Topological Vector Spaces,9780387987262

Topological Vector Spaces

by ;
Edition: 2nd
ISBN13: 9780387987262
ISBN10: 0387987266
Format: Hardcover
Pub. Date: 1999-08-01
Publisher(s): Springer Verlag
List Price: $99.99

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Summary

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

Table of Contents

Preface to the Second Edition v(1)
Preface vi
Prerequisites 1(11)
A. Sets and Order 1(3)
B. General Topology 4(5)
C. Linear Algebra 9(3)
I. TOPOLOGICAL VECTOR SPACES
12(24)
Introduction 12(1)
1 Vector Space Topologies
12(7)
2 Product Spaces, Subspaces, Direct Sums, Quotient Spaces
19(2)
3 Topological Vector Spaces of Finite Dimension
21(3)
4 Linear Manifolds and Hyperplanes
24(1)
5 Bounded Sets
25(3)
6 Metrizability
28(3)
7 Complexification
31(2)
Exercises
33(3)
II. LOCALLY CONVEX TOPOLOGICAL VECTOR SPACES
36(37)
Introduction 36(1)
1 Convex Sets and Semi-Norms
37(3)
2 Normed and Normable Spaces
40(5)
3 The Hahn-Banach Theorem
45(2)
4 Locally Convex Spaces
47(4)
5 Projective Topologies
51(3)
6 Inductive Topologies
54(6)
7 Barreled Spaces
60(1)
8 Bornological Spaces
61(2)
9 Separation of Convex Sets
63(3)
10 Compact Convex Sets
66(2)
Exercises
68(5)
III. LINEAR MAPPINGS
73(49)
Introduction 73(1)
1 Continuous Linear Maps and Topological Homomorphisms
74(2)
2 Banach's Homomorphism Theorem
76(3)
3 Spaces of Linear Mappings
79(3)
4 Equicontinuity. The Principle of Uniform Boundedness and the Banach-Steinhaus Theorem
82(5)
5 Bilinear Mappings
87(5)
6 Topological Tensor Products
92(5)
7 Nuclear Mappings and Spaces
97(9)
8 Examples of Nuclear Spaces
106(2)
9 The Approximation Property. Compact Maps
108(7)
Exercises
115(7)
IV. DUALITY
122(81)
Introduccion 122(1)
1 Dual Systems and Weak Topologies
123(5)
2 Elementary Properties of Adjoint Maps
128(2)
3 Locally Convex Topologies Consistent with a Given Duality. The Mackey-Arens Theorem
130(3)
4 Duality of Projective and Inductive Topologies
133(7)
5 Strong Dual of a Locally Convex Space. Bidual. Reflexive Spaces
140(7)
6 Dual Characterization of Completeness. Metrizable Spaces. Theorems of Grothendieck, Banach-Dieudonne, and Krein-Smulian
147(8)
7 Adjoints of Closed Linear Mappings
155(6)
8 The General Open Mapping and Closed Graph Theorems
161(6)
9 Tensor Products and Nuclear Spaces
167(9)
10 Nuclear Spaces and Absolute Summability
176(9)
11 Weak Compactness. Theorems of Eberlein and Krein
185(5)
Exercises
190(13)
V. ORDER STRUCTURES
203(55)
Introduction 203(1)
1 Ordered Vector Spaces over the Real Field
204(10)
2 Ordered Vector Spaces over the Complex Field
214(1)
3 Duality of Convex Cones
215(7)
4 Ordered Topological Vector Spaces
222(3)
5 Positive Linear Forms and Mappings
225(5)
6 The Order Topology
230(4)
7 Topological Vector Lattices
234(8)
8 Continuous Functions on a Compact Space. Theorems of Stone-Weierstrass and Kakutani
242(8)
Exercises
250(8)
VI. C* -- AND W* -- ALGEBRAS
258(48)
Introduction 258(1)
1 Preliminaries
259(1)
2 C*-Algebras. The Gelfand Theorem
260(7)
3 Order Structure of a C*-Algebra
267(3)
4 Positive Linear Forms. Representations
270(4)
5 Projections and Extreme Points
274(3)
6 W*-Algebras
277(10)
7 Von Neumann Algebras. Kaplansky's Density Theorem
287(5)
8 Projections and Types of W*-Algebras
292(7)
Exercises
299(7)
Appendix. SPECTRAL PROPERTIES OF POSITIVE OPERATORS
306(19)
Introduction 306(1)
1 Elementary Properties of the Resolvent
307(2)
2 Pringsheim's Theorem and Its Consequences
309(7)
3 The Peripheral Point Spectrum
316(9)
Index of Symbols 325(5)
Bibliography 330(9)
Index 339

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