List of Figures ............................................... ix
Preface ....................................................... xi
Acknowledgments ............................................... XV
1 ELEMENTS OF TOPOLOGY ........................................ 1
1.1 Topological Concepts ................................... 3
1.2 Weak, Product and Quotient Topologies ................. 19
1.3 Compactness and Compactification ...................... 28
1.4 Metrizable Spaces ..................................... 41
1.5 Uniform Continuity and Connectedness .................. 58
1.6 Function Spaces ....................................... 68
1.7 Remarks ............................................... 77
1.8 Exercises ............................................. 84
1.9 Solutions to Exercises ................................ 88
2. ELEMENTS OF MEASURE THEORY ............................... 103
2.1 Measures and Measurable Functions .................... 104
2.2 Integration and Convergence Theorems ................. 133
2.3 Signed Measures and the Radon-Nikodym Theorem ........ 154
2.4 Product Measures ..................................... 168
2.5 Measures and Topology ................................ 182
2.6 Polish and Souslin Spaces ............................ 206
2.7 Remarks .............................................. 217
2.8 Exercises ............................................ 224
2.9 Solutions to Exercises ............................... 231
3 BANACH SPACES ............................................. 255
3.1 Hahn-Banach Theorem .................................. 256
3.2 The Three Basic Theorems of Linear Analysis .......... 265
3.3 Separation of Convex Sets ............................ 273
3.4 Weak and Weak* Topologies ............................ 281
3.5 Weak Compactness ..................................... 291
3.6 Reflexive and Separable Banach Spaces ................ 301
3.7 Hilbert Spaces and Compact Linear Operators .......... 309
3.8 Classical Banach Spaces .............................. 326
3.9 Sobolev Spaces ....................................... 337
3.10 Vector-Valued Functions and Bochner Integral ......... 364
3.11 Remarks .............................................. 374
3.12 Exercises ............................................ 383
3.13 Solutions to Exercises ............................... 387
4 SET-VALUED ANALYSIS ....................................... 405
4.1 Continuity of Multifunctions ......................... 407
4.2 Measurability of Multifunctions ...................... 424
4.3 Measurable Selectors ................................. 429
4.4 Continuous Selectors ................................. 436
4.5 Decomposable Sets .................................... 452
4.6 Set-Valued Integration ............................... 466
4.7 Convergence of Sets and Functions .................... 474
4.8 Remarks .............................................. 491
4.9 Exercises ............................................ 498
4.10 Solutions to Exercises ............................... 503
5 NONSMOOTH ANALYSIS ........................................ 517
5.1 Smooth Calculus in Banach Spaces ..................... 518
5.2 Convex, Lower Semicontinuous Functions ............... 528
5.3 Conjugate Functions and Subdifferentials ............. 536
5.4 Optimization and Minimax Theorems .................... 562
5.5 Normal Integrands .................................... 579
5.6 Generalized Subdifferential .......................... 600
5.7 Tangent and Normal Cones ............................. 619
5.8 Remarks .............................................. 630
5.9 Exercises ............................................ 635
5.10 Solutions to Exercises ............................... 641
References ................................................... 665
Index ........................................................ 683
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