Fundamentals of Mathematical Analysis

Cover
Fundamentals of Mathematical Analysis
Copyright
Dedication
Preface
Acknowledgments
The Book in Synopsis
	Part I. Background Material
	Part II. Topology
	Part III. Functional Analysis
	Part IV. Integration Theory
	Appendices
Contents
1: Preliminaries
	1.1 Sets, Functions, and Relations
		Indexed Sets
		Exercises
	1.2 The Real and Complex Number Fields
		Real Numbers
		Complex Numbers
		Exercises
2: Set Theory
	2.1 Finite, Countable, and Uncountable Sets
		Exercises
	2.2 Zorn’s Lemma and the Axiom of Choice
		Exercises
	2.3 Cardinal Numbers
		Cardinal Arithmetic
		The Continuum Hypothesis
		Exercises
3: Vector Spaces
	3.1 Definitions and Basic Properties
		Exercises
	3.2 Independent Sets and Bases
		Exercises
	3.3 The Dimension of a Vector Space
		Exercises
	3.4 Linear Mappings, Quotient Spaces, and Direct Sums
		Quotient Spaces
		Direct Sums
		Linear Functionals and Operators
		Exercises
	3.5 Matrix Representation and Diagonalization
		Matrix Representations of Linear Mappings
		Diagonalization
		Exercises
	3.6 Normed Linear Spaces
		Spaces
		Balls, Lines, and Convex Sets
		Excursion: Convex Hulls and Polytopes
		Exercises
	3.7 Inner Product Spaces
		The Gram-Schmidt Process
		The Spectral Decomposition of a Normal Matrix
		Spectral Theory for Normal Operators
		Exercises
4: The Metric Topology
	4.1 Definitions and Basic Properties
		Exercises
	4.2 Interior, Closure, and Boundary
		Separation by Open Sets
		Subspaces
		The Cantor Set
		Exercises
	4.3 Continuity and Equivalent Metrics
		Homeomorphisms
		Exercises
	4.4 Product Spaces
		Exercises
	4.5 Separable Spaces
		Exercise
	4.6 Completeness
		Exercises
	4.7 Compactness
		Exercises
	4.8 Function Spaces
		Application: A Space-Filling Curve
		Exercise
	4.9 The Stone-Weierstrass Theorem
	4.10 Fourier Series and Orthogonal Polynomials
		Fourier series
		Orthogonal Polynomials: The General Construction
		The Legendre Polynomials
		The Tchebychev Polynomials
		The Hermite Polynomials
		Exercises
5: Essentials of General Topology
	5.1 Definitions and Basic Properties
		Subspace Topology
		Exercises
	5.2 Bases and Subbases
		Exercises
	5.3 Continuity
		Two Important Function Spaces
		Homeomorphisms
		Upper and Lower Semicontinuous Functions
		Exercises
	5.4 The Product Topology: The Finite Case
		Exercises
	5.5 Connected Spaces
		Exercises
	5.6 Separation by Open Sets
		Exercises
	5.7 Second Countable Spaces
		Exercises
	5.8 Compact Spaces
		Compactness and Separation
		Finite Products of Compact Spaces
		Exercises
	5.9 Locally Compact Spaces
		Exercises
	5.10 Compactification
		Exercises
	5.11 Metrization
		Exercises
	5.12 The Product of Infinitely Many Spaces
		Exercises
6: Banach Spaces
	6.1 Finite vs. Infinite-Dimensional Spaces
		Exercises
	6.2 Bounded Linear Mappings
		Exercises
	6.3 Three Fundamental Theorems
		Exercises
	6.4 The Hahn-Banach Theorem
		Exercises
	6.5 The Spectrum of an Operator
		Exercises
	6.6 Adjoint Operators and Quotient Spaces
		Quotient Spaces
		Exercises
	6.7 Weak Topologies
		Exercises
7: Hilbert Spaces
	7.1 Definitions and Basic Properties
		The Completion of an Inner Product Space.
		Exercises
	7.2 Orthonormal Bases and Fourier Series
		Excursion: Inseparable Hilbert Spaces
		Exercises
	7.3 Self-Adjoint Operators
		Normal and Unitary Operators
		Exercises
	7.4 Compact Operators
		The Eigenvalues of a Compact Operator
		The Fredholm Theory
		The Spectral Theorem
		Excursion: Integral Equations
		Exercises
	7.5 Compact Operators on Banach Spaces
		Exercises
8: Integration Theory
	8.1 The Riemann Integral
		Exercises
	8.2 Measure Spaces
		Outer Measures
		Measurable Functions
		Excursion: The Hopf Extension Theorem2
		Exercises
	8.3 Abstract Integration
		Convergence Theorems
		Exercises
	8.4 Lebesgue Measure on Rn
		Preliminaries
		Dicing Rn
		Lebesgue measure: Motivation and Overview
		Lebesgue Measure
		Excursion: Radon Measures
		Exercises
	8.5 Complex Measures
		Exercises
	8.6 ..p Spaces
		Representation of Bounded Linear Functionals on ..p
		Exercises
	8.7 Approximation
		Approximation by ..8 Functions
		Exercises
	8.8 Product Measures
		Products of Measurable Spaces
		Product Measures
		Fubini’s Theorem
		Products of Lebesgue Measures
		Excursion: The Product of Finitely Many Measures
		Exercises
	8.9 A Glimpse of Fourier Analysis
		Fourier Series of 2..-Periodic Functions
		Fourier Series of ..p-Functions
		The Fourier Transform
		Orthogonal Polynomials: One More Time
		Exercises
APPENDIX A: The Equivalence of Zorn’s Lemma, the Axiom of Choice, and the Well Ordering Principle
APPENDIX B: Matrix Factorizations
Bibliography
Glossary of Symbols
Index