Boundedness in generalized Šerstnev PN spaces

Abstract

The motivation of this paper is a suggestion by Höle of comparing the notions of $\D$-boundedness and boundedness in Probabilistic Normed spaces (briefly PN spaces), with non necessarily continuous triangle functions. Such spaces are here called ``pre-PN spaces''. Some results on Šerstnev spaces due to B. Lafuerza, J. A. Rodriguez, and C. Sempi, are here extended to generalized Šerstnev spaces (these are pre-PN spaces satisfying a more general Šerstnev condition). We also prove some facts on PN spaces (with continuous triangle functions). First, a connection between fuzzy normed spaces defined by Felbin and certain Šerstnev PN spaces is established. We further observe that topological vector PN spaces are $F$-normable and paranormable, and also that locally convex topological vector PN spaces are bornological. This last fact allows to describe continuous linear operators between certain generalized Šerstnev spaces in terms of bounded subsets.