Introduction to Hilbert Spaces with Applications: Debnath; Mikusinski: 9780122084386: Book: : eCampus.com

Building on the success of the two previous editions, Introduction to Hilbert Spaces with Applications, 3E, offers an overview of the basic ideas and results of Hilbert space theory and functional analysis. It acquaints students with the Lebesgue integral, and includes an enhanced presentation of results and proofs. Students and researchers will 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 popular chapter on wavelets that has been completely updated. Students and researchers agree that this is the definitive text on Hilbert Space theory.

* Updated chapter on wavelets
* Improved presentation on results and proof
* Revised examples and updated applications
* Completely updated list of references .

Preface to the Third Edition

Preface to the Second Edition

xiii

Preface to the First Edition

CHAPTER 1 Normed Vector Spaces

(38)

1.1 Introduction

(1)

1.2 Vector Spaces

(6)

1.3 Normed Spaces

(11)

1.4 Banach Spaces

(6)

1.5 Linear Mappings

(7)

1.6 Contraction Mappings and the Banach Fixed Point Theorem

(2)

1.7 Exercises

(5)

CHAPTER 2 The Lebesgue Integral

(54)

2.1 Introduction

(1)

2.2 Step Functions

(5)

2.3 Lebesgue Integrable Functions

(3)

2.4 The Absolute Value of an Integrable Function

(2)

2.5 Series of Integrable Functions

(2)

2.6 Norm in L� (R)

(3)

2.7 Convergence Almost Everywhere

(3)

2.8 Fundamental Convergence Theorems

(4)

2.9 Locally Integrable Functions

(2)

2.10 The Lebesgue Integral and the Riemann Integral

(3)

2.11 Lebesgue Measure on R

(4)

2.12 Complex-Valued Lebesgue Integrable Functions

(3)

2.13 The Spaces Lp(R)

(4)

2.14 Lebesgue Integrable Functions on RN

(4)

2.15 Convolution

(2)

2.16 Exercises

(9)

CHAPTER 3 Hilbert Spaces and Orthonormal Systems

(52)

3.1 Introduction

(1)

3.2 Inner Product Spaces

(5)

3.3 Hilbert Spaces

(6)

3.4 Orthogonal and Orthonormal Systems

105

(17)

3.5 Trigonometric Fourier Series

122

(5)

3.6 Orthogonal Complements and Projections

127

(5)

3.7 Linear Functionals and the Riesz Representation Theorem

132

(3)

3.8 Exercises

135

(10)

CHAPTER 4 Linear Operators on Hilbert Spaces

145

(72)

4.1 Introduction

145

(1)

4.2 Examples of Operators

146

(5)

4.3 Bilinear Functionals and Quadratic Forms

151

(7)

4.4 Adjoint and Self-Adjoint Operators

158

(5)

4.5 Invertible, Normal, Isometric, and Unitary Operators

163

(5)

4.6 Positive Operators

168

(7)

4.7 Projection Operators

175

(5)

4.8 Compact Operators

180

(6)

4.9 Eigenvalues and Eigenvectors

186

(10)

4.10 Spectral Decomposition

196

(5)

4.11 Unbounded Operators

201

(10)

4.12 Exercises

211

(6)

CHAPTER 5 Applications to Integral and Differential Equations

217

(70)

5.1 Introduction

217

(1)

5.2 Basic Existence Theorems

218

(6)

5.3 Fredholm Integral Equations

224

(2)

5.4 Method of Successive Approximations

226

(2)

5.5 Volterra Integral Equations

228

(5)

5.6 Method of Solution for a Separable Kernel

233

(3)

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

236

(3)

5.8 Ordinary Differential Equations and Differential Operators

239

(8)

5.9 Sturm�Liouville Systems

247

(6)

5.10 Inverse Differential Operators and Green's Functions

253

(5)

5.11 The Fourier Transform

258

(13)

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

271

(8)

5.13 Exercises

279

(8)

CHAPTER 6 Generalized Functions and Partial Differential Equations

287

(64)

6.1 Introduction

287

(1)

6.2 Distributions

288

(12)

6.3 Sobolev Spaces

300

(3)

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

303

(20)

6.5 Weak Solutions of Elliptic Boundary Value Problems

323

(6)

6.6 Examples of Applications of the Fourier Transform to Partial Differential Equations

329

(14)

6.7 Exercises

343

(8)

CHAPTER 7 Mathematical Foundations of Quantum Mechanics

351

(82)

7.1 Introduction

351

(1)

7.2 Basic Concepts and Equations of Classical Mechanics

352

(11)

Poisson's Brackets in Mechanics

361

(2)

7.3 Basic Concepts and Postulates of Quantum Mechanics

363

(14)

7.4 The Heisenberg Uncertainty Principle

377

(2)

7.5 The Schr�dinger Equation of Motion

379

(16)

7.6 The Schr�dinger Picture

395

(6)

7.7 The Heisenberg Picture and the Heisenberg Equation of Motion

401

(4)

7.8 The Interaction Picture

405

(2)

7.9 The Linear Harmonic Oscillator

407

(5)

7.10 Angular Momentum Operators

412

(8)

7.11 The Dirac Relativistic Wave Equation

420

(3)

7.12 Exercises

423

(10)

CHAPTER 8 Wavelets and Wavelet Transforms

433

(44)

8.1 Brief Historical Remarks

433

(3)

8.2 Continuous Wavelet Transforms

436

(8)

8.3 The Discrete Wavelet Transform

444

(8)

8.4 Multiresolution Analysis and Orthonormal Bases of Wavelets

452

(10)

8.5 Examples of Orthonormal Wavelets

462

(11)

8.6 Exercises

473

(4)

CHAPTER 9 Optimization Problems and Other Miscellaneous Applications

477

(70)

9.1 Introduction

477

(1)

9.2 The Gateaux and Frechet Differentials

478

(12)

9.3 Optimization Problems and the Euler�Lagrange Equations

490

(15)

9.4 Minimization of Quadratic Functionals

505

(2)

9.5 Variational Inequalities

507

(3)

9.6 Optimal Control Problems for Dynamical Systems

510

(7)

9.7 Approximation Theory

517

(5)

9.8 The Shannon Sampling Theorem

522

(4)

9.9 Linear and Nonlinear Stability

526

(4)

9.10 Bifurcation Theory

530

(5)

9.11 Exercises

535

(12)

Hints and Answers to Selected Exercises

547

(18)

Bibliography

565

(6)

Index

571


	Buy Textbooks Sell Textbooks College Apparel Shop by School Virtual Bookstores	Order Status Shipping Rates Return Policy Marketplace Info F.A.S.T.	Contact Us Privacy Policy Legal Notices Site Security Employment	Help Desk eCampus Blog Affiliate Program Bulk Orders College Marketing

Need Help? eService@ecampus.com Copyright© 1999-2008
.