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I am considering Stephen Abbott's Understanding Analysis and Walter Rudin's Principles of Mathematical Analysis. I am looking for a comparison between the two that addresses both of the following matters,

  • Is one of the two substantially more mathematically rigorous than the other?
  • Does one include substantially more challenging problems?
  • Which provides a better introduction to Real Analysis?

Follow-up question:

  • Would someone who has worked through Abbott's book be at a disadvantage compared to someone who has completed Rudin's text?
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    $\begingroup$ Rudin is an industry standard in rigor by far for Real Analysis. $\endgroup$ – Torsten Hĕrculĕ Cärlemän Jul 18 '13 at 20:27
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    $\begingroup$ I don't know Abbott's book, but my money is on Rudin. $\endgroup$ – Daniel Fischer Jul 18 '13 at 20:30
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    $\begingroup$ @Daniel: I can see that you're enthusiastic, but: is it really helpful to voice on opinion on a comparison of two texts when you're not familiar with one of the texts? $\endgroup$ – Pete L. Clark Jul 18 '13 at 20:41
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    $\begingroup$ @PeteL.Clark It's a testimony to the quality of Rudin's books, nothing more. $\endgroup$ – Daniel Fischer Jul 18 '13 at 20:42
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    $\begingroup$ As for exercises, this is sure relevant: abstrusegoose.com/12 $\endgroup$ – Shuhao Cao Jul 18 '13 at 22:11
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They are both rigorous in that they both give complete proofs of their results. Rudin's problems on the other hand are challenging to newcomers. Abbot's problems are on a much lower level than Rudin's. I love Rudin's books, but there are mixed opinions on whether they should be used as introductions. I used Principles after my first year of analysis and loved it. I'd say first work through Abbot because he will likely provide more motivation. Later, get Rudin and push your boundaries of understanding. You might just become an analyst after that approach. It's what happened to me.

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You think you are asking just a simple question but you are adding kerosene to the flame of Pro-Rudins vs Anti-Rudins :) Any ways I have read both books and here are my feelings

(1) Rudin definitely has really good, challenging problems

(2) Rudin goes directly to the point

(3) Rudin gives you the opportunity to think on your feet and fill the gaps

That said coming from a normal math background, Rudin was a tough read for me. Abbott's book was quite a good supplement for some chapters. But for me Pugh's Real Analysis and Apostol Analysis was the best supplement. My strategy was to read and re-read Rudin and give some critical time $t_{c}$ to see if I understand a concept. If that doesn't happen $t>t_{c}$, I refered to the other books. Towards the end of the course, I came to know Terry Tao's book and I have to say it was a pleasure to read. Good luck, you can always pick one book and refer to the others when you need to. You need to have a general idea of where you are going but you don't have to follow every route

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Can't do better than read "Bolzano Bourbaki's" review of Baby Rudin on Amazon.

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    $\begingroup$ For some reason, the older I get the more that review annoys me. I think it's his dismissal (and seeming lack of understanding) of mathematical motivation. Does anyone else feel that way? $\endgroup$ – Alex Lapanowski Jul 18 '13 at 20:45
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    $\begingroup$ I think his understanding of what an "application" is sub-par (true for most mathematicians). However, I think his plan of studying Rudin a good one, and his encouragement to keep going even when it gets hard is sorely needed today. $\endgroup$ – Adrian Keister Jul 18 '13 at 20:47
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    $\begingroup$ Yeah for sure. I suppose most students looking at basic analysis books aren't used declaring war on math books. His pep is amusing. $\endgroup$ – Alex Lapanowski Jul 18 '13 at 20:51
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    $\begingroup$ I'm sorry, that review is childish and offensive, actually. Pity that so many people believe mathematics to be "a contest", or want it to be. $\endgroup$ – paul garrett Jul 18 '13 at 23:25
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    $\begingroup$ "Or you could just change your major back to engineering. It's more money and the books always have lots of nice pictures." hahah $\endgroup$ – Bhaskar Vashishth Aug 23 '15 at 22:55

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