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Topological Vector Spaces : Graduate Texts in Mathematics - H. H. Schaefer

Topological Vector Spaces

Graduate Texts in Mathematics


Published: 24th June 1999
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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.

"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

Preface to the Second Edition
Sets and Orderp. 1
General Topologyp. 4
Linear Algebrap. 9
Topological Vector Spaces
Vector Space Topologiesp. 12
Product Spaces, Subspaces, Direct Sums, Quotient Spacesp. 19
Topological Vector Spaces of Finite Dimensionp. 21
Linear Manifolds and Hyperplanesp. 24
Bounded Setsp. 25
Metrizabilityp. 28
Complexificationp. 31
Locally Convex Topological Vector Spaces
Convex Sets and Semi-Normsp. 37
Normed and Normable Spacesp. 40
The Hahn-Banach Theoremp. 45
Locally Convex Spacesp. 47
Projective Topologiesp. 51
Inductive Topologiesp. 54
Barreled Spacesp. 60
Bornological Spacesp. 61
Separation of Convex Setsp. 63
Compact Convex Setsp. 66
Linear Mappings
Continuous Linear Maps and Topological Homomorphismsp. 74
Banach's Homomorphism Theoremp. 76
Spaces of Linear Mappingsp. 79
Equicontinuity. The Principle of Uniform Boundedness and the Banach-Steinhaus Theoremp. 82
Bilinear Mappingsp. 87
Topological Tensor Productsp. 92
Nuclear Mappings and Spacesp. 97
Examples of Nuclear Spacesp. 106
The Approximation Property. Compact Mapsp. 108
Dual Systems and Weak Topologiesp. 123
Elementary Properties of Adjoint Mapsp. 128
Locally Convex Topologies Consistent with a Given Duality. The Mackey-Arens Theoremp. 130
Duality of Projective and Inductive Topologiesp. 133
Strong Dual of a Locally Convex Space. Bidual. Reflexive Spacesp. 140
Dual Characterization of Completeness. Metrizable Spaces. Theorems of Grothendieck, Banach-Dieudonne, and Krein-Smulianp. 147
Adjoints of Closed Linear Mappingsp. 155
The General Open Mapping and Closed Graph Theoremsp. 161
Tensor Products and Nuclear Spacesp. 167
Nuclear Spaces and Absolute Summabilityp. 176
Weak Compactness. Theorems of Eberlein and Kreinp. 185
Order Structures
Ordered Vector Spaces over the Real Fieldp. 204
Ordered Vector Spaces over the Complex Fieldp. 214
Duality of Convex Conesp. 215
Ordered Topological Vector Spacesp. 222
Positive Linear Forms and Mappingsp. 225
The Order Topologyp. 230
Topological Vector Latticesp. 234
Continuous Functions on a Compact Space. Theorems of Stone-Weierstrass and Kakutanip. 242
C*-- and W*-- Algebras
Preliminariesp. 259
C-Algebras. The Gelfand Theoremp. 260
Order Structure of a C-Algebrap. 267
Positive Linear Forms. Representationsp. 270
Projections and Extreme Pointsp. 274
W-Algebrasp. 277
Von Neumann Algebras. Kaplansky's Density Theoremp. 287
Projections and Types of W-Algebrasp. 292
Spectral Properties of Positive Operators
Elementary Properties of the Resolventp. 307
Pringsheim's Theorem and Its Consequencesp. 309
The Peripheral Point Spectrump. 316
Index of Symbolsp. 325
Bibliographyp. 330
Indexp. 339
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780387987262
ISBN-10: 0387987266
Series: Graduate Texts in Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 349
Published: 24th June 1999
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 1.91
Weight (kg): 1.53
Edition Number: 2
Edition Type: Revised