Research Article | Open Access

Vakeel A. Khan, Nazneen Khan, "On a New *I*-Convergent Double-Sequence Space", *International Journal of Analysis*, vol. 2013, Article ID 126163, 7 pages, 2013. https://doi.org/10.1155/2013/126163

# On a New *I*-Convergent Double-Sequence Space

**Academic Editor:**Wen Xiu Ma

#### Abstract

The sequence space was introduced and studied by Mursaleen (1983). In this article we introduce the sequence space _{2} and study some of its properties and inclusion relations.

#### 1. Introduction and Preliminaries

Let , , and be the sets of all natural, real, and complex numbers, respectively. We write showing the space of all real or complex sequences.

*Definition 1. *A double sequence of complex numbers is defined as a function . We denote a double sequence as where the two subscripts run through the sequence of natural numbers independent of each other [1]. A number is called a double limit of a double sequence if for every there exists some such that
(see [2]).

Let and denote the Banach spaces of bounded and convergent sequences, respectively, with norm . Let denote the space of sequences of bounded variation; that is, where is a Banach space normed by (see [3]).

*Definition 2. *Let be a mapping of the set of the positive integers into itself having no finite orbits. A continuous linear functional on is said to be an invariant mean or -mean if and only if(i) when the sequence has for all ;(ii), where ;(iii) for all .

In case is the translation mapping , a -mean is often called a Banach limit (see [4]), and , the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent sequences (see [5]).

If , then . Then it can be shown that where , . Consider where denote the th iterate of at . The special case of (5) in which was given by Lorentz [5, Theorem 1], and that the general result can be proved in a similar way. It is familiar that a Banach limit extends the limit functional on .

Theorem 3. *A -mean extends the limit functional on in the sense that for all if and only if has no finite orbits; that is to say, if and only if, for all , , (see [3])
**
Put
**
assuming that . A straight forward calculation shows (see [6]) that
*

For any sequence , , and scalar , we have

*Definition 4. *A sequence is of -bounded variation if and only if (i) converges uniformly in ;(ii), which must exist, should take the same value for all .

We denote by , the space of all sequences of -bounded variation (see [7]):

Theorem 5. * is a Banach space normed by
** (see [8]). *

Subsequently, invariant means have been studied by Ahmad and Mursaleen [9], Mursaleen et al. [3, 6, 8, 10–14], Raimi [15], Schaefer [16], Savas and Rhoades [17], Vakeel et al. [18–20], and many others [21–23]. For the first time, *I*-convergence was studied by Kostyrko et al. [24]. Later on, it was studied by Šalát et al. [25, 26], Tripathy and Hazarika [27], Ebadullah et al. [18–20, 28], and Vakeel et al. [1, 29].

*Definition 6 (see [30, 31]). *Let be a nonempty set. Then, a family of sets ( denoting the power set of ) is said to be an ideal in if(i);(ii) is additive; that is, ;(iii) is hereditary that is, , ;

An Ideal is called nontrivial if . A non-trivial ideal is called admissible if .

A non-trivial ideal is maximal if there cannot exist any non-trivial ideal containing as a subset.

For each ideal , there is a filter corresponding to . That is,

*Definition 7 (see [24, 31, 32]). *A double sequence is said to be *-convergent* to a number if for every ,
In this case, we write .

*Definition 8 (see [2]). *A double sequence is said to be -null if . In this case, we write

*Definition 9. *A double sequence is said to be -cauchy if for every there exist numbers , such that

*Definition 10. *A double sequence is said to be -bounded if there exists such that

*Definition 11. *A double-sequence space is said to be solid or normal if implies for all sequence of scalars with for all .

*Definition 12 (see [24, 33]). *A nonempty family of sets is said to be filter on if and only if(i);(ii)for , we have ;(iii)for each and implies .

*Definition 13. *Let be a linear space. A function is called a paranorm, if for all ,(i) if ;(ii);
(iii);
(iv)if is a sequence of scalars with and with , in the sense that , in the sense that .

The concept of paranorm is closely related to that of linear metric spaces. It is a generalization of that of absolute value (see [34, 35]).

#### 2. Main Results

In this paper, we introduce the sequence space

Theorem 14. * is a linear space.*

* Proof. *Let and , be two scalars in . Then for a given , we have
Now let,
be such that . Now consider
this implies that the sequence space
Hence, . Therefore, is a linear space.

Theorem 15. *The space is a paranormed space, paranormed by
*

*Proof. *For , is trivial.

For , , we have(i) 0 for all .(ii) for all .(iii).(iv)Let be a sequence of scalars with and such that
in the sense that
Therefore,
Hence, is a paranormed space.

