Quick description
Functional analysis is the study of infinitedimensional vector spaces. Typically these are spaces of functions such as being the space of functions where
for . These spaces must have a topology, which is often given in terms of a norm such as
The name comes from the notion of functionals which are continuous linear functions where is the space concerned. A major early problem was the identification of ways of representing the functionals of a given infinite dimensional space . For example, the Riesz representation theorem (or perhaps one should say theorems) says that if is the space of continuous functions , then any functional , with the norm on , could be represented in terms of a function of bounded variation with
where the integral is understood in the RiemannStieltjes sense. Related results concern, for example, the functionals on spaces.
Such analysis quickly led to the classification of infinitedimensional spaces: Banach spaces, Hilbert spaces, reflexive spaces, and investigation of their properties.
This is a large area and involves large parts of classical and modern analysis. It is used in a large number of subjects, but especially partial differential equations, geometric analysis, quantum mechanics, and dynamical systems, to give a small sample of relatively applied parts of mathematics that use functional analysis.
Prerequisites
Calculus, basic analysis.
General discussion
Here is a (small) sample of topics:

representation of functionals

dual spaces, topologies

different notions of convergence: strong, weak, weak*

topological vector spaces

operator algebras

spaces of (linear) operators

geometry of Banach spaces

spaces of measures with values in (Banach) spaces (see vector measure)

existence of solutions to operator equations: solve for

fixed point and minimax theorems

Fredholm operators and Fredholm index

compactness of sets in specific spaces

optimization of functions defined on spaces (see Convex and variational analysis)

differential equations on spaces and evolution equations

convex and variational analysis

particular (classes of) spaces: e.g., Sobolev spaces, Orlicz spaces, , ,
Comments
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Sat, 19/06/2010  12:23 — UlagatinHi,
First of all, an excellent site! It's great to have such a compendium of mathematical knowledge online, accessable to all. So a great deal of thanks goes to the creators of this site!
I'm in Year 12 in Australia and I just stumbled upon this page [on functional analysis], and I'd love to see it expanded. It looks to be an interesting field! I assume that capital Omega here is a space/topology, maybe even a set being integrated over? Could an explanation be given of the theory behind L^p(Omega) spaces and how it ties in more broadly to this topic?
I look forward to hearing further (and just as a sidenote, I believe my mathematical background to be good so far  I have a good understanding of convergence of sequences, Maclaurin series, complex numbers, derivatives of inverse trigonometric functions, etc). I have read a little of analysis, to which I have been impressed so far, but that I have only broadly skimmed.
Thanks once again.
Davin