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## Function Spaces, Approximation Theory, and Their Applications

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Research Article | Open Access

Volume 2016 |Article ID 7594031 | https://doi.org/10.1155/2016/7594031

Nazneen Khan, "Classes of -Convergent Double Sequences over -Normed Spaces", Journal of Function Spaces, vol. 2016, Article ID 7594031, 7 pages, 2016. https://doi.org/10.1155/2016/7594031

# Classes of -Convergent Double Sequences over -Normed Spaces

Accepted20 Mar 2016
Published09 May 2016

#### Abstract

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.

#### 1. Introduction

The notion of -convergence was introduced by Kostyrko et al. in [1]. It is known that -convergence is generalization of the statistical convergence which was introduced by Fast [2]. It was further studied by Demirci [3], Das et al. [4], Šalát et al. [5], 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 .

A double sequence space is said to be solid or normal if implies for all sequences of scalars with for all . For more details please see [68].

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 [9] in 1964. Later on it was extended to -normed spaces by Misiak [10]. Since then many mathematicians have worked in this field and obtained many interesting results; for instance see Gunawan [11, 12], Gunawan and Mashadi [13], Mursaleen and Mohiuddine [14], Şahiner et al. [15, 16], and Yamancı and Gürdal [17]. 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 [18] in the year 1953. It was further studied by [7, 8, 1921] 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 [22] 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. [27] as follows: The double lacunary sequence was defined by Savaş and Patterson [28]. 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

#### 4. Conclusion

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.

#### Competing Interests

The author declares that he has no competing interests.

#### Acknowledgments

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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