irreducible unitary representations of compact groups are finite-dimensional
Theorem - If is a unitary representation of a compact topological group in a Hilbert space
, then has a finite-dimensional subrepresentation
(http://planetmath.org/TopologicalGroupRepresentation).
Corollary 1 - If is irreducible (http://planetmath.org/TopologicalGroupRepresentation), then must be finite-dimensional.
Corollary 2 - has an .