Banach-Mazur compactum

The Banach-Mazur metric is a distance on the space of allhttp://planetmath.org/node/Isomorphism2isomorphic Banach spaces. If $B_1,B_2$ are $n$-dimensional Banachspaces, 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 $d(B_1,B_2)=0$ ifand only if $B_1$ and $B_2$ are isometric. The space of isometryhttp://planetmath.org/node/EquivalenceRelationclasses of $n$-dimensional Banach spaces under this metric is a compact metric space, known as a Banach-Mazur compactum.