Editing Peter–Weyl theorem (section)

===Restriction to class functions===
A function <math>f</math> on ''G'' is called a ''class function'' if <math>f(hgh^{-1})=f(g)</math> for all <math>g</math> and <math>h</math> in ''G''. The space of square-integrable class functions forms a closed subspace of <math>L^2(G)</math>, and therefore a Hilbert space in its own right. Within the space of matrix coefficients for a fixed representation <math>\pi</math> is the [[Character (mathematics)|character]] <math>\chi_\pi</math> of <math>\pi</math>, defined by
:<math>\chi_\pi(g)=\operatorname{trace}(\pi(g)).</math>
In the notation above, the character is the sum of the diagonal matrix coefficients:
:<math>\chi_\pi=\sum_{i=1}^{d^{(\pi)}}u_{ii}^{(\pi)}.</math>
An important consequence of the preceding result is the following:
:'''Theorem''': The characters of the irreducible representations of ''G'' form a Hilbert basis for the space of square-integrable class functions on ''G''.
This result plays an important part in Weyl's classification of the [[Compact group#Representation theory of a connected compact Lie group|representations of a connected compact Lie group]].<ref>{{harvnb|Hall|2015}} Chapter 12</ref>