Introduction to Hilbert Spaces with Applications

Contents......Page 5
Preface......Page 9
PART 1: THE RY......Page 13
1.2 Vector Spaces......Page 15
1.3 Linear Independence, Basis, Dimension......Page 21
1.4 Normed Spaces......Page 22
1.5 Banach Spaces......Page 30
1.6 Linear Mappings......Page 34
1.7 Completion of Normed Spaces......Page 39
1.8 Contraction Mappings and the Fixed Point Theorem......Page 41
1.9 Exercises......Page 44
2.1. Introduction......Page 49
2.2 Step Functions......Page 50
2.3 Lebesgue Integrable Functions......Page 55
2.4 Modulus of an Integrable Function......Page 59
2.5 Series of Integrable Functions......Page 61
2.6 Norm in L^1(R)......Page 63
2.7 Convergence Almost Everywhere......Page 66
2.8 Fundamental Theorems......Page 70
2.9 Locally Integrable Functions......Page 74
2.10 The Lebesgue Integral and the Riemann Integral......Page 76
2.11 The Lebesgue Measure......Page 79
2.12 Complex Valued Lebesgue Integrable Functions......Page 83
2.13 The Space L^2(R)......Page 86
2.14 The Spaces L^1(R^N) and L^2(R^N)......Page 87
2.15 Convolution......Page 91
2.16 Exercises......Page 94
3.1 Introduction......Page 99
3.3 Examples of Inner Product Spaces......Page 100
3.4 Norm in an Inner Product Space......Page 102
3.5 Hilbert Spaces-Definition and Examples......Page 105
3.6 Strong and Weak Convergence......Page 109
3.7 Orthogonal and Orthonormal Systems......Page 110
3.8 Properties of Orthonormal Systems......Page 116
3.9 Trigonometric Fourier Series......Page 124
3.10 Orthogonal Complements and Projection Theorem......Page 129
3.11 Linear Functionals and the Riesz Representation Theorem......Page 134
3.12 Separable Hilbert Spaces......Page 136
3.13 Exercises......Page 139
4.1 Introduction......Page 149
4.2 Examples of Operators......Page 150
4.3 Bilinear Functionals and Quadratic Forms......Page 154
4.4 Adjoint and Self-adjoint Operators......Page 161
4.5 Invertible, Normal, Isometric, and Unitary Operators......Page 167
4.6 Positive Operators......Page 173
4.7 Projection Operators......Page 178
4.8 Compact Operators......Page 183
4.9 Eigenvalues and Eigenvectors......Page 188
4.10 Spectral Decomposition......Page 199
4.11 The Fourier Transform......Page 204
4.12 Unbounded Operators......Page 215
4.13 Exercises......Page 224
5.1 Introduction......Page 235
5.2 Basic Existence Theorems......Page 236
5.3 Fredholm Integral Equations......Page 243
5.4 Method of Successive Approximations......Page 246
5.5 Volterra Integral Equations......Page 248
5.6 Method of Solution for a Separable Kernel......Page 254
5.7 Volterra Integral Equations of the First Kind and Abel\'s Integral Equation......Page 258
5.8 Ordinary Differential Equations and Differential Operators......Page 260
5.9 Sturm-Liouville Systems......Page 269
5.10 Inverse Differential Operators and Green\'s Functions......Page 275
5.11 Applications of Fourier Transforms to Ordinary Differential Equations and Integral Equations......Page 280
5.12 Exercises......Page 288
PART 2: APPLICATIONS......Page 233
6.2 Distributions......Page 295
6.3 Fundamental Solutions and Green\'s Functions for Partial Differential Equations......Page 307
6.4 Weak Solutions of Elliptic Boundary Value Problems......Page 322
6.5 Examples of Applications of Fourier Transforms to Partial Differential Equations......Page 328
6.6 Exercises......Page 338
7.2 Basic Concepts and Equations of Classical Mechanics......Page 345
7.3 Basic Concepts and Postulates of Quantum Mechanics......Page 357
7.4 The Heisenberg Uncertainty Principle......Page 371
7.5 The Schrodinger Equation of Motion......Page 373
7.6 The Schrodinger Picture......Page 389
7.7 The Heisenberg Picture and the Heisenberg Equation of Motion......Page 396
7.8 The Interaction Picture......Page 401
7.9 The Linear Harmonic Oscillator......Page 402
7.10 Angular Momentum Operators......Page 408
7.11 Exercises......Page 415
8.1 Introduction......Page 423
8.2 The Gateaux and Frechet Derivatives......Page 424
8.3 Optimization Problems and the Euler-Lagrange Equations......Page 436
8.4 Minimization of a Quadratic Functional......Page 453
8.5 Variational Inequalities......Page 455
8.6 Optimal Control Problems for Dynamical Systems......Page 459
8.7 Approximation Theory......Page 465
8.8 Linear and Nonlinear Stability......Page 472
8.9 Bifurcation Theory......Page 477
8.10 Exercises......Page 483
Hints and Answers to Selected Exercises......Page 491
Bibliography......Page 505
List of Symbols......Page 509
Index......Page 513