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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.

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    $\begingroup$ I am a fan of Stein/Shakarchi's sequence of texts which has clear prose that motivates the concepts well. You would probably be interested in the 3rd book for measure theory, and can follow up with the 1st and 4th for Fourier analysis and functional analysis. $\endgroup$
    – angryavian
    Commented Jun 13, 2022 at 3:53
  • $\begingroup$ Stein and Shakarchi's texts are good. I like services.math.duke.edu/~rtd/PTE/PTE5_011119.pdf for probability theory. $\endgroup$
    – Mason
    Commented Jun 13, 2022 at 4:01
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    $\begingroup$ Given what you said, I think Carothers' book fits you better than the other books mentioned. Two others that fit with how I would classify Carothers' book (see follow-up comments) are Analysis in Euclidean Space by Kenneth Hoffman (same Hoffman of the famous Hoffman/Kunze linear algebra text) and Basic Real and Abstract Analysis by John F. Randolph. The following comments from this answer are (continued) $\endgroup$ Commented Jun 13, 2022 at 6:17
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    $\begingroup$ worth repeating here: Keep in mind that a book no one talks about could still be very good. Most people are only aware of the books they used and perhaps a few other books, and most people commenting online probably have little knowledge of anything that hasn't been widely used in the last 30 years. For example, I think John F. Randolph's Basic Real And Abstract Analysis is one of the best beginning graduate level real analysis texts I know of, but because it appeared in 1968 and because it is an awkward fit for most mathematics programs (continued) $\endgroup$ Commented Jun 13, 2022 at 6:21
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    $\begingroup$ (too advanced for most undergraduate courses, too leisurely for most graduate courses; in this respect it is a bit like Neal L. Carothers' text Real Analysis), there is virtually no discussion of Randolph's book anywhere online. $\endgroup$ Commented Jun 13, 2022 at 6:21

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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!

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Rudin's "Real and Complex Analysis" is highly recommended. I am surprised that no one mentions it. If your interest only lies within in real analysis, then you only need to read the first nine chapters, which cover everything you have listed. You do have the prerequisite as you have read Abott and Bartle.

I finished what you call "higher level analysis course" entirely by studying this textbook. Contrary to Rudin's first book "Principals of Mathematical Analysis", this book is, at least to me and everyone I know, extremely easy to read through. Compared to the first book, the material is better organised and the exercises are in general easier. I can think of only one disadvantage of this book: it does not cover Caratheodory's extension theorem, for which you need to read from other sources.

I do not recommend any textbook that does not start covering measure theory in the general setting. You should avoid textbooks that spends half of the volume covering just measure theory in $\mathbb{R}^n$. As Rudin repeatedly points out in his book, the theory is clearest when presented in the most abstract setting.

If it turns out that a topic in "Real and Complex Analysis" is covered too fast for you, you can gather a few references on the subject (you have already done this) and read all parts of the reference related to the topic.

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    $\begingroup$ Rudin's "Real and Complex Analysis" is highly recommended. I am surprised that no one mentions it. --- No one mentioned it because it is not an appropriate text to recommend to someone who included the following statement: "I remember reading a bit of Rudin's Principles of Mathematical Analysis, but it went a bit too fast for me." The following portion of a comment I made here is also relevant: Rudin's book [Real and Complex Analysis] is almost never used in the U.S., (continued) $\endgroup$ Commented Jun 17, 2022 at 18:03
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    $\begingroup$ but it is a frequently cited book, especially for those preparing for their Ph.D. qualifying exams. I mentioned it because it's famous for being very unsuitable for first courses in either real analysis or complex analysis at the graduate level (i.e. after undergraduate study), but also often considered very instructive for surveying/reviewing the material for those who have already had such courses. $\endgroup$ Commented Jun 17, 2022 at 18:03
  • $\begingroup$ I agree with Dave L. Renfro and do not think this is an appropriate recommendation given what Kman3 said in their post. $\endgroup$
    – Joe
    Commented Jun 22, 2022 at 4:55
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There's an old book by Tom Apostol, "Mathematical Analysis" that I prefer to Baby Rudin (Principles). There's probably an affordable reprint somewhere.

If you supplement Apostol with Duistermaat & Kolk's wonderful, but less well known, 2 volume "Multidimensional Real Analysis" you'll get a slightly more unifying perspective of a broad vista of mathematics - not just real analysis - and a peek of how analysis shows up in other parts of mathematics, particularly geometry.

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