Part Ⅰ General Theory 3
1 Topological Vector Spaces 3
Introduction 3
Separation properties 10
Linear mappings 14
Finite-dimensional spaces 16
Metrization 18
Boundedness and continuity 23
Seminorms and local convexity 25
Quotient spaces 30
Examples 33
Exercises 38
2 Completeness 42
Baire category 42
The Banach-Steinhaus theorem 43
The open mapping theorem 47
The closed graph theorem 50
Bilinear mappings 52
Exercises 53
3 Convexity 56
The Hahn-Banach theorems 56
Weak topologies 62
Compact convex sets 68
Vector-valued integration 77
Holomorphic functions 82
Exercises 85
4 Duality in Banach Spaces 92
The normed dual of a normed space 92
Adjoints 97
Compact operators 103
Exercises 111
5 Some Applications 116
A continuity theorem 116
Closed subspaces of Lp-spaces 117
The range of a vector-valued measure 120
A generalized Stone-Weierstrass theorem 121
Two interpolation theorems 124
Kakutani’s fixed point theorem 126
Haar measure on compact groups 128
Uncomplemented subspaces 132
Sums of Poisson kernels 138
Two more fixed point theorems 139
Exercises 144
Part Ⅱ Distributions and Fourier Transforms 149
6 Test Functions and Dist butions 149
Introduction 149
Test function spaces 151
Calculus with distributions 157
Localization 162
Supports of distributions 164
Distributions as derivatives 167
Convolutions 170
Exercises 177
7 Fourier Transforms 182
Basic properties 182
Tempered distributions 189
Paley-Wiener theorems 196
Sobolev’s lemma 202
Exercises 204
8 Applications to Differential Equations 210
Fundamental solutions 210
Elliptic equations 215
Exercises 222
9 Tauberian Theory 226
Wiener’s theorem 226
The prime number theorem 230
The renewal equation 236
Exercises 239
Part Ⅲ Banach Algebras and Spectral Theory 245
10 Banach Algebras 245
Introduction 245
Complex homomorphisms 249
Basic properties of spectra 252
Symbolic calculus 258
The group of invertible elements 267
Lomonosov’s invariant subspace theorem 269
Exercises 271
11 Commutative Banach Algebras 275
Ideals and homomorphisms 275
Gelfand transforms 280
Involutions 287
Applications to noncommutative algebras 292
Positive functionals 296
Exercises 301
12 Bounded Operators on a Hilbert Space 306
Basic facts 306
Bounded operators 309
A commutativity theorem 315
Resolutions of the identity 316
The spectral theorem 321
Eigenvalues of normal operators 327
Positive operators and square roots 330
The group of invertible operators 333
A characterization of B-algebras 336
An ergodic theorem 339
Exercises 341
13 Unbounded Operators 347
Introduction 347
Graphs and symmetric operators 351
The Cayley transform 356
Resolutions of the identity 360
The spectral theorem 368
Semigroups of operators 375
Exercises 385
Appendix A Compactness and Continuity 391
Appendix B Notes and Comments 397
Bibliography 412
List of Special Symbols 414
Index 417