Introduction to Hilbert Spaces With Applications: Debnath, Lokenath; Mikusinski, Piotr: 9780122084362: Book: Technology: eCampus.com

Continuing on the success of the previous edition, Introduction to Hilbert Spaces with Applications, Second Edition, offers an overview of the basic ideas and results of Hilbert space theory and functional analysis. It acquaints students with the Lebesque integral, and includes an enhanced presentation of results and proofs. Students and researchers benefit from the wealth of revised examples in new, diverse applications as they apply to optimization, variational and control problems, and problems in approximation theory, nonlinear instability, and bifurcation. The text also includes a new, well-researched chapter on wavelets. Students and researchers agree that this is the definitive text on Hilbert Space theory.

* Systematic exposition of the basic ideas and results of Hilbert space theory
* Introduction to the Lebesgue integral
* New chapter on wavelets
* Improved presentation on results and proof
* Revised examples and updated applications
* Completely updated list of references

Preface to the Second Edition

xiii

(2)

Preface to the First Edition

PART 1 THEORY

(222)

CHAPTER 1 Normed Vector Spaces

(36)

1.1 Introduction

(1)

1.2 Vector Spaces

(6)

1.3 Linear Independence, Basis, Dimension

(1)

1.4 Normed Spaces

(8)

1.5 Banach Spaces

(4)

1.6 Linear Mappings

(7)

1.7 Completion of Normed Spaces

(1)

1.8 Contraction Mappings and the Banach Fixed Point Theorem

(3)

1.9 Exercises

(6)

CHAPTER 2 The Lebesgue Integral

(50)

2.1 Introduction

(1)

2.2 Step Functions

(5)

2.3 Lebesgue Integrable Functions

(3)

2.4 The Absolute Value of an Integrable Function

(3)

2.5 Series of Integrable Functions

(2)

2.6 Norm in L(1)(R)

(2)

2.7 Convergence Almost Everywhere

(4)

2.8 Fundamental Theorems

(6)

2.9 Locally Integrable Functions

(1)

2.10 The Lebesgue Integral and the Riemann Integral

(3)

2.11 Lebesgue Measure on R

(4)

2.12 Complex-Valued Lebesgue Integrable Functions

(2)

2.13 The Space L(2)(R)

(1)

2.14 The Spaces L(1)(R(N)) and L(2)(R(N))

(4)

2.15 Convolution

(3)

2.16 Exercises

(7)

CHAPTER 3 Hilbert Spaces and Orthonormal Systems

(52)

3.1 Introduction

(1)

3.2 Inner Product Spaces--Definition and Examples

(2)

3.3 Norm in an Inner Product Space

(3)

3.4 Hilbert Spaces--Definition and Examples

(6)

3.5 Strong and Weak Convergence

(2)

3.6 Orthogonal and Orthonormal Systems

100

(5)

3.7 Properties of Orthonormal Systems

105

(10)

3.8 Trigonometric Fourier Series

115

(5)

3.9 Orthogonal Complements and Projection Theorem

120

(5)

3.10 Linear Functionals and the Riesz Representation Theorem

125

(2)

3.11 Separable Hilbert Spaces

127

(3)

3.12 Exercises

130

(9)

CHAPTER 4 Linear Operators on Hilbert Spaces

139

(84)

4.1 Introduction

139

(1)

4.2 Examples of Operators

140

(4)

4.3 Bilinear Functionals and Quadratic Forms

144

(7)

4.4 Adjoint and Self-Adjoint Operators

151

(6)

4.5 Invertible, Normal, Isometric, and Unitary Operators

157

(5)

4.6 Positive Operators

162

(6)

4.7 Projection Operators

168

(4)

4.8 Compact Operators

172

(6)

4.9 Eigenvalues and Eigenvectors

178

(10)

4.10 Spectral Decomposition

188

(5)

4.11 The Fourier Transform

193

(13)

4.12 Unbounded Operators

206

(8)

4.13 Exercises

214

(9)

PART 2 APPLICATIONS

223

(292)

CHAPTER 5 Applications to Integral and Differential Equations

223

(56)

5.1 Introduction

223

(1)

5.2 Basic Existence Theorems

224

(6)

5.3 Fredholm Integral Equations

230

(3)

5.4 Method of Successive Approximations

233

(2)

5.5 Volterra Integral Equations

235

(4)

5.6 Method of Solution for a Separable Kernel

239

(4)

5.7 Volterra Integral Equations of the First Kind and Abel's Integral Equation

243

(2)

5.8 Ordinary Differential Equations and Differential Operators

245

(9)

5.9 Sturm-Liouville Systems

254

(5)

5.10 Inverse Differential Operators and Green's Functions

259

(5)

5.11 Applications of the Fourier Transform to Ordinary Differential Equations and Integral Equations

264

(8)

5.12 Exercises

272

(7)

Chapter 6 Generalized Functions and Partial Differential Equations

279

(58)

6.1 Introduction

279

(1)

6.2 Distributions

279

(12)

6.3 Fundamental Solutions and Green's Functions for Partial Differential Equations

291

(20)

6.4 Weak Solutions of Elliptic Boundary Value Problems

311

(6)

6.5 Examples of Applications of Fourier Transforms to Partial Differential Equations

317

(13)

6.6 Exercises

330

(7)

CHAPTER 7 Mathematical Foundations of Quantum Mechanics

337

(80)

7.1 Introduction

337

(1)

7.2 Basic Concepts and Equations of Classical Mechanics

337

(12)

7.3 Basic Concepts and Postulates of Quantum Mechanics

349

(14)

7.4 The Heisenberg Uncertainty Principle

363

(2)

7.5 The Schrodinger Equation of Motion

365

(15)

7.6 The Schrodinger Picture

380

(7)

7.7 The Heisenberg Picture and the Heisenberg Equation of Motion

387

(4)

7.8 The Interaction Picture

391

(1)

7.9 The Linear Harmonic Oscillator

392

(6)

7.10 Angular Momentum Operators

398

(7)

7.11 The Dirac Relativistic Wave Equation

405

(4)

7.12 Exercises

409

(8)

CHAPTER 8 Wavelets

417

(30)

8.1 Brief Historical Remarks

417

(3)

8.2 Continuous Wavelet Transforms

420

(7)

8.3 The Discrete Wavelet Transform

427

(7)

8.4 Multiresolution Analysis and Orthonormal Bases of Wavelets

434

(9)

8.5 Exercises

443

(4)

CHAPTER 9 Optimization Problems and Other Miscellaneous Applications

447

(68)

9.1 Introduction

447

(1)

9.2 The Gateaux and Frechet Differentials

448

(12)

9.3 Optimization Problems and the Euler-Lagrange Equations

460

(15)

9.4 Minimization of Quadratic Functionals

475

(2)

9.5 Variational Inequalities

477

(3)

9.6 Optimal Control Problems for Dynamical Systems

480

(7)

9.7 Approximation Theory

487

(5)

9.8 The Shannon Sampling Theorem

492

(4)

9.9 Linear and Nonlinear Stability

496

(4)

9.10 Bifurcation Theory

500

(6)

9.11 Exercises

506

(9)

Hints and Answers to Selected Exercises

515

(16)

Bibliography

531

(6)

List of Symbols

537

(4)

Index

541


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