Weighted Lacunary Statistical Convergence of Double Sequences in Locally Solid Riesz Spaces

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1 Filomat 30:3 206), DOI /FIL60362K Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: Weighted Lacunary Statistical Convergence of Double Sequences in Locally Solid Riesz Spaces Şükran Konca a a Department of Mathematics, Bitlis Eren University, 3000, Bitlis, Turkey Abstract. Recently, the notion of weighted lacunary statistical convergence is studied in a locally solid Riesz space for single sequences by Başarır and Konca [7]. In this work, we define and study weighted lacunary statistical τ-convergence, weighted lacunary statistical τ-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.. Introduction The notion of Riesz space was first introduced by F. Riesz in 928, at the International Mathematical Congress in Bologna, Italy [29]. Soon after, in the mid-thirties, H. Freudenthal [3] and L. V. Kantorovich [5] independently set up the axiomatic foundation and derived a number of properties dealing with the lattice structure of Riesz spaces. From then on the growth of the subject was rapid. In the forties and early fifties the Japanese school led by H. Nakano, T. Ogasawara and K. Yosida, and the Russian school, led by L. V. Kantorovich, A. I. Judin, and B. Z. Vulikh, made fundamental contributions. At the same time a number of books started to appear on the field. The general theory of topological Riesz spaces seems somehow to have been neglected. The recent book by D. H. Fremlin [2] is partially devoted to this subject. Riesz spaces play an important role in analysis, measure theory, operator theory and optimization. They also provide the natural framework for any modern theory of integration. Further, they have some applications in economics [2]. A Riesz space is an ordered vector space which is a lattice at the same time. A locally solid Riesz space is a Riesz space equipped with a linear topology that has a base consisting of solid sets. For recent results related this topic, we may refer, for example, [, 4, 5, 7, 4, 8 20, 22, 32]. The idea of statistical convergence was initially given by Zygmund in 935 [37]. The concept was formally introduced by Fast [] and Steinhaus [35] and later on by Schoenberg [34], and also independently by Buck [9]. By the convergence of a double sequence we mean the convergence in the Pringsheim s sense. A double sequence x = ) x k,l is said to converge to the limit L in Pringsheim s sense shortly, P-convergent to L) if for every ε > 0 there exists an integer N such that xk,l L < ε whenever k, l > N [26]. We shall write this as lim k,l x k,l = L, where k and l tending to infinity independent of each other and L is called the P-limit of 200 Mathematics Subject Classification. Primary 42B5; Secondary 40A35, 40C05, 40G5, 46A40 Keywords. Weighted lacunary double statistical convergence, double sequence, weighted means, statistical convergence, locally solid Riesz spaces Received: 0 July 205; Revised: 0 October 205; Accepted: October 205 Communicated by Ljubiša D.R. Kočinac address: skonca@beu.edu.tr Şükran Konca)

2 Ş. Konca / Filomat 30:3 206), x. A double sequence x = x k,l ) of real or complex numbers is said to be bounded if x = supk,l 0 x k,l <. Note that, in contrast to the case for single sequences, a convergent double sequence need not be bounded. We may refer to [3 5, 8, 0, 7, 9, 2 25, 28, 30 32] for further results related with the concept of double sequence. Başarır and Konca [6] defined a new concept of statistical convergence for single sequences which is called weighted lacunary statistical convergence. In [6] they defined weighted almost lacunary statistical convergence in a real n-normed space. Recently, the notion of weighted lacunary statistical convergence for single sequences is studied in a locally solid Riesz space by Başarır and Konca [7]. In this paper, we define and study weighted lacunary statistical τ-convergence, weighted lacunary statistical τ-bounded of double sequences in a locally solid Riesz space. We also prove some topological results related to these concepts in the framework of locally solid Riesz spaces. Further, we establish some inclusion relations. 2. Definitions and Preliminaries Before beginning of the presentation of the main results, we recall the following basic facts and notations, and some of the basic concepts of Riesz spaces. A topological vector space X, τ) is a vector space, 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. Let X be a real vector space and be a partial order on this space. Then X is said to be an ordered vector space if it satisfies the following properties:. If x, y X and y x, then y + z x + z for each z X. 2. If x, y X and y x, then λy λx for each λ 0. If in addition X is a lattice with respect to the partial ordering, then X is said to be a Riesz space or a vector lattice). For an element x of a Riesz space X, the positive part of x 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 [36]. A subset S of a Riesz space X is said to be solid if y S and x y implies x S. A linear topology τ on a Riesz space X is said to be locally solid 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 [27]. Every linear topology τ on a vector space X has a base N for the neighborhoods of θ satisfying the following properties: T) Each Y N is a balanced set, that is, λx Y holds for all x Y and every λ R with λ. T2) Each Y N is an absorbing set, that is, for every x X, there exists λ > 0 such that λx Y. T3) For each Y N, there exists some E N with E + E Y. Throughout the paper, the symbol N sol will stand for a base at zero consisting of solid sets and satisfying the conditions T), T2), T3) in a locally solid topology. Definition 2.. [3]) Let ) ) p n, pm be sequences of positive numbers and Pn = p + p p n, P m = p + p p m. Then the transformation given by T n,m x) = P n P m n k= m p k p l x k,l l= is called the Riesz mean of double sequence x = x k,l ). If P - limn,m T n,m x) = L, L R, then the sequence x = x k,l ) is said to be Riesz convergent to L. If x = xk,l ) is Riesz convergent to L, then we write PR - lim x = L.

