Bi¸sgin Journal of Inequalities and Applications (2016)2016:304 DOI 10.1186/s13660-016-1252-4 R E S E A R C H Open Access The binomial sequence spaces which include the spaces p and ∞ and geometric properties Mustafa Cemil Bi¸sgin* *Correspondence: [email protected] Abstract Department of Mathematics, r,s r,s Faculty of Arts and Sciences, Recep In this work, we introduce the binomial sequence spaces bp and b∞ which include r,s r,s Tayyip Erdogan˘ University, Zihni the spaces p and ∞, in turn. Moreover, we show that the spaces bp and b∞ are Derin Campus, Rize, 53100, Turkey BK-spaces and prove that these spaces are linearly isomorphic to the spaces p and ∞, respectively. Furthermore, we speak of some inclusion relations and give the r,s Schauder basis of the space bp . Lastly, we determine the α-, β-, and γ -duals of those r,s spaces and give some geometric properties of the space bp . MSC: 40C05; 40H05; 46B45 Keywords: matrix transformations; matrix domain; Schauder basis; α-, β-and γ -duals; matrix classes 1 The basic information and notations The set of all real (or complex) valued sequences is symbolized by w which becomes a vector space under point-wise addition and scalar multiplication. Any vector subspace of w is called a sequence space. The spaces of all bounded, null, convergent, and absolutely p-summable sequences are denoted by ∞, c, c,andp,respectively,where≤ p < ∞. A Banach sequence space is called a BK-space provided each of the maps pn : X −→ C defined by pn = xn is continuous for all n ∈ N []. By considering the notion of BK-space, one can say that the sequence spaces ∞, c,andc are BK-spaces according to their usual | | sup-norm defined by x ∞ = supk∈N xk and p is a BK-space according to its p-norm defined by ∞ p | |p x p = xk , k= where ≤ p < ∞. For an arbitrary infinite matrix A =(ank) of real (or complex) entries and x =(xk) ∈ w, the A-transform of x is defined by ∞ (Ax)n = ankxk (.) k= © The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Bi¸sgin Journal of Inequalities and Applications (2016)2016:304 Page 2 of 15 and is supposed to be convergent for all n ∈ N []. In terms of the ease of use, we prefer that the summation without limits runs from to ∞. Given two sequence spaces X and Y , and an infinite matrix A =(ank), the sequence space XA is defined by XA = x =(xk) ∈ w : Ax ∈ X (.) which is called the domain of an infinite matrix A.Also,by(X : Y), we denote the class of all matrices such that X ⊂ YA.Ifank =fork > n and ann =forall n, k ∈ N, an infinite – matrix A =(ank) is called a triangle. Also, a triangle matrix A uniquely has an inverse A which is a triangle matrix. Let the summation matrix S =(snk) be defined as follows: , ≤ k ≤ n, snk = , k > n for all k, n ∈ N. Then the spaces of all bounded and convergent series are defined by means of the summation matrix such that bs =(∞)S and cs = cS,respectively. The theory of matrix transformation was set in motion by the theory of summability which was developed by Cesàro, Norlund, Riesz, etc. By taking into account this theory, many authors have constructed new sequence spaces. For example, (∞)Nq and cNq in [], r r Xp and X∞ in [], ap and a∞ in []. Furthermore, many authors have used especially the r r r r Euler matrix for defining new sequence spaces. These are e and ec in [], ep and e∞ in r r r r (m) r (m) r (m) r (m) []and[], e(), ec()ande∞()in[], e( ), ec( )ande∞( )in[], e(B ), r (m) r (m) r r r r r ec(B ), and e∞(B )in[], e(, p), ec(, p), and e∞(, p)in[], e(u, p)andec(u, p) in []. r,s r,s In this work, we introduce the binomial sequence spaces bp and b∞ which include the r,s r,s spaces p and ∞, in turn. Moreover, we show that the spaces bp and b∞ are BK-spaces and prove that these spaces are linearly isomorphic to the spaces p and ∞,respectively. Furthermore, we speak of some inclusion relations and give the Schauder basis of the r,s space bp . Lastly, we determine the α-, β-, and γ -duals of those spaces and give some r,s geometric properties of the space bp . 