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#### Book Details

### Abstract

This book provides the reader with a comprehensive introduction to functional analysis. Topics include normed linear and Hilbert spaces, the Hahn-Banach theorem, the closed graph theorem, the open mapping theorem, linear operator theory, the spectral theory, and a brief introduction to the Lebesgue measure. The book explains the motivation for the development of these theories, and applications that illustrate the theories in action. Applications in optimal control theory, variational problems, wavelet analysis and dynamical systems are also highlighted. ‘A First Course in Functional Analysis’ will serve as a ready reference to students not only of mathematics, but also of allied subjects in applied mathematics, physics, statistics and engineering.

Rabindranath Sen is a retired professor and former head of the Department of Applied Mathematics at the University of Calcutta.

‘A First Course in Functional Analysis: Theory and Applications’ provides a comprehensive introduction to functional analysis, beginning with the fundamentals and extending into theory and applications. The volume starts with an introduction to sets and metric spaces and the notions of convergence, completeness and compactness, and continues to a detailed treatment of normed linear spaces and Hilbert spaces. The reader is then introduced to linear operators and functionals, the Hahn-Banach theorem on linear bounded functionals, conjugate spaces and adjoint operators, and the space of linear bounded functionals. Further topics include the closed graph theorem, the open mapping theorem, linear operator theory including unbounded operators, spectral theory, and a brief introduction to the Lebesgue measure. The cornerstone of the book lies in the motivation for the development of these theories, and applications that illustrate the theories in action.

One of the many strengths of this book is its detailed discussion of the theory of compact linear operators and their relationship to singular operators. Applications in optimal control theory, variational problems, wavelet analysis and dynamical systems are highlighted.

This volume strikes an ideal balance between concision of mathematical exposition and offering complete explanatory materials and careful step-by-step instructions. It will serve as a ready reference not only for students of mathematics, but also students of physics, applied mathematics, statistics and engineering.One of the many strengths of the book is the detailed discussion of the theory of compact linear operators and their relationship to singular operators. Applications in optimal control theory, variational problems, wavelet analysis, and dynamical systems are highlighted.

This volume strikes the ideal balance between concision of mathematical exposition, and complete explanatory material accompanied by careful step-by-step instructions intended to serve as a ready reference not only for students of mathematics, but also students of physics, applied mathematics, statistics and engineering.

