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.
