Convergence Structures and Applications to Functional Analysis

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Overview
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Table of Contents

Overview

This text offers a rigorous introduction into the theory and methods of convergence spaces and gives concrete applications to the problems of functional analysis. While there are a few books dealing with convergence spaces and a great many on functional analysis, there are none with this particular focus.

The book demonstrates the applicability of convergence structures to functional analysis. Highlighted here is the role of continuous convergence, a convergence structure particularly appropriate to function spaces. It is shown to provide an excellent dual structure for both topological groups and topological vector spaces.

Readers will find the text rich in examples. Of interest, as well, are the many filter and ultrafilter proofs which often provide a fresh perspective on a well-known result.

Audience: This text will be of interest to researchers in functional analysis, analysis and topology as well as anyone already working with convergence spaces. It is appropriate for senior undergraduate or graduate level students with some background in analysis and topology.

Product Details

ISBN-13: 9781402005664
Publisher: Springer Netherlands
Publication date: 3/31/2002
Edition description: 2002
Edition number: 1
Pages: 264
Product dimensions: 9.21 (w) x 6.14 (h) x 0.69 (d)

	Introduction
1	Convergence spaces	1
1.1	Preliminaries	1
1.2	Initial and final convergence structures	4
1.3	Special convergence spaces, modifications	10
1.4	Compactness	21
1.5	The continuous convergence structure	25
1.6	Countability properties and sequences in convergence spaces	42
1.7	Sequential convergence structures	51
1.8	Categorical aspects	57
2	Uniform convergence spaces	59
2.1	Generalities on uniform convergence spaces	59
2.2	Initial and final uniform convergence structures	67
2.3	Complete uniform convergence spaces	69
2.4	The Arzela-Ascoli theorem	71
2.5	The uniform convergence structure of a convergence group	75
3	Convergence vector spaces	79
3.1	Convergence groups	79
3.2	Generalities on convergence vector spaces	85
3.3	Initial and final vector space convergence structures	88
3.4	Projective and inductive limits of convergence vector spaces	96
3.5	The locally convex topological modification	102
3.6	Countability axioms for convergence vector spaces	109
3.7	Boundedness	111
3.8	Notes on bornological vector spaces	116
4	Duality	119
4.1	The dual of a convergence vector space	119
4.2	Reflexivity	125
4.3	The dual of a locally convex topological vector space	131
4.4	An application of continuous duality	148
4.5	Notes	152
5	Hahn-Banach extension theorems	153
5.1	General results	154
5.2	Hahn-Banach spaces	160
5.3	Extending to the adherence	164
5.4	Strong Hahn-Banach spaces	172
5.5	An application to partial differential equations	178
5.6	Notes	181
6	The closed graph theorem	183
6.1	Ultracompleteness	184
6.2	The main theorems	187
6.3	An application to web spaces	192
7	The Banach-Steinhaus theorem	195
7.1	Equicontinuous sets	196
7.2	Banach-Steinhaus pairs	198
7.3	The continuity of bilinear mappings	204
8	Duality theory for convergence groups	207
8.1	Reflexivity	208
8.2	Duality for convergence vector spaces	215
8.3	Subgroups and quotient groups	217
8.4	Topological groups	224
8.5	Groups of unimodular continuous functions	232
8.6	c- and co-duality for topological groups	240
	Bibliography	247
	List of Notations	257
	Index	259

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Convergence Structures and Applications to Functional Analysis / Edition 1