Countable products of probabilistic normed spaces (original) (raw)

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 mmm-- transforms of several PN spaces. In addition we present a detailed study of alpha\alphaalpha--simple product PN spaces and, finally, the product topologies in PN spaces which are products of countable families of PN spaces.

Some classes of probabilistic normed spaces

The authors’ abstract: “Probabilistic normed spaces (PN spaces) have recently been redefined by C. Alsina, B. Schweizer, and A. Sklar [Aequationes Math. 46, No. 1-2, 91-98 (1993; Zbl 0792.46062)]. The authors begin the study of these spaces by giving several examples; in particular, they (a) present a detailed study of α-simple spaces, (b) construct a PN space on the vector space of (equivalence classes) of random variables, and (c) show that its probabilistic norm alone generates the norms of all L p - and Orlicz spaces”.

Invariant and semi-invariant probabilistic normed spaces

Chaos, Solitons & Fractals, 2009

Probabilistic metric spaces were introduced by Karl Menger. Alsina, Schweizer and Sklar gave a general definition of probabilistic normed space based on the definition of Menger [1]. We introduce the concept of semi-invariance among the PN spaces. In this paper we will find a sufficient condition for some PN spaces to be semi-invariant. We will show that PN spaces are normal spaces. Urysohn's lemma, and Tietze extension theorem for them are proved.

Probabilistic normed Riesz spaces

Acta Mathematica Sinica, English Series, 2012

In this paper, the concepts of probabilistic normed Riesz space and probabilistic Banach lattice are introduced, and their basic properties are studied. In this context, some continuity and convergence theorems are proved.

NORMABILITY OF PROBABILISTIC NORMED SPACES

Relying on Kolmogorov's classical characterization of normable Topological Vector spaces, we study the normability of those Probabilistic Normed Spaces that are also Topological Vector spaces and provide a characterization of normableŠerstnev spaces. We also study the normability of other two classes of Probabilistic Normed Spaces.

A note on the probabilistic N-Banach spaces

arXiv (Cornell University), 2016

In this paper, we define probabilistic n-Banach spaces along with some concepts in this field and study convergence in these spaces by some lemmas and theorem.

On α-Šerstnev probabilistic normed spaces

Journal of Inequalities and Applications, 2011

In this article, the condition a-Š is defined for a ]0, 1[∪]1, +∞[and several classes of a-Šerstnev PN spaces, the relationship between a-simple PN spaces and a-Šerstnev PN spaces and a study of PN spaces of linear operators which are a-Šerstnev PN spaces are given.

Probabilistic n-normed spaces, D-n-compact sets and D-n-bounded sets

In this paper, first we define and study the probabilistic n-normed spaces and D-n-compactness, also we prove some theorems and inequalities. In the next section we define D-n-boundedness and prove some results in relation between D-n-compact and D-nbounded sets in these spaces.