Topological Vector Spaces and Distributions

by John Horvath

Overview

"The most readable introduction to the theory of vector spaces available in English and possibly any other language."—J. L. B. Cooper, MathSciNet Review
Mathematically rigorous but user-friendly, this classic treatise discusses major modern contributions to the field of topological vector spaces. The self-contained treatment includes complete proofs for all necessary results from algebra and topology. Suitable for undergraduate mathematics majors with a background in advanced calculus, this volume will also assist professional mathematicians, physicists, and engineers.
The precise exposition of the first three chapters—covering Banach spaces, locally convex spaces, and duality—provides an excellent summary of the modern theory of locally convex spaces. The fourth and final chapter develops the theory of distributions in relation to convolutions, tensor products, and Fourier transforms. Augmented with many examples and exercises, the text includes an extensive bibliography.

Product Details

ISBN-13:	9780486488509
Publisher:	Dover Publications
Publication date:	07/17/2012
Series:	Dover Books on Mathematics
Edition description:	Reprint
Pages:	464
Product dimensions:	6.10(w) x 9.20(h) x 1.00(d)

About the Author

John Horvath is Professor Emeritus of Mathematics at the University of Maryland in College Park.

Table of Contents

Terminology and Notations

Chapter 1 Banach Spaces

1 The definition of Banach spaces 5

2 Some notions from algebra and topology 17

3 Subspaces 27

4 Linear maps 35

5 Linear forms 40

6 The Hahn-Banach theorem 45

7 The dual space 52

8 The Banach-Steinhaus theorem 62

9 Banach's homomorphism theorem and the closed-graph theorem 68

Chapter 2 Locally Convex Spaces

1 Some notions from topology 71

2 Filters 75

3 Topological vector spaces 79

4 Locally convex spaces 84

5 Linear maps, subspaces, quotient spaces 97

6 Bounded sets, normability, metrizability 108

7 Products and direct sums 117

8 Convergence of filters 124

9 Completeness 128

10 Finite-dimensional and locally compact spaces 141

11 Initial topologies 149

12 Final topologies 157

Chapter 3 Duality

1 The Hahn-Banach theorem 176

2 Pairings 183

3 Polarity 190

4 G-topologies 195

5 The Mackey topology 203

6 Barrelled spaces 211

7 Bornological spaces 220

8 Reflexivity 226

9 Montel spaces 231

10 The Banach-Dieudonné theorem 243

11 Grothendieck's completeness theorem 247

12 The transpose of a linear map 254

13 Duals of subspaces and quotient spaces 260

14 Duals of products and direct sums 266

15 Schwartz spaces 271

16 Distinguished spaces 288

17 The homomorphism theorem and the closed-graph theorem 294

Chapter 4 Distributions

1 The definition of distributions 313

2 Support 317

3 Derivation 323

4 Distributions of finite order 337

5 Integrable distributions 344

6 Multiplication 347

7 Bilinear maps 355

8 Tensor product 365

9 Convolution 381

10 Regularization 401

11 Fourier transform 408

Bibliography 427

Index of notations 437

Table 441

Index 443