# Banach-Mazur compactum

The *Banach-Mazur metric* is a distance on the space of all
http://planetmath.org/node/Isomorphism2isomorphic Banach spaces^{}. If ${B}_{1},{B}_{2}$ are $n$-dimensional Banach
spaces, the distance between them is

$$d({B}_{1},{B}_{2})=\mathrm{ln}inf\{\parallel T\parallel \cdot \parallel {T}^{-1}\parallel :T\in GL({B}_{1},{B}_{2})\}.$$ |

Then $d$ satisfies the triangle inequality^{}, and $d({B}_{1},{B}_{2})=0$ if
and only if ${B}_{1}$ and ${B}_{2}$ are isometric. The space of isometry
http://planetmath.org/node/EquivalenceRelationclasses of $n$-dimensional Banach spaces under this metric is a compact^{} metric space, known as a *Banach-Mazur compactum*.

Title | Banach-Mazur compactum |
---|---|

Canonical name | BanachMazurCompactum |

Date of creation | 2013-03-22 14:55:24 |

Last modified on | 2013-03-22 14:55:24 |

Owner | bbukh (348) |

Last modified by | bbukh (348) |

Numerical id | 5 |

Author | bbukh (348) |

Entry type | Definition |

Classification | msc 52A21 |

Classification | msc 46B20 |

Defines | Banach-Mazur metric |

Defines | Banach-Mazur distance |