#### Abstract

In this paper, we introduce the paranorm Zweier -convergent sequence spaces , , and , a sequence of positive real numbers. We study some topological properties, prove the decomposition theorem, and study some inclusion relations on these spaces.

#### 1. Introduction

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

Let , , and denote the Banach spaces of bounded, convergent, and null sequences, respectively, normed by .

The following subspaces of were first introduced and discussed by Maddox [1]: , , , , where is a sequence of strictly positive real numbers.

After that Lascarides [2, 3] defined the following sequence spaces: where , for all .

Each linear subspace of , for example, , is called a sequence space.

A sequence space with linear topology is called a -space provided each map defined by is continuous for all .

A -space is called an -space provided is a complete linear metric space.

An FK-space whose topology is normable is called a BK-space.

Let and be two sequence spaces and an infinite matrix of real or complex numbers , where . Then we say that defines a matrix mapping from to , and we denote it by writing .

If for every sequence the sequence , the transform of is in , where By , we denote the class of matrices such that .

Thus, if and only if series on the right side of (3) converges for each and every .

The approach of constructing the new sequence spaces by means of the matrix domain of a particular limitation method has been recently employed by Altay et al. [4], Başar and Altay [5], Malkowsky [6], Ng and Lee [7], and Wang [8].

Şengönül[9] defined the sequence which is frequently used as the transform of the sequence , that is, where , and denotes the matrix defined by Following Başar and Altay [5], Şengönül[9] introduced the Zweier sequence spaces and as follows: Here we quote below some of the results due to Şengönül [9] which we will need in order to establish the results of this paper.

Theorem 1 (see [9, Theorem 2.1]). *The sets and are the linear spaces with the coordinate wise addition and scalar multiplication which are the BK-spaces with the norm
*

Theorem 2 (see [9, Theorem 2.2]). *The sequence spaces and are linearly isomorphic to the spaces and , respectively, that is, and .*

Theorem 3 (see [9, Theorem 2.3]). *The inclusions strictly hold for .*

Theorem 4 (see [9, Theorem 2.6]). * is solid.*

Theorem 5 (see [9, Theorem 3.6]). * is not a solid sequence space.*

The concept of statistical convergence was first introduced by Fast [10] and also independently by Buck [11] and Schoenberg [12] for real and complex sequences. Further this concept was studied by Connor [13, 14], Connor et al. [15], and many others. Statistical convergence is a generalization of the usual notion of convergence that parallels the usual theory of convergence. A sequence is said to be statistically convergent to if for a given as The notion of -convergence is a generalization of the statistical convergence. At the initial stage, it was studied by Kostyrko et al. [16]. Later on, it was studied by Šalát et al. [17, 18], Demirci [19], Tripathy and Hazarika [20, 21], and Khan et al. [22–24].

Here we give some preliminaries about the notion of -convergence.

Let X be a nonempty set. Then a family of sets (denoting the power set of ) is said to be an ideal if is additive, that is, , and hereditary, that is, , .

A nonempty family of sets is said to be a filter on if and only if , for we have and for each and implies .

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 I. that is, () = , where .

*Definition 6. *A sequence is said to be -convergent to a number if for every . In this case we write .

The space of all -convergent sequences converging to is given by

*Definition 7. *A sequence is said to be -null if . In this case we write .

*Definition 8. *A sequence is said to be -Cauchy if for every there exists a number such that for all .

*Definition 9. *A sequence is said to be -bounded if there exists such that .

*Definition 10. *Let be two sequences. We say that for *almost all k relative to * (*a.a.k.r.I*), if .

The following lemma will be used for establishing some results of this paper.

Lemma 11. *If and . If , then (see [20, 21]) cf. ([17, 18, 20–24]).*

Recently Khan and Ebadullah [25] introduced the following classes of sequence spaces: We also denote by

In this paper we introduce the following classes of sequence spaces: We also denote by where , is a sequence of positive real numbers.

Throughout the paper, for the sake of convenience now we will denote by for all .

#### 2. Main Results

Theorem 12. *The classes of sequences and are linear spaces.*

*Proof. *We shall prove the result for the space .

The proof for the other spaces will follow similarly.

Let , and let be scalars. Then for a given : we have
where
Let
be such that .

Then
Thus . Hence . Therefore is a linear space. The rest of the result follows similarly.

Theorem 13. *Let . Then and are paranormed spaces, paranormed by where .*

*Proof. *Let .(1)Clearly, if and only if .(2) is obvious.(3)Since and , using Minkowski’s inequality, we have
(4)Now for any complex , we have such that , .

Let such that .

Therefore, , where .

Hence as .

Hence is a paranormed space.

The rest of the result follows similarly.

Theorem 14. * 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 (1) converges to ,(2)if , then .

(1) Since is a Cauchy sequence in then for a given , there exists such that
For a given , we have
Then .

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 .

(2) 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 (23), we have
such that .

Since . Then we have a subset of such that , where
Let , where .

Therefore for each , we have
Then the result follows.

Since the inclusions and are strict, so in view of Theorem 14 we have the following result.

Theorem 15. *The spaces and are nowhere dense subsets of .*

Theorem 16. *The spaces and are not separable.*

*Proof. *We shall prove the result for the space .

The proof for the other spaces will follow similarly.

Let be an infinite subset of of increasing natural numbers such that .

Let
Let .

Clearly is uncountable.

Consider the class of open balls .

Let be an open cover of containing .

Since is uncountable, so cannot be reduced to a countable subcover for .

Thus is not separable.

Theorem 17. *Let and an admissible ideal. Then the following is equivalent.*(a)*;
*(b)*there exists such that , for a.a.k.r.I;*(c)*there exists and such that for all and ;*(d)*there exists a subset of such that and .*

*Proof. *(a) implies (b).

Let . Then there exists such that
Let be an increasing sequence with such that
Define a sequence as
For .
Then and form the following inclusion:
we get , for a.a.k.r.I.

(b) implies (c).

For . Then there exists such that , for a.a.k.r.I.

Let , then .

Define a sequence as
Then and .

(c) implies (d).

Suppose (c) holds.

Let be given.

Let and
Then we have .

(d) implies (a).

Let and .

Then for any , and Lemma 11, we have
Thus .

Theorem 18. *Let and . Then the following results are equivalent.*(a)* and .*(b)* .*

*Proof. *Suppose that and , then the inequalities hold for any and for all .

Therefore the equivalence of (a) and (b) is obvious.

Theorem 19. *Let and be two sequences of positive real numbers. Then if and only if , where such that .*

*Proof. *Let and . Then there exists such that , for all sufficiently large .

Since for a given , we have
Let Then .

Then for all sufficiently large ,
Therefore .

The converse part of the result follows obviously.

Theorem 20. *Let and be two sequences of positive real numbers. Then if and only if , where such that .*

*Proof. *The proof follows similarly as the proof of Theorem 19.

Theorem 21. *Let and be two sequences of positive real numbers. Then if and only if , and , where such that .*

*Proof. *By combining Theorems 19 and 20, we get the required result.

#### 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 this paper.