# 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

 $d(B_{1},B_{2})=\ln\inf\{\,\lVert T\rVert\cdot\lVert T^{-1}\rVert:T\in GL(B_{1}% ,B_{2})\,\}.$

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 BanachMazurCompactum 2013-03-22 14:55:24 2013-03-22 14:55:24 bbukh (348) bbukh (348) 5 bbukh (348) Definition msc 52A21 msc 46B20 Banach-Mazur metric Banach-Mazur distance