Journal of Function Spaces
Volume 2016 (2016), Article ID 7594031, 7 pages
http://dx.doi.org/10.1155/2016/7594031

Research Article

Classes of -Convergent Double Sequences over -Normed Spaces

Nazneen Khan

Department of Mathematics, College of Science and Arts, Taibah University, Madinah 41921, Saudi Arabia

Received 5 February 2016; Accepted 20 March 2016

Academic Editor: Carlo Bardaro

Copyright © 2016 Nazneen Khan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [6–8].

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, 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 [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.

References

P. Kostyrko, T. Šalát, and W. Wilczyński, “I-convergence,” Real Analysis Exchange, vol. 26, no. 2, pp. 669–685, 2000/01.

H. Fast, “Sur la convergence statistique,” Colloquium Mathematicum, vol. 2, pp. 241–244, 1951.

K. Demirci, “I-limit superior and limit inferior,” Mathematical Communications, vol. 6, pp. 165–172, 2001.

P. Das, P. Kostyrko, P. Malik, and W. Wilczyński, “I and I ∗-Convergence of double sequences,” Mathematica Slovaca, vol. 58, pp. 605–620, 2008.

T. Šalát, B. C. Tripathy, and M. Ziman, “On some properties of I-convergence,” Tatra Mountains Mathematical Publications, vol. 28, pp. 279–286, 2004.

M. Mursaleen and S. A. Mohiuddine, “On ideal convergence of double sequences in probabilistic normed spaces,” Mathematical Reports, vol. 12, no. 4, pp. 359–371, 2010.

A. K. Vakeel and N. Khan, “On some I-convergent double sequence spaces defined by a modulus function Engineering,” Scientific Research, vol. 5, no. 5, pp. 35–40, 2013.

A. K. Vakeel and T. Sabiha, “On ideal convergent difference double sequence spaces in 2-normed spaces defined by Orlicz function,” JMI International Journal of Mathematical Sciences, vol. 1, no. 2, pp. 1–9, 2010.

S. Gähler, “Lineare 2-normierte Räume,” Mathematische Nachrichten, vol. 28, no. 1-2, pp. 1–43, 1964.

A. Misiak, “n-inner product spaces,” Mathematische Nachrichten, vol. 140, pp. 299–319, 1989.

H. Gunawan, “The space of p-summable sequence and its natural n-norm,” Bulletin of the Australian Mathematical Society, vol. 64, no. 1, pp. 137–147, 2001.

H. Gunawan, “n-Inner product, n-norms and the Cauchy-Schwartz inequality,” Scientiae Mathematicae Japonicae, vol. 5, pp. 47–54, 2001.

H. Gunawan and M. Mashadi, “On n-normed spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 27, no. 10, pp. 631–639, 2001.

M. Mursaleen and S. A. Mohiuddine, “On ideal convergence in probabilistic normed spaces,” Mathematica Slovaca, vol. 62, no. 1, pp. 49–62, 2012.

A. Şahiner and M. Gürdal, “New sequence spaces in n-normed spaces with respect to an Orlicz function,” The Aligarh Bulletin of Mathematics, vol. 27, no. 1, pp. 53–58, 2008.

A. Şahiner, M. Gürdal, S. Saltan, and H. Gunawan, “Ideal convergence in 2-normed spaces,” Taiwanese Journal of Mathematics, vol. 11, no. 5, pp. 1477–1484, 2007.

U. Yamancı and M. Gürdal, “On lacunary ideal convergence in random n-normed space,” Journal of Mathematics, vol. 2013, Article ID 868457, 8 pages, 2013.

H. Nakano, “Concave modulars,” Journal of the Mathematical Society of Japan, vol. 5, pp. 29–49, 1953.

E. Savaş, “On some generalized sequence spaces defined by a modulus,” Indian Journal of Pure and Applied Mathematics, vol. 30, no. 5, pp. 459–464, 1999.

S. K. Sharma and A. Esi, “Some I-convergent sequence spaces defined by using sequence of moduli and n-normed space,” Journal of the Egyptian Mathematical Society, vol. 21, no. 2, pp. 103–107, 2013.

V. A. Khan and N. Khan, “On some -convergent double sequence spaces defined by a sequence of modulii,” Journal of Mathematical Analysis, vol. 4, no. 2, pp. 1–8, 2013.

W. H. Ruckle, “On perfect symmetric BK spaces,” Mathematische Annalen, vol. 175, pp. 121–126, 1968.

W. H. Ruckle, “Symmetric coordinate spaces and symmetric bases,” Canadian Journal of Mathematics, vol. 19, pp. 828–838, 1967.

W. H. Ruckle, “FK-spaces in which the sequence of coordinate vectors is bounded,” Canadian Journal of Mathematics, vol. 25, no. 5, pp. 973–978, 1973.

E. Kolk, “On strong boundedness and summability with respect to a sequence of moduli,” Acta et Commentationes Universitatis Tartuensis, no. 960, pp. 41–50, 1993.

E. Kolk, “Inclusion theorems for some sequence spaces defined by a sequence of modulii,” Acta et Commentationes Universitatis Tartuensis, vol. 970, pp. 65–72, 1994.

A. R. Freedman, J. J. Sember, and M. Raphael, “Some Cesàro-type summability spaces,” Proceedings of the London Mathematical Society, vol. 37, no. 3, pp. 508–520, 1978.

E. Savaş and R. F. Patterson, “On some double almost lacunary sequence spaces defined by Orlicz functions,” Filomat, vol. 19, pp. 35–44, 2005.