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 -statistical-τ-convergence which is a more general idea of statistical convergence. We also investigate the ideas of double -statistical-τ-boundedness and double -statistical-τ-Cauchy condition of sequences in the framework of locally solid Riesz space endowed with a topology τ and investigate some of their consequences.
MSC: 40G15, 40A35, 46A40.
Keywords:ideal; filter; double ℐ-statistical-τ-convergence; double -statistical-τ-convergence; double -statistical-τ-boundedness; double -statistical-τ-Cauchy condition
The notion of statistical convergence, which is an extension of the idea of usual convergence, was introduced by Fast , Steinhaus  independently in the same year 1951 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 [7,8]; in probabilistic normed spaces [9,10], in intuitionistic fuzzy normed spaces , in random 2-normed spaces . In  Albayrak and Pehlivan studied this notion in locally solid Riesz spaces. Recently, Mohiuddine 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 recently, Das and Savas  introduced the ideas of -convergence, -boundedness and -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, [18-21] where more references can be found).
The idea of statistical convergence was further extended to ℐ-convergence in  using the notion of ideals of ℕ with many interesting consequences. More investigations in this direction and more applications of ideals can be found in [22-31] where many important references can be found.
The notion of a Riesz space was first introduced by Riesz  in 1928, 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 of ℐ-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.
We now recall the following basic facts from .
A family ℐ of subsets of a non-empty set X is said to be an ideal if (i) implies , (ii) , imply . ℐ is called non-trivial if and . ℐ is admissible if it contains all singletons. If ℐ is a proper non-trivial ideal, then the family of sets is a filter on X (where c stands for the complement). It is called the filter associated with the ideal ℐ.
We also recall some of the basic concepts of Riesz spaces.
Definition 2.1 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 and , then for each .
(ii) If and , then for each .
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 , the negative part of x by and the absolute value of x by , where θ is the element zero of L.
A subset S of a Riesz space L is said to be solid if and imply .
A topology τ 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
are continuous, where σ is the usual topology on R. In this case, the pair is called a topological vector space.
Every linear topology τ on a vector space L has a base for the neighborhoods of θ satisfying the following properties:
(a) Each is a balanced set, that is, holds for all and every with .
(b) Each is an absorbing set, that is, for every , there exists a such that .
(c) For each , there exists some with .
Definition 2.2 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 is a Riesz space L equipped with a locally solid topology τ.
will 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 of N, the set of natural numbers. A subset E of N is said to have density if
Note that if is a finite set, then , and for any set , .
Definition 3.1 A sequence is said to be statistically convergent to ℓ if for every ,
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 such that and as . Let and . A sequence of real numbers is said to be lacunary statistically convergent to L (or, -convergent to L) if for any ,
where denotes the cardinality of . In  the relation between lacunary statistical convergence and statistical convergence was established among other things.
We now have the following definitions.
Let be a proper admissible ideal in ℕ. The sequence of elements of ℝ is said to be ℐ-convergent to if for each , the set . The class of all ℐ-statistically convergent sequences will be denoted by .
Definition 3.3 ()
Let θ be a lacunary sequence. A sequence is said to be ℐ-lacunary statistically convergent to L or -convergent to L if for any and ,
In this case, we write . The class of all ℐ-lacunary statistically convergent sequences will be denoted by .
It can be checked, as in the case of statistically and lacunary statistically convergent sequences, that both and are linear subspaces of the space of all real sequences.
Remark 3.1 For , -convergence coincides with lacunary statistical convergence which is defined in .
Let be a two-dimensional set of positive integers and let be the numbers of in K such that and . Then the lower asymptotic density of E is defined as follows:
In the case when the sequence has a limit, we say that E has a natural density and is defined as follows:
For example, let . Then
(i.e., the set E has double natural density zero).
Recently, Mursaleen and Edely  presented the notion of statistical convergence for a double sequence as follows:
A real double sequence is said to be statistically convergent to L provided that for each ,
The double sequence is called double lacunary if there exist two increasing sequences of integers such that
Let us denote , and is determined by .
We have the following.
Definition 3.4 Let be a sequence in a locally solid Riesz space . We say that x is -statistically-τ-convergent to if for every τ-neighborhood U of zero and for ,
In this case, we write (or in brief).
Remark 3.2 For , -statistical-τ-convergence becomes double lacunary statistical τ-convergence in a locally solid Riesz space.
Definition 3.5 Let be a sequence in a locally solid Riesz space . We say that x is -statistically-τ-bounded if for every τ-neighborhood U of zero and , there exists such that
Definition 3.6 Let be a sequence in a locally solid Riesz space . We say that x is -statistically-τ-Cauchy if for every τ-neighborhood U of zero and , there exist such that
Now we are ready to present some basic properties of this new convergence in a locally solid Riesz space.
Theorem 3.1Let be a Hausdorff locally solid Riesz space, and be two sequences inL. Then the following hold:
(a) If and , then .
(b) If , then for each .
(c) If and , then .
Proof (a) Let U be any τ-neighborhood of zero. Then there exists a such that . Take a such that . Let . Since and , we write
Then and for ,
Now write that and cannot be disjoint, for then we will have , which is impossible. So, there is a for which
Thus for every τ-neighborhood U of zero. Since is Hausdorff, the intersection of all τ-neighborhoods of zero is the singleton , and so , i.e., .
(b) Let and let U be an arbitrary τ-neighborhood of zero. Choose such that . For any ,
First let . Since V is balanced, implies that . Therefore
and so ,
which implies that
If and is the smallest integer greater or equal to , choose such that . Again, for , taking
and in view of the fact that
which implies that , proceeding as before, we conclude that
This proves that .
(c) Let U be an arbitrary τ-neighborhood of zero. Then there are such that . Since and , we get, for ,
If , then ,
such that when (say) and (say). Note that
so for all ,
This completes the proof of the theorem. □
Theorem 3.2Let be a locally solid Riesz space. Let , and be three sequences inLsuch that for each . If , then .
Proof Let U be an arbitrary τ-neighborhood of zero. Take such that . Since , so for ,
Hence we observe that ,
Writing and , we see that ,
and as V is solid, so
Clearly, and as in the previous theorem, we show that ,
where and so
This proves that . This completes the proof of the theorem. □
Theorem 3.3An -statisticallyτ-convergent sequence in a locally solid Riesz space is -statisticallyτ-bounded.
Proof Let be -statistically τ-convergent to . Let U be an arbitrary τ-neighborhood of zero. Choose such that . Since W is absorbing, there is a such that . Choose so that . Since W is solid and , we have . Again, as W is balanced, implies that . Now, for any ,
Thus, for all ,
If , then
and so, for all ,
Since , so the set on the right-hand side also belongs to and this proves that is -statistically τ-bounded. □
Theorem 3.4If a sequence in a locally solid Riesz space is -statisticallyτ-convergent, then it is -statisticallyτ-Cauchy.
Proof Let be -statistically τ-convergent to . Let U be an arbitrary τ-neighborhood of zero. Choose such that . Let . Therefore
For all ,
Take and in view of the above, we can choose (since this set cannot be empty). Then . Now observe that if for , then
Hence, as in the earlier proofs, we can prove that
which consequently implies that is -statistically τ-Cauchy.
This completes the proof of the theorem. □
The author declares that they have no competing interests.
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