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

$$d(B_1,B_2)=\ln inf\{\parallel T\parallel \cdot \parallel T^{-1}\parallel :T\in GL(B_1,B_2)\}.$$

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