2 The binomial sequence spaces which include the spaces p and ∞ r,s r,s In this part, we define the binomial sequence spaces bp and b∞ which include the spaces p and ∞, respectively. Furthermore, we show that those spaces are BK-spaces and are linearly isomorphic to the spaces p and ∞. Also, we show that the binomial sequence r,s ≤ ∞ space bp is not a Hilbert space except the case p =,where p < . r,s r,s Let r, s be nonzero real numbers with r + s = . Then the binomial matrix B =(bnk)is defined as follows: n sn–krk,≤ k ≤ n, br,s = (s+r)n k nk , k > n for all k, n ∈ N.Forsr > , one can easily check that the following properties hold for the r,s r,s binomial matrix B =(bnk): (i) Br,s < ∞, Bi¸sgin Journal of Inequalities and Applications (2016)2016:304 Page 3 of 15 r,s lim →∞ ∈ N (ii) n bnk =(each k ), r,s (iii) limn→∞ k bnk =. Thus, the binomial matrix is regular whenever sr > . Here and in the following, unless stated otherwise, we suppose that sr >. r,s r,s By taking into account the binomial matrix B =(bnk), the binomial sequence spaces r,s r,s bp and b∞ are defined by p n n br,s = x =(x ) ∈ w : sn–krkx < ∞ ,≤ p < ∞, p k (s + r)n k k n k= and n r,s n n–k k b∞ = x =(xk) ∈ w : sup s r xk < ∞ . ∈N (s + r)n k n k= r,s r,s By considering the notation of (.), the binomial sequence spaces bp and b∞ can be r,s r,s redefined by the matrix domain of B =(bnk) as follows: r,s r,s bp =(p)Br,s and b∞ =(∞)Br,s .(.) Let us define a sequence y =(yk) as follows: k k Br,sx = y = sk–jrjx (.) k k (s + r)k j j j= for all k ∈ N.ThissequencewillbefrequentlyusedastheBr,s-transform of x. We would like to touch on a point, if we take s + r =,weobtaintheEulermatrixEr = r r,s r,s (enk). So, the binomial matrix B =(bnk) generalizes the Euler matrix. Now, we want to continue with the following theorem which is needed in the next. r,s r,s Theorem . The binomial sequence spaces bp and b∞ are BK-spaces according to their norms defined by ∞ p r,s r,s p x r,s = B x = B x bp p n n= and r,s r,s r,s x b∞ = B x ∞ = sup B x n , n∈N where ≤ p < ∞. Proof We know that the sequence spaces p and ∞ are BK-spaces with their p-norm and sup-norm,respectively,where≤ p < ∞.Furthermore,(.) holds and the binomial r,s r,s matrix B =(bnk) is a triangle matrix. By taking into account these three facts and The- r,s r,s orem .. of Wilansky [], we conclude that the binomial sequence spaces bp and b∞ are BK-spaces, where ≤ p < ∞. This completes the proof of the theorem. Bi¸sgin Journal of Inequalities and Applications (2016)2016:304 Page 4 of 15 r,s r,s Theorem . The binomial sequence spaces bp and b∞ arelinearlyisomorphictothe sequence spaces p and ∞, in turn, where ≤ p < ∞. Proof To refrain from the usage of similar statements, we prove the theorem for only the r,s ≤ ∞ sequence space bp ,where p < . For the proof of the theorem, we need to show the r,s existence of a linear bijection between the spaces bp and p.LetL be a transformation r,s −→ r,s r,s such that L : bp p, L(x)=B x. By the definition of the binomial sequence space bp , ∈ r,s r,s ∈ we conclude that, for all x bp , L(x)=B x p. Furthermore, it is obvious that L is a linear transformation and x = whenever L(x) = . Therefore, L is injective. For given y =(yk) ∈ p,letusdefineasequencex =(xk)suchthat k k x = (–s)k–j(s + r)jy k rk j j j= for all k ∈ N.Thenweget r,s x r,s = B x bp p ∞ p r,s p = B x n n= ∞ p n n p = sn–krkx (s + r)n k k n= k= ∞ p n n k k p = sn–k (–s)k–j(s + r)jy (s + r)n k j j n= k= j= ∞ p p = |yn| n= = y p = L(x) < ∞. p ∈ r,s Hence, we conclude that L is norm preserving and x bp ,namelyL is surjective. As a r,s consequence, L is a linear bijection. This means that the spaces bp and p are linearly r,s ∼ ≤ ∞ isomorphic, that is, bp = p,where p < .