irreducible unitary representations of compact groups are finite-dimensionalIrreducibleUnitaryRepresentationsOfCompactGroupsAreFiniteDimensional2008-05-07 13:06:292008-11-08 19:56:36Theoremunitary representation of compact group has an irreducible subrepresentationunitary group of a complex Hilbert spacenitary representation of compact groupirreducible subrepresentationunitary group of a complex Hilbert space% this is the default PlanetMath preamble. as your knowledge
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{\bf 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 \PMlinkname{subrepresentation}{TopologicalGroupRepresentation}.
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{\bf Corollary 1 -} If $\pi$ is \PMlinkname{irreducible}{TopologicalGroupRepresentation}, then $H$ must be finite-dimensional.
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{\bf Corollary 2 -} $\pi$ has an \PMlinkescapetext{irreducible} \PMlinkescapetext{subrepresentation}.