SEMICONTINUITY OF METRIC PROJECTIONS IN $c_0$-DIRECT SUMS

Abstract

Let $\{X_i : i \in \mathbb{N} \}$ be a family of Banach space and let $Y_i \subseteq X_i$ be a closed subspace in $X_i$ for each $i \in \mathbb{N}$ such that at least two $Y_i'$s are non-trivial. Consider $X = (\oplus_{c_0} X_i)_{i \in \mathbb{N}}$ and $Y = (\oplus_{c_0} Y_i)_{i \in \mathbb{N}}$. We show that $Y$ is strongly proximinal in $X$ if and only if $P_Y$ is upper Hausdorff semi-continuous on $X$ if and only if $Y_i$ is strongly proximinal subspace in $X_i$ for each $i \in \mathbb{N}$. This shows that in [9, Theorem 3.4], strong proximinality of $Y_i$'s is a necessary assumption. We also show that lower semi-continuity of metric projections is stable in $c_0$-direct sums.