On lacunary double statistical convergence in locally solid riesz spaces

R E S E A R C H Open Access

On lacunary double statistical convergence in locally solid Riesz spaces

Ekrem Savas^*

*Correspondence:

ekremsavas@yahoo.com Department of Mathematics, Istanbul Commerce University, Üsküdar, Istanbul, Turkey

Abstract

The concept of statistical convergence is one of the most active areas of research in the field of summability. Most of the new summability methods have relation with this popular method. In this paper, we introduce the concept of double

Iθ-statistical-τ-convergence which is a more general idea of statistical convergence.

We also investigate the ideas of doubleIθ-statistical-τ-boundedness and double Iθ-statistical-τ-Cauchy condition of sequences in the framework of locally solid Riesz space endowed with a topologyτ and investigate some of their consequences.

MSC: 40G15; 40A35; 46A40

Keywords: ideal; filter; double I-statistical-τ-convergence; double Iθ-statistical-τ-convergence; doubleIθ-statistical-τ-boundedness; double Iθ-statistical-τ-Cauchy condition

1 Introduction

The notion of statistical convergence, which is an extension of the idea of usual conver- gence, was introduced by Fast [], Steinhaus [] independently in the same year  and also by Schoenberg []. Its topological consequences were studied first by Fridy [] and Šalát []. The notion has also been defined and studied in different steps, for example, in a locally convex space []; in topological groups [, ]; in probabilistic normed spaces [,

], in intuitionistic fuzzy normed spaces [], in random -normed spaces []. In []

Albayrak and Pehlivan studied this notion in locally solid Riesz spaces. Recently, Mohi- uddine et al. [] studied statistically convergent, statistically bounded and statistically Cauchy for double sequences in locally solid Riesz spaces. Also, in [] Mohiuddine et al.

introduced the concept of lacunary statistical convergence, lacunary statistically bounded and lacunary statistically Cauchy in the framework of locally solid Riesz spaces. Quite re- cently, Das and Savas [] introduced the ideas ofIτ-convergence,Iτ-boundedness and Iτ-Cauchy condition of nets in a locally solid Riesz space.

The more general idea of lacunary statistical convergence was introduced by Fridy and Orhan in []. Subsequently, a lot of interesting investigations have been done on this convergence (see, for example, [–] where more references can be found).

The idea of statistical convergence was further extended toI-convergence in [] using the notion of ideals ofN with many interesting consequences. More investigations in this direction and more applications of ideals can be found in [–] where many important references can be found.

© 2013 Savas; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu- tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Recently in [, ] we used ideals to introduce the concepts ofI^λ-statistical conver- gence andI-lacunary-statistical convergence and investigated their properties.

The notion of a Riesz space was first introduced by Riesz [] in , and since then it has found several applications in measure theory, operator theory, optimization and also in economics (see []). It is well known that a topology on a vector space that makes the operations of addition and scalar multiplication continuous is called a linear topology and a vector space endowed with a linear topology is called a topological vector space. A Riesz space is an ordered vector space which is also a lattice endowed with a linear topology.

Further, if it has a base consisting of solid sets at zero, then it is known as a locally solid Riesz space.

In this paper, we introduce the idea ofI-double lacunary statistical convergence in a locally solid Riesz space and study some of its properties by using the mathematical tools of the theory of topological vector spaces.

2 Preliminaries

We now recall the following basic facts from [].

A familyI of subsets of a non-empty set X is said to be an ideal if (i) A, B ∈ I implies A∪ B ∈I, (ii) A ∈ I, B ⊂ A imply B ∈ I. I is called non-trivial if I = {φ} and X /∈ I. I is admissible if it contains all singletons. IfI is a proper non-trivial ideal, then the family of sets F(I) = {M ⊂ X : M^c∈I} is a filter on X (where c stands for the complement). It is called the filter associated with the idealI.

We also recall some of the basic concepts of Riesz spaces.

