Self-studying real analysis — Tao or Rudin?

Question

The reference requests for analysis books have become so numerous as to blot out any usefulness they could conceivably have had. So here comes another one.

Recently I've began to learn real analysis via Rudin. I would do all the exercises, and if I was unable to do them within a time limit (usually about 30 min) I would look the answers up. Combined with the excellent online lectures by Francis Su, I made rapid progress. Encouraged I now intend to self-study analysis II and function theory. However apart from its uninformative and dry style, Rudin's does not cover everything I intend to study.

After searching for a suitable textbook, I was particularly attracted to Analysis I&II by Terry Tao. His breadth of knowledge and his nack for clear exposition are famous but I particularly like that he starts from the very beginning and builds it up from there, as well as putting real analysis inside a greater unified whole. His books would cover exactly what I intend to study. For instance, he covers fourier series, which Rudin's doesn't.

However after searching for hours I've been unable to find any solutions sets. (apart from a few on the earliest chapters). It is my experience that is almost impossible to self-study a subject thoroughly without solutions or constant feedback, even with an outstanding textbook. Which leaves me with few options:

1. Proceed with Rudin's, perhaps with some supplementary book.
2. Try to work with Terry Tao's Analysis I&II without solutions.
3. Find a different book altogether that is both comprehensive and readable as well as having at least a partial solution set.

I know a lot of people will recommend Rudin but I have to doubt their experience with self-study: yes it is possible to learn directly from Rudin but it's painful and slow. And quite frankly I feel that a lot of people have poured a lot of time and effort in Rudin and feel that more than teach them analysis it has brought them mathematical maturity. That is all well and good but it's not what I'm interested in.

Another idea would be to get both and read Tao, while doing the exercises in Rudin's. I don't think that would be a good idea however, a lot of theorems in Tao are left to the reader and the pace and coverage of both books are very different. In general I dislike getting more than one book.

Does anyone know of an extended (partial) solutions set to Terry's analysis I&II or otherwise a reference for another book that would be suitable?

Answer 1

First of all: you shouldn't give up on problems after 30 minutes. Take a break, try a different problem, maybe wait a few days and try again -- you'll gain a lot more from the problem if you struggle and solve it yourself. Having access to solutions can be helpful, but you don't want to find yourself relying on them. (There's a phrase that gets thrown around a lot: "If you can't solve a problem then there's an easier problem you can't solve; find it").

Baby/Blue Rudin is a great book for an introduction to the basics of analysis (beyond one-variable "advanced calculus"). After that I'd suggest looking at the 'Lectures in Analysis' series written by Elias Stein and Rami Shakarchi (Stein was actually Terrence Tao's advisor). These books cover introductory Fourier analysis, complex analysis, measure theory, and functional analysis. Along the way the authors expose you to all kinds of in-depth and enlightening applications (including PDEs, analytic number theory, additive combinatorics, and probability). Of all the analysis textbooks I've looked at, I feel like I've gained the most from the time I've spent with Stein and Shakarchi's series -- these books will expose you to the "bigger picture" that many classical texts ignore (though the "classics" are still worth looking at).

I've skimmed through parts of Terrence Tao's notes on analysis, and these seem like a good option as well (though I looked at his graduate-level notes, I don't know if this is what you're referring to). I've always gotten a lot out of the expository stuff written by Tao, so you probably can't go wrong with the notes regardless. If you feel like you need more exercises, don't be afraid to use multiple books! Carrying around a pile of books can get annoying, but it's always helpful to see how different authors approach the same subject.

Answer 2

Get Thomas Apostol's book "Mathematical Analysis." I'm studying for a Modern Analysis qualifying exam, and finished a course in "baby" reals a year ago using Rudin and occasionally referring to Apostol on my own for help. Out of the ridiculous set of texts I've checked out from the library to help me study for my Analysis qual ("Baby" and "Mama" Rudin, Aliprantis, Folland, Haaser and Sullivan, Bruckner$^2$ and Thomson, Lang, Bass, Berberian, etc. etc. etc.) it's actually still Apostol's book I'm studying from the most - it not only covers all the topics that are critical to introductory analysis, but it also has important connections to modern analysis that most texts lack. See, for instance, the side-by-side multiple Riemann and Lebesgue integration chapters, and the natural development of Lebesgue integration following his chapter on series of functions.

