Some normed binomial difference sequence spaces related to the lp$\ell_{p}$ spaces

*Correspondence: mengjian0710@163.com Department of Mathematics, Tianjin University of Technology, Tianjin, 300000, P.R. China Abstract The aim of this paper is to introduce the normed binomial sequence spaces br,s p (∇) by combining the binomial transformation and difference operator, where 1 ≤ p ≤ ∞. We prove that these spaces are linearly isomorphic to the spaces p and ∞, respectively. Furthermore, we compute Schauder bases and the α-, βand γ -duals of these sequence spaces.


Introduction and preliminaries
Let w denote the space of all sequences. By p , ∞ , c and c  , we denote the spaces of pabsolutely summable, bounded, convergent and null sequences, respectively, where  ≤ p < ∞. Let Z be a sequence space, then Kizmaz  For an infinite matrix A = (a n,k ) and x = (x k ) ∈ w, the A-transform of x is defined by Ax = {(Ax) n } and is supposed to be convergent for all n ∈ N, where (Ax) n = ∞ k= a n,k x k . For two sequence spaces X, Y and an infinite matrix A = (a n,k ), the sequence space X A is defined by X A = {x = (x k ) ∈ w : Ax ∈ X}, which is called the domain of matrix A in the space X. By (X : Y ), we denote the class of all matrices such that X ⊆ Y A .
The Euler means E r of order r is defined by the matrix E r = (e r n,k ), where  < r <  and The Euler sequence spaces e r p and e r ∞ were defined by Altay, Başar and Mursaleen [] as follows: Altay and Polat [] defined further generalization of the Euler sequence spaces e r  (∇), e r c (∇) and e r ∞ (∇) by for all k, n ∈ N. For sr >  we have (i) B r,s < ∞, (ii) lim n→∞ b r,s n,k =  for each k ∈ N, (iii) lim n→∞ k b r,s n,k = . Thus, the binomial matrix B r,s is regular for sr > . Unless stated otherwise, we assume that sr > . If we take s + r = , we obtain the Euler matrix E r . So the binomial matrix generalizes the Euler matrix. Bişgin [] defined the following spaces of binomial sequences: The main purpose of the present paper is to study the normed difference spaces b r,s p (∇) and b r,s ∞ (∇) of the binomial sequence whose B r,s (∇)-transforms are in the spaces p and ∞ , respectively. These new sequence spaces are the generalization of the sequence spaces defined in [] and []. Also, we compute the bases and α-, β-and γ -duals of these sequence spaces.

The binomial difference sequence spaces
In this section, we introduce the spaces b r,s p (∇) and b r,s ∞ (∇) and prove that these sequence spaces are linearly isomorphic to the spaces p and ∞ , respectively.
We first define the binomial difference sequence spaces b r,s p (∇) and b r,s Let us define the sequence y = (y n ) as the B r,s (∇)-transform of a sequence x = (x k ), that is, for each n ∈ N. Then the binomial difference sequence spaces b r,s p (∇) or b r,s ∞ (∇) can be redefined by all sequences whose B r,s (∇)-transforms are in the space p or ∞ .
Theorem . The sequence spaces b r,s p (∇) and b r,s ∞ (∇) are complete linear metric spaces with the norm defined by Proof The proof of the linearity is a routine verification. It is obvious that where θ is the zero element in b r,s p and α ∈ R. We consider x, z ∈ b r,s p (∇), then we have m= is a Cauchy sequence in the set of real numbers R. Since R is complete, we have lim m→∞ B r,s (∇x m k ) = B r,s (∇x k ) for each k ∈ N. We compute for m > m  . We take i and l → ∞, then the inequality (.) implies that We have Proof Similarly, we only prove the theorem for the space b r,s p (∇). To prove b r,s p (∇) ∼ = p , we must show the existence of a linear bijection between the spaces b r,s p (∇) and p . Consider T : b r,s p (∇) → p by T(x) = B r,s (∇x k ). The linearity of T is obvious and x = θ whenever T(x) = θ . Therefore, T is injective.
Let y = (y n ) ∈ p and define the sequence x = (x k ) by Then we have which implies that x ∈ b r,s p (∇) and T(x) = y. Consequently, T is surjective and is norm preserving. Thus, b r,s p (∇) ∼ = p .

The Schauder basis and α-, βand γ -duals
For a normed space (X, · ), a sequence {x k : x k ∈ X} k∈N is called a Schauder basis [] if for every x ∈ X, there is a unique scalar sequence (λ k ) such that x -n k= λ k x k →  as n → ∞. Next, we shall give a Schauder basis for the sequence space b r,s p (∇). We define the sequence g (k) (r, s) = {g (k) i (r, s)} i∈N by Theorem . The sequence (g (k) (r, s)) k∈N is a Schauder basis for the binomial sequence spaces b r,s p (∇) and every x = (x i ) ∈ b r,s p (∇) has a unique representation by Proof Obviously, B r,s (∇g (k) i (r, s)) = e k ∈ p , where e k is the sequence with  in the kth place and zeros elsewhere for each k ∈ N. This implies that g (k) (r, s) ∈ b r,s p (∇) for each k ∈ N. For x ∈ b r,s p (∇) and m ∈ N, we put By the linearity of B r,s (∇), we have for each k ∈ N. For any given ε > , there is a positive integer m  such that which implies that x ∈ b r,s p (∇) is represented as (.). To prove the uniqueness of this representation, we assume that

Then we have
which is a contradiction with the assumption that λ k (r, This shows the uniqueness of this representation. For the duality theory, the study of sequence spaces is more useful when we investigate them equipped with linear topologies. Köthe and Toeplitz [] first computed duals whose elements can be represented as sequences and defined the α-dual (or Köthe-Toeplitz dual). For the sequence spaces X and Y , define the multiplier space M(X, Y ) by Then the α-, β-and γ -duals of a sequence space X are defined by respectively. We give the following properties: Proof Let u = (u k ) ∈ w and x = (x k ) be defined by (.), then we have Therefore, we deduce that ux = (u k x k ) ∈  whenever x ∈ b r,s  (∇) or b r,s p (∇) if and only if G r,s y ∈  whenever y ∈  or p , which implies that u = (u k ) ∈ Therefore, we deduce that ux = (u k x k ) ∈ c whenever x ∈ b r,s  (∇) if and only if U r,s y ∈ c whenever y ∈  , which implies that u = (u k ) ∈ [b r,s  (∇)] β if and only if U r,s ∈ (  : c). By Lemma .(ii), we obtain [b r,s  (∇)] β = U r,s  ∩ U r,s  . Using Lemma .(i) and (iii)-(viii) instead of (ii), the proof can be completed in a similar way. So, we omit the details.

Conclusion
By considering the definitions of the binomial matrix B r,s = (b r,s n,k ) and the difference operator, we introduce the sequence spaces b r,s p (∇) and b r,s ∞ (∇). These spaces are the natural continuations of [, , ]. Our results are the generalizations of the matrix domain of the Euler matrix of order r. In order to give fully inform the reader on related topics with applications and a possible line of further investigation, the e-book [] is added to the list of references.