On best coapproximations and some special subspaces of function spaces

TL;DR

The paper investigates anti-coproximinal and strongly anti-coproximinal subspaces in Banach spaces, focusing on function spaces and bounded operator spaces. It provides a tractable necessary condition for strong anti-coproximinality in terms of containing all $w$-ALUR points and intersecting maximal faces, and it connects these geometric properties to operator spaces via the BŠ-property, yielding dichotomies for $\mathbb{K}(\mathbb{X},\mathbb{Y})$ in $\mathbb{L}(\mathbb{X},\mathbb{Y})$. The work also establishes that in $\ell_\infty(K)$ and $C_0(K)$, anti-coproximinality and strong anti-coproximinality coincide, and it derives several structural consequences for finite-dimensional and polyhedral spaces, including density results under the Radon–Nikodym Property. These findings link the geometry of the unit ball, such as extreme and w-ALUR points, with best coapproximation behavior, yielding practical criteria and insights for function spaces and operator spaces alike.

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.

Theorems & Definitions (85)

• Definition 1.1
• Lemma 2.1
• Lemma 2.2
• Theorem 2.3
• Theorem 2.4
• proof
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8