On anti-coproximinal and strongly anti-coproximinal subspaces of function spaces

Abstract

The purpose of this article is to study the anti-coproximinal and strongly anti-coproximinal subspaces of the Banach space of all bounded (continuous) functions. We obtain a tractable necessary condition for a subspace to be stronsgly anti-coproximinal. We prove that for a subspace $\mathbb{Y}$ of a Banach space $\mathbb{X}$ to be strongly anti-coproximinal, $\mathbb{Y}$ must contain all w-ALUR points of $\mathbb{X}$ and intersect every maximal face of $B_{\mathbb{X}}.$ We also observe that the subspace $\mathbb{K}(\mathbb{X},\mathbb{Y})$ of all compact operators between the Banach spaces $\mathbb{X}$ and $\mathbb{Y}$ is strongly anti-coproximinal in the space $\mathbb{L}(\mathbb{X},\mathbb{Y})$ of all bounded linear operators between $\mathbb{X}$ and $\mathbb{Y}$, whenever $\mathbb{K}(\mathbb{X},\mathbb{Y})$ is a proper subset of $\mathbb{L}(\mathbb{X},\mathbb{Y}),$ and the unit ball $B_{\mathbb{X}}$ is the closed convex hull of its strongly exposed points.