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).

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

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Corollary 2 - $\pi$ has an .

 Title irreducible unitary representations of compact groups are finite-dimensional Canonical name IrreducibleUnitaryRepresentationsOfCompactGroupsAreFinitedimensional Date of creation 2013-03-22 18:02:44 Last modified on 2013-03-22 18:02:44 Owner asteroid (17536) Last modified by asteroid (17536) Numerical id 13 Author asteroid (17536) Entry type Theorem Classification msc 43A65 Classification msc 22C05 Classification msc 22A25 Synonym unitary representation of a compact group has a finite-dimensional subrepresentation Related topic UnitaryRepresentation Defines unitary representation of compact group has an irreducible subrepresentation Defines unitary group of a complex Hilbert space