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

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
 * 3.9 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