Let be a Lie group and let be a group representation of on (for some natural number ), which is continuous in the sense that the function defined by is continuous. Then for each and each , the function defined by is continuous. The vector space span of all such functions is called the space of representative functions.
The Peter-Weyl theorem says that, if is compact, then
1. The representative functions are dense in the space of all continuous functions, with respect to the supremum norm;
2. The representative functions are dense in the space of all square-integrable functions, with respect to a Haar measure on ;
3. The vector space span of the characters of the irreducible continuous representations of are dense in the space of all continuous functions from into which are constant on each conjugacy class of , with respect to the supremum norm.
This theorem is easy to deduce from the Stone-Weierstrass theorem if it is assumed that is a matrix group. On the other hand, it is a corollary of the Peter-Weyl theorem that every compact Lie group is isomorphic to some matrix group.