Fundamentals of Mathematical Analysis

Description

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. 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