There are TONS of examples, counterexamples (showing WHY certain theorems appear), and topics critical to analysis that aren't given as much attention in other texts, such as dealing thoroughly with double series and differentiation under the integral sign. The big convergence theorems and important consequences are followed by on-the-ground examples and exercises with actual functions, not just "pushing around $f_n$'s." There's even an entire chapter dedicated to Fourier series and integrals, something pushed to the corner comparatively in Rudin.

Rudin is fantastic if you are in lecture IMO, but not self-study. Apostol, in comparison, is an encyclopedic text that practically has an instructor right there on the pages. There are solutions to exercises from the first eight chapters or so somewhere on the 'net; the exercises range from routine to extremely difficult.

The only gripe is the lack of focus on measure theory. The Lebesgue integral is developed, in my opinion, kind of as a natural extension of Riemann, and a few sections of measure theory are thrown in as a side note at the end. I personally don't like that development, but it is worth a glance if you have never encountered Lebesgue integration before. Also, as is usual for almost any classical real analysis text, avoid the last chapter (in this case a way too brief summary of Complex Analysis).

If you're looking for something a little more "toned down," but above the level of, say, Bartle's or Lay's undergrad texts, I'd recommend "Understanding Analysis," by Stephen Abbott. This is another wonderful book that focuses on the ideas currently helping me out the most, and it is far more concise than Apostol. It has Fourier analysis material in the last chapter (just flip past his reproduction of Bartle's plug for the Gauge integral). This book has a solution manual that you can order from the author at a cost if you contact him.

Finally, I'd recommend Pugh's "Real Mathematical Analysis." This is comparatively heavy on topology, and the problems are very difficult (and there is a strange one or two that ask for things like coming up with rhymes about theorems or something), but you learn a lot of geometry and you even get pictures to go along with it. Its downside is that it is extremely chatty.

All in all, you can't go wrong with Apostol's "Mathematical Analysis," probably my favorite math book of all time other than Ahlfors' "Complex Analysis" (which you should also get, even if you have no reason to do so). Apostol's two Calculus texts are excellent companions if you need to "brush up," as well - they are the best Calculus texts ever written IMO, and lead so naturally into "Mathematical Analysis" that I consider them a three-volume set in a way.

Answer 3

I learned most of my analysis from Tao's excellent volumes. The OP is correct that Tao starts at the beginning with defining the number systems. So I believe chapter 5 or Tao is equal to chapter 1 of Rudin, or something like that.

I did not have too much trouble doing the proofs and exercises for the initial chapters of Tao because he really develops the ideas slowly. So that was good. It was only when he started to address the point set topology stuff that I started to have a harder time. So I actually decided that I would do a little more exploration of Point-Set topology on my own. So I read Munkres' book and then did the exercises in the Schaum General Topology outline. That was a good combination.

After I had a better handle on rigorous topology, then I really had not trouble following the rest of Tao's analysis. I think I ready through Tao's analysis and use the Schaum outline in Advanced Calculus, and in Real Variables for doing problems. Once I was able to do the problems in Schaum, then I had much less trouble doing the problems in Tao. In many cases the problems were similar.

But that is just how I approached it. I hope this info helps others pursuing Analysis. It really is a beautiful subject and I fear that people often get psyched out by claims of its difficulty.

Answer 4

I actually study analysis for my college using baby Rudin as a first reference book, but I've chosen another book as supplement e.g. now is watching S. Berberian' s book (the pdf-version) as a good support [http://www.amazon.com/Fundamentals-Analysis-Universitext-Sterling-Berberian/dp/0387984801]. I just can not find solutions...

Answer 5

Personal opinion: You should definitely go with Tao's books!

As others have already mentioned, if you do so, you should be prepared to spend more than hours on separate exercises as some of them are highly non-trivial (needless to say, the payoff is huge).

Should you solve the problem by yourself and want to verify your solution, then some of the solutions are already available for Terence Tao's books as of 2021. You can check them out under this link.