Unbounded Vectorial Cauchy Completion of Vector Metric Spaces

Abstract

A sequence (𝑎𝑛) in a Riesz space E is called uo-convergent (or unbounded orderconvergent) to 𝑎 ∈ 𝐸 if |𝑎𝑛 − 𝑎| ∧ 𝑢𝑜→0 for all 𝑢 ∈ 𝐸+ and unbounded order Cauchy(uo-Cauchy) if |𝑎𝑛 − 𝑎𝑛+𝑝| is uo-convergent to 0. In the first part of this study wedefine 𝑢𝑑,𝐸-convergence (or unbounded vectorial convergence) in vector metric spaces,which is more general than usual metric spaces, and examine relations betweenunbounded order convergence, unbounded vectorial convergence, vectorial convergenceand order convergence. In the last part we construct the unbounded Cauchy completionof vector metric spaces by the motivation of the fact that every metric space has Cauchycompletion. In this way, we have obtained a more general completion of vector metricspaces.