Structure of strongly proximinal subspaces (Q6597312)

A closed linear subspace \(Y\) of a Banach space \(X\) is strongly proximinal if for any \(x\in X\), any minimizing sequence of approximations of \(x\) in \(Y\) gets arbitrarily close to the non-empty set \(P(x)\subset Y\) of best approximants of \(x\). For instance, a hyperplane \(\mathrm{Ker}(x^*)\) is strongly proximinal if and only if \(x^*\) is a point of strong sub-differentiability of the dual norm on \(X^*\). The article under review investigates with great care these notions, and provides several original results. The most important one is probably Theorem~2.5, which delivers a usable sufficient condition for the preservation of strong proximinality under finite intersections of finite-codimensional subspaces, and transitivity of strong proximinality for subspaces of finite codimension. A number of applications of this theorem are given. In particular Theorem 2.8 asserts that all \(L_1\)-preduals \(X\) satisfy the assumption of Theorem~2.5, and this is a sweeping extension of early results on \(X=c_0\) due to V.~Indumathi and the reviewer [\textit{G.~Godefroy} and \textit{V.~Indumathi}, Rev. Mat. Complut. 14, No.~1, 105--125 (2001; Zbl 0993.46004)].