Function Spaces, Approximation Theory, and Their ApplicationsView this Special Issue
Classes of -Convergent Double Sequences over -Normed Spaces
I introduce some new classes of -convergent double sequences defined by a sequence of moduli over -normed space. Study of their algebraic and topological properties and some inclusion relations has also been done.
The notion of -convergence was introduced by Kostyrko et al. in . It is known that -convergence is generalization of the statistical convergence which was introduced by Fast . It was further studied by Demirci , Das et al. , Šalát et al. , and many others.
For a nonempty set , the family of sets , the power set of , is said to be an ideal if(1);(2) is additive; that is, ;(3) is hereditary; that is, .
A nontrivial ideal is called admissible if . is maximal if there cannot exist any nontrivial ideal containing as a subset.
Let , and denote the set of natural, real, and complex numbers, respectively. A double sequence of complex numbers is defined as a function from to . A number is called a limit of a double sequence if for every there exists some such that The set of all double sequences is denoted by . Any subset of the is called double sequence space. A sequence is said to be -convergent to a number if, for every , . In this case we write .
Example 1. Let be the class of all subsets of such that implies that there exists such that .
Then is an ideal of in the usual Pringsheim sense of convergence of double sequences. If is replaced by , the class of finite subsets of , then we get the usual regular convergence of double sequences.
The theory of 2-normed spaces was first introduced by Gähler  in 1964. Later on it was extended to -normed spaces by Misiak . Since then many mathematicians have worked in this field and obtained many interesting results; for instance see Gunawan [11, 12], Gunawan and Mashadi , Mursaleen and Mohiuddine , Şahiner et al. [15, 16], and Yamancı and Gürdal . Let and let be a linear metric space over the field of real or complex numbers of dimension , where . A real valued function on satisfying the following four conditions:(1) if and only if are linearly dependent;(2) is invariant under permutation;(3) for any ;(4)is called an -norm on and the pair is called an -normed space over the field .
Example 2. If we take , equipped with Euclidean -norm spanned by vectors , then the -norm may be given by the formula , where for .
The standard -norm on is defined as
where denotes the inner product on . If , then this -norm is exactly the same as the Euclidean -norm mentioned earlier. For this -norm is the usual norm .
A sequence in an -normed space is said to converge to some if
A sequence in an -normed space is said to be Cauchy if
If every Cauchy sequence in converges to some , then is said to be complete with respect to the -norm. Any complete - normed space is said to be an -Banach space.
The concept of modulus function was introduced by Nakano  in the year 1953. It was further studied by [7, 8, 19–21] and many more. It is defined as a function : satisfying the following conditions:(1) if and only if ,(2) for all ,(3) is increasing,(4) is continuous from the right at zero.Ruckle  used the idea of a modulus function to construct the sequence space The space is closely related to the space which is an space with for all real . Thus Ruckle [23, 24] proved that, for any modulus ,The space is a Banach space with respect to the norm After then Kolk [25, 26] gave an extension of by considering a sequence of modulus functions called the sequence of moduli and defined the sequence space: From the above four properties of modulus function it can be clearly seen that must be continuous everywhere on ). For a sequence of moduli, we have further two properties:(5) for all ;(6) uniformly in and for .
Example 3. Let be a function from to . If we take , then the function is a bounded modulus function and if we take , then is an unbounded modulus function.
By a lacunary sequence ; , where , we mean an increasing sequence of nonnegative integers as . The intervals determined by are denoted by and the ratio will be denoted by . The space of lacunary strongly convergent sequence was defined by Freedman et al.  as follows: The double lacunary sequence was defined by Savaş and Patterson . A double sequence is called double lacunary if there exist two increasing sequences of integers such that The following interval is determined by : The space of double lacunary strongly convergent sequence is defined as follows:
2. New Classes of Double Sequences
Now, we will define the new classes of double sequences.
Let be an admissible ideal, let be a sequence of moduli, let be an -normed space, let be a sequence of positive real numbers, let be a sequence of strictly positive real numbers, and let be the space of all double sequences defined over the -normed space ; then for some and every , we define(i) : ,(ii) :
Case 1. If , then we get(i) : ,(ii) :
Case 2. If , then we get(i) : ,(ii) :
Case 3. If and , then we get(i) : ,(ii) : The following inequality will be used throughout the paper. If , then we have for all and . Also for all
3. Main Results
Theorem 4. The sets and are linear spaces over the field of complex numbers .
Proof. Let , , and ; then for every we can write By the use of inequality (12), we have the following inequality: This inequality says to us that the inclusion holds. From here, since the right side belongs to , the left side also belongs to . This completes the proof.
Lemma 5. Let be a modulus function and let . Then for each , one has
Theorem 6. Let be a sequence of moduli and Then the following statements hold:(i),(ii).
Proof. For some , choose such that . By the continuity of for all , we can choose some such that for every with we have Let ; then for some and for every , we have Therefore for , we have So, by inequality (17), we can write This implies that . Hence The inclusion is strict as for the reverse inclusion we need the condition given in the next theorem. The other part can be proved similarly.
Theorem 7. Let be a sequence of moduli. If for all , then(i),(ii)
Proof. (i) To prove , it is sufficient to show that . Let ; then, by the definition, we get By the given condition for all , we have for all ; that is, From inequalities (21) and (23), we get which consequently implies that That is, This implies that . Hence, from the previous theorem and inclusion (27), we get the required result. The other part can be proved similarly.
Corollary 8. Let and be sequences of moduli. If for all , then(i),(ii).
Theorem 9. Let and be the standard and the Euclidean -norm spaces, respectively. Then
Proof. The proof of this result is easy, so it is omitted.
Theorem 10. Let and be sequences of moduli; then(i),(ii),(iii)⊆,(iv)⊆
Proof. (i) For some , we choose such that . Now as is a sequence of modulus functions which are always continuous, we can choose such that, for every , we get : Then, by the definition, we have Thus, for , we get Now, by the continuity of , we have which further implies that which implies Therefore . This completes the proof.
(iii) Again consider Then by the definition of both the spaces, we get Using the fact that , we have So we get Therefore we have Hence we have
A detailed study of some new classes of -convergent double sequences over -normed spaces has been done. Some algebraic and topological properties and inclusion relations have been proved with supported examples.
The author declares that he has no competing interests.
The author is grateful to the anonymous referees for their careful corrections and comments that improved the presentation of this paper.
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