Functional analysis front page

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

$\int_\Omega |f(x)|^p \, dx < +\infty$

for $1\leq p<\infty$ . These spaces must have a topology, which is often given in terms of a norm such as

$\|f\|_{L^p} = \left[\int_\Omega |f(x)|^p \, dx\right]^{1/p}.$

The name comes from the notion of functionals which are continuous linear functions $X\to\R$ 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 Riesz representation theorem (or perhaps one should say theorems) says that if $C[a,b]$ is the space of continuous functions $[a,b]\to\R$ , then any functional $C[a,b]\to\R$ , 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

$\psi(f) = \int_a^b f(x)\,dm(x),$

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 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
interpolation spaces
particular (classes of) spaces: e.g., Sobolev spaces, Orlicz spaces, $\ell^p$ , $L^p$ , $c_0$

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Sat, 19/06/2010 - 11:23 — Ulagatin

Hi,

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 side-note, 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