Topological Vector Spaces and Distributions

by Horvath, John

• ISBN13: 9780486488509
• ISBN10: 0486488500
• Edition: Reprint
• Format: Paperback
• Copyright: 2012-07-17
• Publisher: Dover Publications

Summary

The precise exposition of this text's first three chapters provides an excellent summary of the modern theory of locally convex modern spaces. The fourth and final chapter develops the theory of distributions in terms of convolutions, tensor products, and Fourier transforms. "The most readable introduction to the theory of vector spaces." J. L. B. Cooper, MathSciNet Review.1966 edition.

Author Biography

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

Table of Contents

Terminology and Notations
Banach Spaces
The definition of Banach spaces	p. 5
Some notions from algebra and topology	p. 17
Subspaces	p. 27
Linear maps	p. 35
Linear forms	p. 40
The Hahn-Banach theorem	p. 45
The dual space	p. 52
The Banach-Steinhaus theorem	p. 62
Banach's homomorphism theorem and the closed-graph theorem	p. 68
Locally Convex Spaces
Some notions from topology	p. 71
Filters	p. 75
Topological vector spaces	p. 79
Locally convex spaces	p. 84
Linear maps, subspaces, quotient spaces	p. 97
Bounded sets, normability, metrizability	p. 108
Products and direct sums	p. 117
Convergence of filters	p. 124
Completeness	p. 128
Finite-dimensional and locally compact spaces	p. 141
Initial topologies	p. 149
Final topologies	p. 157
Duality
The Hahn-Banach theorem	p. 176
Pairings	p. 183
Polarity	p. 190
G-topologies	p. 195
The Mackey topology	p. 203
Barrelled spaces	p. 211
Bornological spaces	p. 220
Reflexivity	p. 226
Montel spaces	p. 231
The Banach-Dieudonné theorem	p. 243
Grothendieck's completeness theorem	p. 247
The transpose of a linear map	p. 254
Duals of subspaces and quotient spaces	p. 260
Duals of products and direct sums	p. 266
Schwartz spaces	p. 271
Distinguished spaces	p. 288
The homomorphism theorem and the closed-graph theorem	p. 294
Distributions
The definition of distributions	p. 313
Support	p. 317
Derivation	p. 323
Distributions of finite order	p. 337
Integrable distributions	p. 344
Multiplication	p. 347
Bilinear maps	p. 355
Tensor product	p. 365
Convolution	p. 381
Regularization	p. 401
Fourier transform	p. 408
Bibliography	p. 427
Index of notations	p. 437
Table	p. 441
Index	p. 443
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