In the mathematical study of functional analysis, the BanachâMazur distance is a way to define a distance on the set of -dimensional normed spaces. With this distance, the set of isometry classes of -dimensional normed spaces becomes a compact metric space, called the BanachâMazur compactum.
If and are two finite-dimensional normed spaces with the same dimension, let denote the collection of all linear isomorphisms Denote by the operator norm of such a linear map — the maximum factor by which it "lengthens" vectors. The BanachâMazur distance between and is defined by
We have if and only if the spaces and are isometrically isomorphic. Equipped with the metric ÎŽ, the space of isometry classes of -dimensional normed spaces becomes a compact metric space, called the BanachâMazur compactum.
Many authors prefer to work with the multiplicative BanachâMazur distance for which and
F. John's theorem on the maximal ellipsoid contained in a convex body gives the estimate:
where denotes with the Euclidean norm (see the article on spaces).
From this it follows that for all However, for the classical spaces, this upper bound for the diameter of is far from being approached. For example, the distance between and is (only) of order (up to a multiplicative constant independent from the dimension ).
A major achievement in the direction of estimating the diameter of is due to E. Gluskin, who proved in 1981 that the (multiplicative) diameter of the BanachâMazur compactum is bounded below by for some universal
Gluskin's method introduces a class of random symmetric polytopes in and the normed spaces having as unit ball (the vector space is and the norm is the gauge of ). The proof consists in showing that the required estimate is true with large probability for two independent copies of the normed space
is an absolute extensor. [2] On the other hand, is not homeomorphic to a Hilbert cube.
In the mathematical study of functional analysis, the BanachâMazur distance is a way to define a distance on the set of -dimensional normed spaces. With this distance, the set of isometry classes of -dimensional normed spaces becomes a compact metric space, called the BanachâMazur compactum.
If and are two finite-dimensional normed spaces with the same dimension, let denote the collection of all linear isomorphisms Denote by the operator norm of such a linear map — the maximum factor by which it "lengthens" vectors. The BanachâMazur distance between and is defined by
We have if and only if the spaces and are isometrically isomorphic. Equipped with the metric ÎŽ, the space of isometry classes of -dimensional normed spaces becomes a compact metric space, called the BanachâMazur compactum.
Many authors prefer to work with the multiplicative BanachâMazur distance for which and
F. John's theorem on the maximal ellipsoid contained in a convex body gives the estimate:
where denotes with the Euclidean norm (see the article on spaces).
From this it follows that for all However, for the classical spaces, this upper bound for the diameter of is far from being approached. For example, the distance between and is (only) of order (up to a multiplicative constant independent from the dimension ).
A major achievement in the direction of estimating the diameter of is due to E. Gluskin, who proved in 1981 that the (multiplicative) diameter of the BanachâMazur compactum is bounded below by for some universal
Gluskin's method introduces a class of random symmetric polytopes in and the normed spaces having as unit ball (the vector space is and the norm is the gauge of ). The proof consists in showing that the required estimate is true with large probability for two independent copies of the normed space
is an absolute extensor. [2] On the other hand, is not homeomorphic to a Hilbert cube.