On some subspaces of vector-valued continuous function space, from the perspective of Best coapproximation

TL;DR

This paper characterizes anti-coproximinal and strongly anti-coproximinal subspaces in vector-valued continuous function spaces and operator spaces, providing new insights into their structure and stability.

Contribution

It offers a complete characterization of strongly anti-coproximinal subspaces in $ C_0(K, ext{X}) $ under specific conditions and analyzes their stability in operator spaces.

Findings

Characterization of strongly anti-coproximinal subspaces in $ C_0(K, ext{X}) $

Stability results for anti-coproximinal subspaces in operator spaces

General characterization of (strong) anti-coproximinal subspaces in Banach spaces

Abstract

This article explores anti-coproximinal and strongly anti-coproximinal subspaces in the spaces of vector-valued continuous functions and operator spaces. We provide a complete characterization of strongly anti-coproximinal subspaces in $ C_0(K, \mathbb{X}) $, under the assumption that the unit ball of $ \mathbb{X}^* $ is the closed convex hull of its weak*-strongly exposed points. Additionally, the work includes a stability analysis of anti-coproximinal and strongly anti-coproximinal subspaces of $ \mathbb{L}(\mathbb{X}, \mathbb{Y}) $ and the space $ \mathbb{Y} $. Beyond these, we present a general characterization of (strong) anti-coproximinal subspaces in the broader context of Banach spaces.