Basic Analysis: Introduction to Real Analysis
This free online textbook (OER more formally) is a course in undergraduate<br>real analysis (somewhere it is called "advanced calculus").<br>The book is meant both for a<br>basic course for students who do not necessarily wish to<br>go to graduate school, but also as a more advanced<br>course that also covers topics such as metric spaces and should prepare<br>students for graduate study.
A prerequisite for the course is a basic proof course.<br>An advanced course could be two semesters long<br>with some of the<br>second-semester topics such as multivariable differential calculus,<br>path integrals, and the multivariable integral using the second volume.<br>There are more topics than can be covered in two semesters, and<br>it can also be reading for beginning graduate students to refresh their<br>analysis or fill in some of the holes.
This book started its life as<br>my lecture notes for Math 444 at the<br>Uni­ver­si­ty of Illi­nois at Ur­ba­na-Cham­paign (UIUC) in the fall semester of 2009. It was later enhanced<br>to teach the Math 521/522 sequence at<br>Uni­ver­si­ty of Wis­con­sin-Madison (UW-Madison) and<br>the Math 4143/4153 sequence at<br>Okla­homa State Uni­ver­si­ty (OSU).
The book (volume I) starts with analysis on the real line, going through<br>sequences, series, and then into continuity, the derivative, and the Rie­mann<br>integral using the Dar­boux approach. There are plenty of available detours<br>along the way, or we can power through towards the metric spaces in chapter 7.<br>The philosophy is that metric<br>spaces are absorbed much better by the students after they have gotten<br>comfortable with basic analysis techniques in the very concrete setting of the<br>real line. As a bonus, the book can be used both by a slower-paced,<br>less abstract course, and a faster-paced more abstract course for future graduate students.<br>The slower course never reaches metric spaces. A nice capstone<br>theorem for such a course is the<br>Picard theorem on existence and uniqueness of ordinary differential equations,<br>a proof which brings together everything one has learned in the course.<br>A faster-paced course would generally reach metric spaces, and as a reward<br>such students can see a streamlined (but more abstract)<br>proof of Picard.
Volume II continues into multivariable analysis.<br>It covers differential calculus,<br>including inverse and implicit function theorems,<br>differentiation under the<br>integral and path integrals, which are often not covered in a course like this,<br>and multivariable Rie­mann integral.<br>Finally, there is also a chapter on power series,<br>Ar­ze­là-As­co­li, Stone-Weier­strass, and Fou­ri­er series. Together the two volumes<br>provide enough material for several different types of year-long sequences.<br>A student who absorbs the first volume and the first three chapters of volume II<br>should be more than prepared for graduate real and complex analysis courses.
I have tried (especially in recent editions) to add many diagrams and graphs<br>to visually illustrate the proofs and make them more accessible.<br>Usually, these are precise and more in-depth versions of the drawings I attempt on the board in class.<br>Together, the two volumes have over a hundred figures.
The aim is to provide a low cost, redistributable, not<br>overly long, high-quality textbook that students will actually keep rather than<br>selling back after the semester is over. Even if the students throw it out,<br>they can always look it up on the net again. You are free to have a local<br>bookstore or copy store make and sell copies for your students. See below<br>about the license.
One reason for making the book freely available<br>is to allow modification and customization for a specific<br>purpose if necessary (as the University of Pittsburgh<br>has done for example).<br>If you do modify these books, make sure to mark them<br>prominently as such to avoid confusion. This aspect is also important for<br>the longevity of the book. The book can be updated and modified even if I happen<br>to drop off the face of the earth. You do not have to depend on any publisher<br>being interested as with traditional textbooks.
Furthermore, errata are fixed promptly, meaning that if you teach the<br>same class next term, all errata that are spotted are most likely already<br>fixed. No need to wait several years for a new edition.<br>Every once in a while I make some major addition and a new major version<br>(edition), and then in between as errata are fixed I make minor version<br>updates (like a corrected printing) usually once or twice a year,<br>depending on the errata discovered. Exercise, chapter, and section numbers<br>are preserved as much as humanly possible.<br>What's added is added at the end with new numbers, so the book is generally<br>compatible even if students (or the instructor) have an older printed copy.<br>The minor updates are totally interchangeable and have very minimal<br>changes, essentially nothing new.
MAA published a<br>review of the...