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My current background in analysis is approximately the material in Folland's Real Analysis. I've also read the Analysis text by Lieb and Loss and I also took a graduate level class on complex analysis, which went up to Big Picard and some Nevanlinna theory. For my own amusement I've thought about furthering my knowledge of general analysis.

I've heard wonderful things about Stein's book on Singular integrals and his Fourier analysis on Euclidean spaces. Would these be an interesting next step? I'm especially interested in learning more about harmonic analysis and especially learning enough to understand the modern language of these fields.

EDIT: Here's maybe a more concise way of phrasing this questions: What's the core knowledge that every graduate student in analysis, regardless of specialization, at a top school is expected to know? What would be a reading list?

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  • $\begingroup$ If you've mastered Folland and Lieb and Loss (these are both much richer than a first course in them would give you), there really isn't much more analysis you need to learn before you can come to grips with research subjects. Stein's books are excellent, as are any number of others (some functional analysis would be useful), but you already have the foundation you need to tackle research problems. From hereon out, learn for fun or as needed for your research, not because you "have" to know it. $\endgroup$ – Ray Yang Apr 16 '13 at 21:37
  • $\begingroup$ I'm just a hobbyist mathematician at this point (I do R&D for a software company). Thus, I'm not really looking at doing research in analysis, just looking at cultivating my knowledge of it. $\endgroup$ – tvk Apr 17 '13 at 7:06
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Graduate courses in my univ. in analysis included 2 semester sequence of measure theory covering real analysis book by Royden, and another 2 semester sequence of Functional analysis covering Reed and Simon part 1.

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