Smallness and the covering of a Banach space
type: Item
instance of: scholarly article
title: Smallness and the covering of a Banach space
zbMATH Open document ID: 1266.46005
author: Pier Luigi Papini
author: Jesús M. F. Castillo
published in: Milan Journal of Mathematics
publication date: 2013-01-15T00:00:00Z
review text: A subset \(A\) of a metric space is small (in \textit{J. Arias de Reyna's} terminology from [Proc. Edinb. Math. Soc., II. Ser. 31, No. 2, 217--229 (1988; Zbl 0662.46021)]) if there is a sequence of balls \(B(x_n, r_n)\) with \(r_n \to 0\) such that for every \(m \in \mathbb N\) one has \(A \subset \bigcup_{n \geq m}B(x_n, r_n)\). The authors give a number of generalizations and simplifications of several previously known results on small sets in Banach spaces. In particular, they prove that a Banach space \(X\) can be equivalently renormed so that the set of extreme points of the dual unit ball \(B_{X^*}\) is small if and only if it can be renormed so that the set of extreme points of \(B_{X^*}\) is countable, thus improving significantly a theorem by \textit{E. Behrends} and \textit{V. M. Kadets} [Stud. Math. 148, No. 3, 275--287 (2001; Zbl 1002.46012)]. Applications to isomorphically polyhedral spaces are given.
reviewed by: Vladimir Kadets
Mathematics Subject Classification ID: 46B03
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 6125711
zbMATH Keywords: small set
zbMATH Keywords: isomorphically polyhedral space
zbMATH Keywords: covering of the unit sphere
zbMATH Keywords: boundary
MaRDI profile type: Publication
full work available at URL: s00032-012-0188-5
OpenAlex ID: W2093721076
cites work: Hausdorff dimension of Banach spaces
cites work: Spaces of largest Hausdorff dimension
cites work: Q5457541
cites work: Acknowledgement of priority Re: My paper ``The small ball property in Banach spaces - quantitative results'' [Rend. Circ. mat. Palermo, Serie II, - Suppl. 76 (2005), 3-16]
cites work: Metric spaces with the small ball property
cites work: Q4475046
cites work: Q3862591
cites work: On Covering the Unit Ball in a Banach Space
cites work: Q3633108
cites work: On some covering properties of metric spaces
cites work: A Simple Norm Inequality
cites work: Extremely weakly unconditionally convergent series
cites work: Massiveness of the set of extreme points of the dual ball of a Banach space. Polyhedral spaces
cites work: Q4521638
cites work: Polyhedral Banach spaces
cites work: Infinite-Dimensional Polyhedrality
cites work: Almost flat locally finite coverings of the sphere
cites work: Q4652145
cites work: Q5590301
cites work: Generalized Hausdorff dimension
cites work: Ultraproducts in Banach space theory.
cites work: Thickness of the unit sphere, $\ell_1$-types, and the almost Daugavet property
cites work: Some Covering Theorems for Infinite Dimensional Vector Spaces
cites work: Q4931992
cites work: Q3134817
cites work: Q3400216
cites work: Q3752078
cites work: Q4521644
cites work: The Size of the Unit Sphere
DOI: 10.1007/S00032-012-0188-5
description: scientific article; zbMATH DE number 6125711