Research Article | Open Access

# A New Double Sequence Space Defined by a Double Sequence of Modulus Functions

**Academic Editor:**Shamsul Qamar

#### Abstract

In this work we introduce new spaces of double sequences defined by a double sequence of modulus functions, and we study some properties of this space.

#### 1. Introduction

In this work, by and , we denote the spaces of single complex sequences and double complex sequences, respectively. and denote the set of positive integers and complex numbers, respectively. If, for all , there is such that where and , then a double sequence is said to be converge (in terms of Pringsheim) to . A real double sequence is nondecreasing, if for . A double series is infinity sum and its convergence implies the convergence by of partial sums sequence , where (see [1–3]).

For , denote the space of sequences such that (see [4]).

A double sequence is said to be bounded if and only if . The space of all bounded double sequences is denoted by . It is known that is Banach space (see [5, 6]).

A double sequence space is said to be normal if whenever for all and .

The double sequence spaces in the various forms were introduced and studies by Khan and Tabassum in [7–14], by Khan in [15], and by Khan et al. in [16, 17].

Now let be a family of subsets having most elements in . Also denote the class of subsets in such that the elements of and are most and , respectively. Besides is taken as a nondecreasing double sequence of the positive real numbers such that (see [18]).

Let be a double sequence. A set is defined by A double sequence space is said to be symmetric if and whenever and .

A BK-space is a Banach sequence space in which the coordinate maps are continuous.

A function is said to be a modulus function if it satisfies the following:(1) if and only if ;(2) for all ;(3) is increasing;(4) is continuous from right at .

It follows that is continuous on . The modulus function may be bounded or unbounded. For example, if we take , then is bounded. But, for , is not bounded.

The BK-spaces , introduced by Sargent in [19], is in the form

Sargent studied some properties of this space and examined relationship between this space and -space.

The space was extended to by Tripathy and Sen [20] as follows: Recently, Raj et al. [21] introduced and studied the following sequence space .

Let be a sequence of modulus functions. Then the space is defined by

In this work we introduce sequence spaces defined by where is a double sequence of modulus functions.

#### 2. Main Results

The result stated in the first theorem is not hard. So, we give it without proof.

Theorem 1. *The sequence space is a linear space.*

Theorem 2. *The sequence spaces are complete.*

*Proof. *Let be a double Cauchy sequence in such that for all . Then
for some and for all . For each , there exists a positive integer such that
for all . Hence
for some and for all . This implies that
for all and for each fixed . Hence is a Cauchy sequence in .

Then, there exists such that as and let us define . From (10), for each fixed ,
for some , for all and .

Letting , we get
for some , for all , and . Thus we obtain
for some and for all . This implies that for all .

Hence . By (14),
for all . This means that as . Hence is a Banach space.

Theorem 3. *The space is a BK-space.*

*Proof. *Suppose that with as . For each there exists such that
for all . Thus
for some and for all . Hence we obtain
for all and for all . This implies as . This completes the proof.

Corollary 4. *The space is a symmetric space.*

*Proof. *Let and let . Then we can write . Thus we obtain

Corollary 5. *The space is a normal space.*

*Proof. *It is obvious.

Theorem 6. *
Consider
*

*Proof. *Suppose that . Then we have
Thus for each fixed and for ,
for some . Hence
for some . This implies that . Hence .

Theorem 7. * if and only if .*

*Proof. *Let . Then for all . If , then
for some . Thus
for some . Hence . This shows that . Conversely, let . We define . Let . Then there exists a subsequence of such that as . Then for we have
for some . This is a contradiction as and this completes the proof.

Proposition 8. *
Consider
*

*Proof. *Clearly, , where for when and by nondecreasing . Then, by Theorem 7, first inclusion is obtained. Suppose . Then
for some . Hence we obtain
for all . Thus and proof is completed.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### References

