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[QUICK DESCRIPTION]
Functional analysis is the study of infinite-dimensional vector spaces.  Typically these are spaces of functions such as $L^p(\Omega)$ being the space of functions $f$ where
[maths]
\int_\Omega |f(x)|^p \, dx < +\infty
[/maths]
for $1\leq p<\infty$.
These spaces must have a topology, which is often given in terms of a norm such as 
[maths]
\|f\|_{L^p} = \left[\int_\Omega |f(x)|^p \, dx\right]^{1/p}.
[/maths]

The name comes from the notion of functionals which are continuous linear functions $\psi:X\to\R$ where $X$ is the space concerned.  A major early problem was the identification of ways of representing the functionals of a given infinite dimensional space $X$.  For example, the [[w:Riesz representation theorem|Riesz representation theorem]] (or perhaps one should say theorems) says that if $C[a,b]$ is the space of continuous functions $f:[a,b]\to\R$, then any functional $\psi:C[a,b]\to\R$, with the norm $\|f\|=\max_{a\leq x\leq b}|f(x)|$ on $C[a,b]$, could be represented in terms of a function $m$ of bounded variation with
[maths]
\psi(f) = \int_a^b f(x)\,dm(x),
[/maths]
where the integral is understood in the Riemann-Stieltjes sense.  Related results concern, for example, the functionals on $L^p(\Omega)$ spaces.

Such analysis quickly led to the classification of infinite-dimensional 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 
** [[Know the Riesz representation theorems!]]
* dual spaces, topologies
** [[Measures on $A$ are in the dual of $C(A)$]]
** [[Know your dual spaces]]
* different notions of convergence: strong, weak, weak* 
** [[Use weak or weak* convergence whenever you can get away with it]]
* topological vector spaces 
* operator algebras
* spaces of (linear) operators $X\to Y$ 
* geometry of Banach spaces
* spaces of measures with values in (Banach) spaces (see [[w:Vector measure|vector measure]])
* existence of solutions to operator equations: solve $Ax=b$ for $x$
* fixed point and minimax theorems 
** [[Look for monotone operators]]
** [[Check compactness in non-obvious spaces]]
* Fredholm operators and Fredholm index 
** [[To compute Fredholm index, use homotopies]]
* compactness of sets in specific spaces
** [[Simon's and Seidman's theorems]]
* optimization of functions defined on spaces (see Convex and variational analysis)
* differential equations on spaces and evolution equations
** [[Use pseudomonotone operators]]
* convex and variational analysis
** [[If in doubt, use the separating hyperplane theorem]]
** [[The Fitzpatrick function is very useful for maximal monotone operators]]
* [[w:Complex interpolation|interpolation spaces]]
* particular (classes of) spaces: e.g., [[w:Sobolev spaces|Sobolev spaces]], Orlicz spaces, $\ell^p$, $L^p$, $c_0$

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