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There have been various comparisons between books on Analysis. I was surprised to find out that Zorich's book on Analysis was not compared anywhere.

Can anyone give a comparison between Zorich and the other books by Rudin, Pugh and Abbott?

A Little Background

I am doing self-study of Analysis on my own. I am a graduate student in Engineering which involves a lot of Mathematics and slowly I am getting in love with Mathematics and thinking of doing a major in Mathematics later. With this in mind, I am determined to consolidate my mathematical background.

So, I have started reading Zorich's texts on Analysis. But when I looked on the internet for reviews, Rudin, Pugh and Abbott had more reviews. Also, Zorich's texts are in two volumes and will take some mighty effort.

In any case, I am actually loving Zorich and would like to continue with this book. Please provide me some comparison between these books.

PS: An additional insight into how should I approach Analysis for self-study and how much time should it take to self-study this course would be much appreciated.

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    $\begingroup$ The title sounds like an ERB. $\endgroup$ – Mark Fantini Nov 3 '14 at 6:30
  • $\begingroup$ The title here is similar: math.stackexchange.com/questions/446852/baby-rudin-vs-abbott. If you, however, think that it is not appropriate please suggest a new title or change accordingly. The question in my view is very genuine, however. $\endgroup$ – shivams Nov 3 '14 at 6:33
  • $\begingroup$ I just find the title amusing, likewise in your related question. :) $\endgroup$ – Mark Fantini Nov 3 '14 at 6:34
  • $\begingroup$ Okay, thanks :) $\endgroup$ – shivams Nov 3 '14 at 6:42
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Shivams, I agree with your assessment that Zorich is more comprehensive than Rudin, taking both volumes of Zorich into consideration. Also, I would say that if you are enjoying this book, there is no reason to switch to reading Rudin instead, whether you intend to use the material for engineering, pure math, or anything else. Just keep reading Zorich.

Rudin's main advantage might be its brevity. This is an advantage only for people who already know most of the material that would be taught in a very rigorous, very complete multivariable calculus course. For some, it might also be an advantage that Rudin introduces topology early and can then use this wherever it helps an argument go more smoothly.

The only major topic I see that is covered by Rudin but not by Zorich is the Stieltjes integral. Rudin's treatment of some topics, such as differential calculus in several variables, is clearly insufficient for practical mastery, even though in some cases you will see the main theorems presented. In the case of multiple integrals, even the theory is insufficient.

Although Zorich starts off more concretely, by the end he includes more abstract material than does Rudin, such as general topological spaces (as opposed to just metric spaces), differential calculus in normed vector spaces (as opposed to finite-dimensional ones), and smooth manifolds, not even touched on in Rudin.

How you study I think depends on the facility you have with the subject. If you can manage the exercises in Zorich's book, just keep reading. If you'd like lots more exercises in analysis with solutions, you can have a look at the problem book by Demidovich, which has an English translation. Kaczor and Nowak also have a problem book that might be worth looking at. One advantage of problems with solutions is that they draw attention to points in which your own solution had an error in it or wasn't detailed enough.

In terms of time, it's very hard to say how long this ought to take. However, I'd say that the content of this book covers almost everything you'd learn in analysis before your final year in a strong BA program in math at a Canadian university, and probably more in depth as well. If it takes you a year to master the content in this book, then you're doing well. Once you've also learned algebra, you'll have an extremely solid foundation for studying more advanced topics.

Correction: There is a chapter at the end of Rudin with a brief introduction to the Lebesgue integral. This has no parallel in Zorich's book. However, I still find Zorich more comprehensive overall.

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  • $\begingroup$ Wow, Mike!! That was some helpful comparison and review. Thanks a lot for answering with such insights. $\endgroup$ – shivams Nov 15 '14 at 18:59
  • $\begingroup$ Very good comparison indeed. (+1) $\endgroup$ – Julien__ Apr 20 '15 at 21:27
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    $\begingroup$ +1 for great advice I couldn't have given better myself of. And I disagree about Demidovich-it's great for beginners. I DO agree,though,the reader would have to supplement it with more theoretical exercises, which one can find in either the wonderful collection by Kaczor and Nowack you mentioned or older collections like Polya/Szebo or Moise. $\endgroup$ – Mathemagician1234 Aug 25 '16 at 20:19
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    $\begingroup$ The latest edition of Zorich's Analysis now has an appendix on the Riemann–Stieltjes integral in Volume 1. $\endgroup$ – shredalert Oct 6 '16 at 20:36
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    $\begingroup$ Honestly, I don't like his definition of Completeness Axiom! :) Anyway, I like the whole work very much due to its lucid, step by step, clear presentation. :) $\endgroup$ – Hosein Rahnama Oct 26 '17 at 22:19
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Not familiar with Abbott, but I haven't seen anyone who excelled Zorich in a comprehensive, clear exposition furnished with intuitive explanations and physical connections.

Rudin explains almost no intuition and I can't recall a single physical application among his examples.

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I am deeply dissatisfied with Analysis books. They do not provide solution manuals in general. I self study this stuff, and its really aggravating not to have solutions. I like to do ALL the problems. I despise this whole Rambo macho mathemtician thing, of spending days if necessary on any particular problem. I dont think its time well spent in general. I think its a kind of thinking that is outdated.

Particularly if this is a side excursion.

I do like Zorich though. I only got it recently, and its a good reference to have. Abbot brushes over the theory and then provides questions without solutions which are absolutely not self contained. I am getting started on Pugh. I just want to have a clear book, with a set of problems, and solutions for if i havent understood.

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