Metrizability of Cone Metric spaces

TL;DR

This paper proves that cone metric spaces are metrizable, showing they are not a true generalization of metric spaces, and that fixed point theorems in cone metric spaces follow directly from classical metric space results.

Contribution

It demonstrates that every cone metric space is metrizable and the equivalent metric preserves the same contractive conditions, clarifying the relationship between cone metric and metric spaces.

Findings

Every cone metric space is metrizable.

Equivalent metrics satisfy the same contractive conditions.

Fixed point theorems in cone metric spaces are direct extensions of metric space results.

Abstract

In 2007 H. Long-Guang and Z. Xian, [H. Long-Guang and Z. Xian, Cone Metric Spaces and Fixed Point Theorems of Contractive Mapping, J. Math. Anal. Appl., 322(2007), 1468-1476], generalized the concept of a metric space, by introducing cone metric spaces, and obtained some fixed point theorems for mappings satisfying certain contractive conditions. The main question was "Are cone metric spaces a real generalization of metric spaces?" Throughout this paper we answer the question in the negative, proving that every cone metric space is metrizable and the equivalent metric satisfies the same contractive conditions as the cone metric. So most of the fixed point theorems which have been proved are straightforward results from the metric case.