Basic Analysis: Introduction to Real Analysis
By: Jiří Lebl (website #1 http://www.jirka.org/ (personal), website #2 http://www.math.ucsd.edu/~jlebl/ (work: ucsd), email: jiri...@gmail.com)
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This free online textbook (e-book in webspeak)
is a one semester course in basic analysis.
These were my lecture notes for teaching Math 444 at the
University of Illinois at Urbana-Champaign (UIUC) in the fall semester of 2009.
The course is a first course in mathematical analysis aimed at students who
do not necessarily wish to continue a graduate study in mathematics. A
prerequisite for the course is a basic proof course.
The course does not cover topics such as metric spaces, which a more
advanced course would. It should be possible to use these notes for a
beginning of a more advanced course, but further material should be added.
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 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. In my view, BS seems to be targeting a different audience than this course, and that is the reason for writing this present book. The generalized Riemann integral is not covered at all.
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. 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.
Table of contents:
Introduction
1. Real Numbers
2. Sequences and Series
3. Continuous Functions
4. The Derivative
5. The Riemann Integral
6. Sequences of Functions
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.
Do let me know (jiri...@gmail.com) if you use the book for teaching a course!
Download:
Download the book as PDF
(August 12th, 2010, 161 pages, 0.8 MB download)
Check 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).
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Buy paperback: Buy a copy at lulu.com for $12.65. I get $2.50, so by buying a copy you will support this project. You will also save your toner cartridge. This copy is the August 12th, 2010 revision. |
Source: 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.
License: 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.
Useful links:
