Research Article | Open Access
Some -Type New Sequence Spaces and Their Geometric Properties
We introduce an -type new sequence space and investigate its some topological properties including and properties. Besides, we examine some geometric properties of this space concerning Banach-Saks type and Gurarii's modulus of convexity.
In general, the -type spaces have many useful applications because of the properties of the spaces and . In  it was shown that the subspaces of Orlicz spaces, which have rich geometric properties, are isomorphic to the space Also since the space is reflexive and convex, it is natural to consider the geometric structure of these spaces.
Recently there has been a lot of interest in investigating geometric properties of sequence spaces besides topological and some other usual properties. In literature, there are many papers concerning the geometric properties of different sequence spaces. For example; in , Mursaleen et al. studied some geometric properties of normed Euler sequence space. Sanhan and Suantai  investigated the geometric properties of Cesáro sequence space ces(p) equipped with Luxemburg norm. Further information on geometric properties of sequence spaces can be found in [4–7].
The main purpose of our work is to introduce an -type new sequence space together with matrix domain and its summability methods. Also we investigate some topological properties of this new space as the paranorm, and properties, and furthermore characterize geometric properties concerning Banach-Saks type and Gurarii's modulus of convexity.
2. Preliminaries and Notations
Let be the space of all real-valued sequences. Each linear subspace of is called a sequence space denoted by . We denote by and absolutely and -absolutely convergent series, respectively.
A sequence space with a linear topology is called a -space provided that each of the maps defined by is continuous for all , where denotes the complex field, and A -space is called -space provided that is a complete linear metric space. An -space whom topology is normable is called -space. An -space is said to have property, if and is a basis for , where is a sequence whose only nonzero term is 1, th place for each , and , the set of all -space, thus implies
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. It is well known that the space is -space where .
Throughout this work, we suppose that is a bounded sequence of strictly positive real numbers with and . Also the summation without limits runs from to . In , the linear space was defined by Maddox (see also Simons  and Nakano ) as follows: which is a complete space paranormed by
Let , be any two sequence spaces, and let be an infinite matrix of real numbers , where . Then we write , the -transform of if converges for each .
A matrix is called a triangle if for and for all . It is trivial that holds for the triangle matrices , and a sequence . Further, a triangle matrix uniquely has an invert which is also a triangle matrix. Then if hold for all
By , we denote the class of all infinite matrices such that The matrix domain of an infinite matrix in a sequence space is defined by which is a sequence space. It is well known that the new sequence space generated by the limitation matrix from a sequence space is the expansion or the contraction of original space
If is triangle, then one can easily observe that the sequence spaces and are linearly isomorphic, that is, . Let be a sequence space. Then the continuous dual of the space is defined by Let be a seminormed space. A set is called fundamental set if the span of is dense in . An application of Hahn-Banach theorem on fundamental set is as follows: if is the subset of a seminormed space and implies for , then is a fundamental set (see ).
By the idea mentioned above, let us give the definitions of some matrices to construct a new sequence space in sequel to this work. We denote and by Malkowsky and Savas  Choudhary and Mishra , and Altay and Baar  have defined the sequence spaces , and respectively. By using the matrix domain, the spaces and may be redefined by and respectively.
If is a sequence space and , -transform with (2.4) corresponds to th partial sum of the series and it is denoted by
By using (2.4) and any infinite lower triangular matrix , we can define two infinite lower triangular matrices and as follows: and . Let be a sequence in By considering the multiplication of infinite lower triangular matrices, we have , that is, Also since we have that is, Now let us write the following equality: It can be seen that for any sequences , and scalar , and We now define new sequence space as follows: For some special cases of the infinite lower triangular matrix and the sequence , we obtain the following spaces.
(i) If for all the space reduces to the normed space denoted by
(ii) If , which is Cesáro matrix order 1, then the space corresponds to the space denoted by where for and
Also if for all , then the spaces and reduce to the spaces and (see ), respectively.
Now let us introduce some definitions of geometric properties of sequence spaces.
