Table Of ContentTOPOLOGICAL UNIFORM
STRUCTURES
TOPOLOGICAL UNIFORM
STRUCTURES
BY WARREN PAGE
New York City Technical College
City University of New York
Do y
VER PUBLICATIONS, INC.' NEW ORK
Copyright © 1978, 1988 by Warren Page.
All rights reserved under Pan American and International Copyright
Conventions.
Published in Canada by General Publishing Company, Ltd., 30 Lesmill
Road, Don Mills, Toronto, Ontario.
Published in the United Kingdom by Constable and Company, Ltd., IO
Orange Street, London WC2H 7EG.
This Dover edition, first published in 1988, is an unabridged, corrected,
slightly revised republication of the work first published by John Wiley &
Sons ("A Wiley-lnterscience Publication'), New York, 1978.
Manufactured in the United States of America
Dover Publications, Inc., 31East2nd Street, Mineola, N.Y. 11501
Library of Congress Cataloging-in-Publication Data
Page, Warren, 1939-
Topological uniform structures/ Warren Page.
p. cm.
Reprint. Originally published: New York: Wiley, c1978.
Bibliography: p.
Includes index.
ISBN 0-486-65808-2 (pbk.)
1. Uniform spaces. I. Title.
[QA61 l.25.P33 1988]
514'.3202-dcl9 88-23261
CIP
To
Esther Page
and
the memory of Irving Page.
They taught me hope, faith, and perseverance.
Preface
This book aims to acquaint the reader with a slice of mathematics that is
interesting, meaningful, and in the mainstream of contemporary mathe
matical developments. Admittedly a number of excellent sources cover, in
part, uniform spaces, topological groups, topological vector spaces,
topological algebras, and abstract harmonic analysis. I believe, however,
that this is the first known text to give a thorough and fully detailed account
of all these topics. It also contains a section on vector-valued measure
spaces and a plethora of problems and examples that have not yet
appeared in book form. The aforementioned topological structures are
worth studying (particularly for the neophyte mathematician), since they
are the juncture points at which many significant mathematical ideas come
into contact.
Most advanced undergraduate and beginning graduate students, con
strained by time and compartmentalized course requirements, do not yet
fully appreciate the true essence of mathematics as a structured, coherent,
and harmonious whole. This, to a large extent, motivated my style and
presentation. The two most salient features of this work are the following:
(1) The overall unifying theme of topologies compatible with increas
ingly enriched algebraic structures.
(2) Its attempt to gradually deepen the reader's appreciation of the
rich interplay among diverse areas of mathematics.
The interested reader can get a better overview by quickly scanning
the chapter perspectives and section summaries. Briefly, (1) may be
described by following the order of topics covered, subject to the require
ment that each structure's algebraic operations be continuous. What
emerges from (2) is a number of striking results that combine and interlace
algebraic, topological, and measure-theoretic properties associated with
the structure under consideration.
A book of this nature could easily become encyclopedic were it not for
the decision to omit topics. This is always heartrending, even more so when
such topics fit so well and clamor for attention. Given freer rein or an
expanded future edition, I would include a section on proximity spaces (as
structures layered between topological and uniform spaces) and some
material on topological fields and generalized topological structures (i.e.,
topological spaces with continuous generalised internal and external
Vil
Vlll PREFACE
operations). The second theme would also be enhanced by a fuller treat
ment of representation theory and an increased cross-pollination with our
Gelfand theory approach to harmonic analysis.
While writing this text I gave readability and motivation prime consi
deration. The pace is leisurely and the contents, highly illustrative. Each
section begins with a brief summary advertising the salient points to
appear. The theory is intended to unfold smoothly through a good deal of
commentary and an abundance of examples. Furthermore, the remarks
(which follow most theorems and examples) are chock full of useful
information, counterexamples, and other points worth noting. In the nota
tion a remark is referred to as u · v or u · v-, depending on whether it
+
follows or precedes item v of Section u. Finally, each section ends with a set
of problems (more than 220 each with several subparts in all). Some test
understanding and give additional insights. A large number, however,
come from findings in recent literature and research papers. Their
arrangement and emphasis I hope will broaden the student's perspective
and whet his appetite for delving further into some of them.
This work is reasonably self-contained and accessible to students with
a background in elementary analysis, linear algebra, and point set
topology. At the same time it covers a good amount of advanced material
without going off into the purple deep. My intent was to design this text
with a maximum of flexibility. Accordingly, it is a set of interconnected
chapters, each of which is a minibook standing at the threshold of several
advanced texts in these areas. A few to which these chapters are well
geared are Uniform Spaces (Cech [16], Isbell [45]); Topologi.cal Groups
(Pontryagin [80], Montgomery and Zippin [64]); Topological Vector Spaces
(Kothe [55], Treves [98], Edwards [28]); Topological Algebras (Naimark
[69], Narici [72], Rickart [82]) and Abstract Harmonic Analysis (Hewitt
and Ross [41], Sugiura [94], and Rudin [85], followed by Dunkl and
Ramirez [27]). In essence, this text is well suited to a variety of one- or
two-semester courses as outlined in A Guide to This Book. I feel that it will
also prove useful as a reference source and as a source of seminar material
at the advanced undergraduate and beginning graduate level.
Finally, I record here my profound gratitude to Professor George
Bachman, who first triggered my impulse to write this book. His valuable
advice, imaginative ideas, and timely infusions of enthusiasm helped me
carry it to completion. I am also indebted to Professor Paul Sally for his
insightful comments and constructive suggestions during the early stages of
this work.
WARREN PAGE
New York, New York
February 1978
A Guide to this Book
I Uniform Spaces
l
II Topological Groups
l
III Topological Vector Spaces V Abstract Harmonic Analysis
l
IV Topological Algebras
One Year Courses Chapters/Sections
Topological Uniform Structures I-IV
Elements of Modern Analysis II-V
One Semester Courses Chapters/Sections
Topology and Groups I, II
Topological Groups and II, 22 & 23, V
Harmonic Analysis
Topological Vector Spaces III
Topological Algebras 13 & 14, IV, 27 & 28
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