Coproximinality for quotient spaces

Abstract

In this paper we study the classical notion of coproximinality, for quotient spaces of Banach spaces. We provide a partial solution to the three space problem, analogous to a classical result of Cheney and Wulbert, by showing that for Z⊂Y⊂X, coproximinality of Z in X and that of Y/Z in X/Z implies the coproximinality of Y in X, when Z is an M-ideal in X. For the space C(K) of continuous functions on a compact extremally disconnected set K we derive the same conclusion under the assumption that Z is an M-ideal in Y.