# Basic Analysis: Introduction to Real Analysis

By: Jiří Lebl
(website #1 http://www.jirka.org/ (personal), website #2 http://www.math.okstate.edu/~lebl/ (work: OSU), email: )

This free online textbook (e-book in webspeak) is a 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. Second semester topics such as multivariable differential calculus, path integrals, and the multivariable integral were added as Volume II so that the entire sequence Math 4143/4153 can be taught at Oklahoma State University (OSU).

The standard book used for the class at UIUC was Bartle and Sherbert, Introduction to Real Analysis third edition (BS from now on). The structure of the book 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. Chapter 7 (Metric Spaces) was added to teach the more advanced course at UW-Madison. The philosophy is that metric spaces are absorbed much better by the students after they have gotten comfortable with basic analysis techniques in the very concrete setting of the real line.

Volume II continues into multivariable analysis. Starting with differential calculus, including inverse and implicit function theorems, then differentiation under the integral and path integrals, which are often not covered in a course like this, and multivariable Riamann integral. There is also a chapter on power series, ArzelĂ -Ascoli, Stone-Weierstrass, and the Fourier series. Together the two volumes provide enough material for several different types of year-long sequences.

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 this book, 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 (less than a year), meaning simply that if you teach the same class next year, all errata that are spotted are most likely already fixed. No need to wait several years for a new edition.

MAA published a review of the book (they looked at the December 2012 edition of Volume I, there was only the first volume then).

Introduction
1. Real Numbers
2. Sequences and Series
3. Continuous Functions
4. The Derivative
5. The Riemann Integral
6. Sequences of Functions
7. Metric Spaces

Volume II:
8. Several Variables and Partial Derivatives
9. One Dimensional Integrals in Several Variables
10. Multivariable Integral (10.7 new 5/7/18)
11. Functions as Limits (new 5/7/18)

There are 528 exercises in Volume I (version 5.0, that is, May 7th 2018 edition).

There are 263 exercises in Volume II (version 2.0, that is, May 7th 2018 edition).

Please let me know at 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 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 your proof is correct. Looking at someone else's proof is a far less effective way of checking your proof than actually checking your proof. It is like going the gym and watching other people work out. The exercises in the book are meant to be a gym for the mind. If you are unsure about the correctness of a solution, then you do not yet have a solution. Furthrermore, the best solution for the student 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 () if you use the book for teaching a course! The book was used, or is being used, as the primary textbook at (other than my courses at UIUC, UCSD, UW-Madison, and OSU) 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, California State University Dominguez Hills, St. John's University of Tanzania, Mary Baldwin College, Ateneo De Manila University, University of New Brunswick Saint John, and many others. See below for a more complete list.

The book has been selected as an Approved Textbook in the American Institute of Mathematics Open Textbook Initiative.

See a list of classroom adoptions for more details.

Lots of minor new additions were added throughout and some changes, and a whole new chapter was added to volume II. See the change log (volume I) (volume II) page to see what is new. If needed, here is the older version 4.0 of volume I, that is the old version 4.0 (February 29th, 2016 version).

Check for any errata (volume I) (volume II) in the current version.

Look at the change log (volume I) (volume II) to see what changed in the newest version (there you can also download old versions if you wish).

I started numbering things with version numbers starting at 4.0 for volume I, and version 1.0 for volume II. The first number is the major number and it really means "edition" and will be raised when substantial changes are made. The second number is raised for corrections only.

I get a bit of money when you buy these (depending on where exactly they are bought). Probably enough to buy me a coffee (as long as it is not a fancy coffee), so by buying a copy you will support this project. You will also save your toner cartridge.

Lulu always has the most up to date version more quickly than amazon, but the difference is usually in terms of days. The paperback copy is on Crown Quatro size (7.44x9.68 inch), and the two versions of it (amazon and lulu) are essentially identical except for cover art (there are those who like the blue). I tested both and they both print quite well, so the quality is approximately the same, and I have seen some of them take quite a bit of beating by students.

Lulu also allows me to make a larger (US letter size) coil bound version which I prefer to get when teaching, as it can easily be opened and kept on a certain page. It may be easier to read, and take notes in as it has larger font and wider margins, though a little less portable. It's only a few dollars more.

### Volume I:

Buy the smaller paperback copy at lulu.com for \$13.20.

Or buy the larger coil-bound copy at lulu.com for \$15.60.

This copy is the version 5.0 (May 7th, 2018) revision of volume I.
No ISBN for the lulu version.

Buy the smaller paperback copy on Amazon for \$13.20.

This copy is the version 5.0 (May 7th, 2018) revision of volume I.
ISBN-13: 978-1718862401
ISBN-10: 1718862407

(The older version 4 of volume I, February 29th, 2016 revision has ISBN 978-1530256747, if you still need it)

### Volume II:

Buy the smaller paperback copy at lulu.com for \$11.00.

Or buy the larger coil-bound copy at lulu.com for \$13.00.

This copy is the version 2.0 (May 7th, 2018) revision of volume II.
No ISBN for the lulu version.

Buy the smaller paperback copy on Amazon for \$11.00.

This copy is the version 2.0 (May 7th, 2018) revision of volume II.
ISBN-13: 978-1718865488
ISBN-10: 1718865481

## Source:

The source is hosted on GitHub: https://github.com/jirilebl/ra (both volumes).

You can get an archive of the source of the released version on github, look under https://github.com/jirilebl/ra/releases, though if you plan to work with it, maybe best to look at just the latest working version as that might have any errata or new additions. Though these might be a work in progress. Perhaps best is to let me know.

Volume I is realanal.tex and volume II is realanal2.tex. I compile the pdf with pdflatex. You need to compile the first volume first before the second volume. You might need to run makeindex (for the index) and makeglossary (for the list of notations) as well, though theoretically it should now be handled automatically. There are scripts publish.sh and publish2.sh, that run everything an obnoxious number of times to make sure it all works.

During the writing of this book, the author was in part supported by NSF grant DMS-0900885 and DMS-1362337.