# Statistical Convergence of Double Sequences in Locally Solid Riesz Spaces

Authors: S. A. Mohiuddine; Abdullah Alotaibi; M. Mursaleen
Journal: Abstract and Applied Analysis
 (2012)

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## 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 statisticalτ-convergence, statistical τ-Cauchy and S∗(τ)-convergence of double sequences in a locally solid Riesz space.


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## Body

## 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 [6–16]. 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.LetX be a real vector space and ≤ a partial order on this space. Then, X is said to be an ordered vector space if it satisfies the following properties:(i)
ifx,y∈X and y≤x, then y+z≤x+z for each z∈X.(ii)
ifx,y∈X and y≤x, then λy≤λx for each λ≥0.If in additionX is a lattice with respect to the partially order ≤, then X is said to be a Riesz space (or a vector lattice) [25].For an elementx of a Riesz space X, the positive part of x is defined by x+=x∨θ=sup{x,θ}, the negative part of x by x-=(-x)∨θ and the absolute value of x by |x|=x∨(-x), where θ is the zero element of X.A subsetS of a Riesz space X is said to be solid if y∈S and |x|≤|y| implies x∈S.Atopological vector space(X,τ) is a vector space X which has a (linear) topology τ such that the algebraic operations of addition and scalar multiplication in X are continuous. Continuity of addition means that the function f:X×X→X defined by f(x,y)=x+y is continuous on X×X, and continuity of scalar multiplication means that the function f:ℂ×X→X defined by f(λ,x)=λx is continuous on ℂ×X.Every linear topologyτ on a vector space X has a base 𝒩 for the neighborhoods of θ satisfying the following properties:(C1)
eachY∈𝒩 is a balanced set, that is, λx∈Y holds for all x∈Y and every λ∈ℝ with |λ|≤1.(C2)
eachY∈𝒩 is an absorbing set, that is, for every x∈X, there exists λ>0 such that λx∈Y.(C3)
for eachY∈𝒩 there exists some E∈𝒩 with E+E⊆Y.A linear topologyτ on a Riesz space X is said to be locally solid [26] if τ has a base at zero consisting of solid sets. A locally solid Riesz space(X,τ) is a Riesz space equipped with a locally solid topology τ.By the convergence of a double sequence we mean the convergence in the Pringsheim's sense [27]. A double sequence x=(xjk) is said to converge to the limitLin Pringsheim's sense (shortly, P-convergent toL) if for every ε>0 there exists an integer N such that |xjk-L|<ε whenever j,k>N. In this case L is called the P-limit of x.A double sequencex=(xjk) of real or complex numbers is said to be bounded if ∥x∥∞=supj,k|xjk|<∞. The space of all bounded double sequences is denoted by ℳu.


