# $Hilb$ category of Hilbert spaces

###### Definition 0.1.

The *category ^{} $\mathrm{H}\mathit{}i\mathit{}l\mathit{}{b}_{f}$ of finite-dimensional Hilbert spaces^{}* is defined as the category whose objects are all finite-dimensional Hilbert spaces ${\mathscr{H}}_{f}$, and whose morphisms are linear maps between ${\mathscr{H}}_{f}$ spaces.
The

*isomorphisms*in $\mathscr{H}il{b}_{f}$ are all isometric isomorphisms.

^{}Furthermore, one also has the following, general definition for any Hilbert space.

###### Definition 0.2.

The *category $\mathrm{H}\mathit{}i\mathit{}l\mathit{}b$ of Hilbert spaces* is defined as the category whose objects are all Hilbert spaces $\mathscr{H}$, and whose morphisms are linear maps between $\mathscr{H}$ spaces.
The *isomorphisms* in $\mathscr{H}ilb$ are all isometric isomorphisms.

###### Remark 0.1.

The category of $\mathscr{H}ilb$ Hilbert spaces has direct sums^{} and is a Cartesian category.

Title | $Hilb$ category of Hilbert spaces |
---|---|

Canonical name | HilbCategoryOfHilbertSpaces |

Date of creation | 2013-03-22 18:25:10 |

Last modified on | 2013-03-22 18:25:10 |

Owner | bci1 (20947) |

Last modified by | bci1 (20947) |

Numerical id | 10 |

Author | bci1 (20947) |

Entry type | Definition |

Classification | msc 46K15 |

Classification | msc 46C05 |

Classification | msc 46C50 |

Classification | msc 46C15 |

Classification | msc 46E20 |

Classification | msc 18-00 |

Synonym | $Hilb$ |

Related topic | DirectSumOfHilbertSpaces |

Related topic | ClassificationOfHilbertSpaces |

Related topic | IndexOfCategories |

Defines | isomorphisms in $Hilb$ |

Defines | Hilbert space morphisms |