3 Ş. Konca / Filomat 30:3 206), The double sequence θ r,s = {k r, l s )} is called double lacunary if there exist two increasing sequences of integers such that k 0 = 0, h r = k r k r as r and l 0 = 0, h s = l s l s as s. Let k r,s = k r l s, h r,s = h r hs and θ r,s is determined by I r,s = {k, l) : k r < k k r and l s < l l s }, q r = k r k r, q s = l s l s and q r,s = q r q s [3]. Using the notations of lacunary sequence and Riesz mean for double sequences, Başarır and Konca [7] have given a new definition: Let θ r,s = {k r, l s )} be a double lacunary sequence and let p k ), p l ) be sequences of positive real numbers such that P kr := k 0,k r ] p k and P ls := l 0,l s ] p l. If the Riesz transformation of double sequences is RHregular it maps every bounded P-convergent sequence into a P-convergent sequence with the same P-limit), then θ r,s = { )} P kr, P ls is a double lacunary sequence, that is; P0 = 0, 0 < P kr < P kr and H r = P kr P kr as r and P 0 = 0, 0 < P ls < P ls and H s = P ls P ls as s. Let P kr,s = P kr P ls, H r,s = H r H s and I r,s = { } k, l) : P kr < k P kr and P ls < l P ls, Qr = P kr P kr, Q s = P ls P ls and Q r,s = Q r Q s. If we take p k =, p l = for all k and l, then H r,s, P kr,s, Q r,s and I r,s reduce to h r,s, k r,s, q r,s and I r,s. Definition 2.2. Let θ r,s = {k r, l s )} be a double lacunary sequence. The double number sequence x = x k,l ) is said to be S R 2,θ r,s) -P-convergent to L provided that for every ε > 0, { P lim k, l) I xk,l r,s : p k p l L } ε = 0. In this case, we write S R 2,θ r,s) -P-lim k,l x k,l = L. Throughout the paper, we will use the limit notation in Pringsheim s sense P-lim r,s brevity. instead of P- lim r,s, for 3. Main Results In this section, we define the concepts of weighted lacunary statistical τ-convergence, weighted lacunary statistical τ-bounded and weighted statistical τ-convergence for double sequences in the framework of locally solid Riesz spaces and prove some topological results related to these concepts. We also examine some inclusion relations. Definition 3.. Let X, τ) be a locally solid Riesz space and θ r,s = {k r, l s )} be a double lacunary sequence. Then, a double sequence x = x k,l ) in X is said to be weighted lacunary statistically τ-convergent or S R 2,θ r,s )τ)-convergent) to the element L X if for every τ-neighborhood U of zero, the set K U H r,s ) = {k, l) N N, k, l) I r,s : p k p l x k,l L) U} has weighted double lacunary τ-density zero or shortly δ R 2,θ r,s )K U H r,s )) = 0, i.e., P lim {k, l) I r,s : p k p l x k,l L) U} = 0. ) In this case, we write S R 2,θ r,s )τ)-p-lim k,l x k,l = L. We denote the set of all weighted lacunary statistically τ-convergent double sequences by S R 2,θ r,s )τ). If we take p k = and p l = for all k, l N in ), then we obtain the definition of lacunary statistical τ-convergence for double sequences given as a special case in [9]), that is; the double sequence x = x k,l ) is said to be lacunary statistically τ-convergent to L X, if for every τ)-neighborhood U of zero P lim {k, l) I r,s : x k,l L) U} = 0. 2) r,s h r,s Now, we give another new definition which is related with the Definition 3..

