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 $Z\subset Y\subset X$, 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 $Y$.