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I'm in search of a mathematical analysis text that covers at least the same material as Walter Rudin's Principles of ... but does so in much more detail, without relegating the important results to the exercises, contrary to what Rudin does. Which one is it, if any?

Do the mathematics students at places like the MIT, Harvard, or UC Berkeley, where Rudin is used, cover this textbook fully, solving each and every problem? If not, then how much of it is taught and in what detail? Is there any university where this book is covered fully in their analysis courses?

Can I access any video lectures based on Rudin?

Is there any TV channel dedicated to higher level mathematics?

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If you're talking about the undergraduate level, then I'd be surprised if a year of coursework can cover all of Rudin's PMA. I took a year of coursework at George Mason University in '96-'97 working through Rudin as fast as we could and we only got to the Implicit Function Theorem. Maybe at MIT they are getting to the end and then doing extras in one year, but it's a very dense book. –  Todd Wilcox Feb 8 '13 at 21:30
    
P.s. There really is a huge amount of personal preference that goes into deciding which book is the "best". For all the books that are being recommended to you, many of them are probably in your schools library or available via inter-library loan. You should check out as many of these recommendations as you can and then decide for yourself which is "best". –  Jim Feb 8 '13 at 22:18
    
What about "Mathematical Analysis", second edition, by Tom M. Apostol? How does that compare with Rudin's "Principoles of Mathematical Analysis", third edition? Do any universities use the former? –  Saaqib Mahmuud Feb 16 '13 at 1:27

6 Answers 6

I went to Berkeley and the real anaylsis class used Elementary Analysis: The Theory of Calculus by Ross. It is a bit simpler than Rubin but much more readable. We did pretty much everything.

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+1 for a great classic that really was the first of it's kind to bridge the gap between watered-down calculus classes and Rudin-level analysis courses. It could also be used, properly supplemented,as an honors calculus textbook and it's an easier read then Spivak. –  Mathemagician1234 Apr 8 '13 at 22:48

I would recommend Bartle's "The Elements of Real Analysis".

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Tao's Analysis I is my favorite. It is very reader friendly and eloquently written.

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I like tao too but OP says "but does so in much more detail, without relegating the important results to the exercises, contrary to what Rudin does" and i think tao is better at it than rudin. –  user45099 Feb 8 '13 at 21:18
    
@user57 Completely agree and I particularly like how Tao explicitly builds the number systems in the first chapter or so of volume 1. This really needs to be done and if you're not too detailed about it,it CAN be done effectively in an analysis course. –  Mathemagician1234 Apr 8 '13 at 22:50

My class is using Intro to Real by Bartle and Sherbert. My previous class (9 years ago) used Introductory Real Analysis by Dangello and Seyfried, which I prefert to my current text. Neither one covers everything in what I would consider "great detail".

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Which university are you at? –  Saaqib Mahmuud Feb 8 '13 at 20:40

Not sure, but "Introductory Real Analysis" by Kolmogorov & Fomin (translation by RA Silverman, publ Dover) is rigorous and extensive and not expensive.

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What about "Mathematical Analysis", second edition, by Tom M. Apostol? How does that compare with Rudin's "Principoles of Mathematical Analysis", third edition? Do any universities use the former? How does the text by Apostol compare with the one by Rudin? –  Saaqib Mahmuud Feb 16 '13 at 1:25

Check the many lecture notes available on the net, e.g. William Chen's. The Trillia Group has textbooks available for free too. Check the AMS for suggestions. MIT has lots of stuff on OCW, and there is now Coursera.

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