Definition . Let L be a real vector space and let ≤ be a partial order on this space. L is said to be an ordered vector space if it satisfies the following properties:

(i) If x, y∈ L and y ≤ x, then y + z ≤ x + z for each z ∈ L.

(ii) If x, y∈ L and y ≤ x, then λy ≤ λx for each λ ≥ .

If in addition L is a lattice with respect to the partial ordering, then L is said to be a Riesz space (or a vector lattice).

For an element x of a Riesz space L, the positive part of x is defined by x⁺= 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 element zero of L.

A subset S of a Riesz space L is said to be solid if y∈ S and |x| ≤ |y| imply x ∈ S.

A topologyτ on a real vector space L that makes the addition and scalar multiplication continuous is said to be a linear topology, that is, when the mappings

(x, y)→ x + y 

from (L× L, τ × τ) → (L, τ) , (λ, x) → λx 

from (R× L, σ × τ) → (L, τ)

are continuous, whereσ is the usual topology on R. In this case, the pair (L, τ) is called a topological vector space.

Every linear topologyτ on a vector space L has a baseN for the neighborhoods of θ satisfying the following properties:

(a) Each V∈N is a balanced set, that is, λx ∈ V holds for all x ∈ V and every λ ∈ R with|λ| ≤ .

(b) Each V∈N is an absorbing set, that is, for every x ∈ L, there exists a λ >  such that λx ∈ V .

(c) For each V∈N , there exists some W ∈ N with W + W ⊂ V .

Definition . A linear topology τ on a Riesz space L is said to be locally solid if τ has a base at zero consisting of solid sets. A locally solid Riesz space (L,τ) is a Riesz space L equipped with a locally solid topologyτ .

Nsolwill stand for a base at zero consisting of solid sets and satisfying the properties (a), (b) and (c) in a locally solid topology.

3 Main results

The notion of statistical convergence depends on the density of subsets ofN, the set of natural numbers. A subset E ofN is said to have density δ(E) if

δ(E) = lim_n→∞ n

n k=

χE(k) exists.

Note that if K⊂ N is a finite set, then δ(K) = , and for any set K ⊂ N, δ(K^C) =  –δ(K).

Definition . A sequence x = (xk) is said to be statistically convergent to if for every ε > ,

δ

k∈ N : |xk– | ≥ ε

= .

In another direction, a new type of convergence called lacunary statistical convergence was introduced in [] as follows. A lacunary sequence is an increasing integer sequence θ = {kr}r∈N∪{} such that k=  and hr= kr– kr–→ ∞ as r → ∞. Let Ir= (kr–, kr] and q_r=_k^kr

r–. A sequence (x_k) of real numbers is said to be lacunary statistically convergent to L (or, S_θ-convergent to L) if for any > ,

r→∞lim

 hr

k∈ Ir:|xk– L| ≥ = ,

where|A| denotes the cardinality of A ⊂ N. In [] the relation between lacunary statistical convergence and statistical convergence was established among other things.

We now have the following definitions.

Definition . (See [, ]) LetI ⊂ ^Nbe a proper admissible ideal inN. The sequence (x_k) of elements ofR is said to beI-convergent to L ∈ R if for each  > , the set A() = {n ∈ N : |xk– L| ≥ } ∈I. The class of all I-statistically convergent sequences will be denoted by S(I).

Definition . ([]) Let θ be a lacunary sequence. A sequence x = (xk) is said to beI- lacunary statistically convergent to L or Sθ(I)-convergent to L if for any >  and δ > ,



r∈ N : 

h_rk∈ Ir:|xk– L| ≥  ≥δ

∈I.

In this case, we write xk→ L(Sθ(I)). The class of all I-lacunary statistically convergent sequences will be denoted by S_θ(I).

It can be checked, as in the case of statistically and lacunary statistically convergent sequences, that both S(I) and Sθ(I) are linear subspaces of the space of all real sequences.

Remark . ForI = Ifin={A ⊆ N : A is a finite subset}, I – Sθ-convergence coincides with lacunary statistical convergence which is defined in [].

