Topics in Functional Analysis
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45 Albert Wilansky Lehigh University, Bethlehem, Pennsylvania
Topics in Functional Analysis Notes by W. D. Laverell
1967
Springer-Verlag· Berlin· Heidelberg· New York
All rights, especially that of translation into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photomechanical means (photostat, microfilm and/or microcard) or by other procedure without writteIJ. permission from Springer Verlag. C by Springer-Verlag Berlin' Heidelberg 1967 Library of Congress Catalog Card Number 67-30514. Printed in Germany. Title No. 7365
Introduction Several topics are touched on in these notes. 1. When spaces are continuously embedded in a fixed Hausdorff space, the identity map between them has a closed graph. Since this allows use of forms of the closed graph theorem, one tries to see when a collection of given spaces can be so embedded. Since they are automatically continuously embedded in their inductive limit, the question of when the inductive limit is separated arises. Since the inductive limit is a quotient of the direct sum, an obvious criterion for separation is at hand. (See Chapter 2, Section 8). For two spaces, this criterion reduces to the familiar one that the identity map between them have closed graph. Thus we obtain the insight that the graph of the identity map should be considered as closed in the direct sum, rather than in the product of two spaces, when it is desired to generalize
this concept to
more than two spaces.
2. To round out the discussion of embedded spaces and closed graph, we have included some material on comparison of topologies (Chapter 1) including a new type of space in which sequential convergence is trivial (Section 4). Also lattice properties and completions of embedded spaces are discussed.
3. For wider applicability of the results, local convexity was not always assumed. This leads to what we call the unrestricted inductive limit which is discussed in Chapter 2. Removing local convexity from the concept of barreled, we have
some ideas introduced by the author and some by Wendy Robertson in Chapter 3. Also shown in Chapter 3 are some connections between bornological and sequential spaces. The digression into sequential spaces was suggested by the trivial sequential convergence shown in Chapter 1,
Section 4.
4. Finally an important result of my former student, A. K. Snyder, on conull FK spaces is presented in Chapter 6. For use in proving Snyder's theorem we present a self-contained version of parts of the two-norm theory recently invented by the Polish school.
Statement of Novelty of Results
Much of the material in these notes is expository. The author claims as new only Theorem 1.4, p. 2; Chapter 1, Section 2; Lemma 3.2, p. 12; Chapter 1, Section 4, except for the first two lines; Chapter 2, Section 3; Chapter 2, Sections 7,8; Example 1.1, p. 45; Theorem 1.1, Example 1.3, and Corollary, pp. 46-48; Examples 2.1, 2.2, pp. 52-53; Theorem 2.4, p. 54; Theorem 2.5, p. 55; the proof on p. 56; pp. 57-58; Example 2.8, p. 60;
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