Compact Linear Operators Between Probabilistic Normed Spaces

Abstract

A pair (X, N) is said to be a probabilistic normed space if X is a real vector space, N is a mapping from X into the set of all distribution functions (for x ∈ X, the distribution function N(x) is denoted by N _x , and N _x (t) is the value N _x at t ∈ ℝ) satisfying the following conditions:

• (N1) N _x (0) = 0

• (N2) N _x (t) = 1 for all t > 0 iff x = 0

• (N3) N _αx (t) = N x $(\frac{t} {{|\alpha |}})$ for all α ∈ ℝ∖0,

• (N4) N _x+y (s + t) ≥ min N _x (s), N _y (t) for all x, y ∈ X, and s, t ∈ ℝ₀ ⁺.

In this article, we study compact linear operators between probabilistic normed spaces.

This research was in part supported by a grant from IPM (No. 86470033).