Basic Analysis: Introduction to Real Analysis

Jiří Lebl

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Source available Yes
Exercises Yes
Answers No
Solution manual No
License CC-BY-SA and CC-BY-NC-SA
  • Two volume text for two semesters of undergraduate real analysis
  • Basic Analyis I: 282 page volume for one semester of undergraduate real analysis with 528 excercises
  • Basic Analyis II: 195 page volume for a second semester with 263 exercises
  • Class tested
  • Paperback copies from Lulu and Amazon; volume I for $13.94 and volume II for $11.16
  • List of course adoptions available
  • For more information and to download

This book originated in 2009 as lecture notes for the undergraduate analysis course at the University of Illinois, whose syllabus was based on the text of Bartle and Sherbert, but it is now a mature textbook. It has been used by a number of instructors at other institutions and they report that they recommend the book and will continue to use it. There is now a second volume, Basic Analysis 2, written for the second semester of a year long course.

There are seven chapters in Basic Analysis I: (1) Real Numbers, (2) Sequences and Series, (3) Continuous Functions, (4) The Derivative, (5) The Riemann Integral, (6) Sequences of Functions, (7) Metric Spaces. For a significant application of the concepts already developed, chapter 6 contains a rigorous treatment of Picard’s theorem on the existence and uniqueness of solutions of ordinary differential equations.

Basic Analysis II has four chapters: (8) Several Variables and Partial Derivatives, (9) One-dimensional Integrals in Several Variables, (10) Multivariable Integral, (11) Functions as Limits.

In order to promote the book’s longevity the author writes:

The book can be updated and modified even if I happen to drop off the face of the earth. You do not have to depend on any publisher being interested as with traditional textbooks. Furthermore, errata are fixed promptly, meaning simply that if you teach the same class next term, all errata that are spotted are already fixed. No need to wait several years for a new edition.