By: Jiří Lebl (website #1 http://www.jirka.org/ (personal), website #2 http://www.math.wisc.edu/~lebl/ (work: UW-Madison), email: jiri...@gmail.com)
This free online textbook (e-book in webspeak) is a one semester course in basic analysis. This book started its life as my lecture notes for Math 444 at the University of Illinois at Urbana-Champaign (UIUC) in the fall semester of 2009, and was later enhanced to teach Math 521 at University of Wisconsin-Madison (UW-Madison). A prerequisite for the course is a basic proof course. 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, but also as a first semester of a more advanced course that also covers topics such as metric spaces.
The standard book used for the class at UIUC is Bartle and Sherbert, Introduction to Real Analysis third edition (BS from now on). The structure of the notes up to chapter 6 mostly follows the syllabus of UIUC Math 444. Some topics covered in BS are covered in slightly different order, some topics differ substantially from BS and some topics are not covered at all. For example, we will define the Riemann integral using Darboux sums and not tagged partitions. The Darboux approach is far more appropriate for a course of this level.
The aim is to provide a low cost, redistributable, not overly long, high quality textbook that students will actually keep rather than selling back after the semester is over. Even if the students throw it out, they can always look it up on the net again. You are free to have a local bookstore or copy store make and sell copies for your students. See below about the license.
One reason for making the book freely available is to allow modification and customization for a specific purpose if necessary (as the University of Pittsburgh has done for example). If you do modify these notes, make sure to mark them prominently as such to avoid confusion. This aspect is also important for longevity of the book. 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.
1. Real Numbers
2. Sequences and Series
3. Continuous Functions
4. The Derivative
5. The Riemann Integral
6. Sequences of Functions
7. Metric Spaces (New in November 18th 2011 edition)
There are 303 exercises (December 16th 2012 edition).
Please let me know at jiri...@gmail.com if you find any typos or have corrections, extra exercises or material, or any other comments. I will always keep all older versions available for download, at least when there are nontrivial updates. When the updates are reasonably minor, I will try to preserve pagination and numbering of sections/examples/theorems/equations/exercises as much as possible.
There is no solutions manual for the exercises. This situation is actually intentional. There is an unfortunately large amount of problems with solutions out there already. Part of learning how to do proofs is to learn how to recognize a correct proof. Looking at someone else's proof is far less effective way of checking your proof than actually checking your proof. If you are unsure about the correctness of a solution, then you do not yet have a solution. Another thing is that the best solution is the one that the student comes up with him(or her)self, not necessarily the one which the professor or the book author comes up with.
Do let me know (jiri...@gmail.com) if you use the book for teaching a course! So far I know that this book was used, or is being used, as the primary textbook at (other than my courses at UIUC, UCSD, and UW-Madison) University of California at Berkeley, University of Pittsburgh, Vancouver Island University, Western Illinois University, Medgar Evers College, San Diego State University, University of Toledo, Oregon Institute of Technology, Iowa State University, and surely others I do not know about.
See a list of classroom adoptions for more details.
Download the book as PDF
(December 16th, 2012, 192 pages, 1 MB download)
You can also look at an expanded draft version (May 14th 2013). See the change log page to see what is new. This is a DRAFT and should be treated as such. Stuff is still subject to change before it becomes final, and there are probably plenty of errors and typos.
Check for any errata in the current version.
Look at the change log to see what changed in the newest version (You can download source files of old versions if you wish).
LaTeX source as a tarball. The main file is realanal.tex. I compile the pdf with pdflatex. You also want to run makeindex to generate the index (I generally run pdflatex realanal three times, then makeindex realanal, and then finally pdflatex realanal again).
During the writing of these notes, the author was in part supported by NSF grant DMS-0900885.
This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License. You can use, print, copy, and share these notes as much as you want. You can base your own notes on these and reuse parts if you keep the license the same. If you plan to use these notes commercially (sell them for more than just production cost), then you need to contact me and we will work something out. If you are printing a course pack for your students, then it is fine if the copy service or bookstore is charging a fee for printing and selling the printed copy. I consider that production cost.