irreducible unitary representations of compact groups are finite-dimensional

Theorem - If $\pi \in rep(G,H)$ is a unitary representation of a compact topological group $G$ in a Hilbert space $H$, then $\pi$ has a finite-dimensional subrepresentation (http://planetmath.org/TopologicalGroupRepresentation).

$$

Corollary 1 - If $\pi$ is irreducible (http://planetmath.org/TopologicalGroupRepresentation), then $H$ must be finite-dimensional.

$$

Corollary 2 - $\pi$ has an .