# 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 |