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