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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    $\begingroup$ math.stackexchange.com/questions/7512/… $\endgroup$ – t.b. Apr 8 '12 at 11:04
  • $\begingroup$ I read that thread already, but apparently the asker was looking for introductory books, whereas I'm looking for more in-depth books. $\endgroup$ – breikj Apr 8 '12 at 11:35
  • $\begingroup$ The books recommended there are amply enough to fill four months full time :) $\endgroup$ – t.b. Apr 8 '12 at 11:45
  • $\begingroup$ 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. $\endgroup$ – t.b. Apr 8 '12 at 11:56
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    $\begingroup$ 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. $\endgroup$ – Jonas Teuwen Apr 8 '12 at 12:28

This question is very old, but I'll write an answer anyway for reference for future readers.

Functional analysis is in some sense the "good" infinite-dimensional analogue of linear algebra that you need to do analysis. Namely, if you study functional analysis you will mainly be confronted with various spaces of functions on some topological spaces (classically, open subsets of $\mathbb{R}^n$).

In order to be able to study functional analysis, you will need knowledge of

  1. Linear algebra: while this is maybe not so fundamental for the subject, it is very important to have strong bases of linear algebra in order to understand the intuition behind many objects and proofs.
  2. (Real) analysis: you will be studying spaces of functions with various properties. In particular, you will need to be familiar with the concepts of continuity, differentiability, smoothness, integration and maybe most importantly Cauchy sequences and convergence of sequences and series.
  3. Basic topology: you will be working on metric spaces, so some basic notions of topology are advised.

Moreover, if you want to go deeper into the study of the subject, you will need many more knowledge of other subjects, such as differential geometry (if you want to do PDEs on manifolds) and others.

As for the references:

  • Personally, I learned functional analysis on these lecture notes by M. Struwe (they are in German, though).
  • Another interesting reference is these notes by Einsiedler and Ward (I haven't read them, but I have been told they are very good; however, I've also heard that they go in some "non-standard" directions and applications).
  • Finally, if you are more of a PDEs person, Evans' book is a classic you must read.

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