Basic Analysis I & II: Introduction to Real Analysis, Volumes I & II

This first volume is a one semester course in basic analysis. With the second volume, it is a year-long course. The book started its life as my lecture notes for teaching Math 444 at the University of Illinois at Urbana-Champaign (UIUC) in the fall semester of 2009. I added the metric space chapter for Math 521 at the University of Wisconsin–Madison (UW). Volume II was added to teach Math 4143/4153 at Oklahoma State University (OSU). A prerequisite for these courses is usually a basic proof course, using, for example, [H], [F], or [DW].

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 (such as UIUC 444), and also as a more advanced one-semester course that also covers topics such as metric spaces (such as UW 521). Here are suggestions for a semester course. A slower course such as UIUC 444:

§0.3, §1.1–§1.4, §2.1–§2.5, §3.1–§3.4, §4.1–§4.2, §5.1–§5.3, §6.1–§6.3

A more rigorous course covering metric spaces that runs quite a bit faster (e.g., UW 521):

§0.3, §1.1–§1.4, §2.1–§2.5, §3.1–§3.4, §4.1–§4.2, §5.1–§5.3, §6.1–§6.2, §7.1–§7.6

It should also be possible to run a faster course without metric spaces covering all sections of chapters 0 through 6. The approximate number of lectures given in the section notes through Chapter 6 are a very rough estimate and were designed for the slower course. The first few chapters of the book can be used in an introductory proofs course, as is done, for example, at Iowa State University Math 201, where this book is used in conjunction with Hammack’s Book of Proof [H].

With volume II, one can run a year-long course that covers multivariable topics. In this scenario, it may make sense to cover most of the first volume in the first semester while leaving metric spaces for the beginning of the second semester.

The structure of the beginning of volume I somewhat follows the standard syllabus of UIUC Math 444 and, therefore, has some similarities with Bartle and Sherbert, Introduction to Real Analysis [BS], which is the standard book at UIUC. A major difference is that we define the Riemann integral using Darboux sums and not tagged partitions. The Darboux approach is far more appropriate for a course of this level.

Our approach allows us to fit a course such as UIUC 444 within a semester and still spend some time on the interchange of limits and end with Picard’s theorem on the existence and uniqueness of solutions of ordinary differential equations. This theorem is a wonderful example that uses many results proved in the book. For more advanced students, material may be covered faster so that we arrive at metric spaces and prove Picard’s theorem using the fixed point theorem as is usual.

Other excellent books exist. My favorite is Rudin’s excellent Principles of Mathematical Analysis [R2] or, as it is commonly and lovingly called, baby Rudin (to distinguish it from his other great analysis textbook, big Rudin). I took a lot of inspiration and ideas from Rudin. However, Rudin is a bit more advanced and ambitious than this present course. For those who wish to continue mathematics, Rudin is a fine investment. An inexpensive and somewhat simpler alternative to Rudin is Rosenlicht’s Introduction to Analysis [R1]. There is also the freely downloadable Introduction to Real Analysis by William Trench [T].

A note about the style of some of the proofs: Many proofs traditionally done by contradiction, I prefer to do by a direct proof or by contrapositive. While the book does include proofs by contradiction, I only do so when the contrapositive statement seemed too awkward or when contradiction follows rather quickly. Contradiction is more likely to get beginning students into trouble, as we are talking about objects that do not exist.

I try to avoid unnecessary formalism where it is unhelpful. Furthermore, the proofs and the language get slightly less formal as we progress through the book, as more and more details are left out to avoid clutter.

As a general rule, I use \(\coloneqq\) instead of \(=\) to define an object rather than to simply show equality. I use this symbol rather more liberally than is usual for emphasis. I use it even when the context is “local,” that is, I may simply define a function \(f(x) \coloneqq x^2\) for a single exercise or example.

Finally, I would like to acknowledge Jana Maříková, Glen Pugh, Paul Vojta, Frank Beatrous, Sönmez Şahutoğlu, Jim Brandt, Kenji Kozai, Arthur Busch, Anton Petrunin, Mark Meilstrup, Harold P. Boas, Atilla Yılmaz, Thomas Mahoney, Scott Armstrong, and Paul Sacks, Matthias Weber, Manuele Santoprete, Robert Niemeyer, Amanullah Nabavi, for teaching with the book and giving me lots of useful feedback. Frank Beatrous wrote the University of Pittsburgh version extensions, which served as inspiration for many more recent additions. I would also like to thank Dan Stoneham, Jeremy Sutter, Eliya Gwetta, Daniel Pimentel-Alarcón, Steve Hoerning, Yi Zhang, Nicole Caviris, Kristopher Lee, Baoyue Bi, Hannah Lund, Trevor Mannella, Mitchel Meyer, Gregory Beauregard, Chase Meadors, Andreas Giannopoulos, Nick Nelsen, Ru Wang, Trevor Fancher, Brandon Tague, Wang KP, Wai Yan Pong, Sam Merat, Judah Nouriyelian, Arnold Cross, Jesse Wallace, Adnan Hashem Mohamed, Nikita Borisov, Bob Strain, an anonymous reader or two, and in general all the students in my classes for suggestions and finding errors and typos. During the writing of this book, the author was in part supported by NSF grant DMS-0900885 and DMS-1362337.

This second volume of “Basic Analysis” is meant to be a seamless continuation. The chapter numbers start where the first volume left off. The book started with my notes for a second-semester undergraduate analysis at University of Wisconsin—Madison in 2012, which I taught more or less with Rudin’s book. Some of the material and some of the proofs are similar to Rudin, though I try to provide more detail and context. In 2016, I taught a second-semester undergraduate analysis at Oklahoma State University, modifying and cleaning up the notes, this time using them as the main text. I have since taught the course several more times, adding chapter 11 (originally written for the Wisconsin course), and making many other smaller improvments.

I plan to eventually add a few more topics. I will try to preserve the numbering in subsequent editions as always. The new topics planned would add chapters onto the end of the book, or add sections to end of existing chapters, and I will try as hard as possible to leave exercise numbers unchanged.

For the most part, this second volume depends on the non-optional parts of volume I, while some of the optional parts are also used. Higher order derivatives (but not Taylor’s theorem itself) are used in 8.6, 9.3, 10.6. Exponentials, logarithms, and improper integrals are used in a few examples and exercises, and they are heavily used in Chapter 11.

An alternate plan for a two-semester course is that some bits of the first volume, such as metric spaces, are covered in the second semester, while some of the optional topics of volume I are covered in the first semester. Leaving metric spaces for the second semester makes the second semester the “multivariable” part of the course.

Several possibilities for things to cover after metric spaces, depending on time are: