Real Analysis and Foundations  book cover
4th Edition

Real Analysis and Foundations

ISBN 9781498777681
Published December 8, 2016 by Chapman and Hall/CRC
430 Pages

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Book Description

The first three editions of this popular textbook attracted

a loyal readership and widespread use. Students find the book to be concise, accessible, and

complete. Instructors find the book to be clear, authoritative, and dependable.

The goal of this new edition is to make real analysis relevant and accessible

to a broad audience of students with diverse backgrounds. Real analysis

is a basic tool for all mathematical scientists, ranging from mathematicians to physicists to

engineers to researchers in the medical profession. This text aims to be the

generational touchstone for the subject and the go-to text for developing young



In this new edition we endeavor to make the book accessible to a broader

audience. This edition includes more explanation, more elementary examples,

and the author stepladders the exercises. Figures are updated and clarified. We make

the sections more concise, and omit overly technical details.

We have updated and augmented the multivariable material in order to bring out

the geometric nature of the topic. The figures are thus enhanced and fleshed out.


  • A renewed enthusiasm for the topic comes through in a revised presentation
  • A new organization removes some advanced topics and retains related ones
  • Exercises are more tiered, offering a more accessible course
  • Key sections are revised for more brevity

Table of Contents

Number Systems

The Real Numbers

Appendix: Construction of the Real Numbers

The Complex Numbers


Convergence of Sequences


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

Change of Variable and Related Ideas

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

Picard’s Existence and Uniqueness Theorem

The Form of a Differential Equation

Picard’s Iteration Technique

Some Illustrative Examples

Estimation of the Picard Iterates

Power Series Methods

Introduction to Harmonic Analysis

The Idea of Harmonic Analysis

The Elements of Fourier Series

An Introduction to the Fourier Transform

Appendix: Approximation by Smooth Functions

Fourier Methods and Differential Equations

Remarks on Different Fourier Notations

The Dirichlet Problem on the Disc

Introduction to the Heat and Wave Equations

Boundary Value Problems

Derivation of the Wave Equation

Solution of the Wave Equation

The Heat Equation

Functions of Several Variables

A New Look at the Basic Concepts of Analysis

Properties of the Derivative

The Inverse and Implicit Function Theorems

Appendix I: Elementary Number Systems

Appendix II: Logic and Set Theory

Appendix III: Review of Linear Algebra

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Steven G. Krantz is a professor of mathematics at Washington University in St. Louis. He has previously taught at UCLA, Princeton University, and Pennsylvania State University. He has written more than 65 books and more than 175 scholarly papers and is the founding editor of the Journal of Geometric Analysis. An AMS Fellow, Dr. Krantz has been a recipient of the Chauvenet Prize, Beckenbach Book Award, and Kemper Prize. He received a Ph.D. from Princeton University.