Preface ...................................................... xiii
Chapter 1 Inner Product Spaces ................................. 1
§1.1 Inner Products ............................................ 3
§1.2 Orthogonality ............................................. 6
§1.3 The Trigonometric System ................................. 10
Exercises ................................................ 11
Chapter 2 Normed Spaces ....................................... 15
§2.1 The Cauchy-Schwarz Inequality and the Space ℓ2 ........... 15
§2.2 Norms .................................................... 18
§2.3 Bounded Linear Mappings .................................. 21
§2.4 Basic Examples ........................................... 23
§2.5 *The ℓρ-Spaces (1 ≤ p < ∞) ............................... 28
Exercises ................................................ 31
Chapter 3 Distance and Approximation .......................... 37
§3.1 Metric Spaces ............................................ 37
§3.2 Convergence .............................................. 39
§3.3 Uniform, Pointwise and (Square) Mean Convergence ......... 41
§3.4 The Closure of a Subset .................................. 47
Exercises ................................................ 50
Chapter 4 Continuity and Compactness .......................... 55
§4.1 Open and Closed Sets ..................................... 55
§4.2 Continuity ............................................... 58
§4.3 Sequential Compactness ................................... 64
§4.4 Equivalence of Norms ..................................... 66
§4.5 Separability and General Compactness ..................... 71
Exercises ................................................ 74
Chapter 5 Banach Spaces ....................................... 79
§5.1 Cauchy Sequences and Completeness ........................ 79
§5.2 Hilbert Spaces ........................................... 81
§5.3 Banach Spaces ............................................ 84
§5.4 Series in Banach Spaces .................................. 86
Exercises ................................................ 88
Chapter 6 The Contraction Principle ........................... 93
§6.1 Banach's Contraction Principle ........................... 94
§6.2 Application: Ordinary Differential Equations ............. 95
§6.3 Application: Google's PageRank ........................... 98
§6.4 Application: The Inverse Mapping Theorem ................ 100
Exercises ............................................... 104
Chapter 7 The Lebesgue Spaces ................................ 107
§7.1 The Lebesgue Measure .................................... 110
§7.2 The Lebesgue Integral and the Space L1(X) ............... 113
§7.3 Null Sets ............................................... 115
§7.4 The Dominated Convergence Theorem ....................... 118
§7.5 The Spaces LP(X) with 1 ≤ p ≤ ∞ ......................... 121
Advice for the Reader ................................... 125
Exercises ............................................... 126
Chapter 8 Hilbert Space Fundamentals ......................... 129
§8.1 Best Approximations ..................................... 129
§8.2 Orthogonal Projections .................................. 133
§8.3 The Riesz-Frechet Theorem ............................... 135
§8.4 Orthogonal Series and Abstract Fourier Expansions ....... 137
Exercises ............................................... 141
Chapter 9 Approximation Theory and Fourier Analysis .......... 147
§9.1 Lebesgue's Proof of Weierstrass' Theorem ................ 149
§9.2 Truncation .............................................. 151
§9.3 Classical Fourier Series ................................ 156
§9.4 Fourier Coefficients of L1-Functions .................... 161
§9.5 The Riemann-Lebesgue Lemma .............................. 162
§9.6 The Strong Convergence Lemma and Fejer's Theorem ........ 164
§9.7 Extension of a Bounded Linear Mapping ................... 168
Exercises ..................................................... 172
Chapter 10 Sobolev Spaces and the Poisson Problem ............ 177
§10.1 Weak Derivatives ........................................ 177
§10.2 The Fundamental Theorem of Calculus ..................... 179
§10.3 Sobolev Spaces .......................................... 182
§10.4 The Variational Method for the Poisson Problem .......... 184
§10.5 Poisson's Problem in Higher Dimensions .................. 187
Exercises ............................................... 188
Chapter 11 Operator Theory I .................................. 193
§11.1 Integral Operators and Fubini's Theorem ................. 193
§11.2 The Dirichlet Laplacian and Hilbert-Schmidt Operators ... 196
§11.3 Approximation of Operators .............................. 199
§11.4 The Neumann Series ...................................... 202
Exercises ............................................... 205
Chapter 12 Operator Theory II ................................. 211
§12.1 Compact Operators ....................................... 211
§12.2 Adjoints of Hilbert Space Operators ..................... 216
§12.3 The Lax-Milgram Theorem ................................. 219
§12.4 Abstract Hilbert-Schmidt Operators ...................... 221
Exercises ............................................... 226
Chapter 13 Spectral Theory of Compact Self-Adjoint
Operators ..................................................... 231
§13.1 Approximate Eigenvalues ................................. 231
§13.2 Self-Adjoint Operators .................................. 234
§13.3 The Spectral Theorem .................................... 236
§13.4 The General Spectral Theorem ............................ 240
Exercises ............................................... 241
Chapter 14 Applications of the Spectral Theorem ............... 247
§14.1 The Dirichlet Laplacian ................................. 247
§14.2 The Schrodinger Operator ................................ 249
§14.3 An Evolution Equation ................................... 252
§14.4 The Norm of the Integration Operator .................... 254
§14.5 The Best Constant in the Poincare Inequality ............ 256
Exercises ............................................... 257
Chapter 15 Baire's Theorem and Its Consequences ............... 261
§15.1 Baire's Theorem ......................................... 261
§15.2 The Uniform Boundedness Principle ....................... 263
§15.3 Nonconvergence of Fourier Series ........................ 266
§15.4 The Open Mapping Theorem ................................ 267
§15.5 Applications with a Look Back ........................... 271
Exercises ............................................... 274
Chapter 16 Duality and the Hahn-Banach Theorem ................ 277
§16.1 Extending Linear Functionals ............................ 278
§16.2 Elementary Duality Theory ............................... 284
§16.3 Identification of Dual Spaces ........................... 289
§16.4 The Riesz Representation Theorem ........................ 295
Exercises ............................................... 299
Historical Remarks ...................................... 305
Appendix A Background ......................................... 311
§A.l Sequences and Subsequences ............................... 311
§A.2 Equivalence Relations .................................... 312
§A.3 Ordered Sets ............................................. 314
§A.4 Countable and Uncountable Sets ........................... 316
§A.5 Real Numbers ............................................. 316
§A.6 Complex Numbers .......................................... 321
§A.7 Linear Algebra ........................................... 322
§A.8 Set-theoretic Notions .................................... 329
Appendix B The Completion of a Metric Space ................... 333
§B.l Uniqueness of a Completion ............................... 334
§B.2 Existence of a Completion ................................ 335
§B.3 The Completion of a Normed Space ......................... 337
Exercises ................................................ 338
Appendix C Bernstein's Proof of Weierstrass' Theorem .......... 339
Appendix D Smooth Cutoff Functions ............................ 343
Appendix E Some Topics from Fourier Analysis .................. 345
§E.l Plancherel's Identity .................................... 346
§E.2 The Fourier Inversion Formula ............................ 347
§E.3 The Carlson-Beurling Inequality .......................... 348
Exercises ................................................ 349
Appendix F General Orthonormal Systems ........................ 351
§F.l Unconditional Convergence ................................ 351
§F.2 Uncountable Orthonormal Bases ............................ 353
Bibliography .................................................. 355
Symbol Index .................................................. 359
Subject Index ................................................. 361
Author Index .................................................. 371
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