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Banach-Mazur compactum
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(Definition)
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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 \begin{equation*} d(B_1,B_2)=\ln\inf \{\,\norm{T}\cdot\norm{T^{-1}} : T\in GL(B_1,B_2)\,\}. \end{equation*}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.
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"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 4359 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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Pending Errata and Addenda
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