On the Ideal Convergence of Double Sequences in Locally Solid Riesz Spaces
The aim of this paper is to define the notions of ideal convergence, -bounded for double sequences in setting of locally solid Riesz spaces and study some results related to these notions. We also define the notion of -convergence for double sequences in locally solid Riesz spaces and establish its relationship with ideal convergence.
1. Introduction and Preliminaries
In 1951, Fast  and Steinhaus  introduced the concept of statistical convergence for single sequences, independently. Some basic and important properties of this concept were studied by Buck , Šalát , Schoenberg , and Fridy . Later, the notion of statistical convergence for single sequences was further defined in various spaces; see Çakalli and Khan [7–9], Di Maio et al. [10, 11], Hazarika [12–14], Maddox , Mohiuddine et al. [16–19], and so forth. Some application of statistical summability methods is presented in [20, 21]. In 2003, the notion of statistical convergence for single sequences has been extended to double sequences by Mursaleen and Edely . Recently, the statistical convergence and statistical Cauchy for double sequences have been defined in the framework fuzzy and intuitionistic normed spaces by Mohiuddine et al.  and Mursaleen and Mohiuddine , respectively, and established some interesting results related to the concept of statistical convergence and statistical Cauchy double sequences. Recently, it was defined and studied by Mohiuddine et al.  in the setting of locally solid Riesz spaces while for single sequences this concept was first studied by Albayrak and Pehlivan  (also see [27–29]). An application of locally solid Riesz spaces in economics can be found in .
The notion of ideal convergence for single sequences, which is a generalization of the concept of statistical convergence, was first defined and studied by Kostyrko et al. . Let us recall the notion of ideal convergence and related concepts by Kostyrko et al.  as follows. Let be a nonempty set. Then a family of sets (power set of ) is said to be an ideal if is additive; that is, and . A family of sets (power sets of ) is called an ideal if and only if, for each , we have and, for each and each , we have . A nonempty family of sets is a filter on if and only if ; for each , we have and for each and each , we have . An ideal is called nontrivial ideal if and . Clearly is a nontrivial ideal if and only if is a filter on . A nontrivial ideal is called admissible if and only if . A nontrivial ideal is maximal if there cannot exist any nontrivial ideal containing as a subset.
We remark that if , then the corresponding convergence coincides with the usual convergence. Also, if , then the corresponding convergence coincides with the statistical convergence (where denotes the natural density of the set ). In the above cases, both and are nontrivial admissible ideals of .
Kumar  defined the notions of and -convergence of double sequence and studied some properties of these notions. Recently, Das et al.  introduced the concepts of and -convergence of double sequences in the setting of metric space and established some relationship between these types of convergence. Quite recently, Mursaleen and Mohiuddine defined and studied the notion of -convergence, -convergence, -limit points, and -cluster points for single and double sequences, in [34, 35], respectively, in probabilistic normed spaces. Şahiner et al.  and Gürdal and Açik  introduced the notion of ideal convergence and -Cauchy sequence in 2-normed spaces, respectively. Mursaleen and Alotaibi  introduced the notion of ideal convergence in random 2-normed spaces and later on it was extended by Mohiuddine et al.  from single to double sequences. For more details on these concepts, one can be referred to [40–52].
Now we recall the definition of locally solid Riesz spaces and some related concepts as follows. Let be a real vector space and let be a partial order on this space. is said to be an ordered vector space if it satisfies the following properties:(1)if and , then for each ;(2)if and , then for each .If, in addition, is a lattice with respect to the partial ordering, then is said to be a Riesz space (or a vector lattice) (see ).
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 is said to be solid if and implies .
A topology on a real vector space that makes the addition and scalar multiplication continuous is said to be a linear topology, that is, when the mappings are continuous, where is the usual topology on . In this case the pair is called a topological vector space.
Every linear topology on a vector space has a base for the neighborhoods of satisfying the following properties.(1)Each is a balanced set; that is, holds for all and every with .(2)Each is an absorbing set; that is, for every , there exists such that .(3)For each there exists some with .
A linear topology on a Riesz space is said to be locally solid (see ) 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 . For more details on these concepts, one can be referred to [55–57].
Throughout the paper, the symbol will stand for a base at zero consisting of solid sets and satisfying conditions (1), (2), and (3) in a locally solid topology. Also we assume that is a nontrivial admissible ideal of .
2. Ideal Convergence of Double Sequences in LSR-Spaces
Throughout the paper will denote the Hausdorff locally solid Riesz space, which satisfies the first axiom of countability. For our convenience, here and in what follows, we will write an LSR-space instead of a locally solid Riesz space.
The notion of convergence for double sequence was first introduced by Pringsheim  as follows. We say that a double sequence of reals is convergent to in Pringsheim’s sense (briefly, -convergent) provided that given there exists a positive integer such that whenever .
