Functional analysis: Difference between revisions

Revision as of 23:37, 16 October 2022

Revision as of 23:38, 16 October 2022

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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 ''A'' be a bounded self-adjoint operator on a Hilbert space ''H''. Then there is a [[measure space]] {{nowrap|(''X'', &Sigma;, &mu;)}} and a real-valued [[ess sup|essentially bounded]] measurable function ''f'' on ''X'' and a unitary operator {{nowrap|''U'':''H'' &rarr; ''L''²_&mu;(''X'')}} such that

:<math> U^* T U = A \;</math>

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:<math> [T \varphi](x) = f(x) \varphi(x). \;</math>

and <math>\|T\| = \|f\|_\infty</math>.</blockquote>

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