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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 719729, 9 pages
http://dx.doi.org/10.1155/2012/719729
Research Article

Statistical Convergence of Double Sequences in Locally Solid Riesz Spaces

1Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Received 26 May 2012; Accepted 13 August 2012

Academic Editor: Ziemowit Popowicz

Copyright © 2012 S. A. Mohiuddine et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 statistical -convergence, statistical -Cauchy and -convergence of double sequences in a locally solid Riesz space.

1. Introduction and Preliminaries

The notion of statistical convergence was introduced by Fast [1] and Steinhaus [2] independently in the same year 1951. Actually, Henry Fast had heard about this concept from Steinhaus, but in fact it was Antoni Zygmund who proved theorems on the statistical convergence of Fourier series in the first edition of his book ([3], pp. 181–188) where he used the term “almost convergence” in place of statistical convergence and at that time this idea was not recognized much. Since the term “almost convergence” was already in use (see Lorentz [4]), Fast had to choose a different name for his concept and “statistical convergence” was mostly the suitable one. Active researches on this topic started after the papers of Fridy [5] and since then a huge amount of literature has appeared. It developed so rapidly like an explosion when the summability theory was at the dying stage. Various extensions, generalizations, variants, and applications have been given by several authors so far, for example [616]. This notion has also been defined and studied in different setups, for example in a locally convex space [17]; in topological groups [18, 19]; in probabilistic normed spaces [20]; in intuitionistic fuzzy normed spaces [21]; in fuzzy/random 2-normed space [22, 23]. Recently, Albayrak and Pehlivan [24] studied this notion in locally solid Riesz spaces. In this paper, we study statistically convergent, statistically bounded, and statistically Cauchy double sequences in locally solid Riesz spaces.

Let be a real vector space and ≤ a partial order on this space. Then, is said to be an ordered vector space if it satisfies the following properties:(i)if and , then for each .(ii)if and , then for each .

If in addition is a lattice with respect to the partially order , then is said to be a Riesz space (or a vector lattice) [25].

For an element of a Riesz space , the positive part of is defined by , the negative part of by and the absolute value of by , where is the zero element of .

A subset of a Riesz space is said to be solid if and implies .

A topological vector space is a vector space which has a (linear) topology such that the algebraic operations of addition and scalar multiplication in are continuous. Continuity of addition means that the function defined by is continuous on , and continuity of scalar multiplication means that the function defined by is continuous on .

Every linear topology on a vector space has a base for the neighborhoods of satisfying the following properties:(C1) each is a balanced set, that is, holds for all and every with .(C2) each is an absorbing set, that is, for every , there exists such that .(C3) for each there exists some with .

A linear topology on a Riesz space is said to be locally solid [26] if has a base at zero consisting of solid sets. A locally solid Riesz space is a Riesz space equipped with a locally solid topology .

By the convergence of a double sequence we mean the convergence in the Pringsheim's sense [27]. A double sequence is said to converge to the limit in Pringsheim's sense (shortly, -convergent to ) if for every there exists an integer such that whenever . In this case is called the -limit of .

A double sequence of real or complex numbers is said to be bounded if . The space of all bounded double sequences is denoted by .

2. Statistical -Convergence

Let be a two-dimensional set of positive integers and let . Then the two-dimensional analogue of natural density can be defined as follows.

In case the sequence has a limit in Pringsheim's sense, then we say that has a double natural density and is defined as

For example, let . Then, That is, the set has double natural density zero, while the set has double natural density .

A real double sequence is said to be statistically convergent (see [2830]) to the number if for each , the set has double natural density zero.

We shall assume throughout this paper that the symbol will denote any base at zero consisting of solid sets and satisfying the conditions , , and in a locally solid topology.

Definition 2.1. Let be a locally solid Riesz space. Then, a double sequence in is said to be statistically -convergent to the number if for every -neighborhood of zero: In this case, we write or .

Definition 2.2. Let be a locally solid Riesz space. We say that a double sequence in is statistically -bounded if for every -neighborhood of zero there exists some , such that the set has double natural density zero.

