The relationship between Goldstine's theorem and the convex point of continuity property
type: Item
instance of: scholarly article
title: The relationship between Goldstine's theorem and the convex point of continuity property
zbMATH Open document ID: 0868.46008
published in: Journal of Mathematical Analysis and Applications
publication date: 1995-03-19T00:00:00Z
review text: Goldstine's theorem says that the natural embedding of the closed unit ball \(B(X)\) of a Banach space \(X\) is weak\(^*\) dense in the second dual ball \(B(X^{**})\). In this paper the author characterizes, in terms of the geometry of \(B(X)\), when the natural embedding of \(B(X)\) into \(B(X^{**})\) is not only weak\(^*\) dense, but also residual. Using this characterization, he shows that a Banach space \(X\) has the convex point of continuity property, if and only if, for each equivalent norm ball \(B(X)\), the natural embedding of \(B(X)\) into \(B(X^{**})\) is residual with respect to the weak\(^*\) topology. He also shows that a Banach space \(X\) has the Radon-Nikodým property if and only if, for each equivalent norm ball \(B(X)\), the set of linear functionals in \(X^*\) which attain their norm on \(B(X)\) is residual in \(X^*\).
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46B22
zbMATH DE Number: 720404
zbMATH Keywords: Goldstine's theorem
zbMATH Keywords: embedding
zbMATH Keywords: second dual
zbMATH Keywords: convex point of continuity property
zbMATH Keywords: Radon-Nikodým property
author: Warren B. Moors
MaRDI profile type: Publication
full work available at URL: jmaa.1994.1465
OpenAlex ID: W2022110014
DOI: 10.1006/JMAA.1994.1465
description: scientific article; zbMATH DE number 720404