Concept in functional analysis
In the mathematical study of functional analysis, the Banach–Mazur 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 Banach–Mazur compactum.
Definitions
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 Banach–Mazur distance between
and
is defined by![{\displaystyle \delta (X,Y)=\log {\Bigl (}\inf \left\{\left\|T\right\|\left\|T^{-1}\right\|:T\in \operatorname {GL} (X,Y)\right\}{\Bigr )}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
We have
if and only if the spaces
and
are isometrically isomorphic. Equipped with the metric δ, the space of isometry classes of
-dimensional normed spaces becomes a compact metric space, called the Banach–Mazur compactum.
Many authors prefer to work with the multiplicative Banach–Mazur distance
for which
and ![{\displaystyle d(X,X)=1.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Properties
F. John's theorem on the maximal ellipsoid contained in a convex body gives the estimate:
[1]
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 Banach–Mazur compactum is bounded below by
for some universal ![{\displaystyle c>0.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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 ![{\displaystyle X(\omega ).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
is an absolute extensor.[2] On the other hand,
is not homeomorphic to a Hilbert cube.
See also
Notes
- ^ Cube
- ^ "The Banach–Mazur compactum is not homeomorphic to the Hilbert cube" (PDF). www.iop.org.
References
- Giannopoulos, A.A. (2001) [1994], "Banach–Mazur compactum", Encyclopedia of Mathematics, EMS Press
- Gluskin, Efim D. (1981). "The diameter of the Minkowski compactum is roughly equal to n (in Russian)". Funktsional. Anal. I Prilozhen. 15 (1): 72–73. doi:10.1007/BF01082381. MR 0609798. S2CID 123649549.
- Tomczak-Jaegermann, Nicole (1989). Banach-Mazur distances and finite-dimensional operator ideals. Pitman Monographs and Surveys in Pure and Applied Mathematics 38. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York. pp. xii+395. ISBN 0-582-01374-7. MR 0993774.
- Banach-Mazur compactum
- A note on the Banach-Mazur distance to the cube
- The Banach-Mazur compactum is the Alexandroff compactification of a Hilbert cube manifold