On Some Properties of New Paranormed Sequence Space of Nonabsolute Type
We introduce some new generalized sequence space related to the space . Furthermore we investigate some topological properties as the completeness, the isomorphism, and also we give some inclusion relations between this sequence space and some of the other sequence spaces. In addition, we compute -, -, and -duals of this space and characterize certain matrix transformations on this sequence space.
In studying the sequence spaces, especially, to obtain new sequence spaces, in general, the matrix domain of an infinite matrix defined by is used. In most cases, the new sequence space generated by a sequence space is the expansion or the contraction of the original space . In some cases, these spaces could be overlapped. Indeed, one can easily see that the inclusion strictly holds for . Similarly one can deduce that the inclusion also strictly holds for , where and are matrix operators.
Recently, in , Mursaleen and Noman constructed new sequence spaces by using matrix domain over a normed space. They also studied some topological properties and inclusion relations of these spaces.
It is well known that paranormed spaces have more general properties than the normed spaces. In this work, we generalize the normed sequence spaces defined by Mursaleen and Noman  to the paranormed spaces. Furthermore we introduce new sequence space over the paranormed space. Next we investigate behaviors of this sequence space according to topological properties and inclusion relations. Finally we give certain matrix transformation on this sequence space and its duals.
In the literature, by using the matrix domain over the paranormed spaces, many authors have defined new sequence spaces. Some of them are as follows. For example, Choudhary and Mishra  have defined the sequence space where the transform is in , Başar and Altay [3, 4] defined the spaces for and , respectively, and Altay and Başar  have defined the spaces . In , Karakaya and Polat defined and examined the spaces , and Karakaya et al.  have recently introduced and studied the spaces , , , where and denote the Riesz and the Euler means, respectively, denotes the band matrix of the difference operators, and , are defined in [1, 8], respectively. Also, the information on matrix domain of sequence spaces can be found in [9–13].
By , we denote the space of all real valued sequences. Any vector subspace of is called a sequence space. By the spaces , and , we denote the spaces of all absolutely convergent series, convergent series, and bounded series, respectively.
A linear topological space over the real field is said to be a paranormed space if there is a subadditivity function such that , , and scalar multiplication is continuous, that is, and imply for all in and in , where is the zero in the linear space .
Let be any two sequence spaces, and let be any infinite matrix of real number , where with . Then we say that defines a matrix mapping from into by writing, if for every sequence , the sequence , the transform of , is in , where By , we denote the class of all matrices such that . Thus, if and only if the series on the right hand side of (1.1) converges for each and every , and we have for all . A sequence is said to be summable to if converges to which is called as the limit of .
Assume here and after that , are bounded sequences of strictly positive real numbers with and , and also let for and for all . The linear space was defined by Maddox  as follows: which are the complete space paranormed by Throughout this work, by Ϝ and , respectively, we will denote the collection of all subsets of and the set of all such that and .
2. The Sequence Space
In this section, we define the sequence space and prove that this sequence space according to its paranorm is complete paranormed linear space. In , Mursaleen and Noman defined the matrix by where is a strictly increasing sequence of positive reals tending to , that is, and as . Now, by using (2.1) we define new sequence space as follows: For any , we define the sequence , which will frequently be used, as the -transform of , that is, , and hence We now may begin with the following theorem.
Theorem 2.1. The sequence space is the complete linear metric space with respect to paranorm defined by
Proof. The linearity of with respect to the coordinatewise addition and scalar multiplication follows from the following inequalities which are satisfied for (see, ): and for any (see, ) It is clear that , for all . Again inequalities (2.5) and (2.6) yield the subadditivity of and hence . Let be any sequence of points such that and also any sequence of scalars such that . Then, since the inequality holds by subadditivity of , we can write that is bounded and we thus have which tends to zero as . Therefore, the scalar multiplication is continuous. Hence is a paranorm on the space . It remains to prove the completeness of the space . Let be any Cauchy sequence in the space , where . Then, for a given , there exists a positive integer such that for all . Using definition of , we obtain for each fixed that for every , which leads us to the fact that is a Cauchy sequence of real numbers for every fixed . Since is complete, it converges, say as . Using these infinitely many limits, we may write the sequence . From (2.9) as , we have for every fixed . By using (2.9) and boundedness of Cauchy sequence, we have Hence, we get . So, the space is complete.
