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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. $\endgroup$ Commented Nov 3, 2014 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
    Commented Nov 3, 2014 at 6:33
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    $\begingroup$ I just find the title amusing, likewise in your related question. :) $\endgroup$ Commented Nov 3, 2014 at 6:34
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    $\begingroup$ Okay, thanks :) $\endgroup$
    – shivams
    Commented Nov 3, 2014 at 6:42

4 Answers 4

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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
    Commented Nov 15, 2014 at 18:59
  • $\begingroup$ Very good comparison indeed. (+1) $\endgroup$
    – Julien__
    Commented Apr 20, 2015 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. $\endgroup$ Commented Aug 25, 2016 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
    Commented Oct 6, 2016 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. :) $\endgroup$ Commented Oct 26, 2017 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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    $\begingroup$ Don't mean to necro but do you think I could have the solutions to pugh as well? I've been trying to find solutions to it but to no avail. $\endgroup$
    – James
    Commented Jun 3, 2020 at 4:30
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    $\begingroup$ Serge Lang's Undergraduate Analysis has a complete solution manual written by Rami Shakarchi: Problems and Solutions for Undergraduate Analysis. He also wrote a similar solutions manual for Lang's Complex Analysis. $\endgroup$
    – KCd
    Commented Dec 11, 2022 at 4:18
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This is not an answer to this question but anyone picking up Zorich should understand that at Russian universities, Mathematical Analysis (as the subject is called) is a 4 semester course (the first 4 semesters of the undergraduate degree) which means that the two volumes of Zorich covers not just Real Analysis but Calculus + Real Analysis, whereas Baby Rudin is pure analysis.

About some omissions in Zorich, it is probably because in MechMat (where professor Zorich was teaching), Volume 1 is covered in Sem 1 and Sem 2, Volume 2 is covered in Sem 3 and Sem 4 but along with this, there is Introduction to Topology in Sem 3 and Real Analysis in Sem 4. Real Analysis in Sem 4 is what covers the Lebesgue Theory.

UPD:

  1. Introduction to Topology has video lectures (in Russian) here and here are the notes prepared by the students. The first two pages contain the contents of the program. You can translate it. One of the recommended books for this course is by Stony Brook professor (+ others) which is Elementary Topology: Problem Textbook. Prof. Viro uses it for his graduate course at Stony Brook.
  2. Real Analysis: Here are the video lectures (in Russian) and the notes can be found here. One of the books in recommended literature is Theory of the Integral - Stanislaw Saks which is available in English.
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  • $\begingroup$ Could you state example textbooks for Introduction to Topology and Real analysis? I am using Zorich's textbook also, and it would be even greater than already to know what do MSU use for the two courses that can be considered supplementary to the normal Mathematical Analysis course. $\endgroup$
    – Yinuo An
    Commented Jul 8 at 9:03
  • $\begingroup$ @OwenAn Sure, I have updated the answer. $\endgroup$
    – Sid
    Commented Jul 8 at 10:46
  • $\begingroup$ Hey sid, I tried translating the intro the topology pdf but the latex becomes a mess. Can you give me some pointers? $\endgroup$
    – Mooonyyy
    Commented Jul 19 at 7:23

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