4 Ş. Konca / Filomat 30:3 206), Definition 3.2. Let X, τ) be a locally solid Riesz space, p k ), pl ) be sequences of positive numbers and P n = p + p p n, P m = p + p p m. Then a double sequence x = x k,l ) in X is said to be weighted statistically τ-convergent or S R 2τ)-convergent) to the element L X if for every τ-neighborhood U of zero, the set K U P n ) = {k, l) N N, k P n and l P m : p k p l x k,l L) U} has weighted τ-density zero or shortly, δ R 2K U P n )) = 0, i.e., P lim {k P n and l P m : p k p l x k,l L) U} = 0. 3) n,m P n P m In this case, we write S R 2τ)-P-lim k,l x k,l = L. We denote the set of all weighted statistically τ-convergent double sequences by S R 2τ). We will investigate the relations between S R 2τ) and S R 2,θ r,s), later. Now, we will introduce the notion of weighted double lacunary statistical τ-bounded in locally solid Riesz spaces. The notion of double lacunary statistical τ-bounded in locally solid Riesz spaces was introduced by Alotaibi et al. see reference [4]). Definition 3.3. Let X, τ) be a locally solid Riesz space and θ r,s = {k r, l s )} be a double lacunary sequence. A double sequence x = x k,l ) in X is said to be weighted lacunary statistically τ-bounded or S R 2,θ r,s )τ)-bounded if for every τ-neighborhood U of zero there exists some α > 0 such that M U := {k, l) N N : αp k p l x k,l U} has weighted double lacunary τ-density zero or δ R 2,θ r,s )M U ) = 0, i.e. P lim {k, l) I r,s : αp k p l x k,l U} = 0. Theorem 3.4. Let X, τ) be a Hausdorff locally solid Riesz space and θ r,s = {k r, l s )} be a double lacunary sequence. Assume that x = x k,l ) and y = y k,l ) are two double sequences in X. Then the followings hold:. If S R 2,θ r,s )τ)-p-lim k,l x k,l = L and S R 2,θ r,s )τ)-p-lim k,l x k,l = L 2 then L = L If S R 2,θ r,s )τ)-p-lim k,l x k,l = L, then S R 2,θ r,s )τ)-p-lim k,l αx k,l = αl, α R. 3. If S R 2,θ r,s )τ)-p-lim k,l x k,l = L and S R 2,θ r,s )τ)-p-lim k,l y k,l = L 2, then S R 2,θ r,s )τ)-p-lim k,l x k,l + y k,l ) = L + L 2. Proof.. Suppose that S R 2,θ r,s )τ)-p-lim k,l x k,l = L and S R 2,θ r,s )τ)-p-lim k,l x k,l = L 2. Let U be any τ- neighborhood of zero. Then, there exists Y N sol such that Y U. Choose any E N sol such that E + E Y. We define the following sets: K = {k, l) N N, k, l) I r,s : p k p l x k,l L ) E}, K 2 = {k, l) N N, k, l) I r,s : p k p l x k,l L 2 ) E}. Since S R 2,θ r,s )τ)-p-lim k,l x k,l = L and S R 2,θ r,s )τ)-p-lim k,l x k,l = L 2, we have δ R 2,θ r,s )K ) = δ R 2,θ r,s )K 2 ) =. Thus, δ R 2,θ r,s )K K 2 ) = and in particular K K 2. Now, let k, l) K K 2. Then p k p l L L 2 ) = p k p l x k,l L 2 ) + p k p l L x k,l ) E + E Y U. Hence, for every τ-neighborhood U of zero, we have p k p l L L 2 ) U. Since p k ), p l ) are sequences of positive reals and X, τ) is Hausdorff, the intersection of all τ-neighborhoods U of zero is the singleton set {θ}. Thus, we get L L 2 = θ, i.e., L = L Let U be an arbitrary τ-neighborhood of zero and S R 2,θ r,s )τ)-p-lim k,l x k,l = L. Then there exists Y N sol such that Y U. Since S R 2,θ r,s )τ)-p-lim k,l x k,l = L, we have P lim {k, l) I r,s : p k p l x k,l L) Y} =. Since Y is balanced, p k p l x k,l L) Y implies αp k p l x k,l L) Y for every α R with α. Hence {k, l) I r,s : p k p l x k,l L) Y} {k, l) I r,s : αp k p l x k,l L) Y}

