目錄
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
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