## 2. Statisticalτ-Convergence

LetK⊆ℕ×ℕ be a two-dimensional set of positive integers and let K(m,n)={(j,k):j≤m,k≤n}. Then the two-dimensional analogue of natural density can be defined as follows.In case the sequence(K(m,n)/mn) has a limit in Pringsheim's sense, then we say that K has a double natural density and is defined as
(2.1)P-limm,nK(m,n)mn=δ2(K).For example, letK={(i2,j2):i,j∈ℕ}. Then,
(2.2)δ2(K)=P-limm,nK(m,n)mn≤P-limm,nmnmn=0,
That is, the set K has double natural density zero, while the set {(i,2j):i,j∈ℕ} has double natural density 1/2.A real double sequencex=(xjk) is said to be statistically convergent (see [28–30]) to the number ℓ if for each ϵ>0, the set
(2.3){(j,k),j≤m,k≤n:|xjk-ℓ|≥ϵ}
has double natural density zero.We shall assume throughout this paper that the symbol𝒩sol will denote any base at zero consisting of solid sets and satisfying the conditions (C1), (C2), and (C3) in a locally solid topology.Definition 2.1.
Let(X,τ) be a locally solid Riesz space. Then, a double sequence x=(xjk) in X is said to be statisticallyτ-convergent to the number ξ∈X if for every τ-neighborhood U of zero:
(2.4)P-limm,n→∞1mn|{(j,k),j≤m,k≤n:xjk-ξ∉U}|=0.
In this case, we write 𝒮(τ)-limx=ξ or xjk→𝒮(τ)ξ.Definition 2.2.
Let(X,τ) be a locally solid Riesz space. We say that a double sequence x=(xjk) in X is statisticallyτ-bounded if for every τ-neighborhood U of zero there exists some λ>0, such that the set
(2.5){(j,k),j≤m,k≤n:λxjk∉U}
has double natural density zero.Theorem 2.3.
Let(X,τ) be a Hausdorff locally solid Riesz space and x=(xjk) and y=(yjk) be two double sequences in X. Then, the following hold:(i)
if𝒮(τ)-limj,kxjk=ξ1 and 𝒮(τ)-limj,kxjk=ξ2, then ξ1=ξ2.(ii)
if𝒮(τ)-limj,kxjk=ξ, then 𝒮(τ)-limj,kαxjk=αξ, α∈ℝ.(iii)
if𝒮(τ)-limj,kxjk=ξ and 𝒮(τ)-limj,kyjk=η, then 𝒮(τ)-limj,k(xjk+yjk)=ξ+η.Proof.
(i) Suppose that𝒮(τ)-limj,kxjk=ξ1 and 𝒮(τ)-limj,kxjk=ξ2. Let U be any τ-neighborhood of zero. Then, there exists Y∈𝒩sol such that Y⊆U. Choose any E∈𝒩sol such that E+E⊆Y. We define the following sets:
(2.6)K1={(j,k),j≤m,k≤n:xjk-ξ1∈E},K2={(j,k),j≤m,k≤n:xjk-ξ2∈E}.
Since 𝒮(τ)-limj,kxjk=ξ1 and 𝒮(τ)-limj,kxjk=ξ2, we have δ2(K1)=δ2(K2)=1. Thus, δ2(K1∩K2)=1, and in particular K1∩K2≠∅. Now, let (j,k)∈K1∩K2. Then
(2.7)ξ1-ξ2=ξ1-xjk+xjk-ξ2∈E+E⊆Y⊆U.
Hence for every τ-neighborhood U of zero, we have ξ1-ξ2∈U. Since (X,τ) is Hausdorff, the intersection of all τ-neighborhoods U of zero is the singleton set {θ}. Thus, we get ξ1-ξ2=θ, that is, ξ1=ξ2.
(ii) LetU be an arbitrary τ-neighborhood of zero and 𝒮(τ)-limj,kxjk=ξ. Then there exists Y∈𝒩sol such that Y⊆U and also
(2.8)P-limm,n→∞1mn|{(j,k),j≤m,k≤n:xjk-ξ∈Y}|=1.
Since Y is balanced, xjk-ξ∈Y implies that α(xjk-ξ)∈Y for every α∈ℝ with |α|≤1. Hence,
