What are some recommended analysis textbooks that complement Rudin's PMA?

[bacte2013]

Dear Physics Forum advisers,

I am a college sophomore with double majors in mathematics and microbiology. I wrote this email to seek your recommendation on selecting the introductory analysis textbook, particularly one that complement with Rudin's PMA well. Starting on this Fall, I will be taking Analysis I course, which uses Rudin's PMA. I will be finishing Calculus II (text: Calculus with Analytic Geometry, George Simmon; computational), and I also have been studying linear algebra (Serge Lang's Introduction to LA) and mathematical proof (Chartrand) book independently and will complete both subjects by mid-May. I was thinking studying Apostol/Spivak during Summer to prepare for Analysis I but I thought it would be best to just enter the analysis with Rudin's PMA and other analysis textbooks since it will be impossible to finish Apostol/Spivak during Summer and that time can be better spent on Rudin's PMA and others. I have enough money to purchase two other analysis textbooks that can complement Rudin's PMA well and help me to learn the analysis. My mind is on Apostol (Mathematical Analysis), Pugh, Ross, Strichartz, Lang, and Abbott, but I am not sure of their contents...unfortunately, those books have been either checked out or in hold at my university's math library. Please give me recommendation of two analysis textbooks that can supplement Rudin's PMA well!

 

[jbunniii]

I think Apostol's Mathematical Analysis is a very good complement to Rudin. It is at a similar level of sophistication but it has more detailed proofs and better coverage of several important topics such as Fourier series.

Of the other ones you listed, I would recommend Pugh, mainly for its excellent, huge collection of challenging problems. His chapter on metric space topology in particular is very good, with over 100 non-trivial exercises, whereas Rudin's treatment of this important topic is rather spare. I don't like Pugh's writing style very much though; I find it too loose and sloppy. But a lot of people like it, so check it out if you can.

The other books, Ross, Strichartz, and Abbott, are all probably too easy for you at this point. Lang's book is nice and it covers a lot of interesting material that is not in Rudin, but I would not recommend it over Apostol or Pugh for your purposes.