# Hilbert space

A *Hilbert space ^{}* is an inner product space

^{}which is http://planetmath.org/node/603complete

^{}under the metric.

In particular, a Hilbert space is a Banach space^{} in the norm by the inner product^{}, since the norm and the inner product both induce the same metric. Any finite-dimensional inner product space is a Hilbert space, but it is worth mentioning that some authors require the space to be infinite dimensional for it to be called a Hilbert space.

Title | Hilbert space |

Canonical name | HilbertSpace |

Date of creation | 2013-03-22 12:19:06 |

Last modified on | 2013-03-22 12:19:06 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 11 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 46C05 |

Related topic | InnerProductSpace |

Related topic | HilbertModule |

Related topic | QuadraticFunctionAssociatedWithALinearFunctional |

Related topic | VectorNorm |

Related topic | RieszSequence |

Related topic | VonNeumannAlgebra |

Related topic | HilbertSpacesAndQuantumGroupsVonNeumannAlgebras |

Related topic | L2SpacesAreHilbertSpaces |

Related topic | QuantumGroupsAndVonNeumannAlgebras |

Related topic | HAlgebra |

Related topic | Ries |