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irreducible unitary representations of compact groups are finite-dimensional (Theorem)

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.

$ \,$

Corollary 1 - If $ \pi$ is irreducible, then $ H$ must be finite-dimensional.

$ \,$

Corollary 2 - $ \pi$ has an irreducible subrepresentation.



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Other names:  unitary representation of a compact group has a finite-dimensional subrepresentation
Also defines:  unitary representation of compact group has an irreducible subrepresentation
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Cross-references: finite-dimensional, Hilbert space, topological group, compact, unitary representation

This is version 3 of irreducible unitary representations of compact groups are finite-dimensional, born on 2008-05-07, modified 2008-05-07.
Object id is 10569, canonical name is IrreducibleUnitaryRepresentationsOfCompactGroupsAreFiniteDimensional.
Accessed 73 times total.

Classification:
AMS MSC22A25 (Topological groups, Lie groups :: Topological and differentiable algebraic systems :: Representations of general topological groups and semigroups)
 22C05 (Topological groups, Lie groups :: Compact groups)
 43A65 (Abstract harmonic analysis :: Representations of groups, semigroups, etc.)

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