It was shown [8] that uniform boundedness in a Serstnev PN space $(V,\nu,\tau,\tau^*)$, (named boundedness in the present setting) of a subset $A\subset V$ 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. Section 2 presents a characterization of those PN spaces, whether they are TV spaces or not, in which the equivalence holds. In Section 3, a characterization of the Archimedeanity of triangle functions $\tau^*$ of the type $\tau_{T,L}$ is given. This work is a partial solution to a problem of comparing the concepts of distributional boundedness ($\D$--bounded in short) and that of boundedness in the sense of associated strong topology.
A study of boundedness in probabilistic normed spaces
SEMPI, Carlo;
2010-01-01
Abstract
It was shown [8] that uniform boundedness in a Serstnev PN space $(V,\nu,\tau,\tau^*)$, (named boundedness in the present setting) of a subset $A\subset V$ 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. Section 2 presents a characterization of those PN spaces, whether they are TV spaces or not, in which the equivalence holds. In Section 3, a characterization of the Archimedeanity of triangle functions $\tau^*$ of the type $\tau_{T,L}$ is given. This work is a partial solution to a problem of comparing the concepts of distributional boundedness ($\D$--bounded in short) and that of boundedness in the sense of associated strong topology.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
