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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