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E.S.E. advisers,

I am a college sophomore with a major in mathematics and an aspiring mathematician in the fields of computation theory and cryptography. I am always curious about the importance of "transition books" from calculus to analysis, such as Spivak, Apostol, and Courant. Do I need to read one of them before jumping into real analysis? I took Calculus I-II course using Serge Lang's A First Course in Calculus, and I am currently studying Artin's Algebra and Hoffman/Kunze's linear Algebra, and preparing myself for upcoming research in the computation theory. I will be taking the theoretical linear algebra and abstract algebra for this academic year, and I will proceed to the analysis on next academic year. However, I want to learn more about calculus since I am preparing for Putnam, and measure/approximation theory will be used on my research. Perhaps it is naturally for me to pick up analysis books of Rudin, Apostol, or Pugh, but it seems that quite many people recommend the advanced calculus books of Apostol, Rudin, and Pugh. Is it necessary for me to read those transition books? Can I just jump right into the analysis books? What are the benefits of reading the transition books first before studying the analysis?

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  • $\begingroup$ In my country, there's no calculus courses outside high school. We start our 1st year with introductory real analysis. $\endgroup$ – Paracosmiste Jul 19 '15 at 16:29
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    $\begingroup$ Can you clarify the distinction you're making between a rigorous calculus course and real analysis? In the US at least, what most universities offer as an introductory real analysis course is a rigorous proof-based calculus course. In particular, I don't know of any "advanced calculus" books by Rudin or Pugh which are not the same as their intro "real analysis" books. $\endgroup$ – Santiago Canez Jul 19 '15 at 20:27
  • $\begingroup$ I notice that you don't say you've studied multivariate calculus. That's probably a more important priority than redoing single variable calculus in a more rigorous fashion, don't you think? One option that comes to mind is Friedman's Advanced Calculus: it covers both single-variable and multivariate calculus and does so in a more rigorous fashion than Lang (e.g. using sup and inf). $\endgroup$ – Nate C-K Jul 20 '15 at 22:43
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    $\begingroup$ Study the basics of analysis (real number system, basic topology, limits, and continuity) from the basic analysis (Rudin-PMA) and jump directly into those real analysis books I mentioned above and study them in a "non-linear, backward" manner, studying the necessary topics from basic analysis books as necessary. He said that he studied on that way and finished both Rudin-PMA and Folland at the same time. Is this actually a possible plan? $\endgroup$ – MathWanderer Jul 21 '15 at 1:47
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    $\begingroup$ Well the real question is, what's the best use of your time? You can read another calculus book now but it will cost you extra time, and it seems like you're in a hurry. Speaking of being in a hurry: does it make sense to self-study real analysis while enrolled at a university that teaches the course? Why not start reading an analysis book right now (not all of them) for your research background and then take the course next year? The first 4 chapters of Apostol's MA cover the topics you mention; start reading and see how you feel. $\endgroup$ – Nate C-K Jul 21 '15 at 14:42

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