Foundations of Analysis: 1st Edition (Paperback) book cover

Foundations of Analysis

1st Edition

By Steven G. Krantz

Chapman and Hall/CRC

312 pages | 41 B/W Illus.

Purchasing Options:$ = USD
New in Paperback: 9781138374928
pub: 2019-06-17
Hardback: 9781482220742
pub: 2014-10-20
eBook (VitalSource) : 9780429159930
pub: 2014-10-20
from $39.98

FREE Standard Shipping!


Foundations of Analysis covers the basics of real analysis for a one- or two-semester course. In a straightforward and concise way, it helps students understand the key ideas and apply the theorems. The book’s accessible approach will appeal to a wide range of students and instructors.

Each section begins with a boxed introduction that familiarizes students with the upcoming topics and sets the stage for the work to be done. Each section ends with several questions that ask students to review what they have just learned. The text is also scattered with notes pointing out places where different pieces of terminology seem to conflict with each other or where different ideas appear not to fit together properly. In addition, many remarks throughout help put the material in perspective.

As with any real analysis text, exercises are powerful and effective learning tools. This book is no exception. Each chapter generally contains at least 50 exercises that build in difficulty, with an exercise set at the end of every section. This allows students to more easily link the exercises to the material in the section.


"… there is a good set of exercises in each section … . If real analysis is to be dealt with in a one-semester course, this book appears to provide a reasonable text for the course."

Mathematical Reviews, April 2015

Table of Contents

Number Systems

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

Sequences and Series of Functions

Convergence of a Sequence of Functions

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

Appendix I: Elementary Number Systems

Appendix II: Logic and Set Theory

Table of Notation




About the Author

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 75 books and more than 195 scholarly papers and is the founding editor of the Journal of Geometric Analysis and Complex Analysis and its Synergies. 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.

Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Functional Analysis