Matches in DBpedia 2014 for { <http://dbpedia.org/resource/Banach–Mazur_compactum> ?p ?o. }
Showing items 1 to 19 of
19
with 100 items per page.
- Banach–Mazur_compactum abstract "In the mathematical study of functional analysis, the Banach–Mazur distance is a way to define a distance on the set Q(n) of n-dimensional normed spaces. If X and Y are two finite-dimensional normed spaces with the same dimension, let GL(X,Y) denote the collection of all linear isomorphisms T : X → Y. The Banach–Mazur distance between X and Y is defined by Equipped with the metric δ, the space Q(n) is a compact metric space, called the Banach–Mazur compactum.Many authors prefer to work with the multiplicative Banach–Mazur distancefor which d(X, Z) ≤ d(X, Y) d(Y, Z) and d(X, X) = 1.F. John's theorem on the maximal ellipsoid contained in a convex body gives the estimate: where ℓn2 denotes Rn with the Euclidean norm (see the article on Lp spaces).From this it follows that d(X, Y) ≤ n for every couple (X, Y) in Q(n). However, for the classical spaces, this upper bound for the diameter of Q(n) is far from being approached. For example, the distance between ℓn1 and ℓn∞ is (only) of order n1/2 (up to a multiplicative constant independent from the dimension n).A major achievement in the direction of estimating the diameter of Q(n) is due to E. Gluskin, who proved in 1981 that the (multiplicative) diameter of the Banach–Mazur compactum is bounded below by c n, for some universal c > 0.Gluskin's method introduces a class of random symmetric polytopes P(ω) in Rn, and the normed spaces X(ω) having P(ω) as unit ball (the vector space is Rn and the norm is the gauge of P(ω)). The proof consists in showing that the required estimate is true with large probability for two independent copies of the normed space X(ω).Q(2) is an absolute extensor. On the other hand, Q(2) is not homeomorphic to a Hilbert cube.".
- Banach–Mazur_compactum wikiPageID "20021256".
- Banach–Mazur_compactum wikiPageRevisionID "551419406".
- Banach–Mazur_compactum first "A.A.".
- Banach–Mazur_compactum id "B/b110100".
- Banach–Mazur_compactum last "Giannopoulos".
- Banach–Mazur_compactum title "Banach–Mazur compactum".
- Banach–Mazur_compactum subject Category:Functional_analysis.
- Banach–Mazur_compactum subject Category:Metric_geometry.
- Banach–Mazur_compactum comment "In the mathematical study of functional analysis, the Banach–Mazur distance is a way to define a distance on the set Q(n) of n-dimensional normed spaces. If X and Y are two finite-dimensional normed spaces with the same dimension, let GL(X,Y) denote the collection of all linear isomorphisms T : X → Y.".
- Banach–Mazur_compactum label "Banach-Mazur-Abstand".
- Banach–Mazur_compactum label "Banach–Mazur compactum".
- Banach–Mazur_compactum label "Compact de Banach-Mazur".
- Banach–Mazur_compactum sameAs Banach%E2%80%93Mazur_compactum.
- Banach–Mazur_compactum sameAs Banach-Mazur-Abstand.
- Banach–Mazur_compactum sameAs Compact_de_Banach-Mazur.
- Banach–Mazur_compactum sameAs Q806050.
- Banach–Mazur_compactum sameAs Q806050.
- Banach–Mazur_compactum wasDerivedFrom Banach–Mazur_compactum?oldid=551419406.