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I'm having a few months of free time, and I decided to do a self-study on functional analysis (in-depth) in the meantime.

I'm aware that functional analysis requires a good deal of foundation from real analysis and linear algebra. How much of them is exactly needed? I've taken courses on analysis and linear algebra which cover Axler's Linear Algebra Done Right and the first 7 chapters of Rudin. Would that be enough?

Also, can you recommend me some books to study functional analysis thoroughly?

Thanks in advance.

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I read that thread already, but apparently the asker was looking for introductory books, whereas I'm looking for more in-depth books. –  breikj Apr 8 '12 at 11:35
    
The books recommended there are amply enough to fill four months full time :) –  t.b. Apr 8 '12 at 11:45
    
Seriously: those books are at the level you say you have. Whether you choose Brézis or Conway, Pedersen or Reed and Simon, all of them will provide much more than what even a very talented student can work through in a few months. Depth is in each of them. Of course: you should do the exercises, too. –  t.b. Apr 8 '12 at 11:56
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I'm really not sure. What are you interested in? If it are partial differential equations and harmonic analysis I'd say you could take a look at the Banach and Hilbert space stuff in e.g. Conway and then look at semigroups (Engel and Nagel for example, or Pazy, but the first is better for these types of applications I think). If you want to study functional analysis in its own right, then well then you could try some operator theory. For example "Banach algebra techniques in operator theory" by Douglas. –  Jonas Teuwen Apr 8 '12 at 12:28

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