- T. Apostol,
*Mathematical Analysis*, Addison-Wesley, London, UK, 1978. - F. Başar,
*Summability Theory and Its Applications*, Bentham Science Publishers, e-books, Monographs, İstanbul, Turkey, 2012. - B. V. Limaye and M. Zeltser, “On the Pringsheim convergence of double series,”
*Proceedings of the Estonian Academy of Sciences*, vol. 58, no. 2, pp. 108–121, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. Başar and Y. Sever, “The space ${L}_{q}$ of double sequences,”
*Mathematical Journal of Okayama University*, vol. 51, pp. 149–157, 2009. View at: Google Scholar | Zentralblatt MATH | MathSciNet - F. Móricz and B. E. Rhoades, “Almost convergence of double sequences and strong regularity of summability matrices,”
*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 104, no. 2, pp. 283–294, 1988. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. Móricz, “Extensions of the spaces $c$ and ${c}_{0}$ from single to double sequences,”
*Acta Mathematica Hungarica*, vol. 57, no. 1-2, pp. 129–136, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - V. A. Khan and S. Tabassum, “Statistically convergent double sequence spaces in 2-normed spaces defined by Orlicz function,”
*Applied Mathematics*, vol. 2, no. 4, pp. 398–402, 2011. View at: Publisher Site | Google Scholar | MathSciNet - V. A. Khan and S. Tabassum, “On some new quasi almost ${\Delta}^{m}$-lacunary strongly P-convergent double sequences defined by Orlicz functions,”
*Journal of Mathematics and Applications*, vol. 34, pp. 45–52, 2011. View at: Google Scholar | MathSciNet - V. A. Khan and S. Tabassum, “On ideal convergent difference double sequence spaces in 2-normed spaces defined by Orlicz function,”
*International Journal of Mathematics and Applications*, vol. 1, no. 2, pp. 26–34, 2011. View at: Google Scholar - V. A. Khan and S. Tabassum, “Some vector valued multiplier difference double sequence spaces in 2-normed spaces defined by Orlicz functions,”
*Journal of Mathematical and Computational Science*, vol. 1, no. 1, pp. 126–139, 2011. View at: Google Scholar | MathSciNet - V. A. Khan and S. Tabassum, “On some new double sequence spaces of invariant means defined by Orlicz functions,”
*Communications de la Faculté des Sciences de l'Université d'Ankara. Séries A1*, vol. 60, no. 2, pp. 11–21, 2011. View at: Google Scholar | Zentralblatt MATH | MathSciNet - V. A. Khan and S. Tabassum, “The strongly summable generalized difference double sequence spaces in 2-normed spaces defined by Orlicz functions,”
*Journal of Mathematical Notes*, vol. 7, no. 2, pp. 45–58, 2011. View at: Google Scholar - V. A. Khan and S. Tabassum, “On some new almost double lacunary ${\Delta}^{m}$-sequence spaces defined by Orlicz functions,”
*Journal of Mathematical Notes*, vol. 6, no. 2, pp. 80–94, 2011. View at: Google Scholar - V. A. Khan and S. Tabassum, “Statistically pre-Cauchy double sequences,”
*Southeast Asian Bulletin of Mathematics*, vol. 36, no. 2, pp. 249–254, 2012. View at: Google Scholar | Zentralblatt MATH | MathSciNet - V. A. Khan, “On ${\Delta}_{v}^{m}$-Cesáro summable double sequences,”
*Thai Journal of Mathematics*, vol. 10, no. 3, pp. 535–539, 2012. View at: Google Scholar | MathSciNet - V. A. Khan, S. Tabassum, and A. Esi, “Statistically convergent double sequence spaces in
*n*-normed spaces,”*Journal of Science and Technology*, vol. 2, no. 10, pp. 991–995, 2012. View at: Google Scholar - V. A. Khan, S. Tabassum, and A. Esi, “${A}_{\sigma}$ double sequence spaces and statistical convergence in 2-normed spaces dened by Orlicz functions,”
*Theory and Applications of Mathematics and Computer Science*, vol. 2, no. 1, pp. 61–71, 2012. View at: Google Scholar - C. Duyar and O. Oğur, “On a new space ${m}^{2}(M,A,\phi ,p)$ of double sequences,”
*Journal of Function Spaces and Applications*, vol. 2013, Article ID 509613, 8 pages, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W. L. C. Sargent, “Some sequence spaces related to the ${l}^{p}$ spaces,”
*Journal of the London Mathematical Society*, vol. 35, pp. 161–171, 1960. View at: Google Scholar | Zentralblatt MATH | MathSciNet - B. C. Tripathy and M. Sen, “On a new class of sequences related to the space ${l}^{p}$,”
*Tamkang Journal of Mathematics*, vol. 33, no. 2, pp. 167–171, 2002. View at: Google Scholar | Zentralblatt MATH | MathSciNet - K. Raj, S. K. Sharma, A. Gupta, and A. Kumar, “A sequence space defined by a sequence of modulus functions,”
*International Journal of Mathematical Analysis*, vol. 5, no. 29-32, pp. 1569–1574, 2011. View at: Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2014 Birsen Sağır et al. 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.