Revision of Functional analysis front page from Mon, 22/06/2009 - 14:43

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
dual spaces, topologies
different notions of convergence: strong, weak, weak*
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
Fredholm operators and Fredholm index
compactness of sets in specific spaces
optimization of functions defined on spaces
differential equations on spaces and evolution equations
convex and variational analysis
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 - 12:23 — 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