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

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## Erwin Kreyszig:

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

Published $\text {1989}$, **Wiley**

- ISBN 9780471504597

### Subject Matter

### 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
- 3.9 Hilbert-Adjoint Operator
- 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]
- 4.5 Adjoint Operator
- 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
- 10.2 Hilbert-Adjoint Operators, Symmetric and Self-Adjoint Linear 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**