Table of Contents

Chapter 1 Lipschitz Functions 1
1.1 Classes of metric spaces 1
1.2 Lipschitz functions 3
1.3 Sums and products of metric spaces 8
1.4 Quotients of metric spaces 14
1.5 Scalar-valued Lipschitz functions 19
1.6 Rademacher’s theorem 27
1.7 Notes 34

Chapter 2 Lipschitz Spaces 35
2.1 Lip and Lip0 spaces 35
2.2 Lip versus Lip0 42
2.3 Composition maps 49
2.4 De Leeuw’s map 55
2.5 Lipschitz gauges 59
2.6 Distortion 65
2.7 Extreme points 73
2.8 Notes 78

Chapter 3 The Predual 81
3.1 The Arens-Eells space 81
3.2 Examples 86
3.3 The mass transfer problem 90
3.4 Uniqueness of the predual 95
3.5 Weak* extreme points (existence) 101
3.6 Weak* extreme points (characterization) 110
3.7 Isometries of Lip spaces 118
3.8 Isometries of Lip0 spaces 120
3.9 Notes 124

Chapter 4 Little Lipschitz Spaces, I 127
4.1 The compact case 127
4.2 The general case 134
4.3 Duality 147
4.4 Extensions of little Lipschitz functions 152
4.5 Notes 161

Chapter 5 Linearization 163
5.1 Linear surrogates 163
5.2 Lipschitz extension constants 171
5.3 Isometric embedding 177
5.4 Nonseparable counterexamples 181
5.5 Notes 185

Chapter 6 Lattice Structure 187
6.1 Linear complete sublattices of Lip0(X) 187
6.2 Complete bands of Lip0(X) 195
6.3 Stone lattices 201
6.4 Complete distributivity 207
6.5 Embedding in cubes 210
6.6 Lipschitz lattices 216
6.7 Completely distributive Lipschitz lattices 221
6.8 Notes 227

Chapter 7 Algebraic Structure 229
7.1 Order complete subalgebras of Lip0(X) 229
7.2 Order complete ideals of Lip0(X) 235
7.3 Spectra 238
7.4 Norm closed ideals 247
7.5 Point derivations 251
7.6 Spectral synthesis 253
7.7 Notes 259

Chapter 8 Little Lipschitz Spaces, II 261
8.1 Little Lipschitz sublattices 261
8.2 Little Lipschitz bands 268
8.3 Little Lipschitz subalgebras 274
8.4 Little Lipschitz ideals 277
8.5 Holder cubes 280
8.6 Finite dimensional Ho lder spaces 286
8.7 Notes 294

Chapter 9 Measurable Metrics 297
9.1 Localizable measure spaces 297
9.2 Measurable pseudometrics 300
9.3 Measurable relations 305
9.4 Lipschitz functions 313
9.5 Lipschitz spaces 322
9.6 Lipschitz gauges 331
9.7 Algebra and lattice structure 336
9.8 Lipschitz lattices 348
9.9 Notes 352

Chapter 10 Derivations 353
10.1 L∞-modules 353
10.2 The derivation theorem 367
10.3 The exterior derivative 377
10.4 Examples 385
10.5 Locally controlled oscillation 391
10.6 Dirichlet forms 395
10.7 The Sierpinski gasket 403
10.8 Wiener spaces 407
10.9 Notes 409

Chapter 11 Quantum Metrics 411
11.1 Quantum relations 411
11.2 Quantum metrics 419
11.3 Examples 429
11.4 Spectral Lipschitz numbers 436
11.5 Commutation Lipschitz numbers 442
11.6 Notes 446

Bibliography 447

Subject Index 453

编辑手记 459

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