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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. Greens Functions 3. Symmetric Operators 4. Examples of Symmetric Operators 5. Sturmiouville Operators 6. Ghrdings 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 Zoms 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