Editing Peter–Weyl theorem (section)

{{More footnotes|date=March 2024}}
{{short description|Basic result in harmonic analysis on compact topological groups}}
In [[mathematics]], the '''Peter–Weyl theorem''' is a basic result in the theory of [[harmonic analysis]], applying to [[topological group]]s that are [[Compact group|compact]], but are not necessarily [[Abelian group|abelian]]. It was initially proved by [[Hermann Weyl]], with his student [[Fritz Peter]], in the setting of a compact [[topological group]] ''G'' {{harv|Peter|Weyl|1927}}. The theorem is a collection of results generalizing the significant facts about the decomposition of the [[regular representation]] of any [[finite group]], as discovered by [[Ferdinand Georg Frobenius]] and [[Issai Schur]].

Let ''G'' be a compact group. The theorem has three parts.  The first part states that the matrix coefficients of [[irreducible representation]]s of ''G'' are dense in the space ''C''(''G'') of continuous [[complex-valued function]]s on ''G'', and thus also in the space ''L''<sup>2</sup>(''G'') of [[square-integrable function]]s.  The second part asserts the complete reducibility of [[unitary representation]]s of ''G''.  The third part then asserts that the regular representation of ''G'' on ''L''<sup>2</sup>(''G'') decomposes as the direct sum of all irreducible unitary representations.  Moreover, the matrix coefficients of the irreducible unitary representations form an [[orthonormal basis]] of ''L''<sup>2</sup>(''G''). In the case that ''G'' is the group of unit complex numbers, this last result is simply a standard result from [[Fourier series]].