On the modulus of noncompact convexity of a Banach space
type: Item
instance of: scholarly article
title: On the modulus of noncompact convexity of a Banach space
zbMATH Open document ID: 0821.46016
DOI: 10.1007/BF01196675
author: Stanisław Prus
published in: Archiv der Mathematik
publication date: 1995-09-28T00:00:00Z
review text: Let \(A\) be a bounded subset of an infinite-dimensional Banach space \(X\). The Hausdorff measure of noncompactness \(\chi_ A\) of the set \(A\) is the infimum of all numbers \(r> 0\) such that \(A\) can be covered by finitely many balls of radius \(r\). For \(\varepsilon\in [0, 1]\), let \[ \Delta(\varepsilon)= \inf\{1- \inf\{\| x\|: x{\i}A\}\}, \] where the infimum is taken over all nonempty closed convex subsets \(A\) of the unit ball \(B_ X\) of \(X\) with \(\chi_ A\geq \varepsilon\). The Kuratowski measure of noncompactness of a bounded subset \(A\) of \(X\) is defined as the infimum of all numbers \(d> 0\) such that \(A\) can be covered by finitely many sets with diameters not exceeding \(d\). Let \(\Delta_ K\) be a modulus whose definition is obtained from the definition of \(\Delta\) by replacing the measure \(\chi\) by the Kuratowski measure. The function \(\Delta_ K\) is defined on the interval \([0, 2]\), and \(\Delta_ K(\varepsilon)\leq \Delta(\varepsilon)\leq \Delta_ K(2\varepsilon)\) for every \(\varepsilon\in [0, 1]\). Some properties of the modulus \(\Delta\) are known in the literature. In this paper the author establishes counterparts of some basic properties of the modulus of convexity for the moduli \(\Delta\) and \(\Delta_ K\) and proves a result which shows a difference between these notions viz. a Hilbert space has the best (i.e. the largest) possible modulus of convexity. He also shows that there is no space with the best modulus of non-compact convexity which gives answers to questions raised by \textit{J. Banas} [Nonlinear Anal., Theory Methods Appl. 16, No. 7/8, 669-682 (1991; Zbl 0724.46019)] and \textit{T. Sekowski} [Rend. Sem. Mat. Fis. Milano 56, 147-153 (1986; Zbl 0655.47050)].
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 47H09
zbMATH DE Number: 701022
zbMATH Keywords: Hausdorff measure of noncompactness
zbMATH Keywords: Kuratowski measure of noncompactness
zbMATH Keywords: modulus
zbMATH Keywords: modulus of convexity
reviewed by: T. D. Narang
MaRDI profile type: Publication
cites work: On Modulus of Noncompact Convexity and Its Properties
cites work: Compactness conditions in the geometric theory of Banach spaces
cites work: Q4751192
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full work available at URL: https://doi.org/10.1007/bf01196675
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description: scientific article; zbMATH DE number 701022