Preface

　Chapter 1 Introduction

　 1. Black Boxes

　 2. Structure of the Plane

　 3. Mathematical Modeling

　 4. The Axiomatic Method. The

　 Process of Abstraction

　 5. Proofs of Theorems

　Chapter 2 Set-Theoretic Structure

　 1. Introduction

　 2. Basic Set Operations

　 3. Cartesian Products

　 4. Sets of Numbers

　 5. Equivalence Relations and

　 Partitions

　 6. Functions

　 7. Inverses

　 8. Systems Types

　Chapter 3 Topological Structure

　 1. Introduction

　 Port A Introduction to Metric Spaces

　 2. Metric Spaces: Definition

　 3. Examples of Metric Spaces

　 4. Subspaces and Product Spaces

　 5. Continuous Functions

　 6. Convergent Sequences

　 7. A Connection Between

　 Continuity and Convergence

　 Part B Some Deeper Metric

　 Space Concepts

　 8. Local Neighborhoods

　 9. Open Sets

　 10. More on Open Sets

　 11. Examples of Homeomorphic

　 Metric Spaces

　 12. Closed Sets and the Closure

　 Operation

　 13. Completeness

　 14. Completion of Metric Spaces

　 15. Contraction Mapping

　 16. Total Boundexlness and

　 Approximations

　 17. Compactness

　Chapter 4 Algebraic Structure

　 1. Introduction

　 Part A Introduction to Linear Spaces

　 2. Linear Spaces and Linear

　 Subspaces

　 3. Linear Transformations

　 4. Inverse Transformations

　 5. Isomorphisms

　 6. Linear Independence and

　 Dependence

　 7. Hamel Bases and Dimension

　 8. The Use of Matrices to Represent

　 Linear Transformations

　 9. Equivalent Linear

　 Transformations

　 Part B Further Topics

　 10. Direct Sums and Sums

　 11. Projections

　 12. Linear Functionals and the Alge-

　 braic Conjugate of a Linear Space

　 13. Transpose of a Linear

　 Transformation

　Chapter 5 Combined Topological

　 and Algebraic Structure

　 1. Introduction

　 Part A Banach Spaces

　 2. Definitions

　 3. Examples of Normal Linear

　 Spaces

　 4. Sequences and Series

　 5. Linear Subspaces

　 6. Continuous Linear

　 Transformations

　 7. Inverses and Continuous Inverses

　 8. Operator Topologies

　 9. Equivalence of Normed Linear

　 Spaces

　 10. Finite-Dimensional Spaces

　 11. Normed Conjugate Space and

　 Conjugate Operator

　 Part B Hilbert Spaces

　 12. Inner Product and HUbert Spaces

　 13. Examples

　 14. Orthogonality

　 15. Orthogonal Complements and the

　 Projection Theorem

　 16. Orthogonal Projections

　 17. Orthogonal Sets and Bases:

　 Generalized Fourier Series

　 18. Examples of Orthonormal Bases

　 19. Unitary Operators and Equiv-

　 alent Inner Product Spaces

　 20. Sums and Direct Sums of

　 Hilbert Spaces

　 21. Continuous Linear Functionals

　 Part C Special Operators

　 22. The Adjoint Operator

　 23. Normal and Self-Adjoint

　 Operators

　 24. Compact Operators

　 25. Foundations of Quantum

　 Mechanics

　Chapter 6 Analysis of Linear Oper-

　 ators (Compact Case)

　 1. Introductioa

　 Part A An Illustrative Example

　 2. Geometric Analysis of Operators

　 3. Geometric Analysis. The Eigen-

　 value-Eigenvector Problem

　 4. A Finite-Dimensional Problem

　 Part B The Spectrum

　 5. The Spectrum of Linear

　 Transformations

　 6. Examples of Spectra

　 7. Properties of the Spectrum

　 Part C Spectral Analysis

　 8. Resolutions of the Identity

　 9. Weighted Sums of Projections

　 10. Spectral Properties of Compact,

　 Normal, and Self-Adjoint

　 Operators

　 11. The Spectral Theorem

　 12. Functions of Operators

　 (Operational Calculus)

　 13. Applications of the Spectral

　 Theorem

　 14. Nonnormal Operators

　Chapter 7 Analysis of Unbounded

　 Operators

　 1. Introduction

　 2. Green's Functions

　 3. Symmetric Operators

　 4. Examples of Symmetric

　 Operators

　 5. Sturmiouville Operators

　 6. Ghrding's Inequality

　 7. EUiptie Partial Differential

　 Operators

　 8. The Dirichlet Problem

　 9. The Heat Equation and Wave

　 Equation

　 10. Self-Adjoint Operators

　 11. The Cayley Transform

　 12. Quantum Mechanics, Revisited

　 13. Heisenberg Uncertainty Principle

　 14. The Harmonic Oscillator

　 Appendix ,4 The H61der, Schwartz,

　 and Minkowski

　 Inequalities

　 Appendix B Cardinality

　 Appendix C Zom's temnm

　 Appendix D Integration and

　 Measure Theory

　 1. Introduction

　 2. The Riemann Integral

　 3. A Problem with the Riemann

　 Integral

　 4. The Space Co

　 5. Null Sets

　 6. Convergence Almost Everywhere

　 7. The Lebesgue Integral

　 8. Limit Theorems

　 9. Miscellany

　 10. Other Definitions of the Integral

　 11. The Lebesgue Spaces,

　 12. Dense Subspaees of

　 13. Differentiation

　 14. The Radon-Nikodym Theorem

　 15. Fubini Theorem

　 Appendix E Probability Spaces and

　 Stochastic Processes

　 1. Probability Spaces

　 2. Random Variables and

　 Probability Distributions

　 3. Expectation

　 4. Stochastic Independence

　 5. Conditional Expectation Operator

　 6. Stochastic Processes

　 Index of Symbols

　 Index