Theorem 2.2. The sequence space of nonabsolute type is linearly isomorphic to the space , where .
Proof. To prove the theorem, we would show the existence of linear bijection between the spaces and . With the notation of (2.3), we define transformation from to by . The linearity of is trivial. Furthermore, it is obvious that whenever and hence is injective.
Let and define the sequence Then, we have Thus, we have that and consequently is surjective. Hence, is a linear bijection and this tells us that the spaces and are linearly isomorphic. This completes the proof.
3. Some Inclusion Relations
In this section, we give some inclusion relations concerning the space . Before giving the theorems about the section, we give a lemma given in .
Lemma 3.1. For any sequence , the equalites hold, where the sequence is defined by
Theorem 3.2. The inclusion holds.
Proof. Let . It can be written . By the definition of the space , as , we obtain . Hence we get .
Theorem 3.3. The inclusion if and only if for every sequence , where .
Proof. We suppose that holds and take any . Then by hypothesis. Thus we obtain from (3.1) that
which yields that .
Conversely, let be given. Then we have by the hypothesis that . Again by using (3.1) which shows that . Hence the inclusion holds. This completes the proof.
Theorem 3.4. (i) If for all , then the inclusion holds.
(ii) If for all , then the inclusion holds.
Proof. (i) If for all , then we write in place of. Let . It is clear that . One can find such that for all . Under condition (i), we have for all . Hence we get .
(ii) We suppose that . Then and there exists such that for all . To obtain the result, we consider the following inequality: for all . So, we get .
4. Some Matrix Transformations and Duals of the Space
In this section, we give the theorems determining the -, -, and -duals of the space . In proving the theorem, we apply the technique used in . Also we give some matrix transformations from the space into paranormed spaces by using the matrix given in .
For the sequence space and , the set defined by is called the multiplier space of and . The , and duals of a sequence space , which are, respectively, denoted by , , and , are defined by We may begin with the following theorem which computes the -dual of the space .
Theorem 4.1. Let and . Define the matrix by Then
Proof. We consider the following equality:
where is defined by (4.3).
From (4.5), it can be obtained that or whenever if and only if or whenever .This means or if and only if or . Hence this completes the proof.
The result of the Theorem above corresponds the Theorem 5.1(0, 8, 12) given in .
As a direct consequence of Theorem 4.1, we have the following.
Corollary 4.2. Let for . Then(i), (ii)
In the following theorem, we characterize the - and -duals of the space .
Theorem 4.3. Let , , and let . Define the sequence , and the matrix by for all . Then
Proof. Consider the equality From (4.9), it can be obtained that or whenever if and only if or whenever . This means that or if and only if or . Hence this completes the proof.
We can write the following corollary from Theorem 4.3.
Corollary 4.4. Let for and for all . Then(i), (ii).
After this step, we can give our theorems on the characterization of some matrix classes concerning the sequence space .
Let be connected by the relation . For an infinite matrix , we have by using (4.9) of Theorem 4.3 that where The necessary and sufficient conditions characterizing the matrix mapping of the sequence space of Maddox have been determined by Grosse-Erdmann . Let and be the natural numbers and define the sets by and and also let us put for and for all . Before giving the theorems, let us suppose that is a nondecreasing bounded sequence of positive real numbers and consider the following conditions:
Theorem 4.5. One has the following:(i) if and only if (4.12), (4.13), (4.14), and (4.25) hold,(ii) if and only if (4.15), (4.16), (4.17), and (4.25) hold,(iii) if and only if (4.18), (4.19), (4.20), (4.21), (4.22), and (4.26) hold,(iv) if and only if (4.23), (4.24), and (4.27) hold.
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