Stability of ball proximinality
In this paper, we show that if E is an order continuous Köthe function space and Y is a separable subspace of X, then E(Y) is ball proximinal in E(X) if and only if Y is ball proximinal in X. As a consequence, E(Y) is proximinal in E(X) if and only if Y is proximinal in X. This solves an open problem of Bandyopadhyay, Lin and Rao. It is also shown that if E is a Banach lattice with a 1-unconditional basis and for each n, Yn is a subspace of Xn, then (⊕Yn)E is ball proximinal in (⊕Xn)E if and only if each Yn is ball proximinal in Xn. © 2014 Elsevier Inc.
Journal of Approximation Theory
Lin, P., Zhang, W., & Zheng, B. (2014). Stability of ball proximinality. Journal of Approximation Theory, 183, 72-81. https://doi.org/10.1016/j.jat.2014.04.008