On Paranorm Zweier -Convergent Sequence Spaces
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.
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 : , , , , 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. , Başar and Altay , Malkowsky , Ng and Lee , and Wang .
Şengönül 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 , Şengönül introduced the Zweier sequence spaces and as follows: Here we quote below some of the results due to Şengönül  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  and also independently by Buck  and Schoenberg  for real and complex sequences. Further this concept was studied by Connor [13, 14], Connor et al. , 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. . Later on, it was studied by Šalát et al. [17, 18], Demirci , 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  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 .
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.
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.
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