Metric spaces with the small ball property

• Autorzy: Ehrhard Behrends , Vladimir M. Kadets
• Języki publikacji: EN

Abstrakty

EN

A metric space (M,d) is said to have the small ball property (sbp) if for every ε₀ > 0 it is possible to write M as the union of a sequence (B(xₙ,rₙ)) of closed balls such that the rₙ are smaller than ε₀ and lim rₙ = 0. We study permanence properties and examples of sbp. The main results of this paper are the following: 1. Bounded convex closed sets in Banach spaces have sbp only if they are compact. 2. Precisely the finite-dimensional Banach spaces have sbp. (More generally: a complete metric group has sbp iff it is separable and locally compact.) 3. Let B be a boundary in the bidual of an infinite-dimensional Banach space. Then B does not have sbp. In particular the set of extreme points in the unit ball of an infinite-dimensional reflexive Banach space fails to have sbp.

Kategorie tematyczne

• 54E35: Metric spaces, metrizability
• 46B20: Geometry and structure of normed linear spaces
• 46B10: Duality and reflexivity

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2001
• Tom: 148
• Numer: 3
• Strony: 275-287

Daty

wydano

2001

Twórcy

autor

Ehrhard Behrends

• I. Mathematisches Institut, Freie Universität Berlin, Arnimallee 2-6, D-14 195 Berlin, Germany

autor

Vladimir M. Kadets

• Department of Mechanics and Mathematics, Kharkov National University, 4 Svobody Sq., Kharkov, 61077 Ukraine

Identyfikatory

DOI

10.4064/sm148-3-6