BASIC THEORY AND APPLICATIONS OF PROBABILISTIC METRIC SPACES (Ⅰ)

Basic theory and applications of probabilistic metric spaces (I)

This paper is devoted to the study of the basic theory and applications of probabilistic metric spaces (PM-space). In this paper the topological structure and properties for PM-space are considered. The conditions of metrization and the form of metric functions for PM-spaces. Menger PM-spaces and probabilistic normed linear spaces (PN-space) are given and the characterizations of various probabilistically bounded sets are presented. As applications we utilize these results obtained in this paper to study the linear operator theory and fixed point theory on PM-spaces. Projects supported by the Science Fund of the Chinese Academy of Sciences.