Let be a normed linear space, and let and be the unit sphere and unit ball of (for the brevity ), respectively. Consider Clarkson's modulus of convexity (Clarkson  and Day ) defined by where . The inequality for all characterizes the uniformly convex spaces.
In , Gurariĭ's modulus of convexity is defined by where . It is easily shown that for any Also if , then is uniformly convex, and if , then is strictly convex.
A Banach space is said to have the Banach-Saks property if every bounded sequence in admits a subsequence such that the sequence is convergent in the norm in (see ), where
Let A Banach space is said to have the Banach-Saks type or property , if every weakly null sequence has a subsequence such that for some for all (see ).
3. Some Topological Properties of the Space
In this section, we investigate some topological properties of the sequence space as the paranorm property and property. Let us begin the following theorem.
Theorem 3.1. (i) The space is complete linear metric space with respect to the paranorm defined by
(ii) If the sequence is constant sequence and , then is a Banach space normed by
Proof. The proof of (ii) is routine verification by using standard techniques and hence it is omitted.
The proof of (i) is that the linearity of with respect to the coordinatewise addition and scalar multiplication follows from the following inequalities which are satisfied for : and for any (see ). After this step, we must show that the space holds the paranorm property and the completeness with respect to given paranorm. It is easy to show that , and for all . Besides, from (3.3) we obtain for all . To complete the paranorm conditions for the space , it remains to show the continuity of the scalar multiplication. Let be any sequence in such that and let be also any sequence of scalars such that . From subadditivity of , we give the inequality . Hence is bounded and we have which tends to zero as . Consequently we obtain that is a paranorm over the space . To prove the completeness of the space let us take any Cauchy sequence in the space . Then for a given , there exists a positive integer such that for all . By using the definitions of the Cauchy sequence and the paranorm, we have, for each fixed , for every . Hence we obtain that the sequence is a Cauchy sequence of real numbers for every fixed . Since is complete, it converges, that is, as where . Now let us choose such that for each and . By taking and for every , we get Again taking and for every , it is obtained that . We write the following equality: By using (3.7) and Minkowski's inequality, we get which implies . It follows as . Consequently, since is any Cauchy sequence, we obtain that the space is complete. This completes the proof.
Theorem 3.2. The space is linearly isomorphic to the space .
Proof. Let us define -transform between the spaces and such that . We have to show that the transformation is linear, injective and surjective. The linearity of is obvious. Moreover it is injective because of whenever . For the surjective property, let . From (2.3) and (2.7), there exists a matrix such that . We have Hence we obtain that the transformation is surjective. Consequently, the spaces and are linearly isomorphic spaces.
Theorem 3.3. The space has property.
Proof. Let . Then for some . Since has AK property and where , for some . Also since and the inclusion holds, we have . For any and , we have where is transpose of the matrix . Hence from Hahn-Banach theorem, is dense in if and only if for implies . Besides, since the null space of the operator on is , has property. Hence the proof is completed.
4. Some Geometric Properties of the space
In this section, we give some geometric properties for the space .
Theorem 4.1. The space has the Banach-Saks of type .
Proof. Let be a sequence of positive numbers for which . Let be a weakly null sequence in . Set and . Then there exists such that
Since is a weakly null sequence implies that with respect to the coordinatwise, there is an such that
where . Set . Then there exists an such that
By using the fact that (coordinatwise), there exists an such that
If we continue this process, we can find two increasing subsequences and such that for each and where . Hence, On the other hand, since and , it can be seen that Therefore . We have Hence we obtain By using the fact for all and , we have Hence has the Banach-Saks type . This completes the proof.
Theorem 4.2. Gurarii's modulus of convexity for the normed space is where
Proof. We have Then we have Let and consider the following sequences: where is the inverse of the matrix . Since and we have By using sequences given above, we obtain the following equalities: To complete the conditions of for Gurarii's modulus of convexity, it remains to show the infimum of for We have Consequently we get for This is the desired result. Hence the proof is completed.
Corollary 4.3. (i) If then and hence is strictly convex.
(ii) If , then and hence is uniformly convex.
Corollary 4.4. If then
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