(2.9){(j,k),j≤m,k≤n:xjk-ξ∈Y}⊆{(j,k),j≤m,k≤n:αxjk-αξ∈Y}⊆{(j,k),j≤m,k≤n:αxjk-αξ∈U}.
Thus, we obtain
(2.10)P-limm,n→∞1mn|{(j,k),j≤m,k≤n:αxjk-αξ∈U}|=1,
for each τ-neighborhood U of zero. Now let |α|>1 and [|α|] be the smallest integer greater than or equal to |α|. There exists E∈𝒩sol such that [|α|]E⊆Y. Since 𝒮(τ)-limj,kxjk=ξ, the set
(2.11)K={(j,k),j≤m,k≤n:xjk-ξ∈E}
has double natural density zero. Therefore,
(2.12)|αξ-αxjk|=|α||ξ-xjk|≤[|α|]|ξ-xjk|∈[|α|]E⊆Y⊆U.
Since the set Y is solid, we have αξ-αxjk∈Y. This implies that αξ-αxjk∈U. Thus,
(2.13)P-limm,n→∞1mn|{(j,k),j≤m,k≤n:αxjk-αξ∈U}|=1,
for each τ-neighborhood U of zero. Hence, 𝒮(τ)-limj,kαxjk=αξ.
(iii) LetU be an arbitrary τ-neighborhood of zero. Then there exists Y∈𝒩sol such that Y⊆U. Choose E in 𝒩sol such that E+E⊆Y. Since 𝒮(τ)-limj,kxjk=ξ and 𝒮(τ)-limj,kyjk=η, we have δ2(H1)=1=δ2(H2), where
(2.14)H1={(j,k),j≤m,k≤n:xjk-ξ∈E},H2={(j,k),j≤m,k≤n:yjk-η∈E}.
Let H=H1∩H2. Hence, we have δ2(H)=1 and
(2.15)(xjk+yjk)-(ξ+y)=(xjk-x)+(yjk-η)∈E+E⊆Y⊆U.
Therefore,
(2.16)P-limm,n→∞1mn|{(j,k),j≤m,k≤n:(xjk+yjk)-(ξ+η)∈U}|=1.
Since U is arbitrary, we have 𝒮(τ)-limj,k(xjk+yjk)=ξ+η.
This completes the proof of the theorem.Theorem 2.4.
Let(X,τ) be a locally solid Riesz space. If a double sequence x=(xjk) is statistically τ-convergent, then it is statistically τ-bounded.Proof.
Supposex=(xjk) is statistically τ-convergent to the point ξ∈X and let U be an arbitrary τ-neighborhood of zero. Then, there exists Y∈𝒩sol such that Y⊆U. Let us choose E∈𝒩sol such that E+E⊆Y. Since 𝒮(τ)-limj,k→∞xjk=ξ, the set
(2.17)K={(j,k),j≤m,k≤n:xjk-ξ∉E}
has double natural density zero. Since E is absorbing, there exists λ>0 such that λξ∈E. Let α be such that α≤1 and α≤λ. Since E is solid and |αξ|≤|λξ|, we have αξ∈E. Since E is balanced, xjk-ξ∈E implies that α(xjk-ξ)∈E. Then, we have
(2.18)αxjk=α(xjk-ξ)+αξ∈E+E⊆Y⊆U,
for each (j,k)∈ℕ×ℕ∖K. Thus
(2.19)P-limm,n→∞1mn|{(j,k),j≤m,k≤n:αxjk∉U}|=0.
Hence, (xjk) is statistically τ-bounded.
This completes the proof of the theorem.Theorem 2.5.
Let(X,τ) be a locally solid Riesz space. If (xjk), (yjk), and (zjk) are three double sequences such that(i)
xjk≤yjk≤zjk for all j,k∈ℕ;(ii)
𝒮(τ)-limj,kxjk=ξ=𝒮(τ)-limj,kzjk;
then𝒮(τ)-limj,kyjk=ξ.Proof.
LetU be an arbitrary τ-neighborhood of zero, there exists Y∈𝒩sol such that Y⊆U. Choose E∈𝒩sol such that E+E⊆Y. From condition (ii), we have δ2(A)=1=δ2(B), where
(2.20)A={(j,k),j≤m,k≤n:xjk-ξ∈E}B={(j,k),j≤m,k≤n:zjk-ξ∈E}.
Now, let Q=A∩B. Then, from (i), we have
(2.21)xjk-ξ≤yjk-ξ≤zjk-ξ,
this implies
(2.22)|yjk-ξ|≤|xjk-ξ|+|zjk-ξ|∈E+E⊆Y
for each (j,k)∈Q. Since Y is solid, we have yjk-ξ∈Y⊆U. Thus,
(2.23)P-limm,n→∞1mn|{(j,k),j≤m,k≤n:yjk-ξ∈U}|=1,
for each τ-neighborhood U of zero. Hence 𝒮(τ)-limj,kyjk=ξ.
This completes the proof of the theorem.


