Introduction to Banach spaces : analysis and probability

목차 (Table of Contents)

• [volume. vol.1]----------
• CONTENTS
• Contents of Volume 2 = x
• Preface = xiii
• Preliminary Chapter Weak and Weak* Topologies. Filters, Ultrafilters. Ordinals = 1
• Ⅰ Introduction = 1
• Ⅱ Weak and Weak* Topologies = 1
• Ⅲ Filters, Ultrafilters. Ordinals = 7
• Ⅳ Exercises = 12
• 1 Fundamental Notions of Probability = 13
• Ⅰ Introduction = 13
• Ⅱ Convergence = 15
• Ⅲ Series of Independent Random Variables = 21
• Ⅳ Khintchine's Inequalities = 30
• Ⅴ Martingales = 35
• Ⅵ Comments = 42
• Ⅶ Exercises = 43
• 2 Bases in Banach Spaces = 46
• Ⅰ Introduction = 46
• Ⅱ Schauder Bases : Generalities = 46
• Ⅲ Bases and the Structure of Banach Spaces = 59
• Ⅳ Comments = 74
• Ⅴ Exercises = 76
• 3 Unconditional Convergence = 83
• Ⅰ Introduction = 83
• Ⅱ Unconditional Convergence = 83
• Ⅲ Unconditional Bases = 90
• Ⅳ The Canonical Basis of c₀ = 94
• Ⅴ The James Theorems = 96
• Ⅵ The Gowers Dichotomy Theorem = 101
• Ⅶ Comments = 110
• Ⅷ Exercises = 111
• 4 Banach Space Valued Random Variables = 117
• Ⅰ Introduction = 117
• Ⅱ Definitions. Convergence = 117
• Ⅲ The Paul Lévy Symmetry Principle and Applications = 129
• Ⅳ The Contraction Principle = 133
• Ⅴ The Kahane Inequalities = 138
• Ⅵ Comments = 151
• Ⅶ Exercises = 152
• 5 Type and Cotype of Banach Spaces. Factorization through a Hilbert Space = 159
• Ⅰ Introduction = 159
• Ⅱ Complements of Probability = 159
• Ⅲ Complements on Banach Spaces = 172
• Ⅳ Type and Cotype of Banach Spaces = 177
• Ⅴ Factorization through a Hilbert Space and Kwapień's Theorem = 193
• Ⅵ Some Applications of the Notions of Type and Cotype = 200
• Ⅶ Comments = 203
• Ⅷ Exercises = 205
• 6 p-Summing Operators. Applications = 210
• Ⅰ Introduction = 210
• Ⅱ p-Summing Operators = 211
• Ⅲ Grothendieck's Theorem = 217
• Ⅳ Some Applications of p-Summing Operators = 227
• Ⅴ Sidon Sets = 231
• Ⅵ Comments = 258
• Ⅶ Exercises = 260
• 7 Some Properties of L$$His^{p}$$-Spaces = 266
• Ⅰ Introduction = 266
• Ⅱ The Space L¹= 267
• Ⅲ The Trigonometric System = 277
• Ⅳ The Haar Basis in L$$His^{p}$$ = 284
• Ⅴ Another Proof of Grothendieck's Theorem = 296
• Ⅵ Comments = 305
• Ⅶ Exercises = 315
• 8 The Space ℓ₁= 326
• Ⅰ Introduction = 326
• Ⅱ Rosenthal's ℓ₁Theorem = 326
• Ⅲ Further Results on Spaces Containing ℓ₁= 341
• Ⅳ Comments = 350
• Ⅴ Exercises = 353
• Annex Banach Algebras. Compact Abelian Groups = 357
• Ⅰ Introduction = 357
• Ⅱ Banach Algebras = 357
• Ⅲ Compact Abelian Groups = 364
• References = 382
• Notation Index for Volume 1 = 413
• Author Index for Volume I = 415
• Subject Index for Volume 1 = 419
• Notation Index for Volume 2 = 425
• Author Index for Volume 2 = 426
• Subject Index for Volume 2 = 429
• [volume. vol.2]----------
• CONTENTS
• Contents of Volume 1 = ix
• Preface = xiii
• 1 Euclidean Sections = 1
• Ⅰ Introduction = 1
• Ⅱ An Inequality of Concentration of Measure = 1
• Ⅲ Comparison of Gaussian Vectors = 8
• Ⅳ Dvoretzky's Theorem = 18
• Ⅴ The Lindenstrauss - Tzafriri Theorem = 40
• Ⅵ Comments = 45
• Ⅶ Exercises = 46
• 2 Separable Banach Spaces without the Approximation Property = 51
• Ⅰ Introduction and Definitions = 51
• Ⅱ The Grothendieck Reductions = 53
• Ⅲ The Counterexamples of Enflo and Davie = 59
• Ⅳ Comments = 68
• Ⅴ Exercises = 70
• 3 Gaussian Processes = 72
• Ⅰ Introduction = 72
• Ⅱ Gaussian Processes = 72
• Ⅲ Brownian Motion = 76
• Ⅳ Dudley's Majoration Theorem = 79
• Ⅴ Fernique's Minoration Theorem for Stationary Processes = 85
• Ⅵ The Elton - Pajor Theorem = 95
• Ⅶ Comments = 122
• Ⅷ Exercises = 123
• 4 Reflexive Subspaces of L¹= 127
• Ⅰ Introduction = 127
• Ⅱ Structure of Reflexive Subspaces of L¹= 128
• Ⅲ Examples of Reflexive Subspaces of L¹= 142
• Ⅳ Maurey's Factorization Theorem and Rosenthal's Theorem = 150
• Ⅴ Finite-Dimensional Subspaces of L¹= 157
• Ⅵ Comments = 176
• Ⅶ Exercises = 180
• 5 The Method of Selectors. Examples of Its Use = 193
• Ⅰ Introduction = 193
• Ⅱ Extraction of Quasi-Independent Sets = 193
• Ⅲ Sums of Sines and Vectorial Hilbert Transforms = 217
• Ⅳ Minoration of the K-Convexity Constant = 223
• Ⅴ Comments = 228
• Ⅵ Exercises = 230
• 6 The Pisier Space of Almost Surely Continuous Functions. Applications = 234
• Ⅰ Introduction = 234
• Ⅱ Complements on Banach-Valued Variables = 235
• Ⅲ The C$$His^{as}$$ Space = 243
• Ⅳ Applications of the Space C$$His^{as}$$ = 261
• Ⅴ The Bourgain-Milman Theorem = 268
• Ⅵ Comments = 282
• Ⅶ Exercises = 287
• Appendix A : News in the Theory of Infinite-Dimensional Banach Spaces in the Past 20 Years = 290
• Appendix B : An Update on Some Problems in High-Dimensional Convex Geometry and Related Probabilistic Results = 297
• Appendix C : A Few Updates and Pointers = 307
• Appendix D : On the Mesh Condition for Sidon Sets = 316
• References = 324
• Notation Index for Volume 2 = 355
• Author Index for Volume 2 = 356
• Subject Index for Volume 2 = 359
• Notation Index for Volume 1 = 363
• Author Index for Volume 1 = 365
• Subject Index for Volume 1 = 369