Editing Peter–Weyl theorem (section)

==Matrix coefficients==
A '''[[matrix coefficient]]''' of the group ''G'' is a complex-valued function <math>\varphi</math> on ''G'' given as the composition

:<math>\varphi = L\circ \pi</math>

where π&nbsp;:&nbsp;''G''&nbsp;→&nbsp;GL(''V'') is a finite-dimensional ([[continuous function|continuous]]) [[group representation]] of ''G'', and ''L'' is a [[linear functional]] on the vector space of [[endomorphism]]s of ''V'' (e.g. trace), which contains GL(''V'') as an open subset.  Matrix coefficients are continuous, since representations are by definition continuous, and linear functionals on finite-dimensional spaces are also continuous.

The first part of the Peter–Weyl theorem asserts ({{harvnb|Bump|2004|loc=§4.1}}; {{harvnb|Knapp|1986|loc=Theorem 1.12}}):

<blockquote>'''Peter–Weyl Theorem (Part I).''' The set of matrix coefficients of ''G'' is [[dense set|dense]] in the space of [[continuous functions on a compact Hausdorff space|continuous complex functions]] C(''G'') on ''G'', equipped with the [[uniform norm]].</blockquote>

This first result resembles the [[Stone–Weierstrass theorem]] in that it indicates the density of a set of functions in the space of all continuous functions, subject only to an ''algebraic'' characterization.  In fact, the matrix coefficients form a unital algebra invariant under complex conjugation because the product of two matrix coefficients is a matrix coefficient of the tensor product representation, and the complex conjugate is a matrix coefficient of the dual representation. Hence the theorem follows directly from the Stone–Weierstrass theorem if the matrix coefficients separate points, which is obvious if ''G'' is a [[matrix group]] {{harv|Knapp|1986|p=17}}. Conversely, it is a consequence of the theorem that any compact [[Lie group]] is isomorphic to a matrix group {{harv|Knapp|1986|loc=Theorem 1.15}}.

A corollary of this result is that the matrix coefficients of ''G'' are dense in ''L''<sup>2</sup>(''G'').