Normability of Probabilistic Normed Spaces

Duality of $κ$-normed topological vector spaces and their applications

A duality of $κ$-normed topological vector spaces is defined and investigated. For such spaces the analog of the Mackey-Arens theorem is proved. There are investigated cases, when $κ$-normability of a topological vector space implies its local convexity. There are given applications of $κ$-normed spaces for resolutions of differential equations and for approximations of functions in mathematical economy.

$\pmb{D}$-boundedness and $\pmb{D}$-compactness in finite dimensional probabilistic normed spaces

In this paper, we prove that in a finite dimensional probabilistic normed space, every two probabilistic norms are equivalent and we study the notion of $D$-compactness and $D$-boundedness in probabilistic normed spaces.

Finite and countable infinite products of Probabilistic Normed Spaces

In this work we first give for PN spaces results parallel to those obtained by Egbert for the product of PM spaces, and generalize results by Alsina and Schweizer in order to study non-trivial products and the product of $m$-- transforms of several PN spaces. In addition we present a detailed study of $α$--simple product PN spaces and, finally, the product topologies in PN spaces which are products of countable families of PN spaces.

概率赋范空间中函数的$I^K_ν$-收敛性

$I^K_ν$-Convergence of functions in probabilistic normed spaces

本文研究了概率赋范空间中函数关于概率范数$ν$的$I^K$-收敛性，作为$I^*_ν$-收敛性的推广，并探讨了相应的$I^K$-Cauchy函数和$I^K$-极限点。

Translation-invariant generalized topologies induced by probabilistic norms

In this paper we consider probabilistic normed spaces as defined by Alsina, Sklar, and Schweizer, but equipped with non necessarily continuous triangle functions. Such spaces endow a generalized topology that is Fréchet-separable, translation-invariant and countably generated by radial and circled 0-neighborhoods. Conversely, we show that such generalized topologies are probabilistically normable.

Completeness with respect to the probabilistic Pompeiu-Hausdorff metric

The aim of the present paper is to prove that the family of all closed nonempty subsets of a complete probabilistic metric space $L$ is complete with respect to the probabilistic Pompeiu-Hausdorff metric $H$. The same is true for the families of all closed bounded, respectively compact, nonempty subsets of $L$. If $L$ is a complete random normed space in the sense of Šerstnev, then the family of all nonempty closed convex subsets of $L$ is also complete with respect to $H$.