Topological Vector Spaces

By: M.P. Wolff, H.H. Schaefer

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Hardcover | 1 July 1999 | Edition Number 2

At a Glance

Hardcover
368 Pages

Edition Type
Revised

Dimensions(cm)
23.39 x 15.6 x 2.06

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Intended as a systematic text on topological vector spaces, this text assumes familiarity with the elements of general topology and linear algebra. Similarly, the elementary facts on Hilbert and Banach spaces are not discussed in detail here, since the book is mainly addressed to those readers who wish to go beyond the introductory level. Each of the chapters is preceded by an introduction and followed by exercises, which in turn are devoted to further results and supplements, in particular, to examples and counter-examples, and hints have been given where appropriate. This second edition has been thoroughly revised and includes a new chapter on C^* and W^* algebras.

Industry Reviews

"The book has firmly established itself both as a superb introduction to the subject and as a very common source of reference. It is beccoming evident that the book itself will only become irrelevant and pale into insignificance when (and if!) the entire subject of topological vector spaces does. An attractive feature of the book is that it is essentially self-contained, and thus perfectly suitable for senior students having a basic training in the area of elementary functional analysis and set-theoretic topology. My view - let even possibly biased for sentimental resasons - is that the book under review would make for a very practical and useful addition to every matahemtaician's personal office collection."
Vladimir Pestov in Nesletter of the New Zealand Mathematical Society, August 2000

Second Edition

H.H. Schaefer and M.P. Wolff

Topological Vector Spaces

"The reliable textbook, highly esteemed by several generations of students since its first edition in 1966 . . . The book contains a large number of interesting exercises . . . the book of Schaefer and Wolff is worth reading."--ZENTRALBLATT MATH

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Preface to the Second Edition
Preface
Prerequisites
Sets and Order	p. 1
General Topology	p. 4
Linear Algebra	p. 9
Topological Vector Spaces
Vector Space Topologies	p. 12
Product Spaces, Subspaces, Direct Sums, Quotient Spaces	p. 19
Topological Vector Spaces of Finite Dimension	p. 21
Linear Manifolds and Hyperplanes	p. 24
Bounded Sets	p. 25
Metrizability	p. 28
Complexification	p. 31
Locally Convex Topological Vector Spaces
Convex Sets and Semi-Norms	p. 37
Normed and Normable Spaces	p. 40
The Hahn-Banach Theorem	p. 45
Locally Convex Spaces	p. 47
Projective Topologies	p. 51
Inductive Topologies	p. 54
Barreled Spaces	p. 60
Bornological Spaces	p. 61
Separation of Convex Sets	p. 63
Compact Convex Sets	p. 66
Linear Mappings
Continuous Linear Maps and Topological Homomorphisms	p. 74
Banach's Homomorphism Theorem	p. 76
Spaces of Linear Mappings	p. 79
Equicontinuity. The Principle of Uniform Boundedness and the Banach-Steinhaus Theorem	p. 82
Bilinear Mappings	p. 87
Topological Tensor Products	p. 92
Nuclear Mappings and Spaces	p. 97
Examples of Nuclear Spaces	p. 106
The Approximation Property. Compact Maps	p. 108
Duality
Dual Systems and Weak Topologies	p. 123
Elementary Properties of Adjoint Maps	p. 128
Locally Convex Topologies Consistent with a Given Duality. The Mackey-Arens Theorem	p. 130
Duality of Projective and Inductive Topologies	p. 133
Strong Dual of a Locally Convex Space. Bidual. Reflexive Spaces	p. 140
Dual Characterization of Completeness. Metrizable Spaces. Theorems of Grothendieck, Banach-Dieudonne, and Krein-Smulian	p. 147
Adjoints of Closed Linear Mappings	p. 155
The General Open Mapping and Closed Graph Theorems	p. 161
Tensor Products and Nuclear Spaces	p. 167
Nuclear Spaces and Absolute Summability	p. 176
Weak Compactness. Theorems of Eberlein and Krein	p. 185
Order Structures
Ordered Vector Spaces over the Real Field	p. 204
Ordered Vector Spaces over the Complex Field	p. 214
Duality of Convex Cones	p. 215
Ordered Topological Vector Spaces	p. 222
Positive Linear Forms and Mappings	p. 225
The Order Topology	p. 230
Topological Vector Lattices	p. 234
Continuous Functions on a Compact Space. Theorems of Stone-Weierstrass and Kakutani	p. 242
C-- and W-- Algebras
Preliminaries	p. 259
C-Algebras. The Gelfand Theorem	p. 260
Order Structure of a C-Algebra	p. 267
Positive Linear Forms. Representations	p. 270
Projections and Extreme Points	p. 274
W-Algebras	p. 277
Von Neumann Algebras. Kaplansky's Density Theorem	p. 287
Projections and Types of W-Algebras	p. 292
Spectral Properties of Positive Operators
Elementary Properties of the Resolvent	p. 307
Pringsheim's Theorem and Its Consequences	p. 309
The Peripheral Point Spectrum	p. 316
Index of Symbols	p. 325
Bibliography	p. 330
Index	p. 339
Table of Contents provided by Blackwell. All Rights Reserved.

Topological Vector Spaces

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Hardcover

Industry Reviews

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