5 Thus, we obtain P lim {k, l) I r,s : αp k p l x k,l L) Y} =, Ş. Konca / Filomat 30:3 206), {k, l) I r,s : αp k p l x k,l L) U}. for each τ-neighborhood U of zero. Now, let α > and [ α ] be the smallest integer greater than or equal to α. There exists E N sol such that [ α ]E Y. Since S R 2,θ r,s )τ)-p-lim k,l x k,l = L, we have δ R 2,θ r,s )K) = where K = {k, l) N N, k, l) I r,s : p k p l x k,l L) E}. Then we have, αp k p l x k,l L) = α p k p l x k,l L) [ α ] p k p l x k,l L) [ α ]E Y U. Since the set Y is solid, we have αp k p l x k,l L) Y and this implies that αp k p l x k,l L) U. Thus, P lim {k, l) I r,s : αp k p l x k,l L) U} =, for each τ-neighborhood U of zero. Hence, S R 2,θ r,s )τ)-p-lim k,l αx k,l = αl for every α R. 3. Let U be an arbitrary τ-neighborhood of zero. Then there exists Y N sol such that Y U. Choose E in N sol such that E + E Y. Since S R 2,θ r,s )τ)-p-lim k,l x k,l = L and S R 2,θ r,s )τ)-p-lim k,l y k,l = L 2, we have S R 2,θ r,s )H ) = = S R 2,θ r,s )H 2 ) where H = {k, l) N N, k, l) I r,s : p k p l x k,l L ) E}, H 2 = {k, l) N N, k, l) I r,s : p k p l y k,l L 2 ) E}. Let H = H H 2. Hence, we have δ R 2,θ r,s )H) = and p k p l x k,l + y k,l ) L + L 2 )) = p k p l x k,l L ) + p k p l y k,l L 2 ) E + E Y U. Thus, we get P lim {k, l) I r,s : p k p l x k,l + y k,l ) L + L 2 )) U} =. Since U is arbitrary, we have S R 2,θ r,s )τ)-p-lim k,l x k,l + y k,l ) = L + L 2. Theorem 3.5. Let X, τ) be a locally solid Riesz space and θ r,s = {k r, l s )} be a double lacunary sequence. If a double sequence x = x k,l ) is weighted lacunary statistically τ-convergent and p k,l ) is bounded where p k,l := p k p l, then the sequence x = x k,l ) is weighted lacunary statistically τ-bounded. Proof. Suppose that the double sequence x = x k,l ) is weighted lacunary statistically τ-convergent to L X and p k,l ) is a bounded sequence. Let U be an arbitrary τ-neighborhood of zero. Then there exists Y N sol such that Y U. Let us choose E N sol such that E + E Y. Since S R 2,θ r,s )τ)-p-lim k,l x k,l = L, δ R 2,θ r,s )K) = 0 where K = {k, l) N N, k, l) I r,s : p k p l x k,l L) E}. Since E is absorbing, there exists λ > 0 such that λl E. Let α be such that 0 < α. Since p k,l ) is bounded, then there exists a M = λ α > 0 such that p k,l = p k p l M for all k, l N. Then, we can write αp k p l λ for all k, l N. Since E is solid and αp k p l L λl, we have αp k p l L E. Since E is balanced, p k p l x k,l L) E implies αp k p l x k,l L) E. Then we have αp k p l x k,l = αp k p l x k,l L) + αp k p l L E + E Y U for each k, l) N N)\K. Thus, P lim k, l) I r,s : αp k p l x k,l U = 0. Hence, the double sequence x k,l ) is weighted lacunary statistically τ-bounded.

