A remark on the duality mapping on \(\ell^{\infty}\) (Q1072058)

Let S be the unit ball of \(\ell^{\infty}\) and sm S the set of smooth points of S. The duality mapping \(F_ 0\) from S to the dual \(\ell^{\infty*}\) of \(\ell^{\infty}\) is defined as \(F_ 0(v)=\{\lambda \in \ell^{\infty*}: \lambda (v)=\ell =\| \lambda \| \}\), \(v\in S\). Let ext \(F_ 0(v)\) denote the set of extremal points of \(F_ 0(v)\). In the paper of \textit{I. Hada, K. Hashimoto} and \textit{S. Oharu} [Tokyo J. Math. 2, 71-97 (1979; Zbl 0417.46008)] the following question is raised: ''Given \(v\in S\setminus sm S\) and \(\lambda\in ext F_ 0(v)\), does there exist a sequence \(\{v_ n\}\subset sm S\) such that \(\| v_ n-v\| \to 0\) and that \(\lambda\) is a weak* cluster point of the sequence \(\{F_ 0(v_ n)\}?''\) The note under review answers this question negatively. Namely, two propositions are formulated, from which a counterexample for this question can be constructed easily.