Peter-Weyhl Theorem

The Peter-Weyl Theorem states that:

Let G be a compact topological group. For any G-invariant subspace V in L2(G), where G acts on the left, we consider the image of G in GL(V). It is closed, since G is compact, and a subgroup of the Lie group GL(V). It follows by a basic theorem (of Élie Cartan) that the image of G is also a Lie group. From this follows that:

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 L^2(G) of all square-integrable functions, with respect to a Haar measure on G;
3. The vector space span of the characters of the irreducible continuous representations of G are dense in the space of all continuous functions from G into C which are constant on each conjugacy class of G, with respect to the supremum norm.

References

Peter-Weyl Theorem from Wolfram MathWorld