6 Ş. Konca / Filomat 30:3 206), Theorem 3.6. Let X, τ) be a locally solid Riesz space and θ r,s = {k r, l s )} be a double lacunary sequence. If x = x k,l ), y = y k,l ) and z = z k,l ) are sequences such that. x k,l y k,l z k,l for all k, l N. 2. S R 2,θ r,s )τ)-p-lim k,l x k,l = L =S R 2,θ r,s )τ)-p-lim k,l z k,l then S R 2,θ r,s )τ)-p-lim k,l y k,l = L. Proof. Let U be an arbitrary τ-neighborhood of zero, then there exists Y N sol such that Y U. Choose E N sol such that E + E Y. From the condition 2), we have δ R 2,θ r,s )K ) = = δ R 2,θ r,s )τ)k 2 ), where K = {k, l) N N, k, l) I r,s : p k p l x k,l L) E}, K 2 = {k, l) N N, k, l) I r,s : p k p l z k,l L) E}. Also, we get δ R 2,θ r,s )K K 2 ) = and from ) we have p k p l x k,l L) p k p l y k,l L) p k p l z k,l L) for all k, l N. This implies that for all k, l) K K 2, we get p k p l y k,l L) p k p l z k,l L) + p k p l x k,l L) E + E Y. Since Y is solid, we have p k p l y k,l L) Y U. Thus, P lim r,s H r,s {k, l) I r,s : p k p l y k,l L) U} = for each τ-neighborhood U of zero. Hence, S R 2,θ r,s )τ)-p-lim k,l y k,l = L. This completes the proof of the theorem. Theorem 3.7. Let x = x k,l ) be a double sequence in a locally solid Riesz space X, τ) and θ r,s = {k r, l s )} be a double lacunary sequence. If liminf r Q r > and liminf s Q s > then S R 2τ) S R 2,θ r,s )τ). Proof. Suppose that liminf r Q r > and limsup s Q s >, then there exists a δ > 0 such that Q r + δ and Q s + δ for sufficiently large values of r and s, which implies that H r = P k r P kr = Q r δ +δ and H s P ls = P l s P ls = Q s δ +δ. Let S R 2 )τ)-p-lim k,l x k,l = L and let U be an arbitrary τ)-neighborhood of zero. Then for all r > r 0 and s > s 0, we have P kr P ls { k Pkr and l P ls : p k p l xk,l L ) U } {P kr < k P kr and P ls < l P ls : p k p l x k,l L) U} P kr P ls = H r,s { k, l) I P kr P ls H r,s : p k p l x k ζ) U} ) r,s ) δ 2 { k, l) I + δ H r,s : p k p l xk,l L ) U }. r,s Since S R 2τ)-P-lim k,l x k,l = L, then the inequality above implies that S R 2,θ r,s )τ)-p-lim k,l x k,l = L. Hence, S R 2τ) S R 2,θ r,s )τ). Theorem 3.8. Let X, τ) be a locally solid Riesz space and x = x k,l ) be a double sequence in X. For any double lacunary sequence θ r,s = {k r, l s )}, if limsup r Q r < and limsup s Q s <, then S R 2,θ r,s )τ) S R 2τ). P kr

7 Ş. Konca / Filomat 30:3 206), Proof. If limsup r Q r < and limsup s Q s <, then there exists a K > 0 such that Q r K for all r N and Q s K for all s N. Let x S R 2,θ r,s )τ) with S R 2,θ r,s )τ)-p-lim k,l x k,l = L and U be an arbitrary τ-neighborhood of zero. We write N r,s := { k, l) I r,s : p k p l xk,l L ) U }. 4) By 4) and from the definition of weighted lacunary statistical convergence for double sequences, for a given ε > 0, there exist positive integers r 0 and s 0 such that N r,s H r,s < ε K 2 for all r > r 0 and s 0. Now, let M := max{n r,s : r r 0 and r r 0 } 5) and let n and m be any integers satisfying k r < n k r and l s < m l s. Thus, we have the following { P n P m k Pn and l P m : p k p l xk,l L ) U } P kr P ls { k Pkr and l P ls : p k p l xk,l L ) U } r 0,s 0 = P kr P ls t,u=, Mr 0s 0 P kr P ls + ε K 2 P kr P ls N t,u + P kr P ls N t,u r 0 <t r) s 0 <u s) H t,u r 0 <t r) s 0 <u s) Mr 0s 0 P kr P ls + ε P kr P ls K 2 P kr P ls = Mr 0s 0 P kr P ls + ε Q K 2 r Q s Mr 0s 0 P kr P ls + ε. Since P kr and P ls as r, s, it follows that P n P m { k Pn and l P m : p k p l xk,l L ) U } 0 as n, m. Therefore x S R 2τ) with S R 2τ)-P-lim k,l x k,l = L. The following corollary is a result of Theorem 3.7 and Theorem 3.8. Corollary 3.9. Let X, τ) be a locally solid Riesz space and x = x k,l ) be a double sequence in X. For any lacunary sequence θ r,s = {k r, l s )}, if < limin f r Q r limsup r Q r < and < limin f s Q s limsup s Q s <, then S R 2,θ r,s )τ) = S R 2τ) and S R 2,θ r,s )τ)-p-lim k,l x k,l = S R 2τ)-P-lim k,l x k,l = L. Theorem 3.0. Let X, τ) be a locally solid Riesz space and x = x k,l ) be a double sequence in X. For any double lacunary sequence θ r,s = {k r, l s )}, the following statements are true:. If p k and p l for all k, l N, then S θr,s τ) S R 2,θ r,s )τ) and S θr,s τ)- P -lim k,l x k,l = S R 2,θ r,s )τ)- P-lim k,l x k,l = L. 2. If p k and p l for all k, l N and H r h r ), H s h s ) are upper bounded, then S R 2,θ r,s )τ) S θr,s τ) with S R 2,θ r,s )τ)-p-lim k,l x k,l = S θr,s τ)-p-lim k,l x k,l = L. Proof.. If p k for all k N and p l for all l N, then H r h r for all r N and H s h s for all s N. So, there exist M and M 2 constants such that 0 < M H r h r for all r N and 0 < M 2 H s h s for all s N.

