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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:

Comments

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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