Convergence and Completeness in L_2 (P) with respect to a Partial Metric

Abstract

Metric spaces can be generalized to be partial metric spaces. Partial metric spaces have a unique concept related to a distance. In usual case, there is no distance from two same points. But, we can obtain the distance from two same points in partial metric spaces. It means that the distance is not absolutely zero. Using the basic concept of partial metric spaces, we find analogy between metric spaces and partial metric spaces. We define a metric d^p formed by a partial metric p, with applying characteristics of metric and partial metric. At the beginning, we implement the metric d^p to determine sequences in L_2 (P). We then ensure the convergence and completeness in L_2 [a,b] can be established in L_2 (P). In this study, we conclude that the convergence and completeness in L_2 [a,b]  can be established in L_2 (P) by constructing a partial metric p_2 induced by a metric d^p.