8 Ş. Konca / Filomat 30:3 206), Let x = x k,l ) be a double sequence in a locally solid Riesz space X, τ) and assume that S θr,s τ)-p-lim k,l x k,l = L. Let U be an arbitrary τ-neighborhood of zero, then we have { H r,s k, l) I r,s : p k p l xk,l L ) U } = H r H s { Pkr < k P kr and P ls < l P ls : p k p l xk,l L ) U } M M 2 h r hs { kr < k k r andl s < l l s : x k,l L ) U } = M,2 h r,s { k, l) Ir,s : x k,l L ) U } where M,2 = M M 2. Hence, we obtain the result by taking the P-limit as r, s. 2. If p k and p l for all k, l N, then h r H r and h s H s for all r, s N. Since H r h r ), H s ) are upper h s bounded, there exist N and N 2 constants such that H r h r N < and H s h s N 2 < for all r, s N. Assume that x S R 2,θ r,s )τ) with S R 2,θ r,s )τ)-p-lim k,l x k,l = L. Let U be an arbitrary τ-neighborhood of zero. Then we have { h r,s k, l) Ir,s : x k,l L ) U } = h r hs { kr < k k r and l s < l l s : x k,l L ) U } N N 2 H r H s { Pkr < k P kr and P ls < l P ls : p k p l xk,l L ) U } = N,2 H r,s { k, l) I r,s : p k p l xk,l L ) U } where N,2 = N N 2. Hence, the result is obtained by taking the P-limit as r, s. Acknowledgement The author is grateful to anonymous referees for their careful reading of the paper which improved it greatly. References [] H. Albayrak, S. Pehlivan, Statistical convergence and statistical continuity on locally solid Riesz spaces, Topology and its Applications ) [2] C.D. Aliprantis, O. Burkinshaw, Locally Solid Riesz Spaces with Applications to Economics second ed.), American Mathematical Society, [3] A.M. Alotaibi, C. Çakan, The Riesz convergence and Riesz core of double sequences, Journal of Inequalities and Applications 20256) 202). [4] A. Alotaibi, B. Hazarika, S.A. Mohiuddine, On lacunary statistical convergence of double sequences in locally solid Riesz spaces, Journal of Computational Analysis and Applications 7 204) [5] A. Alotaibi, B. Hazarika, S.A. Mohiuddine, On the ideal convergence of double sequences in locally solid Riesz spaces, Abstract and Applied Analysis ), Article ID , 6 pages. [6] M. Başarır, Ş. Konca, On some spaces of lacunary convergent sequences derived by Norlund-type mean and weighted lacunary statistical convergence, Arab Journal of Mathematical Sciences ) [7] M. Başarır, Ş. Konca, Weighted lacunary statistical convergence in locally solid Riesz spaces, Filomat ) [8] M. Başarır, Ş. Konca, On some lacunary almost convergent double sequence spaces and Banach limits, Hindawi Publishing Corporation, Abstract and Applied Analysis 202) Article ID , 7 pages. [9] R.C. Buck, Generalized asymptotic density, American Journal of Mathematics ) [0] C. Çakan, B. Altay, H. Coşkun, Double lacunary density and lacunary statistical convergence of double sequences, Studia Scientiarum Mathematicarum Hungarica ) [] H. Fast, Sur la convergence statistique, Colloquium Mathematicum 2 95) [2] D.H. Fremlin, Topological Riesz Spaces and Measure Theory, Cambridge University Press, London and New York, 974.

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