A counterexample to the Bishop-Phelps theorem in complex spaces
type: Item
instance of: scholarly article
title: A counterexample to the Bishop-Phelps theorem in complex spaces
zbMATH Open document ID: 0954.46009
DOI: 10.1007/BF02810578
published in: Israel Journal of Mathematics
publication date: 2001-02-08T00:00:00Z
review text: In 1961 \textit{E. Bishop} and \textit{R. R. Phelps} [Bull. Am. Math. Soc. 67, 97-98 (1961; Zbl 0098.07905)] proved that the set of support functionals for a closed bounded convex subset \(S\) of a real Banach space \(E\) is norm dense in \(E^*\). The natural question of whether this statement is true in a complex Banach space for any closed bounded convex subset. In 1977 \textit{J. Bourgain} [Israel J. Math. 28, 265-271 (1977; Zbl 0365.46021)] proved that it is correct if a Banach space \(E\) has the Radon-Nikodým property. In the present paper the author constructs a complex Banach space \(E\) and a closed bounded convex subset \(S\) of \(E\) such that the set of the support points of \(S\) is empty.
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46B22
zbMATH DE Number: 1436096
zbMATH Keywords: Bishop-Phelps theorem
zbMATH Keywords: support functionals
zbMATH Keywords: Radon-Nikodým property
zbMATH Keywords: support points
author: Victor Lomonosov
reviewed by: Jarosław Górnicki
MaRDI profile type: Publication
Wikidata QID: Q124982645
cites work: A proof that every Banach space is subreflexive
cites work: On dentability and the Bishop-Phelps property
cites work: Q5727906
cites work: On a Question of Bishop and Phelps
cites work: On operators which attain their norm
cites work: Q4027031
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description: scientific article; zbMATH DE number 1436096