Well, it seems as though I just failed my analysis prelim for the second time... I have one more try in about $5$ months.
I'm failing to build up a framework for how to think about analysis problems. When I am confronted with an exam-type problem in, for example, algebraic topology, I can usually immediately see the ideas and theorems that may be relevant, why they are connected, and how to go between them. (In part this is due to the phenomenal professor who taught the subject!) Too bad my department doesn't offer a topology exam as an alternative... Real analysis, on the other hand, appears to me as a jumbled toolbox with no clear common purpose or function, and my studying is reduced to pure memorization, no matter how many exercises and practice exams I solve.
The textbook my department uses is Hunter and Nachtergaele's Applied Analysis, and the exam covers Banach spaces, Hilbert spaces, linear operators on these spaces, spectral theory (of operators), Fourier analysis, distributions, Sobolev embedding theorems... and of course the various connections between these things, which is what I'm really struggling to figure out how to learn.
So, what, if any, textbooks, would you recommend to a struggling graduate student? I am looking for a book that is more than just a list of theorems and definitions, which is essentially what our book is, if such a book exists.