Let E⊆ N × N be a two-dimensional set of positive integers and let Em,nbe the numbers of (i, j) in K such that i≤ n and j ≤ m. Then the lower asymptotic density δ(E) of E is defined as follows:

lim inf

m,n

E_m,n

mn =δ(K ).

In the case when the sequence (^E_mn^m,n)^∞,∞_m,n=,has a limit, we say that E has a natural density and is defined as follows:

limm,n

E_m,n

mn =δ(E).

For example, let E ={(i^, j^) : (i, j)∈ N × N}. Then

δ(E) = lim

m,n

E_m,n mn ≤ lim

m,n

√m√ n mn =  (i.e., the set E has double natural density zero).

Recently, Mursaleen and Edely [] presented the notion of statistical convergence for a double sequence x = (xkl) as follows:

A real double sequence x = (x_kl) is said to be statistically convergent to L provided that for each > ,

limm,n



mn(k, l) : k≤ m and l ≤ n, |xk,l– L| ≥ = .

The double sequenceθr,s={(kr, ls)} is called double lacunary if there exist two increasing sequences of integers such that

k= , hr= kr– kk–→ ∞ as r → ∞

and

l_= , ¯h_s= l_s– l_s–→ ∞ as s → ∞.

Let us denote kr,s = krls, hr,s = hr¯hs and θr,s is determined by Ir,s ={(k, l) : kr–< k ≤ krand ls–< l≤ ls}.

We have the following.

Definition . Let (xkl) be a sequence in a locally solid Riesz space (L,τ). We say that x is Iθr,s-statistically-τ -convergent to xif for everyτ -neighborhood U of zero and for δ > ,



(r, s)∈ N × N : 

h_rs(k, l)∈ Irs: xkl– x∈ U ≥δ/

∈I.

In this case, we writeIθr,s– st_τ– lim xkl= x(or xkl I_θr,s–stτ

––––––→ xin brief ).

Remark . ForI = Ifin,Iθr,s-statistical-τ -convergence becomes double lacunary statis- ticalτ -convergence in a locally solid Riesz space.

Definition . Let (xkl) be a sequence in a locally solid Riesz space (L,τ). We say that x is Iθrs-statistically-τ -bounded if for every τ -neighborhood U of zero and δ > , there exists α >  such that



(r, s)∈ N × N :  hrs

(k, l)∈ Irs:αxkl∈ U ≥δ/

∈I.

Definition . Let (xkl) be a sequence in a locally solid Riesz space (L,τ). We say that x isIθr,s-statistically-τ -Cauchy if for every τ -neighborhood U of zero and δ > , there exist p, q∈ N such that



(r, s)∈ N × N :  hrs

(k, l)∈ Irs: xkl– xpq∈ U ≥δ/

∈I.

Now we are ready to present some basic properties of this new convergence in a locally solid Riesz space.

Theorem . Let (L, τ) be a Hausdorff locally solid Riesz space, x = (xkl) and y = (y_kl) be two sequences in L. Then the following hold:

(a) IfIθr,s– st_τ– lim x_kl= y_andIθr,s– st_τ– lim x_kl= z_, then y= z_. (b) IfIθr,s– st_τ – lim xkl= x, thenIθr,s– st_τ– limαxkl=αxfor each α ∈ R.

(c) IfIθr,s– st_τ– lim xkl= xandIθr,s– st_τ– lim ykl= y, then Iθr,s– st_τ– lim(xkl+ ykl) = x+ y.

Proof (a) Let U be anyτ -neighborhood of zero. Then there exists a V ∈Nsol such that V⊂ U. Take a W ∈Nsolsuch that W + W⊂ V . Let δ =^_. SinceIθr,s– st_τ– lim x_kl= y_and I_θr,s– st_τ– lim x_kl= z_, we write

K_=



(r, s)∈ N × N :  hrs

(k, l)∈ Irs: x_kl– y_∈ W/ <δ

∈ F(I)

and K_=



(r, s)∈ N × N :  hrs

(k, l)∈ Irs: x_kl– z_∈ W/ <δ

∈ F(I).

Then K = K∩ K∈ F(I) and for r, s ∈ K,

 hrs

(k, l)∈ Irs: x_kl– y_∈ W/ <δ,

i.e.,

 hrs

(k, l)∈ Irs: x_kl– y_∈ W>  –δ =

. ()

Similarly,

 hrs

(k, l)∈ Irs: x_kl– z_∈ W>

. ()

Now write that{(k, l) ∈ Irs: x(k,l)– y∈ W} and {(k, l) ∈ Irs: xkl– z∈ W} cannot be disjoint, for then we will have _h^

rs|{(k, l) ∈ Irs}| >^_, which is impossible. So, there is a (kr, ls)∈ Irsfor which

xkr,ls– y∈ W and xkr,ls– z∈ W.

Then

x– z= y– xkrls+ xkrls– z∈ W + W ⊂ V ⊂ U.

Thus y– z∈ U for every τ -neighborhood U of zero. Since (L, τ) is Hausdorff, the in- tersection of allτ -neighborhoods of zero is the singleton {θr,s}, and so y– z=θ, i.e., y= z.

(b) LetIθr,s– st_τ– lim xk= xand let U be an arbitraryτ -neighborhood of zero. Choose V∈Nsolsuch that V⊂ U. For any  > δ > ,

K =



(r, s)∈ N × N : 

h_rs(k, l)∈ Irs: xkl– x∈ V/ <δ

∈ F(I),

i.e.,∀r, s ∈ K,



h_rs(k, l)∈ Irs: xkl– x∈ V>  –δ.

First let|α| ≤ . Since V is balanced, xkl– x_∈ V implies that α(xkl– x_)∈ V . Therefore

(k, l)∈ Irs:αxkl–αx∈ V

⊃

(k, l)∈ Irs: x_kl– x_∈ V ,

and so∀r, s ∈ K,



h_rs(k, l)∈ Irs:αxkl–αx∈ V ≥ 

h_rs(k, l)∈ Irs: xkl– x∈ V>  –δ, which implies that



(r, s)∈ N × N :  hrs

(k, l)∈ Irs:αxkl–αx∈ V/ <δ

⊃ K

and finally



(r, s)∈ N × N : 

h_rs(k, l)∈ Irs:αxkl–αx∈ V/ <δ

∈ F(I).

If|α| >  and [|α|] is the smallest integer greater or equal to |α|, choose W ∈Nsolsuch that [|α|]W ⊂ V . Again, for  > δ > , taking

K =



(r, s)∈ N × N :  hrs

(k, l)∈ Irs: xkl– x∈ W/ <δ

∈ F(I)

and in view of the fact that

|αx–αxkl| = |α||x– xkl| ≤

|α|

|xnm– x| ∈

|α|

W⊂ V ⊂ Um,

which implies thatαx–αxkl∈ V ⊂ U, proceeding as before, we conclude that



(r, s)∈ N × N :  hrs

(k, l)∈ Irs:αxkl–αx∈ U/ <δ

∈ F(I).

This proves that I_θr,s– st_τ– limαxkl=αx.

(c) Let U be an arbitraryτ -neighborhood of zero. Then there are V, W ∈Nsolsuch that W + W⊂ V ⊂ U. Since Iθ– st_τ– lim xkl= xand I_θr,s– st_τ– lim ykl= y, we get, for  <δ < ,

K_=



(r, s)∈ N × N :  hrs

(k, l)∈ Irs: x_kl– x_∈ W/ <δ



∈ F(I)

and

K=



(r, s)∈ N × N :  hrs

(k, l)∈ Irs: ykl– y∈ W/ <δ



∈ F(I).

