This is a graduate text on functional analysis. After presenting the fundamental function spaces and their duals, the authors study topics in operator theory and finally develop the theory of distributions up to significant applications such as Sobolev spaces and Dirichlet problems. Along the way, the reader is presented with a truly remarkable assortment of well formulated and interesting exercises, which test the understanding as well as point out many related topics. The answers and hints that are not already contained in the statements of the exercises are collected at the end of the book.
Author(s): Francis Hirsch, Gilles Lacombe, S. Levy
Series: Graduate texts in mathematics 192
Edition: 1
Publisher: Springer
Year: 1999
Language: English
Pages: 412
Front Cover......Page 1
Title......Page 4
Copyright......Page 5
Preface......Page 6
Contents......Page 10
Notation......Page 14
1 Countability......Page 16
2 Separability......Page 22
3 The Diagonal Procedure......Page 27
4 Bounded Sequences of Continuous Linear Maps......Page 33
I FUNCTION SPACES AND THEIR DUALS......Page 40
1 The Space of Continuous Functions on a Compact Set......Page 42
1 Generalities......Page 43
2 The Stone-Weierstrass Theorems......Page 46
3 Ascoli's Theorem......Page 57
1 Locally Compact Spaces......Page 64
2 Daniell's Theorem......Page 72
3 Positive Radon Measures......Page 83
3A Positive Radon Measures on R and the Stieltjes Integral......Page 86
3B Surface Measure on Spheres in R^d......Page 89
4 Real and Complex Radon Measures......Page 101
1 Definitions, Elementary Properties, Examples......Page 112
2 The Projection Theorem......Page 120
3 The Riesz Representation Theorem......Page 126
3A Continuous Linear Operators on a Hilbert Space......Page 127
3B Weak Convergence in a Hilbert Space......Page 129
4 Hilbert Bases......Page 138
1 Definitions and General Properties......Page 158
2 Duality......Page 174
3 Convolution......Page 184
II OPERATORS......Page 200
1 Operators on Banach Spaces......Page 202
2 Operators in Hilbert Spaces......Page 216
2A Spectral Properties of Hermitian Operators......Page 218
2B Operational Calculus on Hermitian Operators......Page 220
1 General Properties......Page 228
1A Spectral Properties of Compact Operators......Page 232
2 Compact Selfadjoint Operators......Page 249
2A Operational Calculus and the Fredholm Equation......Page 253
2B Kernel Operators......Page 255
III DISTRIBUTIONS......Page 270
1A Notation......Page 272
1B Convergence in Function Spaces......Page 274
1C Smoothing......Page 276
1D C^{infty} Partitions of Unity......Page 277
2A Definitions......Page 282
2B First Examples......Page 283
2C Restriction and Extension of a Distribution to an Open Set......Page 286
2E Principal Values......Page 287
2F Finite Parts......Page 288
3A Distributions of Finite Order......Page 295
3C Distributions with Compact Support......Page 296
1 Multiplication......Page 302
2 Differentiation......Page 307
3 Fundamental Solutions of a Differential Operator......Page 321
3A The Laplacian......Page 322
3B The Heat Operator......Page 325
3C The Cauchy-Riemann Operator......Page 326
1 Tensor Product of Distributions......Page 332
2A Convolution in mathcal{E}'......Page 339
2B Convolution in mathcal{D}'......Page 340
2C Convolution of a Distribution with a Function......Page 347
3A Primitives and Sobolev's Theorem......Page 352
3B Regularity......Page 355
3D The Algebra mathcal{D}_{+}......Page 358
1 The spaces H^1{Omega} and H_0^1(Omega) (S2)......Page 364
2 The Dirichlet Problem......Page 378
2A The Dirichlet Problem......Page 381
2B The Heat Problem......Page 382
2C The Wave Problem......Page 383
Answers to the Exercises......Page 394
Index......Page 402
Back Cover......Page 412
