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The short version of this question is this:

I like functional analysis and want to learn more. I've taken a class on it and I've read the books by Brezis and Conway. Where can I go from here? Do you have recommendations on what books might be good for someone looking to go more deeply into the subject? Pointers to related fields that you have found interesting along with an explanation of why are also appreciated.

Here is some additional background to the question:

  • I came from a PDE background but slowly realized that I actually liked the functional analysis tools more than the PDEs themselves. So I'm not necessarily looking for PDE books applying the ideas.
  • I liked the operator theory that came in the second half of the Conway book. I'm sure there have to be some books out there pursuing these topics further.
  • I'm not necessarily looking for books that are "more advanced" or contain only "futher topics." Since there does not seem to be one encyclopedic treatment of functional analysis, I am also interested in other introductory texts that emphasize a different collection of topics or stress noticeably different viewpoints than the ones I've mentioned.
  • The non-linear stuff looks cool too. However, after casually browsing around a little (eg. flipping through Deimling) I realized that I don't seem to have a very good understanding of the prerequisites needed here. So any introductory texts in this area or books containing the prerequisites for proper study are good answers as well.
  • Not sure it its helpful or necessary to mention, but my analysis background more or less culminated in the study of Real and Complex Analysis - Rudin, the first half or so of Complex Analysis - Stein & Shakarchi and some PDE books along the lines of Evans. So it would be nice if your answers took this into account or added additional references for topics that one should be familiar with in order to be capable of proceeding to your recommendation.

Basically, I'm just looking for an overview of what other topics are out there what resources might be helpful in getting started. I also realize the question is a bit unmanageable as it is, so any suggestions on what restrictions or further information might be helpful are much appreciated. Thank you for all your thoughts and inputs in advance.

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I didn't read the books you mention and so my recommendation comes qualified, but I found Luenberger's Optimization by Vector Space Methods (http://www.amazon.com/Optimization-Vector-Space-Methods-Professional/dp/047118117X) very enjoyable. You certainly have the necessary background from what you mention you have read (Rudin alone is more than enough). Its obvious focus is optimization and control theory, a lot is done by projection in general linear spaces, so you'd need to decide if that interests you. In that field, it's a classic; and I like Luenberger's style quite a bit. I should add that its excellent for self-study, which, it appears, you plan on.

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