# Book:Erwin Kreyszig/Introductory Functional Analysis with Applications/Second Edition

## Erwin Kreyszig: Introductory Functional Analysis with Applications (2nd Edition)

Published $\text {1989}$, Wiley

ISBN 9780471504597.

### Subject Matter

Functional Analysis

### Contents

Preface

Contents

Notations

Chapter 1. Metric Spaces
1.1 Metric Space
1.2 Further Examples of Metric Spaces
1.3 Open Set, Closed Set, Neighborhood
1.4 Convergence, Cauchy Sequence, Completeness
1.5 Examples. Completeness Proofs
1.6 Completion of Metric Spaces
Chapter 2. Normed Spaces. Banach Spaces
2.1 Vector Space
2.2 Normed Space. Banach Space
2.3 Further Properties of Normed Spaces
2.4 Finite Dimensional Normed Spaces and Subspaces
2.5 Compactness and Finite Dimension
2.6 Linear Operators
2.7 Bounded and Continuous Linear Operators
2.8 Linear Functionals
2.9 Linear Operators and Functionals on Finite Dimensional Spaces
2.10 Normed Spaces of Operators. Dual Space
Chapter 3. Inner Product Saces. Hilbert Spaces
3.1 Inner Product Space. Hilbert Space
3.2 Further Properties of Inner Product Spaces
3.3 Orthogonal Complements and Direct Sums
3.4 Orthonormal Sets and Sequences
3.5 Series Related to Orthonormal Sequences and Sets
3.6 Total Orthonormal Sets and Sequences
3.7 Legendre, Hermite and Laguerre Polynomials
3.8 Representation of Functionals on Hilbert Spaces
3.10 Self-Adjoint, Unitary and Normal Operators
Chapter 4. Fundamental Theorems for Normed and Banach Spaces
4.1 Zorn's Lemma
4.2 Hahn-Banach Theorem
4.3 Hahn-Banach Theorem for Complex Vector Spaces and Normed Spaces
4.4 Application to Bounded Linear Functionals on C[a, b]
4.6 Reflexive Spaces
4.7 Category Theorem. Uniform Boundedness Theorem
4.8 Strong and Weak Convergence
4.9 Convergence of Sequences of Operators and Functionals
4.10 Application to Summability of Sequences
4.11 Numerical Integration and Weak* Convergence
4.12 Open Mapping Theorem
4.13 Closed Linear Operators. Closed Graph Theorem
Chapter 5. Further Applications: Banach Fixed Point Theorem
5.1 Banach Fixed Point Theorem
5.2 Application of Banach's Theorem to Linear Equations
5.3 Application of Banach's Theorem to Differential Equations
5.4 Application of Banach's Theorem to Integral Equations
Chapter 6. Further Applications: Approximation Theory
6.1 Approximation in Normed Spaces
6.2 Uniqueness, Strict Convexity
6.3 Uniform Approximation
6.4 Chebyshev Polynomials
6.5 Approximation in Hilbert Space
6.6 Splines
Chapter 7. Spectral Theory of Linear Operators in Normed Spaces
7.1 Spectral Theory in Finite Dimensional Normed Spaces
7.2 Basic Concepts
7.3 Spectral Properties of Bounded Linear Operators
7.4 Further Properties of Resolvent and Spectrum
7.5 Use of Complex Analysis IN Spectral Theory
7.6 Banach Algebras
7.7 Further Properties of Banach Algebras
Chapter 8. Compact Linear Operators on Normed Spaces and Their Spectrum
8.1 Compact Linear Operators on Normed Spaces
8.2 Further Properties of Compact Linear Operators
8.3 Spectral Properties of Compact Linear Operators on Normed Spaces
8.4 Further Spectral Properties of Compact Linear ()perators
8.5 Operator Equations Involving Compact Linear Operators
8.6 Further Theorems of Fredholm Type
8.7 Fredholm Alternative
Chapter 9. Spectral Theory of Bounded Self-Adjoint Linear Operators
9.1 Spectral Properties of Bounded Self-Adjoint Linear Operators
9.2 Further Spectral Properties of Bounded Self-Adjoint Linear Operators
9.3 Positive Operators
9.4 Square Roots of a Positive Operator
9.5 Projection Operators
9.6 Further Properties of Projections
9.7 Spectral Family
9.8 Spectral Family of a Bounded Self-Adjoint Linear Operator
9.9 Spectral Representation of Bounded Self-Adjoint Linear Operators
9.10 Extension of the Spectral Theorem to Continuous Functions
9.11 Properties of the Spectral Family of a Bounded Self-Adjoint Linear Operator
Chapter 10. Unbounded Linear Operators in Hilbert Space
10.1 Unbounded Linear Operators and their Hilbert-Adjoint Operators
10.3 Closed Linear Operators and Closures
10.4 Spectral Properties of Self-Adjoint. Linear Operators
10.5 Spectral Representation of Unitary Operators
10.6 Spectral Representation of Self-Adjoint Linear operators
10.7 Multiplication Operator and Differentiation Operator
Chapter 11. Unbounded Linear Operators in Quantum Mechnics
11.1 Basic Ideas. States, Observables, Position Operator
11.2 Momentum Operator. Heisenberg Uncertainty Principle
11.3 Time-Independent Schrodinger Equation
11.4 Hamilton Operator
11.5 Time-Dependent SchrOdinger Equation
Appendix 1: Some Material for Review and Reference
A1.1 Sets
A1.2 Mappings
A1.3 Families
A1.4 Equivalence Relations
A1.5 Compactness
A1.6 Supremum and Infimum
A1.7 Cauchy Convergence Criterion
A1.8 Groups
Appendix 2: Answers to Odd-Numbered Problems
Appendix 3: References
Index