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.