If K = K_∩ K, then∀r, s ∈ K,



h_rs(k, l)∈ Irs: xkl– x∈ W/ <δ

, i.e.,



h_rs(k, l)∈ Irs: xkl– x∈ W>  –δ

 and also

 hrs

(k, l)∈ Irs: ykl– y∈ W/ <δ

. But

(xkl+ ykl) – (x+ y) = (xkl– x) + (ykl– y)∈ W + W ⊂ V ⊂ U

∀(k, l) ∈ Irs such that k, l ∈ A ∩ B when {(k, l) ∈ Irs : xkl – x ∈ W} = A (say) and {(k, l) ∈ Irs: y_kl– y_∈ W} = B (say). Note that

|A| = |A ∩ B| + |A\B| ≤ |A ∩ B| +B^c,

i.e.,



hrs|A| ≤ 

hrs|A ∩ B| +  hrs

B^c

< 

hrs|A ∩ B| + δ

, i.e.,



hrs|A ∩ B| =  hrs

(k, l)∈ Irs: xkl– x∈ W ∧ ykl– y∈ W

>  hrs

(k, l)∈ Irs: xkl– x∈ W–δ



>  –δ

–δ



>  –δ.

Since

(k, l)∈ Irs: (xkl+ ykl) – (x+ y)∈ U

⊃ A ∩ B,

so for all r, s∈ K,



h_rs(k, l)∈ Irs: (xkl+ ykl) – (x+ y)∈ U ≥ 

h_rs|A ∩ B| >  – δ, i.e.,



h_rs(k, l)∈ Irs: (xkl+ ykl) – (x+ y) /∈ U<δ.

Hence

K⊂



(r, s)∈ N × N : 

h_rs(k, l)∈ Irs: (xkl+ ykl) – (x+ y) /∈ U<δ

and so



(r, s)∈ N × N :  hrs

(k, l)∈ Irs: (xkl+ ykl) – (x+ y)∈ U<δ

∈ F(I).

This completes the proof of the theorem. 

Theorem . Let (L, τ) be a locally solid Riesz space. Let x = {xkl}, y = {ykl} and z = {zkl} be three sequences in L such that xkl≤ ykl≤ zklfor each (k, l)∈ N × N. IfIθr,s– st_τ – lim xkl= a =Iθr,s– st_τ – lim znm, thenIθr,s– st_τ– lim ykl= a.

Proof Let U be an arbitraryτ -neighborhood of zero. Take V, W ∈Nsolsuch that W + W⊂ V⊂ U. SinceIθr,s– st_τ– lim xkl= a =Iθr,s– st_τ– lim zkl, so for  <δ < ,

K=



(r, s)∈ N × N : 

h_rs(k, l)∈ Irs: xkl– a /∈ W<δ



∈ F(I)

and K=



(r, s)∈ N × N :  hrs

(k, l)∈ Irs: zkl– a /∈ W< δ



∈ F(I).

Hence we observe that∀r, s ∈ K,



h_rs(k, l)∈ Irs: xkl– a /∈ W< δ

, i.e.,

 hrs

(k, l)∈ Irs: x_kl– a∈ W>  –δ

 and



h_rs(k, l)∈ Irs: zkl– a /∈ W<δ

.

Writing A ={(k, l) ∈ Irs: xkl– a∈ W} and B = {(k, l) ∈ Irs: zkl– a∈ W}, we see that ∀k, l ∈ A∩ B,

xkl≤ ykl≤ zkl,

x_kl– a≤ ykl– a≤ zkl– a,

|ykl– a| ≤ |xkl– a| + |zkl– a| ∈ W + W ⊂ V,

and as V is solid, so y_kl– a∈ V ⊂ U.

Clearly,{(k, l) ∈ Irs: y_kl– a∈ U} ⊃ A ∩ B and as in the previous theorem, we show that

∀r, s ∈ K,

 hrs

(k, l)∈ Irs: y_kl– a∈ U ≥ 

hrs|A ∩ B| >  – δ, i.e.,

 hrs

(k, l)∈ Irs: y_kl– a /∈ U<δ.

Hence



(r, s)∈ N × N :  hrs

(k, l)∈ Irs: y_kl– a /∈ U<δ

⊃ K,

where K∈ F(I) and so



(r, s)∈ N × N : 

h_rs{k∈ Irs: ykl– a /∈ U} ≥δ

∈I.

