A counterexample to the Bishop-Phelps theorem in complex spaces
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.
scientific article; zbMATH DE number 1436096
scientific article