Introduction to Hilbert Spaces with Applications - 3rd Edition - ISBN: 9780122084386, 9780080455921

Introduction to Hilbert Spaces with Applications

3rd Edition

Authors: Lokenath Debnath Piotr Mikusinski
Hardcover ISBN: 9780122084386
eBook ISBN: 9780080455921
Imprint: Academic Press
Published Date: 29th September 2005
Page Count: 600
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Description

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.

Key Features

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

Readership

2-semester course on Functional Analysis or Hilbert space course for junior-senior-grad math students, Also researchers and others interested in math theory.

Table of Contents

Dedication

Preface to the Third Edition

Preface to the Second Edition

Preface to the First Edition

Chapter 1: Normed Vector Spaces

1.1 Introduction

1.2 Vector Spaces

1.3 Normed Spaces

1.4 Banach Spaces

1.5 Linear Mappings

1.6 Contraction Mappings and the Banach Fixed Point Theorem

1.7 Exercises

Chapter 2: The Lebesgue Integral

2.1 Introduction

2.2 Step Functions

2.3 Lebesgue Integrable Functions

Definition 2.3.1. (Lebesgue integrable function)

Lemma 2.3.2.

2.4 The Absolute Value of an Integrable Function

2.5 Series of Integrable Functions

2.6 Norm in L1()

Definition 2.6.1. (L1-norm)

Definition 2.6.2. (Null function)

Theorem 2.6.3.

Definition 2.6.4. (Convergence in norm)

Theorem 2.6.5.

Theorem 2.6.6.

2.7 Convergence Almost Everywhere

2.8 Fundamental Convergence Theorems

2.9 Locally Integrable Functions

2.10 The Lebesgue Integral and the Riemann Integral

2.11 Lebesgue Measure on

2.12 Complex-Valued Lebesgue Integrable Functions

2.13 The Spaces Lp()

2.14 Lebesgue Integrable Functions on N

2.15 Convolution

2.16 Exercises

Chapter 3: Hilbert Spaces and Orthonormal Systems

3.1 Introduction

3.2 Inner Product Spaces

3.3 Hubert Spaces

3.4 Orthogonal and Orthonormal Systems

3.5 Trigonometric Fourier Series

3.6 Orthogonal Complements and Projections

3.7 Linear Functional and the Riesz Representation Theorem

3.8 Exercises

Chapter 4: Linear Operators on Hilbert Spaces

4.1 Introduction

4.2 Examples of Operators

4.3 Bilinear Functional and Quadratic Forms

4.4 Adjoint and Self-Adjoint Operators

4.5 Invertible, Normal, Isometric, and

Details

No. of pages:
600
Language:
English
Copyright:
© Academic Press 2006
Published:
Imprint:
Academic Press
Hardcover ISBN:
9780122084386
eBook ISBN:
9780080455921

About the Author

Lokenath Debnath

Lokenath Debnath is Professor of the Department of Mathematics and Professor of Mechanical and Aerospace Engineering at the University of Central Florida in Orlando. He received his M.Sc. and Ph.D. degrees in pure mathematics from the University of Calcutta, and obtained D.I.C. and Ph.D. degrees in applied mathematics from the Imperial College of Science and Technology, University of London. He was a Senior Research Fellow at the University of Cambridge and has had visiting appointments to several universities in the United States and abroad. His many honors and awards include two Senior Fulbright Fellowships and an NSF Scientist award to visit India for lectures and research. Dr. Debnath is author or co-author of several books and research papers in pure and applied mathematics, and serves on several editorial boards for scientific journals. He is the current and founding Managing Editor of the International Journal of Mathematics and Mathematical Sciences.

Affiliations and Expertise

University of Central Florida, Orlando, U.S.A.

Piotr Mikusinski

Piotr Mikusinski received his Ph.D. in mathematics from the Institute of Mathematics of the Polish Academy of Sciences. In 1983, he became visiting lecturer at the University of California at Santa Barbara, where he spent two years. He is currently a member of the faculty in the Department of Mathematics at the University of Central Florida in Orlando. His main research interests are the theory of generalized functions and real analysis. He has published many research articles and is co-author with his father, Jan Mikusinski, of An Introduction to Analysis: From Number to Integral.

Affiliations and Expertise

University of Central Florida, Orlando, U.S.A.

Reviews

"...this is a very useful and good book and it can find a place in the library of anybody interested in functional analysis, particularly Hilbert Spaces and their applications." -MAA REVIEWS