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I am a college sophomore in US with a major in mathematics and an aspiring mathematician in the computation theory and cryptography. I recently got an undergraduate research in the computation theory, which is heavily based on the algebra but also requires the extensive knowledge in the measure theory, approximation theory, and other bits of real analysis. On this upcoming academic year, I will be taking the year-long abstract algebra, linear algebra with proofs, applied linear algebra, combinatorics, and multivariable calculus (chug-and-plug level; taking on Spring 2016). I won't be taking any analysis course until next academic year. This means that I will be self-studying the real analysis by myself.

My research adviser recommended me to read the following books: Rudin (RCA), Folland (Real Analysis), Royden (Real Analysis), Stein-Shakarchi (Real Analysis), Cohn (Measure Theory), Halmos (Measure Theory), and Trefethen (Approximation Theory). If my memory serves me correct, those books are considered as advanced real analysis books that even touch a functional analysis. I told my adviser that I never took a mathematical analysis course, let alone any analysis book. He then advised me to self-study the real analysis, which I am motivated and excited to do so, and he devised a following self-studying plan for real analysis:

Learn the basics of analysis (real number system, basic topology, limit, continuity, and basics of series/sequence) from the elementary analysis books (such as Rudin's PMA, Apostol's Mathematical Analysis, and Pugh's RMA), and jump directly into the real analysis books mentioned on the last paragraph, and study them in a "non-linear, backward" style. That means that I will study those advanced real analysis books and learn the necessary topics from basic analysis books as needed. He said that method will allow me to learn the proper real analysis, gain deeper understanding, and prepare myself for the upcoming research and even Putnam competition. He even gave me an advise that the mathematics is usually learned in a nonlinear style, and it is always a good method to learn the materials from the advanced, comprehensive books.

I like his plan but I fear that perhaps it is too risky. Perhaps my adviser overestimated my ability. Do you like the plan? What is your suggestion? What other advanced real analysis books do you recommend to learn with what I mentioned? Will my lack of knowledge in the multivariable calculus be a significant problem to understand those texts?

MY BACKGROUND: I am currently studying and enjoying Artin's Algebra and Hoffman/Kunze's Linear Algebra.

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  • $\begingroup$ Will you clarify what you mean by "I recently got an undergraduate research in the computation theory"? An undergraduate research fellowship or tutorial? For research when? And in computation theory meaning computability theory or computational complexity theory or numerical computation? -- i.e. computing with recursive functions or integers or real numbers? $\endgroup$ – user173 Jul 20 '15 at 1:32
  • $\begingroup$ Dear Matt F.: I apologize for the confusion. The type of undergraduate research I got is a supervised one where my research adviser (math-CS professor) gave me a project for me to investigate. My project is in the computational complexity theory where I investigate the squares of real numbers and their binary operations (such as sums, products, etc.). The real investigation will start at early September, which I registered as an independent study with him. I can do research as long as I want, preferably until my graduation. $\endgroup$ – MathWanderer Jul 20 '15 at 1:54
  • $\begingroup$ I like the nonlinear backward approach cuz I frequently use it in my courses with B.Sc maths and physicists with outstanding results $\endgroup$ – janmarqz Sep 20 '15 at 17:48
  • $\begingroup$ Is the nonlinear approach where you learn the subject of your interest without studying its prerequisite first, and learn the necessary materials as you learn that subject or book? $\endgroup$ – MathWanderer Sep 20 '15 at 20:59
  • $\begingroup$ I actually have been learning the complex analysis (required for my undergraduate research) and I found that learning the required topics from real analysis as I go is better than formally studying the real analysis. It is surprising to me since I often hear that the mathematics must be learned linearly due to the prerequisites and their influence to learn new materials. $\endgroup$ – MathWanderer Sep 20 '15 at 21:02
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I can not imagine a good research project for a sophomore that would use measure theory. If you're looking for fast algorithms to square real numbers, I don't think measure theory or other analysis will help you much.

The comment that "it is always a good method to learn the materials from the advanced, comprehensive books" seems harmful.

Your lack of knowledge of multivariable calculus may not be a problem in learning analysis. However, have you ever had to manipulate $\forall x \exists \epsilon$ vs $\exists \epsilon \forall x$ in proofs? If not, that lack of exposure to rigor in analysis may be a problem in reading the books. I recommend learning that rigor in a classroom setting.

So I don't like the plan. I suggest that you enjoy the summer without the analysis books.

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  • $\begingroup$ Then would you like to see a copy of the foundational paper for my research? $\endgroup$ – MathWanderer Jul 20 '15 at 2:39
  • $\begingroup$ @MathWanderer: If you post a link to it (or email me privately if anonymity is important), I'll take a look at it and see if there is anything useful I can add to what you and others have said. $\endgroup$ – Dave L Renfro Jul 21 '15 at 16:21
  • $\begingroup$ Dear Dave L. Renfro: thank you very much for the offer. Could you inform me your email? I prefer to send the paper via email form. Also will my lack of knowledge in the multivariable calculus be a problem for me to tackle Rudin-PMA, Apostol (Mathematical Analysis), Pugh, Folland, Stein/Shakarchi, and Royden? $\endgroup$ – MathWanderer Jul 21 '15 at 16:43
  • $\begingroup$ @MathWanderer: See my Profile. (You might have to look at the one at Math StackExchange.) $\endgroup$ – Dave L Renfro Jul 21 '15 at 16:56
  • $\begingroup$ I have been searching for your contact but I could not find it. $\endgroup$ – MathWanderer Jul 21 '15 at 17:27
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In my opinion measure theory does not depend much on multivariable calculus. So it is okay to do measure theory directly after studying real analysis.

My short answer: Just do it! ;-) You will shortly note whether you can study real analysis by your own or whether you need external help in form of a course. In case you have a question you can use forums like math.stackexchange.com.

As you already may know: While reading a mathematics textbook you will find passages where you stuck for a long time. That's totally normal. So: Don't give up to early.

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