scholarly journals Ideal convergence in locally solid Riesz spaces

Filomat ◽  
2014 ◽  
Vol 28 (4) ◽  
pp. 797-809 ◽  
Author(s):  
Bipan Hazarika
Keyword(s):  
Bounded Sequence ◽  
Riesz Space ◽  
Ideal Convergence ◽  
Riesz Spaces ◽  
Basic Properties ◽  
Locally Solid Riesz Space ◽  
Positive Integers ◽  
The Ideal

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.

scholarly journals On generalized statistical convergence and boundedness of Riesz space-valued sequences

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 4989-5002
Author(s):  
Sudip Pal ◽  
Sagar Chakraborty
Keyword(s):  
Statistical Convergence ◽  
Necessary Conditions ◽  
Convergent Sequence ◽  
Bounded Sequence ◽  
Riesz Space ◽  
Riesz Spaces ◽  
Natural Density

We consider the notion of generalized density, namely, the natural density of weight 1 recently introduced in [4] and primarily study some sufficient and almost converse necessary conditions for the generalized statistically convergent sequence under which the subsequence is also generalized statistically convergent. Also we consider similar types of results for the case of generalized statistically bounded sequence. Some results are further obtained in a more general form by using the notion of ideals. The entire investigation is performed in the setting of Riesz spaces extending the recent results in [13].

Modes of ideal continuity and the additive property in the Riesz space setting

2014 ◽  
Vol 20 (1) ◽  
Author(s):  
Antonio Boccuto ◽  
Xenofon Dimitriou ◽  
Nikolaos Papanastassiou ◽  
Władysław Wilczyński
Keyword(s):  
Riesz Space ◽  
Ideal Convergence ◽  
Additive Property ◽  
Basic Properties ◽  
Space Setting ◽  
Comparison Results ◽  
Different Types

Abstract.In this paper we present some different types of ideal convergence/divergence and of ideal continuity for Riesz space-valued functions, and prove some basic properties and comparison results. We investigate the relations among different modes of ideal continuity and present a characterization of the (

scholarly journals On the Ideal Convergence of Double Sequences in Locally Solid Riesz Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
A. Alotaibi ◽  
B. Hazarika ◽  
S. A. Mohiuddine
Keyword(s):  
Double Sequences ◽  
Ideal Convergence ◽  
Riesz Spaces ◽  
The Ideal

The aim of this paper is to define the notions of ideal convergence,I-bounded for double sequences in setting of locally solid Riesz spaces and study some results related to these notions. We also define the notion ofI*-convergence for double sequences in locally solid Riesz spaces and establish its relationship with ideal convergence.

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

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 621-629
Author(s):  
Şükran Konca
Keyword(s):  
Statistical Convergence ◽  
Riesz Space ◽  
Double Sequences ◽  
Riesz Spaces ◽  
Lacunary Statistical Convergence ◽  
Inclusion Relations ◽  
Locally Solid Riesz Space ◽  
Statistical Boundedness ◽  
Weighted Lacunary Statistical Convergence

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.

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

1988 ◽  
Vol 31 (4) ◽  
pp. 477-486 ◽  
Author(s):  
G. Buskes ◽  
I. Labuda
Keyword(s):  
Riesz Space ◽  
Space Theory ◽  
Fatou Property ◽  
Riesz Spaces ◽  
Classical Object ◽  
The Status ◽  
Locally Solid Riesz Space ◽  
Condition B

AbstractLet (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.

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

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
S. A. Mohiuddine ◽  
Bipan Hazarika ◽  
Abdullah Alotaibi
Keyword(s):  
Decomposition Theorem ◽  
Riesz Space ◽  
Riesz Spaces ◽  
Inclusion Relations ◽  
Locally Solid Riesz Space ◽  
Inclusion Results ◽  
Convergent Sequences

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.

scholarly journals Statistical Convergence of Double Sequences in Locally Solid Riesz Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
S. A. Mohiuddine ◽  
Abdullah Alotaibi ◽  
M. Mursaleen
Keyword(s):  
Statistical Convergence ◽  
Riesz Space ◽  
Double Sequences ◽  
Riesz Spaces ◽  
Locally Solid Riesz Space

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.

Sequentially Relatively Uniformly Complete Riesz Spaces and Vulikh Algebras

1972 ◽  
Vol 24 (6) ◽  
pp. 1110-1113 ◽  
Author(s):  
C. T. Tucker
Keyword(s):  
Positive Integer ◽  
Cauchy Sequence ◽  
Convergent Sequence ◽  
Zero Element ◽  
Riesz Space ◽  
Riesz Spaces ◽  
Weak Unit ◽  
Uniformly Convergent ◽  
Positive Integers ◽  
Unique Limit

Throughout this paper V will denote an Archimedean Riesz space with a weak unit e and a zero element θ. A sequence f1,f2,f3, … of points of V is said to converge relatively uniformly to a point f (with regulator the point g of V) if, for each ∈ > 0, there is a number N such that, if n is a positive integer and n > N, then |f — fn| < ∈g. In an Archimedean Riesz space a relatively uniformly convergent sequence has a unique limit. The sequence f1, f2, f3, … is called a relatively uniform Cauchy sequence (with regulator g) if, for each ∈ > 0, there is a number N such that if n and m are positive integers and n, m > N, then |fn — fm| < eg. A subset M of V is said to be sequentially relatively uniformly complete, written s.r.u.-complete, whenever every relatively uniform Cauchy sequence of points of M (with regulator in V) converges to a point of M. This property was defined by Luxemburg and Moore in [4] and some related conditions were derived.

Small Riesz spaces

1989 ◽  
Vol 105 (3) ◽  
pp. 523-536 ◽  
Author(s):  
G. Buskes ◽  
A. van Rooij
Keyword(s):  
Representation Theory ◽  
Direct Method ◽  
Riesz Space ◽  
Pictorial Representation ◽  
Representation Theorems ◽  
Riesz Spaces ◽  
Number Of Authors

Many facts in the theory of general Riesz spaces are easily verified by thinking in terms of spaces of functions. A proof via this insight is said to use representation theory. In recent years a growing number of authors has successfully been trying to bypass representation theorems, judging them to be extraneous. (See, for instance, [9,10].) In spite of the positive aspects of these efforts the following can be said. Firstly, avoiding representation theory does not always make the facts transparent. Reading the more cumbersome constructions and procedures inside the Riesz space itself one feels the need for a pictorial representation with functions, and one suspects the author himself of secret heretical thoughts. Secondly, the direct method leads to repeating constructions of the same nature over and over again.

scholarly journals A Fubini Theorem in Riesz spaces for the Kurzweil-Henstock Integral

2011 ◽  
Vol 9 (3) ◽  
pp. 283-304 ◽  
Author(s):  
A. Boccuto ◽  
D. Candeloro ◽  
A. R. Sambucini
Keyword(s):  
Type Theorem ◽  
Riesz Space ◽  
Fubini Theorem ◽  
Real Plane ◽  
Henstock Integral ◽  
Riesz Spaces

A Fubini-type theorem is proved, for the Kurzweil-Henstock integral of Riesz-space-valued functions defined on (not necessarily bounded) subrectangles of the “extended” real plane.

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