A Readable yet Rigorous Approach to an Essential Part of Mathematical Thinking
Back by popular demand, Real Analysis and Foundations, Third Edition bridges the gap between classic theoretical texts and less rigorous ones, providing a smooth transition from logic and proofs to real analysis.. Along with the basic material, the text covers Riemann-Stieltjes integrals, Fourier analysis, metric spaces and applications, and differential equations.
New to the Third Edition
Offering a more streamlined presentation, this edition moves elementary number systems and set theory and logic to appendices and removes the material on wavelet theory, measure theory, differential forms, and the method of characteristics. It also adds a chapter on normed linear spaces and includes more examples and varying levels of exercises.
Extensive Examples and Thorough Explanations Cultivate an In-Depth Understanding
This best-selling book continues to give students a solid foundation in mathematical analysis and its applications. It prepares them for further exploration of measure theory, functional analysis, harmonic analysis, and beyond.
Number Systems
The Real Numbers
The Complex Numbers
Sequences
Convergence of Sequences
Subsequences
Limsup and Liminf
Some Special Sequences
Series of Numbers
Convergence of Series
Elementary Convergence Tests
Advanced Convergence Tests
Some Special Series
Operations on Series
Basic Topology
Open and Closed Sets
Further Properties of Open and Closed Sets
Compact Sets
The Cantor Set
Connected and Disconnected Sets
Perfect Sets
Limits and Continuity of Functions
Basic Properties of the Limit of a Function
Continuous Functions
Topological Properties and Continuity
Classifying Discontinuities and Monotonicity
Differentiation of Functions
The Concept of Derivative
The Mean Value Theorem and Applications
More on the Theory of Differentiation
The Integral
Partitions and the Concept of Integral
Properties of the Riemann Integral
Another Look at the Integral
Advanced Results on Integration Theory
Sequences and Series of Functions
Partial Sums and Pointwise Convergence
More on Uniform Convergence
Series of Functions
The Weierstrass Approximation Theorem
Elementary Transcendental Functions
Power Series
More on Power Series: Convergence Issues
The Exponential and Trigonometric Functions
Logarithms and Powers of Real Numbers
Differential Equations
Picards Existence and Uniqueness Theorem
Power Series Methods
Introduction to Harmonic Analysis
The Idea of Harmonic Analysis
The Elements of Fourier Series
An Introduction to the Fourier Transform
Fourier Methods and Differential Equations
Functions of Several Variables
A New Look at the Basic Concepts of Analysis
Properties of the Derivative
The Inverse and Implicit Function Theorems
Advanced Topics
Metric Spaces
Topology in a Metric Space
The Baire Category Theorem
The Ascoli-Arzela Theorem
Normed Linear Spaces
What Is This Subject About?
What Is a Normed Linear Space?
Finite-Dimensional Spaces
Linear Operators
The Three Big Results
Applications of the Big Three
Appendix I: Elementary Number Systems
Appendix II: Logic and Set Theory
Appendix III: Review of Linear Algebra
Table of Notation
Glossary
Bibliography
Index
Exercises are included at the end of each section.
(0 Comentarios)
