Norm-to-weak upper semi-continuity of the duality and pre-duality mappings
type: Item
instance of: scholarly article
title: Norm-to-weak upper semi-continuity of the duality and pre-duality mappings
zbMATH Open document ID: 1042.46008
DOI: 10.1023/A:1020662918653
author: Gilles Godefroy
author: V. Indumathi
published in: Set-Valued Analysis
publication date: 2003-03-12T00:00:00Z
review text: For an element \(x\) in the unit sphere of a Banach space \(X\), the duality mapping is the (set-valued function) given by \(J_{X^\ast}(x)= \{ x^\ast \in X^\ast : 1= \| x^\ast \| = x^\ast (x) \}\). In the case of a dual space, the preduality mapping at an element \(x^\ast \in X^\ast \) is just \(J_{X^{\ast \ast}} \cap X\). The authors use the concept of norm to weak upper semi-continuity for the duality mapping as it appears in [\textit{J. R. Giles}, \textit{D. A. Gregory} and \textit{B. Sims}, Pac. J. Math. 79, 99--109 (1979; Zbl 0399.46012)]. The present paper provides some characterizations of the norm to weak upper semicontinuity of the preduality mapping of a Banach space. For instance, the authors obtain the following result: if the duality mapping is n-w upper-semicontinuous at a point \(x^\ast\) on the unit sphere of a dual Banach space, then \(x^\ast\) attains its norm and the preduality mapping has the same kind of semicontinuity at \(x^\ast\) (Theorem 2.3). A point \(x \) in the unit sphere of \(X\) is said to be very smooth if there is just one element \(\varphi \in X^{\ast \ast \ast }\) satisfying \(\| \varphi \| = 1 = \varphi (x)\). The authors exhibit several assumptions of different nature for which very smoothness goes up from the space to the bidual. For instance, this happens if \(X\) is an \(M\)-ideal in its bidual, a fact that was probably known before. Also, there is an assumption of isomorphic nature implying the same result: the quotient \(X^{\ast \ast }\backslash X\) is a Grothendieck space (see Proposition 3.4). As a consequence, the authors show that for spaces isomorphic to \(C^\ast\)-algebras the above situation holds (Corollary 3.7 and comments that follow). In Section 4, it is proved that the previous results related to very smooth points are no longer true in general by giving an example, whose proof is also suggestive. To summarize, this paper gives a new and interesting view of the norm-weak upper semi-continuity of the duality mapp…
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 54C60
zbMATH DE Number: 1880353
zbMATH Keywords: duality mapping
zbMATH Keywords: subdifferential maps
zbMATH Keywords: norm-weak upper semi-continuity
zbMATH Keywords: very smooth norms
MaRDI profile type: Publication
full work available at URL: a:1020662918653
OpenAlex ID: W82838069
description: scientific article; zbMATH DE number 1880353