New Type of Riesz Sequence Space of Non-absolute Type

respectively; where x=(xk), y=(yk) ∈ ω and α=C. By a sequence space we understand a linear subspace of ω i.e., the sequence space is the set of scalar sequences(real or complex) which is closed under co-ordinate wise addition and scalar multiplication. Throughout the paper N, R and C denotes the set of non-negative integers, the set of real numbers and the set of complex numbers, respectively. Let l∞, c and c0, respectively, denotes the space of all bounded sequences , the space of convergent sequences and the sequences converging to zero. Also, by l1, l(p), cs and bs we denote the spaces of all absolutely, p-absolutely convergent, convergent and bounded series, respectively.


Preliminaries, Background and Notation
We denote the set of all sequences with complex terms by ω. It is a routine verification that ω is a linear space with respect to the coordinate wise addition and scalar multiplication of sequences which are defined, as usual, by respectively; where x=(x k ), y=(y k ) ∈ ω and α=C. By a sequence space we understand a linear subspace of ω i.e., the sequence space is the set of scalar sequences(real or complex) which is closed under co-ordinate wise addition and scalar multiplication. Throughout the paper N, R and C denotes the set of non-negative integers, the set of real numbers and the set of complex numbers, respectively. Let l ∞ , c and c 0 , respectively, denotes the space of all bounded sequences , the space of convergent sequences and the sequences converging to zero. Also, by l 1 , l(p), cs and bs we denote the spaces of all absolutely, p-absolutely convergent, convergent and bounded series, respectively.
Let X, Y be two sequence spaces and let A=(a nk ) be an infinite matrix of real or complex numbers a nk , where n, k ∈ N. Then, the matrix A defines the A-transformation from X into Y, if for every sequence x=(x k ) ∈ X the sequence Ax={(Ax) n }, the A-transform of x exists and is in Y; where ( ) = nk k n k Ax a x ∑ . For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞. By A ∈ (X:Y) we mean the characterizations of matrices from X to Y i.e., A:X →Y. A sequence x is said to be A-summable to l if Ax converges to l which is called as the A-limit of x.
For a sequence space X, the matrix domain X A of an infinite matrix A is defined as The theory of matrix transformations is a wide field in summability; it deals with the characterizations of classes of matrix mappings between sequence spaces by giving necessary and sufficient conditions on the entries of the infinite matrices.
The classical summability theory deals with a generalization of convergence of sequences and series. One original idea was to assign a limit to divergent sequences or series. Toeplitz [1] was the first to study summability methods as a class of transformations of complex sequences by complex infinite matrices.
Let A=(a nk ) be any matrix. Then a sequence x is said to be summable to l, written x k → l, if and only if exists for each n and A n x → l (n → ∞). For example, if I is the unit matrix, then x k → l(I) means precisely that x k → l (k → ∞), in the ordinary sense of convergence.
We denote by (A) the set of all sequences which are summable A. The set (A) is called summability field of the matrix A. Thus, if Ax=(a n (x)), then (A)={x:Ax ∈ c}, where c is the set of convergent sequences. For example, (I)=c.
A infinite matrix A=(a nk ) is said to be regular [2] if and only if the following conditions hold: The Riesz mean (R, q n ) is regular if and only if Q n → ∞ as n → ∞ [2].

The Riesz Sequence Space
In the present section, we introduce Riesz sequence space ( , ) q r u p ∞ , prove that these spaces are complete paranormed linear space and show that the ( , ) q r u p ∞ are linearly isomorphic to the space l ∞ (p). We also compute α-, β-and γ-duals of these spaces. Finally, we give basis for the spaces A linear Topological space X over the field of real numbers R is said to be a paranormed space if there is a sub-additive function h: X → R such that h(θ)=0, h(-x)=h(x) and scalar multiplication is continuous, that is, | | 0 n α α − → and h(x n -x) → 0 imply h(a n x n -ax) → 0 for all α's in R and x's in X, where θ is a zero vector in the linear space X. Assume here and after that (p k ) be a bounded sequence of strictly positive real numbers with sup k p k= H and M=max{1,H}. Then, the linear spaces l(p) and l ∞ (p) were defined by Maddox [8] and [18] as follows : which is complete spaces paranormed by We shall assume throughout that Following Basar and Altay [3], Choudhary and Mishra [4], Edermann [5], Mursaleen et al. [12][13][14], Neyaz and Hamid [15] we define the spaces ( , ) q r u p ∞ as the set of all sequences whose q u R -transform is in the spaces l ∞ (p) i.e., Define the sequence y=(y k ), which will be used, by the q u R -transform of a sequence x=(x k ), i.e., Now, we begin with the following theorem which is essential in the text.  such that g(x n -x) → 0 and (α n ) is a sequence of scalars such that α n → α. Then, since the inequality, holds by subadditivity of g, {g(x n )} is bounded and we thus have which tends to zero as n → ∞. That is to say that the scalar multiplication is continuous. Hence, g is paranorm on the space ( , ). . Then, for a given ε > 0 there exists a positive integer n 0 (ε) such that for all i,j ≥ n 0 (ε). Using definition of g and for each fixed k ∈ ℕ that for i,j ≥ n 0 (ε), which leads us to the fact that is a Cauchy sequence of real numbers for every fixed k ∈ ℕ. Since R is complete, it converges,say, ( as I → ∞. Using these infinitely many limits for all k, i.e., Finally, taking ε=1 in (7) and letting I ≥ n 0 (1). we have by Minkowski's inequality for each m ∈ ℕ that for all i ≥ n 0 (ε), it follows that x i → x as i → ∞, hence we have shown that ( , ) q r u p ∞ is complete, hence the proof.
Note that one can easily see the absolute property does not hold on the space ( , ) for atleast one sequence in the spaces ( , ) q r u p ∞ and consequently we see that the space ( , ) q r u p ∞ is a sequence space of non-absolute type.

Theorem 2:
The sequence spaces ( , ) q r u p ∞ of non-absolute type is linearly isomorphic to the spaces l ∞ (p).

Proof :
To prove the theorem, we should show the existence of a linear bijection between the spaces ( , ) q r u p ∞ and l ∞ (p). With the notation of (3), define the transformation T from ( , ) q r u p ∞ to l ∞ (p) by x → y=Tx. The linearity of T is trivial. Further, it is obvious that x=θ whenever Tx=θ and hence T is injective.
Let y ∈ l ∞ (p) and define the sequence x=(x k ) by 1  Where n, k ∈ ℕ. Thus we deduce from Lemma 3 with (11) that ax=(a n x n ) ∈ cs whenever = ( ) ( , ) if and only if Dy ∈ c whenever y ∈ l(p). Therefore, we derive fom (8)  if and only if Dy ∈ l ∞ whenever y=(y k ) ∈ l(p). Therefore, we again obtain the condition (12) which means that and the proof of the theorem is complete [19][20][21].