\(M\)-ideals and the Bishop-Phelps theorem (Q2912194)

Let \(X\) be a real Banach space. A closed subspace \(Y \subset X\) is said to be an \(M\)-ideal if there is a linear projection \(P: X^\ast \rightarrow X^\ast\) such that \(\ker(P) = Y^\bot\) and \(\|P(x^\ast)\| + \|x^\ast - P(x^\ast)\| = \|x^\ast\|\) for all \(x^\ast \in X^\ast\). A well-known result of \textit{E. Behrends} and \textit{P. Harmand} [Stud. Math. 81, 159--169 (1985; Zbl 0529.46015)] describes these objects in terms of the weak\(^\ast\)-closure of the set of best approximations, being a ball in the second dual.NEWLINENEWLINEIn this paper the author provides an alternative proof using the Bishop-Phelps support point theorem.