Theorem 2.3. Let be a Hausdorff locally solid Riesz space and and be two double sequences in . Then, the following hold:(i)if and , then .(ii)if , then , .(iii)if and , then .

Proof. (i) Suppose that and . Let be any -neighborhood of zero. Then, there exists such that . Choose any such that . We define the following sets: Since and , we have . Thus, , and in particular . Now, let . Then Hence for every -neighborhood of zero, we have . Since is Hausdorff, the intersection of all -neighborhoods of zero is the singleton set . Thus, we get , that is, .
(ii) Let be an arbitrary -neighborhood of zero and . Then there exists such that and also Since is balanced, implies that for every with . Hence, Thus, we obtain for each -neighborhood of zero. Now let and be the smallest integer greater than or equal to . There exists such that . Since , the set has double natural density zero. Therefore, Since the set is solid, we have . This implies that . Thus, for each -neighborhood of zero. Hence, .
(iii) Let be an arbitrary -neighborhood of zero. Then there exists such that . Choose in such that . Since and , we have , where Let . Hence, we have and Therefore, Since is arbitrary, we have .
This completes the proof of the theorem.

Theorem 2.4. Let be a locally solid Riesz space. If a double sequence is statistically -convergent, then it is statistically -bounded.

Proof. Suppose is statistically -convergent to the point and let be an arbitrary -neighborhood of zero. Then, there exists such that . Let us choose such that . Since , the set has double natural density zero. Since is absorbing, there exists such that . Let be such that and . Since is solid and , we have . Since is balanced, implies that . Then, we have for each . Thus Hence, is statistically -bounded.
This completes the proof of the theorem.

Theorem 2.5. Let be a locally solid Riesz space. If , , and are three double sequences such that(i) for all ;(ii);
then .

Proof. Let be an arbitrary -neighborhood of zero, there exists such that . Choose such that . From condition (ii), we have , where Now, let . Then, from (i), we have this implies for each . Since is solid, we have . Thus, for each -neighborhood of zero. Hence .
This completes the proof of the theorem.

3. Statistically -Cauchy and -Convergence

Definition 3.1. Let be a locally solid Riesz space. A double sequence in is statistically -Cauchy if for every -neighborhood of zero there exist such that for all , , the set has double natural density zero.

Theorem 3.2. Let be a locally solid Riesz space. If a double sequence is statistically -convergent, then it is statistically -Cauchy.

Proof. Suppose that . Let be an arbitrary -neighborhood of zero, there exists such that . Choose such that . Then, Also, we have for all , where Therefore, the set For every -neighborhood of zero there exist such that for all , , the set and has double natural density zero. Hence, is statistically -Cauchy.
This completes the proof of the theorem.

In [30], it was shown that a real double sequence is statistically convergent to a number if and only if there exists a subset such that and .

This fact suggests defining further another type of convergence in locally solid Riesz spaces.

Definition 3.3. A sequence in a locally solid Riesz space is said to -convergent to if there exists a set , with such that . In this case, we write .

Theorem 3.4. A double sequence is statistically -convergent to a number if it is -convergent to in a locally solid Riesz space .

Proof. Let be an arbitrary -neighborhood of . Since is -convergent to , there is a set , with and ,   such that ,  , and imply that . Then, Therefore, Hence, is statistically -convergent to .
This completes the proof of the theorem.

Note that the converse holds for first countable space.

Recall that a first countable space is a topological space satisfying the “first axiom of countability.” Specifically, a space is said to be first countable if each point has a countable neighbourhood basis (local base), that is, for each point in there exists a sequence of open neighbourhoods of such that for any open neighbourhood of there exists an integer with contained in .

Theorem 3.5. Let be a first countable locally solid Riesz space. If a double sequence is statistically -convergent to a number , then it is -convergent to .

Proof. Let be statistically -convergent to a number . Fix a countable local base at . For every , put Then, and(1); (2).
Now we have to show that for is convergent to . Suppose that is not convergent to . Therefore, for infinitely many terms. Let Then,(3), and by (1), . Hence, which contradicts (2). Therefore, is convergent to . Hence by Definition 3.3, is -convergent to .
This completes the proof of the theorem.

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