On some properties of new paranormed sequence space of non-absolute type

In this work, we introduce some new generalized sequence space related to the space l(p). 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 alpha-, beta- and gamma-duals of this space, and characterize certain matrix transformations on this sequence space.


Introduction
In studying the sequence spaces, especially, to obtain new sequence spaces, in general, the matrix domain μ A of an infinite matrix A defined by μ A {x x k ∈ w : Ax ∈ μ} is used. In most cases, the new sequence space μ A generated by a sequence space μ is the expansion or the contraction of the original space μ. In some cases, these spaces could be overlapped. Indeed, one can easily see that the inclusion μ S ⊂ μ strictly holds for μ ∈ { ∞ , c, c 0 }. Similarly one can deduce that the inclusion μ ⊂ μ Δ also strictly holds for μ ∈ { ∞ , c, c 0 }, where S and Δ are matrix operators.
Recently, in 1 , 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 1 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 2 have defined the sequence space p where the S-transform is in p , Başar and Altay 3, 4 defined the spaces λ u, v; p {λ p } G for λ ∈ { ∞ , c, c 0 } and u, v; p { p } G , respectively, and Altay and Başar 5 have defined the spaces r t ∞ p , r t c p , r t 0 p . In 6 , Karakaya and Polat defined and examined the spaces e r 0 Δ; p , e r Δ; p , e r ∞ Δ; p , and Karakaya et al. 7 have recently introduced and studied the spaces ∞ λ, p , c λ, p , c 0 λ, p , where R t and E r denote the Riesz and the Euler means, respectively, Δ denotes the band matrix of the difference operators, and Λ, G are defined in 1, 8 , respectively. Also, the information on matrix domain of sequence spaces can be found in 9-13 . By w, we denote the space of all real valued sequences. Any vector subspace of w is called a sequence space. By the spaces 1 , cs, and bs, we denote the spaces of all absolutely convergent series, convergent series, and bounded series, respectively.
A linear topological space X over the real field R is said to be a paranormed space if there is a subadditivity function h : X → R such that h θ 0, h x h −x , and scalar multiplication is continuous, that is, |α n − α| → 0 and h x n − x → 0 imply h α n x n − αx → 0 for all α in R and x in X, where θ is the zero in the linear space X.
Let μ, ν be any two sequence spaces, and let A a nk be any infinite matrix of real number a nk , where n, k ∈ N with N {0, 1, 2, . . .}. Then we say that A defines a matrix mapping from μ into ν by writingA : By μ, ν , we denote the class of all matrices A such that A : μ → ν. Thus, A ∈ μ, ν if and only if the series on the right hand side of 1.1 converges for each n ∈ N and every x ∈ μ, and we have Ax ∈ ν for all x ∈ μ. A sequence x is said to be A-summable to a if Ax converges to a which is called as the A-limit of x. Assume here and after that p k , q k are bounded sequences of strictly positive real numbers with sup p k H and M max 1, H , and also let ' p k p k / p k − 1 for 1 < p k < ∞ and for all k ∈ N. The linear space p was defined by Maddox 14 as follows: Throughout this work, by and N k , respectively, we will denote the collection of all subsets of N and the set of all n ∈ N such that n ≥ k and e 1, 1, 1, . . . . Abstract and Applied Analysis 3

The Sequence Space λ, p
In this section, we define the sequence space λ, p and prove that this sequence space according to its paranorm is complete paranormed linear space. In 1 , Mursaleen and Noman defined the matrix Λ λ nk ∞ n,k 0 by where λ λ k ∞ k 0 is a strictly increasing sequence of positive reals tending to ∞, that is, 0 < λ 0 < λ 1 < · · · and λ k → ∞ as k → ∞. Now, by using 2.1 we define new sequence space as follows: For any x x n ∈ w, we define the sequence y y n , which will frequently be used, as the Λ-transform of x, that is, y Λ x , and hence We now may begin with the following theorem.

2.4
Proof. The linearity of λ, p with respect to the coordinatewise addition and scalar multiplication follows from the following inequalities which are satisfied for x, t ∈ λ, p see, 15 : for every i,j > m 0 ε which leads us to the fact that {Λ n x 0 , Λ n x 1 , Λ n x 2 , . . .} is a Cauchy sequence of real numbers for every fixed n ∈ N. Since R is complete, it converges, say Λ n x i − Λ n x as i → ∞. Using these infinitely many limits, we may write the sequence {Λ 0 x , Λ 1 x , Λ 2 x , . . .}. From 2.9 as i → ∞, we have for every fixed n ∈ N. By using 2.9 and boundedness of Cauchy sequence, we have Hence, we get x ∈ λ, p . So, the space λ, p is complete. Proof. To prove the theorem, we would show the existence of linear bijection between the spaces λ, p and p . With the notation of 2.3 , we define transformation T from λ, p to p by x → y Tx. The linearity of T is trivial. Furthermore, it is obvious that x θ whenever Tx θ and hence T is injective. Thus, we have that x ∈ λ, p and consequently T is surjective. Hence, T is a linear bijection and this tells us that the spaces λ, p and p are linearly isomorphic. This completes the proof.

Some Inclusion Relations
In this section, we give some inclusion relations concerning the space λ, p . Before giving the theorems about the section, we give a lemma given in 1 .

Lemma 3.1.
For any sequence x x k ∈ w, the equalites Theorem 3.2. The inclusion λ, p ⊂ c 0 λ, p holds.
Proof. Let x ∈ λ, p . It can be written Λx ∈ p . By the definition of the space p , Λ n x → ∞ as n → ∞, we obtain Λx ∈ c 0 . Hence we get x ∈ c 0 λ, p . Proof. i If p p n for all n ∈ N, then we write λ p in place of λ, p . Let x ∈ λ p . It is clear that Λ x ∈ p . One can find m ∈ N such that |Λ n x | < 1 for all n ≥ m. Under condition i , we have |Λ n x | p n < |Λ n x | for all n ≥ m. Hence we get x ∈ λ, p .
ii We suppose that x ∈ λ, p . Then Λ x ∈ p and there exists m ∈ N such that |Λ n x | p n < 1 for all n ≥ m. To obtain the result, we consider the following inequality: for all n ≥ m. So, we get x ∈ λ p .

Some Matrix Transformations and Duals of the Space λ, p
In this section, we give the theorems determining the α-, β-, and γ-duals of the space λ, p .
In proving the theorem, we apply the technique used in 3 . Also we give some matrix transformations from the space λ, p into paranormed spaces q by using the matrix given in 1 .

4.2
We may begin with the following theorem which computes the α-dual of the space λ, p .