Basic Analysis: Introduction to Real Analysis

This book is a one semester course in basic analysis. It started its life as my lecture notes for teaching Math 444 at the University of Illinois at Urbana-Champaign (UIUC) in Fall semester 2009.

Later I added the metric space chapter to teach Math 521 at University of Wisconsin–Madison (UW). A prerequisite for this course is a basic proof course, using for example [H], [F], or [DW]. It should be possible to use the book for both a basic course for students who do not necessarily wish to go to graduate school (such as UIUC 444), but also as a more advanced one-semester course that also covers topics such as metric spaces (such as UW 521).

It should also be possible to run a faster course without metric spaces covering all sections of chapters 0 through 6. The approximate number of lectures given in the section notes through chapter 6 are a very rough estimate and were designed for the slower course. The first few chapters of the book can be used in an introductory proofs course as is for example done at Iowa State University Math 201, where this book is used in conjunction with Hammack’s Book of Proof [H].

The book normally used for the class at UIUC is Bartle and Sherbert, Introduction to Real Analysis third edition [BS]. The structure of the beginning of the book somewhat follows the standard syllabus of UIUC Math 444 and therefore has some similarities with [BS]. A major difference is that we define the Riemann integral using Darboux sums and not tagged partitions. The Darboux approach is far more appropriate for a course of this level.