## 3. Statisticallyτ-Cauchy and 𝒮*(τ)-Convergence

Definition 3.1.
Let(X,τ) be a locally solid Riesz space. A double sequence x=(xjk) in X is statisticallyτ-Cauchy if for every τ-neighborhood U of zero there exist N,M∈ℕ such that for all j,p≥N, k,q≥M, the set
(3.1){(j,k),j≤m,k≤n:xjk-xpq∉U}
has double natural density zero.Theorem 3.2.
Let(X,τ) be a locally solid Riesz space. If a double sequence x=(xjk) is statistically τ-convergent, then it is statistically τ-Cauchy.Proof.
Suppose that𝒮(τ)-limj,k→∞xjk=ξ. Let U be an arbitrary τ-neighborhood of zero, there exists Y∈𝒩sol such that Y⊆U. Choose E∈𝒩sol such that E+E⊆Y. Then,
(3.2)P-limm,n→∞1mn|{(j,k),j≤m,k≤n:xjk-ξ∉E}|=0.
Also, we have
(3.3)xjk-xpq=xjk-ξ+ξ-xpq∈E+E⊆Y⊆U,
for all (j,k),(p,q)∈ℕ×ℕ∖K, where
(3.4)K={(j,k),j≤m,k≤n:xjk-ξ∉E}.
Therefore, the set
(3.5){(j,k),j≤m,k≤n:xjk-xpq∉U}⊆K.
For every τ-neighborhood U of zero there exist N,M∈ℕ such that for all j,p≥N, k,q≥M, the set {(j,k),j≤m and k≤n:xjk-xpq∉U} has double natural density zero. Hence, (xjk) is statistically τ-Cauchy.
This completes the proof of the theorem.In [30], it was shown that a real double sequence x=(xjk) is statistically convergent to a number ℓ if and only if there exists a subset K={(j,k)}⊆ℕ×ℕ,j,k=1,2,… such that δ2(K)=1 and limj,k→∞xjk=ℓ.This fact suggests defining further another type of convergence in locally solid Riesz spaces.Definition 3.3.
A sequence(xjk) in a locally solid Riesz space (X,τ) is said to 𝒮*(τ)-convergent to ξ∈X if there exists a set K={(j,k)}⊆ℕ×ℕ,j,k=1,2,…, with δ2(K)=1 such that limj,k→∞xjk=ξ. In this case, we write ξ=𝒮*(τ)-limx.Theorem 3.4.
A double sequencex=(xjk) is statistically τ-convergent to a number ξ if it is 𝒮*(τ)-convergent to ξ in a locally solid Riesz space (X,τ).Proof.
LetU be an arbitrary τ-neighborhood of ξ. Since x=(xjk) is 𝒮*(τ)-convergent to ξ, there is a set K={(j,k)}⊆ℕ×ℕ,j,k=1,2,…, with δ2(K)=1 and j0=j0(U),  k0=k0(U) such that j≥j0,  k≥k0, and (j,k)∈K imply that xjk-ξ∈U. Then,
(3.6)KU={(j,k)∈ℕ×ℕ:xjk-ξ∉U}⊆ℕ×ℕ-{(jN+1,kN+1),(jN+2,kN+2),…}.
Therefore,
(3.7)δ2(KU)≤1-1=0.
Hence, x is statistically τ-convergent to ξ.
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 spaceX is said to be first countable if each point has a countable neighbourhood basis (local base), that is, for each point x in X there exists a sequence U1,U2,… of open neighbourhoods of x such that for any open neighbourhood V of x there exists an integer i with Ui contained in V.Theorem 3.5.
Let(X,τ) be a first countable locally solid Riesz space. If a double sequence x=(xjk) is statistically τ-convergent to a number ξ, then it is 𝒮*(τ)-convergent to ξ.Proof.
Letx be statistically τ-convergent to a number ξ. Fix a countable local base U1⊃U2⊃U3⊃⋯ at ξ. For every i∈ℕ, put
(3.8)Ki={(j,k)∈ℕ×ℕ:xjk-ξ∉Ui},Mi={(j,k)∈ℕ×ℕ:xjk-ξ∈Ui}(r=1,2,3,…).
Then, δ2(Ki)=0 and(1)
M1⊃M2⊃⋯⊃Mi⊃Mi+1⊃⋯;(2)
δ2(Mi)=1,i=1,2,3,….
Now we have to show that for(j,k)∈Mi,(xjk) is convergent to ξ. Suppose that (xjk) is not convergent to ξ. Therefore, xjk∉Ui for infinitely many terms. Let
(3.9)Mr={(j,k)∈ℕ×ℕ:xjk-ξ∈Ur}(r>i).
Then,(3)
δ2(Mr)=0, and by (1), Mi⊂Mr. Hence, δ2(Mi)=0 which contradicts (2). Therefore, (xjk) is convergent to ξ. Hence by Definition 3.3, x is 𝒮*(τ)-convergent to ξ.
This completes the proof of the theorem.



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