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# Banach-Mazur compactum

The *Banach-Mazur metric* is a distance on the space of all
isomorphic 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
classes of $n$-dimensional Banach spaces under this metric is a compact metric space, known as a *Banach-Mazur compactum*.

Defines:

Banach-Mazur metric, Banach-Mazur distance

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

52A21*no label found*46B20

*no label found*

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