I will be finishing up Advanced Calculus 2 soon and I would like to continue self studying Analysis. I want to learn Real and Complex Analysis, Measure Theory and all that other good stuff. but I am not exactly sure what book to use.

My class used the text An Introduction to Analysis by William R. Wade, although I would consider this book rather easy because it's exercises are quite simple. The homework problems my professor gave were much more difficult and often took problems from Rudin's Principles of Mathematical Analysis book. Another thing worth mention is that the whole reason I got into math was from reading about the first half of Micheal Spivak's Calculus completely on my own. So (I think) I can say that I feel more comfortable with the theorem-proof format than most students who went through a class similar to mine.

Anyway, I think Rudin's Real and Complex Analysis would probably be too hard for me. Basically I have been looking at the Table of Contents from books on amazon. I saw the book Analysis by Leib and Loss, but it seems geared towards students of Physics. The table of contents of DiBenedetto's Real Analysis seem to be what I am looking for but I think the book may be too advanced for me as it looks like it just dives right into the deep end. Knapp's Basic Real Analysis is the book I am leaning towards, but since I have very little money, I would really like some advice before shelling out $70 on the book.

Any help picking out a book appropriate for my level would be greatly appreciated.

  • $\begingroup$ He has a few analysis books, which book exactly? His Analysis II book? $\endgroup$ – Eric Feb 1 '13 at 13:29
  • $\begingroup$ sorry, I missed some points in your question. $\endgroup$ – user45099 Feb 1 '13 at 13:36

You might keep the well known graduate analysis textbooks by Folland and Royden in mind, but it looks like they are out of your price range. You might be more interested in Stein and Shakharchi, though I don't own it.

I own Lieb and Loss's book and I don't think it is appropriate for a student just finished with undergrad analysis--more like a second semester graduate textbook since it really deals with functional analysis. I also own Knapp. It will contain a review of advanced calc in the beginning and then you'll learn pretty much everything covered in a first semester graduate analysis course.

But as MSRoris said, doing little Rudin thoroughly is definitely worthwhile. If you want to be serious about this stuff, take the grad student approach and do every problem!

  • $\begingroup$ Would doing Rudin and Knapp at the same time be too much? Say, solidifying the old while learning new material? $\endgroup$ – Eric Feb 1 '13 at 13:32
  • $\begingroup$ It may be, it may not. Knapp actually has plenty of exercises you can do that will help you review. However, many of the Rudin problems are "famous" and their results become part of your common knowledge and tools after you do them. I am of the opinion that learning analysis takes a long time anyways, so you might as well pick somewhere to start and work on it until you take a course. $\endgroup$ – abnry Feb 1 '13 at 13:33
  • $\begingroup$ In that case, whatever book I use I will certainly try to do both. If I fail then I may default to Rudin. $\endgroup$ – Eric Feb 1 '13 at 13:36

Despite doing rudins regular analysis problems, it might be worth considering going all the way through. Its fairly rigorous and coveres a pretty wide range of topics if you want to supplement your current analysis knowledge.

If you want to move forward a little deeper into analysis maybe Munkres : analysis on manifolds could be interesting. It relys on toplogy and metric spaces which it reviews in the beginning. im going through the book right now and i find it pleasant, yet challanging.

  • $\begingroup$ I have heard similar advice from others. Some graduate students I know have told me to go back and learn analysis "the right way," but my professor actually advised me not to do that. He said after his analysis class I would be best served by pushing forward. $\endgroup$ – Eric Feb 1 '13 at 13:28
  • $\begingroup$ I think the idea is to not solely focus on just one phase of analysis until you have seen more branches of math. there are many, and they all have their appeals - finding the right one for you is important. If you like analysis try the munkres book its further into analysis so it will test how much you actually like it :) $\endgroup$ – faith_in_facts Feb 1 '13 at 13:43

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