (C (X),β) and (C (X),β ) with the spaces
b o b b 1
M (X), M (X) and M (X) of tight, τ-additive and σ-additive measures,
t τ σ
respectively, on X (see LeCam (1957), Varadarajan (1965) and Wheeler
(1983) for detail of these spaces of measures). Subsequently, several au-
thors have further explored the properties of β ,β and β for both C (X)
o 1 b
and C (X,E), where E is a normed space or a locally convex space; see,
b
e.g. Mosiman and Wheeler (1972), Wheeler (1973), Summers (1972),
Haydon (1976), Cooper (1971), Choo (1979), Fontenot (1974), Katsaras
(1975), Khurana (1978a, 1978b), Morishita and Khan (1997), Zafarani
(1986, 1988), and also the survey papers by Hirschfeld (1978), Collins
(1976) and Wheeler (1983), and monographs by Cooper (1978), Prolla
(1977), Schmets (1983), Singh and Manhas (1993).
In1967,Nachbinstudiedindetailthemoregeneralnotionofweighted
topology ω on certain subspaces CV (X) and CV (X) of C(X) and
V b o
β,k, and u were shown to be the special cases of ω for suitable choices
V
of V consisting of non-negative upper-semicontinuous functions on X.
Further work on ω has been done by Summers (1969, 1971), Prolla
V
x PREFACE
(1971a, 1971b), Bierstedt (1973, 1975), and others. Singh and Summers
(1988) and Singh and Manhas (1991, 1992) made an extensive study
of composition and multiplication operators, respectively on CV (X,E).
b
Another useful topology on C (X) is the σ-compact-open topology σ
b
whichwasintroducedbyGulick(1972)andfurther studiedbyGulickand
Schmets (1972). The strict and related topologies have also been studied
on not necessarily ‘function spaces’. For instance, Wiwegar (1961) and
Cooper (1971, 1978) defined it in the form of ‘mixed topology’ on a
normed space; Busby (1968) considered it on the double centralizer (or
multiplier) algebra M (A) of Banach algebra A; Sentilles and Taylor
d
(1969) defined it on a Banach A-module; Ruess (1977) studied it on an
arbitrary locally convex space (see the survey paper by Collins (1976)).
In all the above mentioned investigations about C (X,E), E has
b
been assumed to be the scalar field or a locally convex space. The
case of E a general topological vector space has been first considered by
Shuchat in(1972a, 1972b)where heestablished several useful approxima-
tion results for (C (X,E),u) and characterized the dual of (C (X,E),u),
b b
with X a compact space and without assuming the local convexity of
E (see also Bierstedt (1973) and Wealbroeck (1971, 1973)). In 1979,
Khan defined the β and other related topologies on C (X,E), where
b
X is a Hausdorff space and E any Hausdorff topological vector space,
and showed that β has almost all the properties of the “strict topol-
ogy” studied by the above authors. Further work in this direction has
been done in later years by Khan (1980 through 2011), Khan-Rowlands
(1981, 1991), Katsaras (1981, 1983), Kalton (1983), Nawrocki (1985,
1989), Abel (1987, 2004), Prolla (1993b), Khan-Thaheem (1997, 2002),
Manhas-Singh (1998), Khan-Mohammad-Thaheem (1999, 2005), Khan-
Oubbi (2005), Katsaras-Khan-Khan (2011), Katsaras (2011) and others.
The purpose of this monograph is to present some results of these au-
thors regarding the strict, weighted, and related topologies on C (X,E)
b
andCV (X,E)inthenon-locallyconvex setting. Wealsostudy thestrict
o
and related topologies on any topological modules. Several examples and
counter-examples are included in the text. .
The monograph consists of twelve chapters which are organized as
follows.
In Chapter 1, we introduce and study the strict topology β (and the
relatedonessuch astheuniformtopologyu, compact-opentopologyk, σ-
compact-open topology σ) and also the more general notion of weighted
topology ω on the function spaces C (X,E) (resp. CV (X,E)) with E
V b b