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There are many theorems known as the [[spectral theorem]], but one in particular has many applications in functional analysis.
<blockquote>'''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> 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^* T U = A \;</math>
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