A study of boundedness in probabilistic normed spaces

作者：Lafuerza-Guillen, Bernardo; Sempi, Carlo^*; Zhang, Gaoxun

来源：Nonlinear Analysis-Theory Methods & Applications, 2010, 73(5): 1127-1135.

DOI：10.1016/j.na.2009.12.037

• Probabilistic normed spaces
• Boundedness
• Archimedean triangle function

It was shown in Lafuerza-Guillen, Rodriguez-Lallena and Sempi (1999) [8] that uniform boundedness in a Serstnev PN space. (V, nu, tau, tau*), (named boundedness in the present setting) of a subset A subset of 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 Archimedeanity of triangle functions tau* of 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.

• 出版日期2010-9-1
• 单位西南财经大学