#### 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, 1014], Raimi [15], Schaefer [16], Savas and Rhoades [17], Vakeel et al. [1820], and many others [2123]. 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. [1820, 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.