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I want to know good books I can use to learn real analysis by myself. I only have the upcoming Christmas holidays were I get 2-3 weeks off. Any suggestions?

J. W. Tanner
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Juan Pablo Arango Renteria
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    I'm a huge fan of "Understanding Analysis' by Abbott. He does a great job of explaining things and simplifying concepts. I read this after only having taken calc 1 & 2, and was still able to understand the material with relative ease. – mm8511 Oct 15 '19 at 16:18
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    There are many posts similar to yours on this site. Have you had a look at any of them? – Theoretical Economist Oct 15 '19 at 16:24
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    Yes I have looked through them. Just wanted first hand suggestions – Juan Pablo Arango Renteria Oct 15 '19 at 16:35
  • You won't learn real analysis in three weeks. The basics usually require two years (I mean more or less the contents of Apostol's calculus). What's your background/level? What level do you want to reach? The answer will depend on this, among others. Are you afraid of formal proofs? Are you interested in a specific subject? – Jean-Claude Arbaut Oct 15 '19 at 16:43
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    I have done a course in applied complex Analysis, and I have some PDE, ODE, and Computational math courses under my belt – Juan Pablo Arango Renteria Oct 15 '19 at 16:48
  • Usually someone with that background already knows quite some real analysis (if only to understand those courses). So, what part do you want to explore? I might suggest some Lebesgue integration and maybe Yeh's "Real Analysis:Theory of Measure and Integration" (very good for self study, in my opinion). But that's really just a guess. – Jean-Claude Arbaut Oct 15 '19 at 16:51
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    That’s not a good enough reason to post a new question, unfortunately. As it stands your post should probably be closed as a duplicate of those many other questions. If you don’t want that to happen, you should probably edit your post to include an explanation of why this isn’t just a duplicate question. Otherwise, you’ll just get the exact same answers as the other questions have, which doesn’t really help you, nor does it add anything to the site (and only adds to the clutter). – Theoretical Economist Oct 15 '19 at 16:53
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    I hope my directness doesn’t make you feel unwelcome. You are very much welcome here, but please do adhere to the site’s conventions while you’re here. – Theoretical Economist Oct 15 '19 at 16:53
  • I agree with the recommendation of Abbott's Understanding Analysis. It's a great introduction that is accessible for self-study. Obviously you won't get very far in three weeks, but you'll probably be motivated to carve out more time for further reading. –  Oct 15 '19 at 16:54

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Although I have never read them, I've heard many good words about

  • Stephen Abbott's Understanding Analysis, and
  • Victor Bryant's Yet Another Introduction to Analysis.

From various user reviews, it seems that these two books are super user-friendly and full of motivations, albeit a bit pricey. Abbott's book covers the contents of a typical first course, while Bryant's book is more elementary.

However, I would like to recommend Robert Ash's Real Variables with Basic Metric Space Topology. Ash was a great expositor. Many of his textbooks are well-written and I believe this one on analysis is among his best. While this book may not be as friendly as two books mentioned in the above, it is still gentler to beginners than most in the wild are. The presentation is lucid and its format and pacing are suitable for private study. Its coverage of topics is broader than Bryant's but slightly narrower than Abbott's. As a Dover publication, the book is cheap (its list price is only USD 11.95 at the time of writing).

My only complaint about this book is that its discussion on interchange of certain operations is not deep enough. In particular, interchange of order of summation in double sequences/series, differentiation under the integral sign and Fubini's theorem are left out. However, as a first introduction to real analysis, it is a competent piece of work.

Ramen Nii-chan
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