Birkhoff-James extensions of continuous functions on metric spaces

TL;DR

This paper introduces Birkhoff-James extensions of continuous functions on metric spaces, exploring their properties in compact and non-compact contexts and applying these concepts to orthogonality in function spaces.

Contribution

It extends Birkhoff-James orthogonality concepts to continuous functions on metric spaces, providing new insights and detailed analysis in both compact and non-compact cases.

Findings

Defined Birkhoff-James extensions for continuous functions.

Analyzed properties in compact and non-compact metric spaces.

Applied concepts to orthogonality in $C(X)$ spaces.

Abstract

In this paper, we extend the investigations regarding Birkhoff-James orthogonality of linear operators to bounded continuous functions on metric spaces. We introduce Birkhoff-James extensions of continuous functions and study them in detail, in the separate contexts of compact and non-compact metric spaces. We conclude by discussing an application of our ideas to the study of Birkhoff-James orthogonality in $C(X)$ with the supremum norm, where $X$ is a compact metric space.