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Which parts of real-analysis are worth studying if you have already taken several calculus courses? I know that real-analysis is more 'rigurous', but still I wonder whether it is worth to again go over a lot of subjects that I already know from through calculus.

Should I redo all of the subjects from real analysis that I already know of from calculus or are there specific topics I should choose to study and skip the rest? I read that a lot of people advise the book by Terence Tao, but would it be useful to go through this whole book given what I already know? Or are there chapters of it that I should read, and or are there other references which only cover the topics that I do not yet know of?

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You have to read the whole book. –  user29743 May 18 '13 at 13:04
The point is that you will not spend nearly as much time relearning things from a new perspective when you read chapters covering stuff you know as you will spend getting frustrated when you skip chapters and then bump into symbols and concepts that were introduced earlier which you are unfamiliar with. –  user29743 May 18 '13 at 13:05
@countinghaus Thank you for the advice. Would you also say that the book by Tao is the best book out there or is there a different one that you would advise? –  dreamer May 18 '13 at 13:07
I don't know about his book, but his notes (availible for free on his ucla site) are probably the best resource I've found for real analysis. –  DepeHb May 18 '13 at 13:13
If you don't want to spend time relearning calculus stuffs then read the first ten chapthers of Rudin's Real and Complex Analysis. –  user01123581321345589144... May 18 '13 at 13:17

1 Answer 1

up vote 5 down vote accepted

When I first entered university, shortly before classes began, I met with an professor whose task was to advise me on which classes to take in my first semester. After hearing me describe my background, which included passing the college-credit calculus exam at age fifteen, he suggested that I take real analysis.

“But I took that already,” I protested. “I had a two-semester course in real analysis at Columbia University last year. We used the little blue Rudin book. I got A’s.”

But the professor said to me that analysis was a deep enough and rich enough subject that I would not be wasting my time to take it again, and that I would not be bored. I thought about this a little bit, and then I agreed that he was probably right. So I took the analysis course again. We used the same textbook, but I was not bored, and it was not a waste of time. It was an extremely good use of time; I have never regretted it.

So that's my answer about which topics of real analysis should be studied if you have already done calculus: all of them. You will not be bored, and it will not be a waste of time, because the answer is the same even if you have already taken real analysis.

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Thanks a lot for your advice! That is a very clear answer. Then I know what to do :). Do you also have advice for me on which book I should go for? You mentioned Rudin's book but I often read that people describe it as very 'concise'. Do you know how it compares to Tao's book? –  dreamer May 19 '13 at 20:00
I have not read the Tao book. The Rudin book is extremely concise. I don't think I would recommend it for self-study. –  MJD May 19 '13 at 23:13
Thank you @MJD :) –  dreamer May 20 '13 at 9:40

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