Hilbert Spaces with Applications

Contents......Page 8
Preface to the Third Edition......Page 12
Preface to the Second Edition......Page 14
Preface to the First Edition......Page 16
Introduction......Page 20
Vector Spaces......Page 21
Normed Spaces......Page 27
Banach Spaces......Page 38
Linear Mappings......Page 44
Banach Fixed Point Theorem......Page 51
Exercises......Page 53
Introduction......Page 58
Step Functions......Page 59
Lebesgue Integrable Functions......Page 64
The Absolute Value of an Integrable Function......Page 67
Series of Integrable Functions......Page 69
Norm in L1(R)......Page 71
Convergence Almost Everywhere......Page 74
Fundamental Convergence Theorems......Page 77
Locally Integrable Functions......Page 81
The Lebesgue Integral and the Riemann Integral......Page 83
Lebesgue Measure on R......Page 86
Complex-Valued Lebesgue Integrable Functions......Page 90
The Spaces Lp(R)......Page 93
Lebesgue Integrable Functions on RN......Page 97
Convolution......Page 101
Exercises......Page 103
Introduction......Page 112
Inner Product Spaces......Page 113
Hilbert Spaces......Page 118
Orthogonal and Orthonormal Systems......Page 124
Trigonometric Fourier Series......Page 141
Orthogonal Complements and Projections......Page 146
Riesz Representation Theorem......Page 151
Exercises......Page 154
Introduction......Page 164
Examples of Operators......Page 165
Bilinear Functionals and Quadratic Forms......Page 170
Adjoint and Self-Adjoint Operators......Page 177
Normal, Isometric, and Unitary Operators......Page 182
Positive Operators......Page 187
Projection Operators......Page 194
Compact Operators......Page 199
Eigenvalues and Eigenvectors......Page 205
Spectral Decomposition......Page 215
Unbounded Operators......Page 220
Exercises......Page 230
Introduction......Page 236
Basic Existence Theorems......Page 237
Fredholm Integral Equations......Page 243
Method of Successive Approximations......Page 245
Volterra Integral Equations......Page 247
Method of Solution for a Separable Kernel......Page 252
Abel's Integral Equation......Page 255
Ordinary Differential Equations......Page 258
Sturm-Liouville Systems......Page 266
Inverse Differential Operators......Page 272
The Fourier Transform......Page 277
Applications of the Fourier Transform......Page 290
Exercises......Page 298
Introduction......Page 306
Distributions......Page 307
Sobolev Spaces......Page 319
Fundamental Solutions......Page 322
Elliptic Boundary Value Problems......Page 342
Applications of the Fourier Transform......Page 348
Exercises......Page 362
Introduction......Page 370
Basic Concepts and Equations......Page 371
Postulates of Quantum Mechanics......Page 382
The Heisenberg Uncertainty Principle......Page 396
The Schrödinger Equation of Motion......Page 398
The Schrödinger Picture......Page 414
The Heisenberg Picture......Page 420
The Interaction Picture......Page 424
The Linear Harmonic Oscillator......Page 426
Angular Momentum Operators......Page 431
The Dirac Relativistic Wave Equation......Page 439
Exercises......Page 442
Brief Historical Remarks......Page 452
Continuous Wavelet Transforms......Page 455
The Discrete Wavelet Transform......Page 463
Multiresolution Analysis......Page 471
Examples of Orthonormal Wavelets......Page 481
Exercises......Page 492
Introduction......Page 496
The Gateaux and Fréchet Differentials......Page 497
Optimization Problems......Page 509
Minimization of Quadratic Functionals......Page 524
Variational Inequalities......Page 526
Optimal Control Problems......Page 529
Approximation Theory......Page 536
The Shannon Sampling Theorem......Page 541
Linear and Nonlinear Stability......Page 545
Bifurcation Theory......Page 549
Exercises......Page 554
1.7 Exercises......Page 566
2.16 Exercises......Page 567
3.8 Exercises......Page 569
5.13 Exercises......Page 570
6.7 Exercises......Page 572
7.12 Exercises......Page 575
8.6 Exercises......Page 579
9.11 Exercises......Page 580
Bibliography......Page 584
Index......Page 590