Students preparing for courses in real analysis often encounter either very exacting theoretical treatments or books without enough rigor to stimulate an in-depth understanding of the subject. Further complicating this, the field has not changed much over the past 150 years, prompting few authors to address the lackluster or overly complex dichotomy existing among the available texts. The enormously popular first edition of Real Analysis and Foundations gave students the appropriate combination of authority, rigor, and readability that made the topic accessible while retaining the strict discourse necessary to advance their understanding. The second edition maintains this feature while further integrating new concepts built on Fourier analysis and ideas about wavelets to indicate their application to the theory of signal processing. The author also introduces relevance to the material and surpasses a purely theoretical treatment by emphasizing the applications of real analysis to concrete engineering problems in higher dimensions. Expanded and updated, this text continues to build upon the foundations of real analysis to present novel applications to ordinary and partial differential equations, elliptic boundary value problems on the disc, and multivariable analysis. These qualities, along with more figures, streamlined proofs, and revamped exercises make this an even more lively and vital text than the popular first edition.

Contenuti

PREFACE TO THE SECOND EDITION
PREFACE TO THE FIRST EDITION
LOGIC AND SET THEORY
Introduction
"And" and "Or"
"Not" and "If-Then"
Contrapositive, Converse, and "Iff"
Quantifiers
Set Theory and Venn Diagrams
Relations and Functions
Countable and Uncountable Sets
Exercises
NUMBER SYSTEMS
The Natural Numbers
Equivalence Relations and Equivalence Classes
The Integers
The Rational Numbers
The Real Numbers
The Complex Numbers
Exercises
SEQUENCES
Convergence of Sequences
Subsequences
Limsup and Liminf
Some Special Sequences
Exercises
SERIES OF NUMBERS
Convergence of Series
Elementary Convergence Tests
Advanced Convergence Tests
Some Special Series
Operations on Series
Exercises
BASIC TOPOLOGY
Open and Closed Sets
Further Properties of Open and Closed Sets
Compact Sets
The Cantor Set
Connected and Disconnected Sets
Perfect Sets
Exercises
LIMITS AND CONTINUITY OF FUNCTIONS
Definition and Basic Properties of the Limit of a Function
Continuous Functions
Topological Properties and Continuity
Classifying Discontinuities and Monotonicity
Exercises
DIFFERENTIATION OF FUNCTIONS
The Concept of Derivative
The Mean Value Theorem and Applications
More on the Theory of Differentiation
Exercises
THE INTEGRAL
Partitions and The Concept of Integral
Properties of the Riemann Integral
Another Look at the Integral
Advanced Results on Integration Theory
Exercises
SEQUENCES AND SERIES OF FUNCTIONS
Partial Sums and Pointwise Convergence
More on Uniform Convergence
Series of Functions
The Weierstrass Approximation Theorem
Exercises
ELEMENTARY TRANSCENDENTAL FUNCTIONS
Power Series
More on Power Series: Convergence Issues
The Exponential and Trigonometric Functions
Logarithms and Powers of Real Numbers
The Gamma Function and Stirling's Formula
Exercises
APPLICATIONS OF ANALYSIS TO DIFFERENTIAL EQUATIONS
Picard's Existence and Uniqueness Theorem
The Method of Characteristics
Power Series Methods
Exercises
INTRODUCTION TO HARMONIC ANALYSIS
The Idea of Harmonic Analysis
The Elements of Fourier Series
An Introduction to the Fourier Transform
Fourier Methods in the Theory of Differential Equations
Exercises
FUNCTIONS OF SEVERAL VARIABLES
Review of Linear Algebra
A New Look at the Basic Concepts of Analysis
Properties of the Derivative
The Inverse and Implicit Function Theorems
Differential Forms
Exercises
ADVANCED TOPICS
Metric Spaces
Topology in a Metric Space
The Baire Category Theorem
The Ascoli-Arzela Theorem
The Lebesgue Integral
A Taste of Probability Theory
Exercises
A GLIMPSE OF WAVELET THEORY
Localization in the Time and Space Variables
A Custom Fourier Analysis
The Haar Basis
Some Illustrative Examples
Closing Remarks
Exercises
BIBLIOGRAPHY
INDEX

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