### Table of Contents

Section Title | Page | Action | Price |
---|---|---|---|

FRONT MATTER\r | i | ||

Half Title | i | ||

Title | iii | ||

Copyright\r | iv | ||

Preface | vii | ||

Contents | xiii | ||

Introduction | xvii | ||

CHAPTER 1 PRELIMINARIES | xix | ||

1.1 Set | xix | ||

1.2 Function, Mapping | 3 | ||

1.3 Linear Space | 4 | ||

1.4 Metric Spaces | 10 | ||

1.5 Topological Spaces | 43 | ||

1.6 Continuity, Compactness | 50 | ||

CHAPTER 2 NORMED LINEAR SPACES | 57 | ||

2.1 Definitions and Elementary Properties | 57 | ||

2.2 Subspace, Closed Subspace | 72 | ||

2.3 Finite Dimensional Normed Linear Spaces and Subspaces | 75 | ||

2.4 Quotient Spaces | 84 | ||

2.5 Completion of Normed Spaces\r | 87 | ||

CHAPTER 3 HILBERT SPACE | 90 | ||

3.1 Inner Product Space, Hilbert Space | 90 | ||

3.2 Cauchy-Bunyakovsky-Schwartz (CBS) Inequality | 93 | ||

3.3 Parallelogram Law | 94 | ||

3.4 Orthogonality | 99 | ||

3.5 Orthogonal Projection Theorem | 101 | ||

3.6 Orthogonal Complements, Direct Sum | 106 | ||

3.7 Orthogonal System | 110 | ||

3.8 Complete Orthonormal System | 121 | ||

3.9 Isomorphism between Separable Hilbert Spaces | 127 | ||

CHAPTER 4 LINEAR OPERATORS | 129 | ||

4.1 Definition: Linear Operator | 129 | ||

4.2 Linear Operators in Normed Linear Spaces | 135 | ||

4.3 Linear Functionals | 146 | ||

4.4 The Space of Bounded Linear Operators | 149 | ||

4.5 Uniform Boundedness Principle | 153 | ||

4.6 Some Applications | 158 | ||

4.7 Inverse Operators | 163 | ||

4.8 Banach Space with a Basis | 174 | ||

CHAPTER 5 LINEAR FUNCTIONALS | 178 | ||

5.1 Hahn-Banach Theorem | 178 | ||

5.2 Hahn-Banach Theorem for Complex Vector and Normed Linear Space | 185 | ||

5.3 Application to Bounded Linear Functionals on C([a, b]) | 193 | ||

5.4 The General Form of Linear Functionals in Certain Functional Spaces | 199 | ||

5.5 The General Form of Linear Functionals in Hilbert Spaces | 203 | ||

5.6 Conjugate Spaces and Adjoint Operators | 205 | ||

CHAPTER 6 SPACE OF BOUNDED LINEAR FUNCTIONALS | 220 | ||

6.1 Conjugates (Duals) and Transposes (Adjoints) | 220 | ||

6.2 Conjugates (Duals) of Lp([a, b]) and C([a, b]) | 227 | ||

6.3 Weak∗ and Weak Convergence | 239 | ||

6.4 Reflexivity | 251 | ||

6.5 Best Approximation in Reflexive Spaces | 260 | ||

CHAPTER 7 CLOSED GRAPH THEOREM AND ITS CONSEQUENCES | 266 | ||

7.1 Closed Graph Theorem | 266 | ||

7.2 Open Mapping Theorem | 271 | ||

7.3 Bounded Inverse Theorem | 275 | ||

7.4 Applications of the Open Mapping Theorem | 279 | ||

CHAPTER 8 COMPACT OPERATORS ON NORMED LINEAR SPACES | 281 | ||

8.1 Compact Linear Operators | 281 | ||

8.2 Spectrum of a Compact Operator | 289 | ||

8.3 Fredholm Alternative | 300 | ||

8.4 Approximate Solutions | 310 | ||

CHAPTER 9 ELEMENTS OF SPECTRAL THEORY OF SELF-ADJOINT OPERATORS IN HILBERT SPACES | 322 | ||

9.1 Adjoint Operators | 322 | ||

9.2 Self-Adjoint Operators | 324 | ||

9.3 Quadratic Form | 326 | ||

9.4 Unitary Operators, Projection Operators | 328 | ||

9.5 Positive Operators, Square Roots of a Positive Operator | 333 | ||

9.6 Spectrum of Self-Adjoint Operators | 340 | ||

9.7 Invariant Subspaces | 344 | ||

9.8 Continuous Spectra and Point Spectra | 346 | ||

CHAPTER 10 MEASURE AND INTEGRATION IN Lp SPACES | 353 | ||

10.1 The Lebesgue Measure on R | 353 | ||

10.2 Measurable and Simple Functions | 359 | ||

10.3 Calculus with the Lebesgue Measure | 366 | ||

10.4 The Fundamental Theorem for Riemann Integration | 367 | ||

10.5 The Fundamental Theory for Lebesgue Integration | 368 | ||

10.6 Lp Spaces and Completeness | 369 | ||

10.7 Lp Convergence of Fourier Series | 377 | ||

CHAPTER 11 UNBOUNDED LINEAR OPERATORS | 380 | ||

11.1 Definition: An Unbounded Linear Operator | 381 | ||

11.2 States of a Linear Operator | 381 | ||

11.3 Definition: Strictly Singular Operators | 385 | ||

11.4 Relationship between Singular and Compact Operators | 388 | ||

11.5 Perturbation by Bounded Operators | 389 | ||

11.6 Perturbation by Strictly Singular Operators | 392 | ||

CHAPTER 12 THE HAHN-BANACH THEOREM AND OPTIMIZATION PROBLEMS | 399 | ||

12.1 The Separation of a Convex Set | 399 | ||

12.2 Minimum Norm Problem and the Duality Theory | 401 | ||

12.3 Application to Chebyshev Approximation | 405 | ||

12.4 Application to Optimal Control Problems | 406 | ||

CHAPTER 13 VARIATIONAL PROBLEMS | 409 | ||

13.1 Minimization of Functionals in a Normed Linear Space | 409 | ||

13.2 Gateaux Derivative \r | 409 | ||

13.3 Frechet Derivative\r | 416 | ||

13.4 Equivalence of the Minimizing Problem for Solving Variational Inequality | 419 | ||

13.5 Distributions | 423 | ||

13.6 Sobolev Space | 424 | ||

CHAPTER 14 THE WAVELET ANALYSIS | 429 | ||

14.1 An Introduction to Wavelet Analysis | 429 | ||

14.2 The Scalable Structure of Information | 430 | ||

14.3 Algebra and Geometry of Wavelet Matrices | 432 | ||

14.4 One-Dimensional Wavelet Systems | 438 | ||

CHAPTER 15 DYNAMICAL SYSTEMS | 442 | ||

15.1 A Dynamical System and Its Properties | 442 | ||

15.2 Homeomorphism, Diffeomorphism, Riemannian Manifold | 446 | ||

15.3 Stable Points, Periodic Points and Critical Points | 448 | ||

15.4 Existence, Uniqueness and Topological Consequences | 450 | ||

15.5 Bifurcation Points and Some Results | 451 | ||

END MATTER\r | 453 | ||

List of Symbols | 453 | ||

Abbreviations | 457 | ||

Bibliography | 458 | ||

Index | 462 |