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.
Best Answer
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.