PlanetMath: Banach-Mazur compactum

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Banach-Mazur compactum

(Definition)

The Banach-Mazur metric is a distance on the space of all isomorphic Banach spaces. If are -dimensional Banach spaces, the distance between them is

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

Then

satisfies the triangle inequality, and

if and only if

and

are isometric. The space of isometry classes of

-dimensional Banach spaces under this metric is a compact metric space, known as a Banach-Mazur compactum.

"Banach-Mazur compactum" is owned by bbukh.

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Also defines:

Banach-Mazur metric, Banach-Mazur distance

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Cross-references: metric space, compact, metric, isometry, isometric, triangle inequality, Banach spaces

This is version 2 of Banach-Mazur compactum, born on 2005-01-02, modified 2005-01-03.
Object id is 6611, canonical name is BanachMazurCompactum.
Accessed 2894 times total.

Classification:

AMS MSC:	52A21 (Convex and discrete geometry :: General convexity :: Finite-dimensional Banach spaces )
	46B20 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Geometry and structure of normed linear spaces)

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