Bounded linear operators in probabilistic normed space (original) (raw)

Strongly Bounded and Classes of Linear Operators in Probabilistic Normed Space

2005

In this paper we redefined the definition of a bounded linear operator in probabilistic normed space by introducing the notion of strongly ψbounded linear maps. We then show that this new definition of boundedness implies all contraction functions in probabilistic normed space are bounded. Also, we introduce the classes of linear operators in probabilistic normed space, as the set of all certainly bounded Lc(V, V ), D-bounded L D (V, V ), strongly B-bounded L B (V, V ), and strongly ψ-bounded L ψ (V, V ) we then prove they are linear spaces. 2000 Mathematics Subject Classification: 54E70

New Types of Continuous Linear Operator in Probabilistic Normed Space

Bulletin of the Malaysian Mathematical Sciences Society, 2009

In this paper, new types of continuous linear operator, such as continuous, strongly continuous, weakly continuous and sequentially continuous linear operators, in probabilistic normed space are introduced. Also, the relation between the boundedness and continuity of these linear operators in probabilistic normed spaces is studied.

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.

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”.

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.

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.

A common fixed point for operators in probabilistic normed spaces

Chaos, Solitons & Fractals, 2009

Probabilistic Metric spaces was introduced by Karl Menger. Alsina, Schweizer and Sklar gave a general definition of probabilistic normed space based on the definition of Menger [Alsina C, Schweizer B, Sklar A. On the definition of a probabilistic normed spaces. Aequationes Math 1993;46:91-8].