Weighted lacunary statistical convergence of double sequences in locally solid Riesz spaces

Recently, the notion of weighted lacunary statistical convergence is studied in a locally solid Riesz space for single sequences by Ba?ar?r and Konca [7]. In this work, we define and study weighted lacunary statistical ?-convergence, weighted lacunary statistical ?-boundedness of double sequences in locally solid Riesz spaces. We also prove some topological results related to these concepts in the framework of locally solid Riesz spaces and give some inclusion relations.

Ideal convergence in locally solid Riesz spaces

An ideal I is a family of subsets of positive integers N which is closed under taking finite unions and subsets of its elements. In this paper, we introduce the concepts of ideal ?-convergence, ideal ?-Cauchy and ideal ?-bounded sequence in locally solid Riesz space endowed with the topology ?. Some basic properties of these concepts has been investigated. We also examine the ideal ?-continuity of a mapping defined on locally solid Riesz space.

Some Remarks on the Algebraic Sum of Ideals and Riesz Subspaces

Following ideas used by Drewnowski and Wilansky we prove that if I is an infinite dimensional and infinite codimensional closed ideal in a complete metrizable locally solid Riesz space and I does not contain any order copy of ℝN then there exists a closed, separable, discrete Riesz subspace G such that the topology induced on G is Lebesgue, I ∩ G = {0}, and I + G is not closed.

Double Lacunary Density and Some Inclusion Results in Locally Solid Riesz Spaces

We define the notions of double statistically convergent and double lacunary statistically convergent sequences in locally solid Riesz space and establish some inclusion relations between them. We also prove an extension of a decomposition theorem in this setup. Further, we introduce the concepts of doubleθ-summable and double statistically lacunary summable in locally solid Riesz space and establish a relationship between these notions.

Statistical Convergence of Double Sequences in Locally Solid Riesz Spaces

Recently, the notion of statistical convergence is studied in a locally solid Riesz space by Albayrak and Pehlivan (2012). In this paper, we define and study statisticalτ-convergence, statisticalτ-Cauchy andS∗(τ)-convergence of double sequences in a locally solid Riesz space.

On Levi-Like Properties and some of Their Applications in Riesz Space Theory

Let (L, λ) be a locally solid Riesz space. (L, λ) is said to have the Levi property if for every increasing λ-bounded net (xα) ⊂ L+, sup xα exists. The Levi property, appearing in literature also as weak Fatou property (Luxemburg and Zaanen), condition (B) or monotone completeness (Russian terminology), is a classical object of investigation. In this paper we are interested in some variations of the property, their mutual relationships and applications in the theory of topological Riesz spaces. In the first part of the paper we clarify the status of two problems of Aliprantis and Burkinshaw. In the second part we study ideal-injective Riesz spaces.