Editing Functional analysis (section)

===Spectral theorem===
{{main|Spectral theorem}}
There are many theorems known as the [[spectral theorem]], but one in particular has many applications in functional analysis.

{{math theorem | name = Spectral theorem<ref>{{Cite book|last=Hall|first=Brian C.|url={{google books |plainurl=y |id=bYJDAAAAQBAJ|page=147}}|title=Quantum Theory for Mathematicians|date=2013-06-19|publisher=[[Springer Science & Business Media]]|isbn=978-1-4614-7116-5|page=147|language=en}}</ref>
|math_statement = Let <math>A</math>  be a bounded self-adjoint operator on a Hilbert space <math>H</math>. Then there is a [[measure space]] <math>(X,\Sigma,\mu)</math> and a real-valued [[ess sup|essentially bounded]] measurable function <math>f</math> on <math>X</math> and a unitary operator <math>U:H\to L^2_\mu(X)</math> such that
<math display="block"> U^* T U = A </math>
where ''T'' is the [[multiplication operator]]:
<math display="block"> [T \varphi](x) = f(x) \varphi(x). </math>
and <math>\|T\| = \|f\|_\infty</math>.}}

This is the beginning of the vast research area of functional analysis called [[operator theory]]; see also the [[spectral measure#Spectral measure|spectral measure]].

There is also an analogous spectral theorem for bounded [[normal operator]]s on Hilbert spaces. The only difference in the conclusion is that now <math>f</math> may be complex-valued.