A CHARACTERIZATION OF u-UNIFORMLY COMPLETENESS OF RIESZ SPACES IN TERMS OF STATISTICAL u-UNIFORMLY PRE-COMPLETENESS

Abstract

In this paper we introduce statistically u-uniformly convergent sequences in Riesz spaces (vector lattices) and then we give a characterization of u-uniformly completeness of Riesz spaces. The notion of statistical convergence of sequences was introduced by Steinhauss [6] at a conference held at Wroclaw University, Poland, in 1949 (see also [1]). A sequence (xn) of real numbers is said to be statistically convergent to a real number x if lim n→∞ 1 n |{k : k ≤ n, | xk − x |≥ }| = 0 for each 0 < , where the vertical bars denote the cardinality of the set which they enclose. Maddox [4] has generalized the notion statistical convergent sequence for locally convex spaces as follows: A sequence (xn) in a locally convex space X which determined by the seminorms (qi)i∈I , is said to be statistical convergent to x ∈ X if: lim n→∞ 1 n |{k : k ≤ n, qi(xn − x) ≥ }| = 0 for each 0 < and i ∈ I. A vector space X with a partial order ≤ is called an ordered vector space if αx + z ≤ αy + z for each z ∈ X whenever x ≤ y, 0 ≤ α ∈ R. An ordered vector space X is called a Riesz space (or vector lattice) if supremum of ∗2000 Mathematics Subject Classification. Primary:40A05,46A40.