Английская Википедия:Banach–Mazur compactum
Шаблон:Short description Шаблон:Distinguish Шаблон:Cleanup bare URLs In the mathematical study of functional analysis, the Banach–Mazur distance is a way to define a distance on the set <math>Q(n)</math> of <math>n</math>-dimensional normed spaces. With this distance, the set of isometry classes of <math>n</math>-dimensional normed spaces becomes a compact metric space, called the Banach–Mazur compactum.
Definitions
If <math>X</math> and <math>Y</math> are two finite-dimensional normed spaces with the same dimension, let <math>\operatorname{GL}(X, Y)</math> denote the collection of all linear isomorphisms <math>T : X \to Y.</math> Denote by <math>\|T\|</math> the operator norm of such a linear map — the maximum factor by which it "lengthens" vectors. The Banach–Mazur distance between <math>X</math> and <math>Y</math> is defined by <math display="block">\delta(X, Y) = \log \Bigl( \inf \left\{ \left\|T\right\| \left\|T^{-1}\right\| : T \in \operatorname{GL}(X, Y) \right\} \Bigr).</math>
We have <math>\delta(X, Y) = 0</math> if and only if the spaces <math>X</math> and <math>Y</math> are isometrically isomorphic. Equipped with the metric δ, the space of isometry classes of <math>n</math>-dimensional normed spaces becomes a compact metric space, called the Banach–Mazur compactum.
Many authors prefer to work with the multiplicative Banach–Mazur distance <math display="block">d(X, Y) := \mathrm{e}^{\delta(X, Y)} = \inf \left\{ \left\|T\right\| \left\|T^{-1}\right\| : T \in \operatorname{GL}(X, Y) \right\},</math> for which <math>d(X, Z) \leq d(X, Y) \, d(Y, Z)</math> and <math>d(X, X) = 1.</math>
Properties
F. John's theorem on the maximal ellipsoid contained in a convex body gives the estimate:
- <math>d(X, \ell_n^2) \le \sqrt{n}, \,</math> [1]
where <math>\ell_n^2</math> denotes <math>\R^n</math> with the Euclidean norm (see the article on <math>L^p</math> spaces). From this it follows that <math>d(X, Y) \leq n</math> for all <math>X, Y \in Q(n).</math> However, for the classical spaces, this upper bound for the diameter of <math>Q(n)</math> is far from being approached. For example, the distance between <math>\ell_n^1</math> and <math>\ell_n^{\infty}</math> is (only) of order <math>n^{1/2}</math> (up to a multiplicative constant independent from the dimension <math>n</math>).
A major achievement in the direction of estimating the diameter of <math>Q(n)</math> is due to E. Gluskin, who proved in 1981 that the (multiplicative) diameter of the Banach–Mazur compactum is bounded below by <math>c\,n,</math> for some universal <math>c > 0.</math>
Gluskin's method introduces a class of random symmetric polytopes <math>P(\omega)</math> in <math>\R^n,</math> and the normed spaces <math>X(\omega)</math> having <math>P(\omega)</math> as unit ball (the vector space is <math>\R^n</math> and the norm is the gauge of <math>P(\omega)</math>). The proof consists in showing that the required estimate is true with large probability for two independent copies of the normed space <math>X(\omega).</math>
<math>Q(2)</math> is an absolute extensor.[2] On the other hand, <math>Q(2)</math>is not homeomorphic to a Hilbert cube.
See also
Notes
References
- Шаблон:Springer
- Шаблон:Cite journal
- Шаблон:Cite book
- https://planetmath.org/BanachMazurCompactum
- A note on the Banach-Mazur distance to the cube
- The Banach-Mazur compactum is the Alexandroff compactification of a Hilbert cube manifold
Шаблон:Banach spaces Шаблон:Functional analysis Шаблон:Topological vector spaces
