Metrizability of Cone Metric Spaces Via Renorming the Banach Spaces

TL;DR

This paper demonstrates that by renorming an ordered Banach space, cone metric spaces can be converted into standard metric spaces, allowing all metric space theorems to apply to cone metric spaces.

Contribution

It introduces a method to renorm Banach spaces so that cone metric spaces become equivalent to metric spaces, simplifying their analysis.

Findings

Cone metric spaces can be transformed into metric spaces via renorming.

All metric space theorems are applicable to cone metric spaces after renorming.

The approach ensures the cone P becomes a normal cone with constant K=1.

Abstract

In this paper we show that by renorming an ordered Banach space, every cone P can be converted to a normal cone with constant K = 1 and consequently due to this approach every cone metric space is really a metric one and every theorem in metric space valid for cone metric space automatically.