Foundations of Analysis  book cover
1st Edition

Foundations of Analysis

ISBN 9780429159930
Published October 20, 2014 by Chapman and Hall/CRC
312 Pages

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Book Description

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.

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




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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 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.


"… 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