# Real Analysis and Foundations

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## Book Description

Through four editions 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 primary goal of this new edition remains the same as in previous editions. It is to make real analysis relevant and accessible to a broad audience of students with diverse backgrounds while also maintaining the integrity of the course. This text aims to be the generational touchstone for the subject and the go-to text for developing young scientists.

This new edition continues the effort to make the book accessible to a broader audience. Many students who take a real analysis course do not have the ideal background. The new edition offers chapters on background material like set theory, logic, and methods of proof. The more advanced material in the book is made more apparent.

This new edition offers a new chapter on metric spaces and their applications. Metric spaces are important in many parts of the mathematical sciences, including data mining, web searching, and classification of images.

The author also revised the material on sequences and series adding examples and exercises that compare convergence tests and give additional tests.

The text includes rare topics such as wavelets and applications to differential equations. The level of difficulty moves slowly, becoming more sophisticated in later chapters. Students have commented on the progression as a favorite aspect of the textbook.

The author is perhaps the most prolific expositor of upper division mathematics. With over seventy books in print, thousands of students have been taught and learned from his books.

## Table of Contents

Preface

0 Background Material

0.1 Number Systems

0.1.1 The Natural Numbers

0.1.2 The Integers

0.1.3 The Rational Numbers

02 Logic and Set

0.2.1 And” and “Or”

0.2.2 “not” and “if then”

0.2.3 Contrapositive, Converse, and “Iff”

0.2.4 Quantifiers

0.2.5 Set Theory and Venn Diagrams

0.2.6 Relations and Functions

0.2.7 Countable and Uncountable Sets

1 Real and Complex Numbers

1.1 The Real Numbers

Appendix: Construction of the Real Numbers

1.2 The Complex Numbers

2 Sequences 71

2.1 Convergence of Sequences

2.2 Subsequences

2.3 Limsup and Liminf

2.4 Some Special Sequences

3 Series of Numbers

3.1 Convergence of Series

3.2 Elementary Convergence Tests

3.3 Advanced Convergence Tests

3.4 Some Special Series

3.5 Operations on Series

4 Basic Topology

4.1 Open and Closed Sets

4.2 Further Properties of Open and Closed Sets

4.3 Compact Sets

4.4 The Cantor Set

4.5 Connected and Disconnected Sets

4.6 Perfect Sets

5 Limits and Continuity of Functions

5.1 Basic Properties of the Limit of a Function

5.2 Continuous Functions

5.3 Topological Properties and Continuity

5.4 Classifying Discontinuities and Monotonicity

6 Differentiation of Functions

6.1 The Concept of Derivative

6.2 The Mean Value Theorem and Applications

6.3 More on the Theory of Differentiation

7 The Integral

7.1 Partitions and the Concept of Integral

7.2 Properties of the Riemann Integral

7.3 Change of Variable and Related Ideas

7.4 Another Look at the Integral

7.5 Advanced Results on Integration Theory

8 Sequences and Series of Functions

8.1 Partial Sums and Pointwise Convergence

8.2 More on Uniform Convergence

8.3 Series of Functions

8.4 The Weierstrass Approximation Theorem

9 Elementary Transcendental Functions

9.1 Power Series

9.2 More on Power Series: Convergence Issues

9.3 The Exponential and Trigonometric Functions

9.4 Logarithms and Powers of Real Numbers

10 Functions of Several Variables

10.1 A New Look at the Basic Concepts of Analysis

10.2 Properties of the Derivative

10.3 The Inverse and Implicit Function Theorems

11 Advanced Topics

11.1 Metric Spaces

11.2 Topology in a Metric Space

11.3 The Baire Category Theorem

11.4 The Ascoli-Arzela Theorem

12 Differential Equations

12.1 Picard’s Existence and Uniqueness Theorem

12.1.1 The Form of a Differential Equation

12.1.2 Picard’s Iteration Technique

12.1.3 Some Illustrative Examples

12.1.4 Estimation of the Picard Iterates

12.2 Power Series Methods

13 Introduction to Harmonic Analysis

13.1 The Idea of Harmonic Analysis

13.2 The Elements of Fourier Series

13.3 An Introduction to the Fourier Transform

Appendix: Approximation by Smooth Functions

13.4 Fourier Methods and Differential Equations

13.4.1 Remarks on Different Fourier Notations

13.4.2 The Dirichlet Problem on the Disc

13.4.3 Introduction to the Heat and Wave Equations

13.4.4 Boundary Value Problems

13.4.5 Derivation of the Wave Equation

13.4.6 Solution of the Wave Equation

13.5 The Heat Equation

Appendix: Review of Linear Algebra

Table of Notation

Glossary

Bibliography

Index

## Author(s)

### Biography

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 130 books and more than 250 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.