Functional Analysis for the Applied Sciences

Preface......Page 7
Contents......Page 9
 1.1 Sets......Page 13
 1.3 Real Numbers......Page 15
 1.4 Complex Numbers......Page 27
 1.5 Linear Spaces......Page 28
 1.6 Exercises......Page 38
 2.1 Definitions......Page 43
 2.2 Completeness......Page 46
 2.3 Compact Sets......Page 52
 2.4 Continuous Functions on Compact Sets......Page 56
 2.5 The Banach Contraction Principle......Page 67
 2.6 Exercises......Page 70
 3.1 Measurable Sets in Rk......Page 76
 3.2 Measurable Functions......Page 82
 3.3 The Lebesgue Integral......Page 86
 3.4 Lp Spaces......Page 93
 3.5 Exercises......Page 97
 4.1 Definitions, Examples, Operator Norm......Page 100
 4.2 Main Principles of Functional Analysis......Page 104
 4.3 Compact Linear Operators......Page 107
 4.4 Linear Functionals, Dual Spaces, Weak Topologies......Page 108
 4.5 Exercises......Page 115
 5.1 Test Functions......Page 118
 5.2 Friedrichs\' Mollification......Page 123
 5.3 Scalar Distributions......Page 130
  5.3.1 Some Operations with Distributions......Page 132
  5.3.2 Convergence in Distributions......Page 133
  5.3.3 Differentiation of Distributions......Page 136
  5.3.4 Differential Equations for Distributions......Page 142
 5.4 Sobolev Spaces......Page 154
 5.5 Bochner\'s Integral......Page 160
 5.6 Vector Distributions, Wmp(a,b;X) Spaces......Page 166
 5.7 Exercises......Page 171
 6.1 Examples......Page 176
 6.2 Jordan–von Neumann Characterization Theorem......Page 179
 6.3 Projections in Hilbert Spaces......Page 182
 6.4 The Riesz Representation Theorem......Page 186
 6.5 Lax–Milgram Theorem......Page 191
 6.6 Fourier Series Expansions......Page 197
 6.7 Exercises......Page 206
 7.1 The Adjoint of a Linear Operator......Page 211
 7.2 Adjoints of Operators on Hilbert Spaces......Page 214
  7.2.1 The Case of Compact Operators......Page 215
 7.3 Symmetric Operators and Self-adjoint Operators......Page 219
 7.4 Exercises......Page 222
 8.1 Definition and Examples......Page 227
 8.2 Main Results......Page 229
 8.3 Eigenvalues of -Delta Under the Dirichlet BoundaryCondition......Page 236
 8.4 Eigenvalues of -Delta Under the Robin BoundaryCondition......Page 238
 8.5 Eigenvalues of -Delta Under the Neumann BoundaryCondition......Page 240
 8.6 Some Comments......Page 242
 8.7 Exercises......Page 249
9 Semigroups of Linear Operators......Page 253
 9.1 Definitions......Page 254
 9.2 Some Properties of C0-Semigroups......Page 256
 9.3 Uniformly Continuous Semigroups......Page 262
 9.4 Groups of Linear Operators. Definitions and Linkto Operator Semigroups......Page 264
 9.5 Translation Semigroups......Page 267
 9.6 The Hille–Yosida Generation Theorem......Page 270
 9.7 The Lumer–Phillips Theorem......Page 275
 9.8 The Feller–Miyadera–Phillips Theorem......Page 278
 9.9 A Perturbation Result......Page 281
 9.10 Approximation of Semigroups......Page 283
 9.11 The Inhomogeneous Cauchy Problem......Page 289
  9.12.1 The Heat Equation......Page 293
  9.12.2 The Wave Equation......Page 296
  9.12.3 The Transport Equation......Page 298
  9.12.4 The Telegraph System......Page 301
 9.13 Exercises......Page 303
 10.1 First Order Linear EvolutionEquations......Page 307
 10.2 Second Order Linear EvolutionEquations......Page 314
 10.3 Examples......Page 318
 10.4 Exercises......Page 319
 11.1 Volterra Equations......Page 324
 11.2 Fredholm Equations......Page 334
 11.3 Exercises......Page 345
 12.1 Answers to Exercises for Chap.1......Page 349
 12.2 Answers to Exercises for Chap.2......Page 351
 12.3 Answers to Exercises for Chap.3......Page 362
 12.4 Answers to Exercises for Chap.4......Page 367
 12.5 Answers to Exercises for Chap.5......Page 373
 12.6 Answers to Exercises for Chap.6......Page 383
 12.7 Answers to Exercises for Chap.7......Page 391
 12.8 Answers to Exercises for Chap.8......Page 398
 12.9 Answers to Exercises for Chap.9......Page 406
 12.10 Answers to Exercises for Chap.10......Page 415
 12.11 Answers to Exercises for Chap.11......Page 425
Bibliography......Page 436