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Haase M. Functional analysis: an elementary introduction. - Providence: American mathematical society, 2014. - xviii, 372 p.: ill. - (Graduate studies in mathematics; vol.156). - Incl. bibl. ref. and ind. - Auth. ind.: p.371-372. - ISBN 978-0-8218-9171-1
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/ Contents
 
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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