This proves thatIθr,s– st_τ– lim ykl= a. This completes the proof of the theorem. 

Theorem . AnIθr,s-statisticallyτ -convergent sequence (xkl) in a locally solid Riesz space (L,τ) isIθr,s-statisticallyτ -bounded.

Proof Let (xkl) be I_θr,s-statistically τ -convergent to x∈ L. Let U be an arbitrary τ - neighborhood of zero. Choose V , W∈Nsolsuch that W + W⊂ V ⊂ U. Since W is ab- sorbing, there is aμ >  such that μx∈ W. Choose α ≤  so that α ≤ μ. Since W is solid and|λx| ≤ |μx|, we have αx∈ W. Again, as W is balanced, xkl– x∈ W implies that α(xnm– x)∈ W. Now, for any  < δ < ,

K =



(r, s)∈ N × N : 

h_rs(k, l)∈ Irs: xkl– x∈ W/ <δ

∈ F(I).

Thus, for all r, s∈ K,



h_rs(k, l)∈ Irs: xkl– x∈ W/ <δ, i.e.,



h_rs(k, l)∈ Irs: xkl– x∈ W>  –δ.

If B_rs={(k, l) ∈ Irs: x_kl– x_∈ W}, then ∀k, l ∈ Brs

αxkl=α(xkl– x_) +αx∈ W + W ⊂ V ⊂ U,

and so, for all r, s∈ K,



h_rs(k, l)∈ Irs:αxkl∈ W ≥ 

h_rs(k, l)∈ Irs: xkl– x∈ W

>  –δ

i.e.,

 hrs

(k, l)∈ Irs:αxkl∈ W/ <δ.

Hence

K⊂



(r, s)∈ N × N :  hrs

(k, l)∈ Irs:αxkl∈ W/ <δ

.

Since K∈ F(I), so the set on the right-hand side also belongs to F(I) and this proves that

(x_kl) isIθr,s-statisticallyτ -bounded. 

Theorem . If a sequence (xkl) in a locally solid Riesz space (L,τ) isIθr,s-statistically τ -convergent, then it isIθr,s-statisticallyτ -Cauchy.

Proof Let (x_kl) be Iθr,s-statistically τ -convergent to x ∈ L. Let U be an arbitrary τ - neighborhood of zero. Choose V , W ∈Nsol such that W + W⊂ V ⊂ U. Let  < δ < .

Therefore

K =



(r, s)∈ N × N :  hrs

(k, l)∈ Irs: xkl– x∈ W/ <δ

∈ F(I).

For all r, s∈ K,

 hrs

(k, l)∈ Irs: x_kl– x_∈ W/ <δ,

i.e.,

 hrs

(k, l)∈ Irs: x_kl– x_∈ W>  –δ.

Take r, s∈ K and in view of the above, we can choose p, q ∈ {(k, l) ∈ Irs : xkl– x ∈ W} (since this set cannot be empty). Then xpq – x_ ∈ W. Now observe that if for (k, l)∈ Irs, x_kl– x_∈ W, then

xkl– xpq= xkl– x+ x– xpq∈ W + W ⊂ V ⊂ U.

Hence, as in the earlier proofs, we can prove that

K⊂



(r, s)∈ N × N : 

h_rs(k, l)∈ Irs: xkl– xpq∈ W/ <δ

,

which consequently implies that (xkl) isIθr,s-statisticallyτ -Cauchy.

This completes the proof of the theorem. 

It should be noted that single and double case of I_λ-statistical convergence in locally solid Riesz spaces are introduced in [] and [] respectively.

Competing interests

The author declares that they have no competing interests.

Received: 24 October 2012 Accepted: 15 February 2013 Published: 13 March 2013 References

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doi:10.1186/1029-242X-2013-99

Cite this article as: Savas: On lacunary double statistical convergence in locally solid Riesz spaces. Journal of Inequalities and Applications 2013 2013:99.