Boundedness in generalized \v{S}erstnev PN spaces

Abstract

The motivation of this paper is a suggestion by H\"ole 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 \v{S}erstnev spaces due to B. Lafuerza, J. A. Rodriguez, and C. Sempi, are here extended to generalized \v{S}erstnev spaces (these are pre-PN spaces satisfying a more general \v{S}erstnev 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 \v{S}erstnev 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 \v{S}erstnev spaces in terms of bounded subsets.