Self-studying real analysis — Tao or Rudin?

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

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.

Attribution
Source : Link , Question Author : Lee Wang , Answer Author : A. Barron

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