The statistically unbounded τ-convergence on locally solid Riesz spaces

Abstract

A sequence (xn) in a locally solid Riesz space (E, τ) is said to be statistically unbounded τ-convergent to x ∈ E if, for every zero neighborhood U , 1 n {k ≤ n : |x k − x| ∧ u / ∈ U } → 0 as n → ∞. In this paper, we introduce the concept of the stuτ-convergence and give the notions of stuτ-closed subset, stuτ-Cauchy sequence, stuτ-continuous and stuτ-complete locally solid vector lattice. Also, we give some relations between the order convergence and the stuτ-convergence.

The Statistical Multiplicative Order Convergence in Riesz Algebras

“…Another important concept of functional analysis is vector lattice (or, Riesz spaces) which was introduced by F. Riesz [13]. We refer the reader for applications of Riesz spaces to [1,2,3,4,5,18]. We aim to combine concepts of the order and the statistical convergence, and the multiplicative on Riesz algebras, and so, we introduce the convergence on Riesz algebras without topological structure.…”

“…Several applications and generalizations about the statistical convergence have been investigated by several authors (cf. [3,7,8,11,16,17]). In this paper, we abbreviate the cardinality of subsets in the vertical bar.…”

Statistically multiplicative convergence on locally solid Riesz algebras

“…Recently, several generalizations and applications of this concept have been investigated by several authors in series of recent papers (c.f. [3,8,10,11]).…”

“…The statistical convergence in a locally solid Riesz space was introduced and studied by Aydın in [3] with respect to unbounded order convergence on locally solid Riesz spaces. Let (x n ) be a sequence in a locally solid vector lattice E is said to be statistically unbounded τ -convergent to x ∈ E if, for every zero neighborhood U , it satisfies…”

(V, Λ)-Order SUMMABLE IN RIESZ SPACES

“…But, statistical convergence on Riesz spaces has not been studied extensively. Only a few studies have been conducted on this recently; see for example [10][11][12][13]. They give relations between the order convergence and the statistical convergence on Riesz spaces.…”