Fundamentals of Mathematical Analysis explores real and functional analysis with a substantial component on topology. The three leading chapters furnish background information on the real and complex number fields, a concise introduction to set theory, and a rigorous treatment of vector spaces.
Fundamentals of Mathematical Analysis is an extensive study of metric spaces, including the core topics of completeness, compactness and function spaces, with a good number of applications. The later chapters consist of an introduction to general topology, a classical treatment of Banach and Hilbert spaces, the elements of operator theory, and a deep account of measure and integration theories. Several courses can be based on the book. This book is suitable for a two-semester course on analysis, and material can be chosen to design one-semester courses on topology or real analysis. It is designed as an accessible classical introduction to the subject and aims to achieve excellent breadth and depth and contains an abundance of examples and exercises. The topics are carefully sequenced, the proofs are detailed, and the writing style is clear and concise. The only prerequisites assumed are a thorough understanding of undergraduate real analysis and linear algebra, and a degree of mathematical maturity.
Author(s): Adel N. Boules
Publisher: OUP Oxford
Year: 2021
Language: English
Commentary: True PDF
Pages: 480
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
