In the second edition of the MAA classic, exploration continues to be an essential component. More than 60 new exercises have been added, and the chapters on Infinite Summations, Differentiability and Continuity, and Convergence of Infinite Series have been reorganized to make it easier to identify the key ideas. A Radical Approach to Real Analysis is an introduction to real analysis, rooted in and informed by the historical issues that shaped its development. It can be used as a textbook, as a resource for the instructor who prefers to teach a traditional course, or as a resource for the student who has been through a traditional course yet still does not understand what real analysis is about and why it was created. The book begins with Fourier’s introduction of trigonometric series and the problems they created for the mathematicians of the early 19th century. It follows Cauchy’s attempts to establish a firm foundation for calculus, and considers his failures as well as his successes. It culminates with Dirichlet’s proof of the validity of the Fourier series expansion and explores some of the counterintuitive results Riemann and Weierstrass were led to as a result of Dirichlet’s proof.
Table of Contents
Crisis in Mathematics: Fourier’s Series
Differentiability and Continuity
The Convergence of Infinite Series
Understanding Infinite Series
Return to Fourier Series
Explorations of the Infinite
Hints to Selected Exercises
About the Author
David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College. He served in the Peach Corps, teaching math and science at the Clare Hall School in Antigua, West Indies before studying with Emil Grosswald at Temple and then teaching at Penn State for 17 years, eight of them as full professor. He chaired the mathematics department at Macalester from 1995 until 2001. He has held visiting positions at the Institute for Advanced Study, the University of Wisconsin-Madison, the University of Minnesota, Université Louis Pasteur (Strasbourg, France), and the State College Area High School. He has received the MAA Distinguished Teaching Award (Allegheny Mountain Section), the MAA Beckenbach Book Prize for Proofs and Confirmations, and has been a Pólya Lecturer for the MAA. He is a recipient of Macalester’s Jefferson Award. He has published over 50 research articles in number theory, combinatorics, and special functions. Other books include Factorization and Primality Testing, Second Year Calculus from Celestial Mechanics to Special Relativity, the first and second editions of A Radical Approach to Real Analysis, and, with Stan Wagon, A Course in Computational Number Theory. David Bressoud chairs the MAA Committee on the Undergraduate Program in Mathematics. He has chaired the AP Calculus Development Committee and has served as Director of FIPSE-supported program Quantitative Methods for Public Policy. He has been active in the activities and programs of both the Mathematical Association of America and the American Mathematical Society.
My introduction to real analysis came from G.H. Hardy’s A Course of Pure Mathematics, which, in 1908, became the first rigorous English university text on analysis. However, as with many subsequent books on analysis, it conforms well to the description given by David Bressoud, who says:
"The traditional course begins with a discussion of the properties of the real numbers, moves on to continuity, then differentiability, integrability, and finally infinite series, culminating in a rigorous proof of Talyor series…This is the right way to view analysis, but it is not the right way to teach it."
Bressoud believes that a better way (the best way?) to teach analysis is rooted in the historical issues that have shaped its development, which is why he includes the word ‘radical’ in the title. The aim is to dispel the myth that mathematical ideas and methods are perfectly formed in the moment of their conception. Accordingly, Bressoud’s ‘Darwinian’ approach reveals how, over long periods of time, the central ideas of analysis have evolved by processes of experiment and scholarly disputation (which, of course, is true of mathematics as a whole). Continued...