Preface to the Third Edition xi Preface to the Second Edition xiii Preface to the First Edition xv
CHAPTER 1 Normed Vector Spaces 1 1.1 Introduction 1 1.2 Vector Spaces 2 1.3 Normed Spaces 8 1.4 Banach Spaces 19 1.5 Linear Mappings 25 1.6 Contraction Mappings and the Banach Fixed Point Theorem 32 1.7 Exercises 34
CHAPTER 2 The Lebesgue Integral 39 2.1 Introduction 39 2.2 Step Functions 40 2.3 Lebesgue Integrable Functions 45 2.4 The Absolute Value of an Integrable Function 48 2.5 Series of Integrable Functions 50 2.6 Norm in 52 2.7 Convergence Almost Everywhere 55 2.8 Fundamental Convergence Theorems 58 2.9 Locally Integrable Functions 62 2.10 The Lebesgue Integral and the Riemann Integral 64 2.11 Lebesgue Measure on 67 2.12 Complex-Valued Lebesgue Integrable Functions 71 2.13 The Spaces 74 2.14 Lebesgue Integrable Functions on78 2.15 Convolution 82 2.16 Exercises 84
CHAPTER 3 Hilbert Spaces and Orthonormal Systems 93 3.1 Introduction 93 3.2 Inner Product Spaces 94 3.3 Hilbert Spaces 99 3.4 Orthogonal and Orthonormal Systems 105 3.5 Trigonometric Fourier Series 122 3.6 Orthogonal Complements and Projections 127 3.7 Linear Functionals and the Riesz Representation Theorem 132 3.8 Exercises 135
CHAPTER 4 Linear Operators on Hilbert Spaces 145 4.1 Introduction 145 4.2 Examples of Operators 146 4.3 Bilinear Functionals and Quadratic Forms 151 4.4 Adjoint and Self-Adjoint Operators 158 4.5 Invertible, Normal, Isometric, and Unitary Operators 163 4.6 Positive Operators 168 4.7 Projection Operators 175 4.8 Compact Operators 180 4.9 Eigenvalues and Eigenvectors 186 4.10 Spectral Decomposition 196 4.11 Unbounded Operators 201 4.12 Exercises 211 CHAPTER 5 Applications to Integral and Differential Equations 217 5.1 Introduction 217 5.2 Basic Existence Theorems 218 5.3 Fredholm Integral Equations 224 5.4 Method of Successive Approximations 226 5.5 Volterra Integral Equations 228 5.6 Method of Solution for a Separable Kernel 233 5.7 Volterra Integral Equations of the First Kind and Abel’s Integral Equation 236 5.8 Ordinary Differential Equations and Differential Operators 239 5.9 Sturm–Liouville Systems 247 5.10 Inverse Differential Operators and Green’s Functions 253 5.11 The Fourier Transform 258 5.12 Applications of the Fourier Transform to Ordinary Differential Equations and Integral Equations 271 5.13 Exercises 279
CHAPTER 6 Generalized Functions and Partial Differential Equations 287 6.1 Introduction 287 6.2 Distributions 288 6.3 Sobolev Spaces 300 6.4 Fundamental Solutions and Green’s Functions for Partial Differential Equations 303 6.5 Weak Solutions of Elliptic Boundary Value Problems 323 6.6 Examples of Applications of the Fourier Transform to Partial Differential Equations 329 6.7 Exercises 343
CHAPTER 7 Mathematical Foundations of Quantum Mechanics 351 7.1 Introduction 351 7.2 Basic Concepts and Equations of Classical Mechanics 352 Poisson’s Brackets in Mechanics 361 7.3 Basic Concepts and Postulates of Quantum Mechanics 363 7.4 The Heisenberg Uncertainty Principle 377 7.5 The Schrödinger Equation of Motion 379 7.6 The Schrödinger Picture 395 7.7 The Heisenberg Picture and the Heisenberg Equation of Motion 401 7.8 The Interaction Picture 405 7.9 The Linear Harmonic Oscillator 407 7.10 Angular Momentum Operators 412 7.11 The Dirac Relativistic Wave Equation 420 7.12 Exercises 423
CHAPTER 8 Wavelets and Wavelet Transforms 433 8.1 Brief Historical Remarks 433 8.2 Continuous Wavelet Transforms 436 8.3 The Discrete Wavelet Transform 444 8.4 Multiresolution Analysis and Orthonormal Bases of Wavelets 452 8.5 Examples of Orthonormal Wavelets 462 8.6 Exercises 473
CHAPTER 9 Optimization Problems and Other Miscellaneous Applications 477 9.1 Introduction 477 9.2 The Gateaux and Fréchet Differentials 478 9.3 Optimization Problems and the Euler–Lagrange Equations 490 9.4 Minimization of Quadratic Functionals 505 9.5 Variational Inequalities 507 9.6 Optimal Control Problems for Dynamical Systems 510 9.7 Approximation Theory 517 9.8 The Shannon Sampling Theorem 522 9.9 Linear and Nonlinear Stability 526 9.10 Bifurcation Theory 530 9.11 Exercises 535
Hints and Answers to Selected Exercises 547 Bibliography 565 Index 571
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