A study of boundedness in probabilistic normed spaces(II)

Abstract

It was shown in Lafuerza-Guillén, Rodríguez-Lallena and Sempi (1999)that uniform boundedness in a Serstnev PN space (V,\un,\tau,\tau^*), named boundedness in the present setting, of a subset A in V with respect to the strong topology is equivalent to the fact that the probabilistic radius R_A of A is an element of D^+.Here we extend the equivalence just mentioned to a larger class of PN spaces, namely those PN spaces that are topological vector spaces (briefly TV spaces), but are not Serstnev PN spaces. We present a characterization of those PN spaces, whether they are TV spaces or not,in which the equivalence holds. Then, a characterization of the Archimedeanmity of triangle function \tau^* of type \tau_{T,L} is given. This work is a partial solution to a problema of comparing the concepts of distributional boundedness (D-bounded in short) and that of boundedness in the sense of associated strong topology.