Let and denotes the number of in such that and (see ). Then the lower natural density of is defined by . In this case, the sequence has a limit in Pringsheim’s sense; then we say that has a double natural density and is defined by .
In the recent past, Mohiuddine et al.  introduced the notion of statistical convergence of double sequences in LSR-space as follows. Let be a LSR-space. A double sequence of points in is said to be -convergent to an element if for each -neighborhood of zero Now we introduce the notions of -convergence and -bounded double sequences in LSR-spaces.
Definition 1. Let be a LSR-space. A double sequence of points in is said to be -convergent to an element of if for each -neighborhood of zero That is, In this case, one writes or .
Definition 2. Let be a LSR-space. Then, a double sequence of points in is said to be -bounded in if, for each -neighborhood of zero, there is some ,
Definition 3. Let be a LSR-space. One says that a double sequence is -Cauchy in if, for each -neighborhood of zero, there exist such that, for all and ,
Definition 4. Let be a LSR-space. Then, a double sequence in is said to be -convergent to if there is a set , , with such that . In this case, one writes -.
Theorem 5. Let be a LSR-space. Every -convergent sequence in has only one limit.
Proof. Suppose that is a double sequence in such that - and -. Let be any -neighborhood of zero. Also for each -neighborhood of zero there is a set such that . Let in be such that . We define the sets and as follows: Since and , we get . Now, let . Then we have As we know, intersection of all -neighborhoods of zero is the singleton set because is Hausdorff. Hence ; that is, .
Theorem 6. Let be a LSR-space and let and be two double sequences of points in . Then,(i)if - and -, then -;(ii)if -, then - for .
Proof. Assume that - and -. Suppose that is an arbitrary -neighborhood of zero. Then there exists such that . Let such that . Thus, we can write
Then we have .
Let . Hence we have and Therefore Since is arbitrary, we have .
(ii) Suppose that - and also suppose that is an arbitrary -neighborhood of zero. Then there exists such that , so we have Since is balanced, holds for all and for every with . Therefore Thus, we have for each -neighborhood of zero. Now let and be the smallest integer greater than or equal to . Then there exists such that . From our assumption that -, we obtain that Therefore Since is solid, . It follows that . Thus, for each -neighborhood of zero. We conclude that -.
Theorem 7. Let be a LSR-space. If a double sequence in is -convergent, then it is -bounded.
Proof. Assume that -. Suppose is an arbitrary -neighborhood of zero. Then, there exists such that . Let such that . Using our assumption, we obtain that Since is absorbing, there exists such that . Let be such that and . Since is solid and , we have . Also, since is balanced, implies . Then we have Thus Hence is -bounded.
Theorem 8. Let be a LSR-space and let , , and be three double sequences of points in such that(i), for all ,(ii)--.Then -.
Proof. Suppose that the given conditions (i) and (ii) hold for the double sequences , , and . Suppose is an arbitrary -neighborhood of zero. Then, there exists such that . Let such that . It follows from (ii) that , where Also from the given condition (i), we have Since is solid, we have . Thus, for each -neighborhood of zero. Thus -.
Theorem 9. Let be a LSR-space. A double sequence is -convergent to in if and only if for each -neighborhood of zero there exists a subsequence of such that and
Proof. Suppose that . Also, suppose that is an arbitrary -neighborhood of zero. Let be a sequence of nested base of -neighborhoods of zero. For each , put
Then, and . Let and be such that and , respectively. Then . For such that and , choose ; that is, . In general, choose and such that and hold. Then . Therefore for all which satisfy and , choose ; that is, . Hence, it follows that .
Since is an arbitrary -neighborhood of zero, there exists such that . Let such that . Now Also and imply that Next suppose for an arbitrary -neighborhood of zero that there exists a subsequence of such that and Since is any -neighborhood of zero, we choose such that . Then we have That is, Therefore
Theorem 10. If and , then .
Proof. Let be any -neighborhood of 0. Then there exists such that . Let such that . Since , then there exist integers such that implies that . Hence By the assumption , . Thus That is, This implies that .
Theorem 11. Let be a LSR-space and let be a double sequence in . If there is a -convergent sequence in such that then is also -convergent.
Proof. Suppose that and . Then for an arbitrary -neighborhood of zero, we have Now, Therefore, we have
Theorem 12. Let be a LSR-space. If a double sequence is -convergent to , then it is -convergent to .
Proof. Suppose that -. Let be an arbitrary -neighborhood of zero. Since -, there is a set , with such that , and implies . Then Therefore Hence is -convergent to .
Theorem 13. The sequential method is regular.
Proof of the theorem is straightforward, so it is omitted.
From Theorem 12, we can easily obtain the following useful result.
Theorem 14. The sequential method is subsequential.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.
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