Theorem 16. * is a closed subspace of .*

* Proof. *Let be a cauchy sequence in such that . We show that . Since , then there exists such that

We need to show that (i) converges to .(ii)If , then .

Since is a cauchy sequence in , then for a given , there exists such that

For a given , we have
Then , and . Let where . Then . We choose , then for each , we have
Then is a cauchy sequence of scalars in , so there exists a scalar such that , as .

For the next step, let be given. Then, we show that if
then . Since , then there exists such that
which implies that . The number , can be so chosen that together with (32), we have
such that . Since , then we have a subset of such that , where
Let , where

Therefore, for each , we have
Hence, the result follows.

Theorem 17. *The space is nowhere dense subsets of .*

*Proof. *Proof of the result follows from the previous theorem.

Theorem 18. *The space is solid and monotone.*

*Proof. *Let and be a sequence of scalars with for all . Then, we have
The space is solid follows from the following inclusion relation:
Also a sequence space is solid implies monotone. Hence, the space is monotone.

Theorem 19. * and the inclusions are proper.*

*Proof. *Let . Then, we have . Since , implies

Now let,
be such that . As , taking supremum over we get . Hence, .

Next we show that the inclusion is proper

(i) First for . Consider , then by the definition
we have
where
Therefore,
On solving, we get
As is a translation map, that is, , we have
Taking , we have
Since , therefore which implies that . Hence, we get that the inclusion is proper.

(ii) Second for .

The result of this part follows from the proof of Theorem 18.

Theorem 20. * and the inclusions are proper.*

*Proof. *Let . Then, we have
Since , which implies implies

Now let,
be such that . As
taking over , we get . Hence,

Next, we show that the inclusion is proper

(i) First for . We show that .

Let , then by the definition
We have, . We say that the .

Now considering the case when , then
when , then we have . Therefore we get,
Hence, . Hence, the inclusion is proper.

(ii) Second for .

The result follows from the proof of Theorem 18.

#### Acknowledgment

The authors would like to record their gratitude to the reviewer for his careful reading and making some useful corrections which improved the presentation of the paper.

