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Can anyone suggest some good books for analysis for self-study? I was following Rudin but it lacks proof intuition and examples.

Currently, I want a good book which contains very nice examples on metric space, open set, closed set, compact set etc. I have just started to do a basic course in analysis in my university.

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"Analysis" is really broad, can you specify? You can try Spivak and Apostol for the basics (Calculus) then Apostol's Mathematical Analysis for more advanced stuff. Rudin is probably a book you want to read after you know a little of everything of the others. For example, he gives a "crash course" in topology of the Euclidean space one wouldn't get anything off if hasn't already bumped into such ideas. –  Pedro Tamaroff Aug 24 '13 at 3:46
    
What level of rigor are you interested in? Introductory real analysis or advanced (similar to the level of Rudin)? –  Cameron Williams Aug 24 '13 at 3:49
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@Peter Tamaroff: Currently, I want a good book which contains very nice examples on metric space, open set, closed set, compact set etc. –  podu Aug 24 '13 at 4:12
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Charles Chapman Pugh's REAL MATHEMATICAL ANALYSIS. There you go. –  Mathemagician1234 Aug 24 '13 at 4:16

4 Answers 4

Understanding Analysis by Abbot is a really great book.

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I enjoy R. Creighton Buck's Advanced Calculus.

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If you want an interesting alternative that goes deep into why things work out as they do in real analysis, then you can have a look at analysis textbooks (apart from the books already mentioned in the comments) such as

A course of pure mathematics - Hardy

Introductory Real Analysis - A. Kolmogorov, S. Fomin

Real Mathematical Analysis- Charles Chapman

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While we're at it, Shilov has a nice book, too. –  Pedro Tamaroff Aug 24 '13 at 3:58

This set of free lecture notes by Fields Medal winner Vaughan Jones are excellent. In that they are virtually verbatim, you get the full presentation as if he was speaking directly to you rather than a summary.They are very accessible, and assume no prior knowledge. They build from the beginning adding insight along the way.

As one of the great mathematicians, he uses his insight in presenting his own treatment of the material and makes his train of thought very clear.

https://sites.google.com/site/math104sp2011/lecture-notes

This would be my first choice covering what you ask for and more.

For additional consideration: Pugh's "Real Mathematical Analysis" (as mentioned above) is also excellent, likewise with examples and pictures which are quit beneficial.

Here is a short review from Promys:

An absolutely fantastic introduction to Analysis, it has excellent exposition and is full of great examples and over 500 (good) exercises. A counterpart to Rudin, Pugh always builds up machinery first and uses it to provide very clear proofs that grant a good sense of "why" something is true. Judging by page count and the amount of material it covers, it seems that it must be as concise as Rudin, yet it reads very easily.

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