A study of boundedness in probabilistic normed spaces

It was shown in Lafuerza-Guillén, Rodríguez-Lallena and Sempi (1999) [8] that uniform boundedness in a Šerstnev PN space (V,ν,τ,τ∗)(V,ν,τ,τ∗), (named boundedness   in the present setting) of a subset A⊂VA⊂V with respect to the strong topology is equivalent to the fact that the probabilistic radius RARA of AA is an element of D+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 Šerstnev 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 Archimedeanity of triangle functions τ∗τ∗ of type τT,LτT,L is given. This work is a partial solution to a problem of comparing the concepts of distributional boundedness (DD-bounded in short) and that of boundedness in the sense of associated strong topology.