The Topology of Uniform Convergence on Order-Bounded Sets
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531 Yau-Chuen Wong
The Topology of Uniform Convergence on Order-Bounded Sets
Springer-Verlag Berlin.Heidelberg
9New York1976
Author Yau-Chuen Wong Department of Mathematics United College The Chinese University of Hong Kong Shatin, N.T./Hong Kong
Library of Congress Cataloging in Publication Data
Wong, Yau-ehu~n. The topology of uniform convergence on order-bounded sets.
(Lecture notas in mathematics ; 531) Bibliography: p. Includes index. i. Linear topological spaces. 2. Convergence. 3. Duality theory (Mathematics) I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 531. QA3.L28 vol. 531 [QA322~ 515'.73 76-~6481
AMS Subject Classifications (1970): 06A65, 46A05, 46A15, 46A20, 46A35, 46A40, 46A45, 46A99, 47 B55, 47D15 ISBN 3-540-07800-2 Springer-Verlag Berlin Heidelberg 9 New 9 York ISBN 0-387-07800-2 Springer-Verlag New York Heidelberg 9 Berlin 9 This .v~ork is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 9 by Springer-Verlag Berlin - Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
CONTENTS
V
INTRODUCTION CHAPTER 1. 1.1
A SURVEY OF ORDERED V E C T ~ SPACES
Duality theorems
2
1.2
Seminorms on ~d~red vector spaces
20
1.3
Topologies on ~dered vector spaces
28
CHAPTER 2.
CRDERS AND TOPOLOGIES ON SPACES CONSISTING OF FAMILIES
2.1
Summability of families
45
2.2
Locally solid topologies on spaces consisting of families
2.3
The topological dual of
2.4
The topological dual of
CHAPTER 3.
54 ~ < A , E> mo(A , E)
80 and of
mo,2(A,E )
92
SOME CHARACTERIZATIONS OF T ~ TOPOLOGY OF UNI/ORM CONVERGENCE ON 0EDER-BOU~DED SETS
3.1
Cone-absolutely s,,mm~.~ mappin6s
IO~
3.2
Some special classes of semincrm8
126
3.3
Cone-prenuo lear mappings
141
BIBLIOGRAPHY
156
INDEX AND SYMBOLS
160
INTRODUCr ION In studying ordered topological vector spaces, particularly important roles are played by two intrinsic topologies:
the order-bounded (or order) topology and the
topology ~S of uniform convergence on all order-bounded sets. The order-bound topology was studied independently by Schaefer El] and Namioka [1] , while the topology ~S
was studied by Nakano [i] and Dieudonn& in the special case of locally
convex Riesz spaces, and by Peressini [3] in a fairly general setting (he used the notation
o(E, E')). A remarkable theorem of Nakano [I] (asserting that, for topolo-
gical Riesz spaces, topological completeness follows from certain order completeness assumption) is one of the deepest results in the theory of locally convex Riesz spaces; the author showed, in 1969, that ~S verse of Nakano's theorem.
is relevant for establ
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