#### References

- A. K. Vakeel 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 A*, vol. 60, no. 2, pp. 11–21, 2011. View at: Google Scholar | MathSciNet - E. D. Habil, “Double sequences and double series,”
*The Islamic University Journal, Series of Natural Studies and Engineering*, vol. 14, pp. 1–32, 2006. View at: Google Scholar - M. Mursaleen, “On some new invariant matrix methods of summability,”
*The Quarterly Journal of Mathematics*, vol. 34, no. 133, pp. 77–86, 1983. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Banach,
*Theorie des Operations Lineaires*, Warszawa, Poland, 1932. - G. G. Lorentz, “A contribution to the theory of divergent sequences,”
*Acta Mathematica*, vol. 80, pp. 167–190, 1948. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Mursaleen, “Matrix transformations between some new sequence spaces,”
*Houston Journal of Mathematics*, vol. 9, no. 4, pp. 505–509, 1983. View at: Google Scholar | Zentralblatt MATH | MathSciNet - V. A. Khan, “On a new sequence space defined by Orlicz functions,”
*Communications de la Faculté des Sciences de l'Université d'Ankara Séries A*, vol. 57, no. 2, pp. 25–33, 2008. View at: Google Scholar | Zentralblatt MATH | MathSciNet - M. Mursaleen and S. A. Mohiuddine, “Some new double sequence spaces of invariant means,”
*Glasnik Matematički*, vol. 45, no. 1, pp. 139–153, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. U. Ahmad and Mursaleen, “An application of Banach limits,”
*Proceedings of the American Mathematical Society*, vol. 103, no. 1, pp. 244–246, 1988. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Mursaleen and A. Alotaibi, “On $I$-convergence in random 2-normed spaces,”
*Mathematica Slovaca*, vol. 61, no. 6, pp. 933–940, 2011. View at: Publisher Site | Google Scholar | MathSciNet - M. Mursaleen, S. A. Mohiuddine, and O. H. H. Edely, “On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces,”
*Computers & Mathematics with Applications*, vol. 59, no. 2, pp. 603–611, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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. View at: Google Scholar | Zentralblatt MATH | MathSciNet - M. Mursaleen and S. A. Mohiuddine, “On ideal convergence in probabilistic normed spaces,”
*Mathematica Slovaca*, vol. 62, no. 1, pp. 49–62, 2012. View at: Publisher Site | Google Scholar | MathSciNet - M. Mursaleen, A. Alotaibi, and M. A. Alghamdi, “$I$-summability and $I$-approximation through invariant mean,”
*Journal of Computational Analysis and Applications*, vol. 14, no. 6, pp. 1049–1058, 2012. View at: Google Scholar | MathSciNet - R. A. Raimi, “Invariant means and invariant matrix methods of summability,”
*Duke Mathematical Journal*, vol. 30, pp. 81–94, 1963. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. Schaefer, “Infinite matrices and invariant means,”
*Proceedings of the American Mathematical Society*, vol. 36, pp. 104–110, 1972. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. Savas and B. E. Rhoades, “On some new sequence spaces of invariant means defined by Orlicz functions,”
*Mathematical Inequalities & Applications*, vol. 5, no. 2, pp. 271–281, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. K. Vakeel, K. Ebadullah, and S. Suantai, “On a new I-convergent sequence space,”
*Analysis*, vol. 32, no. 3, pp. 199–208, 2012. View at: Publisher Site | Google Scholar | MathSciNet - A. K. Vakeel and K. Ebadullah, “On some I-Convergent sequence spaces defined by a modullus function,”
*Theory and Applications of Mathematics and Computer Science*, vol. 1, no. 2, pp. 22–30, 2011. View at: Google Scholar - A. K. Vakeel and K. Ebadullah, “I-convergent difference sequence spaces defined by a sequence of moduli,”
*Journal of Mathematical and Computational Science*, vol. 2, no. 2, pp. 265–273, 2012. View at: Google Scholar | MathSciNet - A. Komisarski, “Pointwise $I$-convergence and $I$-convergence in measure of sequences of functions,”
*Journal of Mathematical Analysis and Applications*, vol. 340, no. 2, pp. 770–779, 2008. View at: Publisher Site | Google Scholar | MathSciNet - V. Kumar, “On
*I*and ${I}^{\ast}$-convergence of double sequences,”*Mathematical Communications*, vol. 12, no. 2, pp. 171–181, 2007. View at: Google Scholar | MathSciNet - 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. View at: Google Scholar | Zentralblatt MATH | MathSciNet - P. Kostyrko, T. Šalát, and W. Wilczyński, “$I$-convergence,”
*Real Analysis Exchange*, vol. 26, no. 2, pp. 669–685, 2000. View at: Google Scholar | MathSciNet - T. Šalát, B. C. Tripathy, and M. Ziman, “On some properties of $I$-convergence,”
*Tatra Mountains Mathematical Publications*, vol. 28, pp. 279–286, 2004. View at: Google Scholar | MathSciNet - T. Šalát, B. C. Tripathy, and M. Ziman, “On $I$-convergence field,”
*Italian Journal of Pure and Applied Mathematics*, no. 17, pp. 45–54, 2005. View at: Google Scholar | MathSciNet - B. C. Tripathy and B. Hazarika, “Paranorm $I$-convergent sequence spaces,”
*Mathematica Slovaca*, vol. 59, no. 4, pp. 485–494, 2009. View at: Publisher Site | Google Scholar | MathSciNet - V. A. Khan and K. Ebadullah, “On a new difference sequence space of invariant means defined by Orlicz functions,”
*Bulletin of the Allahabad Mathematical Society*, vol. 26, no. 2, pp. 259–272, 2011. View at: Google Scholar | MathSciNet - 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. View at: Google Scholar - P. Das, P. Kostyrko, W. Wilczyński, and P. Malik, “
*I*and ${I}^{\ast}$-convergence of double sequences,”*Mathematica Slovaca*, vol. 58, no. 5, pp. 605–620, 2008. View at: Publisher Site | Google Scholar | MathSciNet - M. Gurdal and M. B. Huban, “On
*I*-convergence of double sequences in the topology induced by random 2-norms,”*Matematički Vesnik*, vol. 65, no. 3, pp. 1–13, 2013. View at: Google Scholar - M. Gürdal and A. Şahiner, “Extremal $I$-limit points of double sequences,”
*Applied Mathematics E-Notes*, vol. 8, pp. 131–137, 2008. View at: Google Scholar | MathSciNet - S. Roy, “Some new type of fuzzy I-convergent double difference sequence spaces,”
*International Journal of Soft Computing and Engineering*, vol. 1, pp. 429–431, 2012. View at: Google Scholar - I. J. Maddox,
*Elements of Functional Analysis*, Cambridge University Press, 1970. View at: MathSciNet - H. Nakano, “Concave modulars,”
*Journal of the Mathematical Society of Japan*, vol. 5, pp. 29–49, 1953. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2013 Vakeel A. Khan and 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.