My university does not offer a real analysis course for math major students in the later stages of their undergrad; these are reserved for honours students. I had an excellent professor in my final introductory analysis course and I wish to study some higher level real analysis on my own. I was looking for a book that might help me in this endeavour.
In my analysis courses, we covered sequences and series, limits including $\limsup$ and $\liminf$, continuity, differentiability, and Riemann integration (Darboux's approach only). Our main textbook was Abbott's Understanding Analysis, though I found Bartle's Introduction to Real Analysis much more helpful. I remember reading a bit of Rudin's Principles of Mathematical Analysis, but it went a bit too fast for me.
I looked at the syllabi for the honours courses and they involve topics like point-set topology, some introductory measure theory, and Lebesgue integration, with a brief foray into functional analysis and Fourier analysis only at the end of the year.
Some potential books that I looked at were Royden's Real Analysis, Carothers' Real Analysis, and Axler's Measure, Integration, & Real Analysis. Royden seems to be used by my university, but upon a quick glance I may need a bit more mathematical maturity before I attempt to self-study from it. Carothers and Axler both seem to match my pace, but I am interested as to your thoughts on either the books I mentioned or some other books outside of these three that you believe might suit me best.
Best Answer
TL;DR: Carothers Real Analysis is the book for you.
First, let me tell you a little bit of my background so that you have a sense of where I’m coming from. My first real analysis class was a two semester sequence taught out of Carothers’s Real Analysis. I then took a graduate real analysis class taught out of Folland’s Real Analysis. Years later when I was prepping for graduate school entrance exams, I used Rudin’s Principles of Mathematical Analysis and Royden’s Real Analysis to prepare. Then I took a graduate real analysis class taught out of Stein & Shakarchi’s Real Analysis. Then I taught an undergraduate course out of Marsden’s Elementary Classical Analysis. Then I taught an undergraduate course out of a combination of texts with no single designated “book” (which was a mistake). I think the comment that Dave Renfro made that “Most people are only aware of the books they used and perhaps a few other books” is insightful. I cannot claim to have great knowledge beyond the books I mentioned above, plus a few others I combed through when assembling the chaotic book-less analysis course.
If Rudin’s Principles of Mathematical Analysis went too fast for you, I would encourage you to ignore what 温泽海 said in their answer. There is a reason that Principles is referred to as “baby Rudin” and Real and Complex Analysis is referred to as “papa Rudin”. If you found baby Rudin to be too fast, I do not think you would find reading papa Rudin to be a better experience. I mean no disrespect to 温泽海. I’m sure they had a great experience with papa Rudin! However given what you said in your question, I do not think this would be an appropriate book for you.
Out of all the books I mentioned above, I personally think that Carothers’s text would be best for you. It is very clear, very complete, and oozes with personality. Sadly, I usually have a hard time recommending that book to people, but it sounds like it would be a good fit for you. Carother’s is a little difficult for a first course in real analysis if a student has no prior experience with real analysis at all. In particular, for many students, the first real analysis class is also an “intro to proofs” class. Carothers assumes the reader is already comfortable with proof writing. Additionally, Chapter One, “Calculus Review” is challenging for some students. In about ten pages, Carothers covers the least-upper-bound property, the epsilon-delta definition of a limit, liminf and limsup, Cauchy sequences, the Bolzano Weierstrass theorem, continuity, and basic continuity properties of monotone functions. For many, this is not review! It certainly wasn’t review when I took a class out of this book.
For you, though, having already taken a real analysis class, this will actually be review. This would allow you to quickly get into the rest of the book.
Carothers’s book is long, and here are my thoughts. I will mention parts that I believe can be skipped in the interest of time.
Carothers’s book is divided into three parts. Part 1 is, in my humble opinion, pedagogically inspired. Carothers makes the unusual choice of teaching the point-set topology of metric spaces, with the real numbers as a special case serving as an illuminating source of examples. Chapter 9 can be skipped. The Baire Category theorem is interesting (especially if you’re interested in descriptive set theory or functional analysis), but is not essential for the remainder of the book.
Part two is on spaces of functions. Here Carothers again makes an unusual choice in presenting the Weierstrass approximation theorem (and the more general Stone-Weiestrass theorem) without a discussion of Taylor series. While I understand why Carothers does it this way, I think this could be considered an omission. Chapter 12 (the Stone-Weierstrass theorem) can be skipped. This is a very powerful tool, but it is not used much in the rest of the book. Chapter 14 presents another anomaly in Carothers’s text. It teaches the Riemann-Stieltjes integral instead of the “ordinary” Riemann integral. You may find this an interesting generalization of the integral you learned in your first Real Analysis class. In some regards, I would say this chapter is safely skippable, but in part three Carothers defines the Lebesgue integral and compares it to the Riemann integral. Without the specific facts about the Riemann integral presented in chapter 14, some parts of these comparisons may be hard to understand. Perhaps you could reference chapter 14 later as needed. Chapter 15 (Fourier series) can safely be skipped.
Part three is on Lebesgue’s integration theory. Carothers’s exposition of Lebesuge measure is very thorough. Chapters 19 and 20 (additional topics & Lebesgue differentiation) feel rushed to me. I would recommend skipping these chapters and consulting a different source. Finally, “abstract integration” theory is a notable omission from the text. Chapter 6 of Stein Shakarchi’s Real Analysis would be a fine source for this topic.
The topics I suggested possibly skipping are very interesting topics. If you have the time, you should read them! I’m merely pointing out that some chapters can be skipped in the interest of time without impacting the “flow” of the text.
Finally, I would highly encourage you, if you use Carothers’s text, to do all the exercises with little sideways triangles next to them. These exercises are essential for learning.
All in all, Carothers text is really good. It has some idiosyncrasies that make it hard to recommend as a first course in real analysis, but I think it would be a perfect book for you based on what you said. I hope you enjoy the book!