Let G be a Lie group and let ρ be a group representation of G on C^n (for some natural number n), which is continuous in the sense that the function G×C^n->C^n defined by (g, v)↦ρ(g)(v) is continuous. Then for each v element C^n and each α element (C^n)^*, the function G->C defined by g↦α(ρ(g)(v)) is continuous. The vector space span of all such functions is called the space of representative functions. The Peter-Weyl theorem says that, if G is compact, then 1.

Lie group | matrix group | norm | Stone-Weierstrass theorem | supremum norm