Unbounded Vectorial Cauchy Completion of Vector Metric Spaces

Abstract

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