Schwartz Spaces, Nuclear Spaces and Tensor Products
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726 Yau-Chuen Wong
Schwartz Spaces, Nuclear Spaces and Tensor Products
Springer-Verlag Berlin Heidelberg New York 1979
Author Yau-Chuen Wong Department of Mathematics United College The Chinese University of Hong Kong Shatin, N.T./Hong Kong
AMS Subject Classifications (1970): 46A05, 46A 15, 46A45, 46 M05, 47B10, 47015 ISBN 3-540-09513-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-09513-6 Springer-Verlag New York Heidelberg Berlin Library of Congress Cataloging in Publication Data Wong, Yau-chuen. Schwartz spaces, nuclear spaces, and tensor products. (Lecture notes in mathematics; 726) Bibliography: p. Includes index. 1. Schwartz spaces. 2. Nuclear spaces (Functional analysis) 3. Tensor products. I. Title. II. Series: Lecture notes in mathematics (Berlin); 726. OA3.L28 no. 726 [0A322)510'.8s [515'.73) 79-16330
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TO MY WIFE
Judy, Dick-Ha Nei
INTRODUCTION
Nuclear spaces were discovered by A. Grothendieck, and the most important part of the theory of nuclear spaces was developed in his famous article (see Grothendieck [ 2]).
Unfortunately, the machinery of tensor
products used there is very cumbersome; this makes the theory some unnecessary difficulties and complicacy.
Pietsch [ 1] was the first to simplify the theory
of nuclear spaces by using locally convex spaces of summable and absolutely summable families, instead of locally convex tensor products.
It is
~
impression
that the idea of using seminorms satisfying some expected properties to investigate some classes of locally convex spaces, would make the development of the whole theory easier. the topology
() S
This has been done by the author [1] for stu~ing
of uniform convergence on ordeJ:"'-bounded sets.
Thus, one of
the purposes of these lecture notes is to embark upon a unifying treatment of Schwartz spaces, nuclear spaces, and A-nuclear spaces by using this idea. Therefore, this monograph is actually a continuation of that written by the author [ 1]. These lecture notes contain six chapters, starting with Chapter 0 in which terminology and some elementary results in topological vector spaces, that will be used through these notes, are given.
Chapter 1 deals with Schwartz
spaces by means of precompactness of seminorms. nuclear spaces in terms of
prenucleari~
Chapter 3 is concerned with
of seminorms; in order to show that
our definition of nuclearity coincides with the usual one, we use the concept of summabili ty wh
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