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I have completed two courses in real analysis that covered up to chapter 9 in Rudin's Principle of Mathematical Analysis (and one on complex analysis). So if I am interested in continuing on in analysis (real analysis and not complex analysis), what would be a good direction to go from here? What would be a good book to learn from? As my interest is primarily number theory I was wondering if there is a direction in analysis that would be helpful in this aspect.

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Complex analysis is certainly useful when dealing with number theory. –  Fernando Martin Jun 14 '12 at 1:37
    
@FernandoMartin As you can see in the question, I've done some complex analysis and I'm looking to foray into real analysis. –  Eugene Jun 14 '12 at 1:38
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Perhaps learn enough ergodic theory to be able to handle its number-theoretic application. –  André Nicolas Jun 14 '12 at 1:45
    
@AndréNicolas That is a very good suggestion. I didn't think of that. Thanks! –  Eugene Jun 14 '12 at 1:52
    
@AndréNicolas I really like your suggestion on ergodic theory best. Do you mind putting your comment as an answer so that I can accept it? –  Eugene Jun 15 '12 at 2:47
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2 Answers 2

I suggest Folland's Real Analysis: Modern Techniques and Their Applications. It covers all you need to know about measure theory and Lebesgue information, and has chapters on probability, distributions, Fourier analysis, and lots of information about functional analysis.

Other people will recommend "Big Rudin," Royden, and Stein and Shakarchi's series of books, but I find Folland the most clearly written and comprehensive.

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Would the downvoter please say why they don't like this answer? –  Potato Nov 15 '13 at 14:45
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If you are interested in number theory, you really should get further into complex analysis. Analytic number theory uses some ideas from complex analysis like the gamma and zeta functions. A good book for this subject, especially the number theory aspect of complex analysis is Stein and Shakarchi's Complex Analysis.

If you want to go further into Real Analysis, you should continue with measure theory and functional analysis. (I can't remember if the first 9 nine chapter of the first Rudin covers measure theory.) A good book for a first introduction to functional analysis and measure theory would be Rudin's Real and Complex Analysis. For more advance book on functional analysis, I would recommend Brezis Functional Analysis, Sobolev Spaces, and Partial Differential Equation$.

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@DylanMoreland Thanks for letting me know. –  William Jun 14 '12 at 1:53
    
@William Thanks for the complex analytic suggestions but I am already familiar with the gamma and zeta functions. –  Eugene Jun 14 '12 at 1:55
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