Stability of ball proximinality (Q2451173)

Let \(X\) be a real Banach space. A closed subspace \(Y \subset X\) is said to be ball proximinal if the closed unit ball \(Y_1\) is a proximinal set in \(X\). Such a subspace is proximinal and being ball proximinal is a stronger notion. Properties and stability results of this class were studied by the reviewer in collaboration with \textit{P. Bandyopadhyay} and \textit{B.-L. Lin} [in: Banach spaces and their applications in analysis. Proceedings of the international conference, Miami University, Oxford, OH, USA, May 22--27, 2006. In honor of Nigel Kalton's 60th birthday. Berlin: Walter de Gruyter. 251--264 (2007; Zbl 1135.41011)]. Motivated by this, the present authors study the stability of this and a stronger notion, called strong ball proximinality, for Köthe-Bochner function spaces. For an order continuous function space \(E\) and for a separable ball proximinal subspace \(Y \subset X\), the authors show that \(E(Y)\) is ball proximinal in \(E(X)\). In the case of Bochner integrable functions, this answers a question raised in [loc.\,cit.]. They note that the result is not true without the separability hypothesis. The authors also show that strong ball proximinality is stable under \(c_0\)-sums.