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Starting Real Analysis

  1. Apr 25, 2008 #1
    I want to start analysis this summer, because I think it'd give me a heads up on upper year quantum mechanics and differential geometry.

    About me? I'm entering 3rd year physics, and have done intermediate level calculus 1,2,3+vector analysis, linear algebra 1&2 (up to Jordan canonical), and abstract math (ie. number theory, cantor sets, impossibility proofs). I especially have a strong intuitive feel for calculus and vectors thanks to electromagnetics.

    I don't know if I consider myself a "strong" mathemetician. I found linear algebra really really hard, and the proof questions scared me. While I get the main concepts, quadratic forms and all the way one can diagonilize them was confusing. On the other hand, I always found calculus proofs straight forward. Mind you, this wasn't honors calculus. It was "intermediate". We proved all the theorems, so I've seen my share of epsilon delta. But I didn't deal with set theory or anything like that. The books used were Salas & Etgen Calculus (which is basically rigorous stewart) and Folland Advanced Calculus (which is quite theoretical).

    I was wondering... would it be better to reaquaint myself with Calculus via Spivak or Apostol then proceed to Real Analysis? Or go straight into Analysis, because I've heard all of calc is generalized there. I was thinking of using Pugh's Real Mathematical Analysis and see how far I'd get. I was told to stay away from Rudin, and by several people.

    Pugh can be seen here:
    Last edited by a moderator: Apr 23, 2017
  2. jcsd
  3. Apr 25, 2008 #2
    A very friendly real analysis book is Elementary Analysis by Kenneth Ross. Despite its friendly and readable nature, it still covers analysis rigorously and pretty thoroughly and is a great first book in my opinion.

    Here is the amazon link: amazon.c om/Elementary-Analysis-Calculus-Kenneth-Ross/dp/038790459X

    These appear to be lecture notes for a course which used this book; it actually looks like an abridged version of the book itself: math.ust.h k/~makyli/201/201_04-05Sp/201_SylLectNt.pdf

    Note: You'll have to reconstruct those two urls since I was unable to post them whole. I put a space in the "com" and the "hk".
  4. Apr 25, 2008 #3
    Thanks. Judging by the contents though, it looks like review. Aside from the starred metric sections, I've learned all of that pretty well.
  5. Apr 25, 2008 #4
    there are so many books, it's hard to say because it depends what lvl you are at

    i'm assuming you have access to a school library, so i'd recommend checking out rudin, bucks, and bartle and sherberts and maybe some others you see that look readable

    i do think rudin is worth owning though, but again, it's hard to read if you are new to calculus proofs, he is terse on purpose and it is hard on the beginner

    and whatever book you get , no matter how tempted you may be, always always try to do the proofs on your own first, you learn math by thinking about it, it's the only way
    Last edited: Apr 25, 2008
  6. Apr 26, 2008 #5
    Baby Rudin FTW
  7. Apr 26, 2008 #6
    Like I've said, I'm not new to calculus proofs. We've proved everything like inverse function and EVT. I just heard that Rudin is lacks any motivation. I was told Pugh is identical to Rudin but actually readable (just check out reviews). I'll give both books a shot, but my main question is this: Do I proceed to real analysis or do I re-start calc with the likes of Spivak.
  8. Apr 26, 2008 #7
    If you have already done everything in the book that I recommended (Ross), then it seems like you've already seen most/all of elementary analysis. What exactly, then, are you looking for? Do you want generalizations to higher dimensions? If so, a great book is Analysis on Manifolds by James Munkres.

    amazon.co m/Analysis-Manifolds-James-R-Munkres/dp/0201315963
  9. Apr 26, 2008 #8
    This manifolds book looks too complex for what I'm doing haha.

    I want to start Real Analysis. Maybe I know elemntary analysis by ross, but the stuff by Rudin is all new. I am just wondering if I have enough knowledge to take on Rudin or Pugh. Or should I re-learn all of calculus using Spivak.
  10. Apr 26, 2008 #9
    well then spivak is too easy
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