Some p -Type New Sequence Spaces and Their Geometric Properties

and Applied Analysis 3 of the space λA is defined by λA {f : f g ◦ A; g ∈ λ′}. Let X be a seminormed space. A set Y ⊂ X is called fundamental set if the span of Y is dense in X. An application of HahnBanach theorem on fundamental set is as follows: if Y is the subset of a seminormed space X and f Y 0 implies f 0 for f ∈ X′, then Y is a fundamental set see 11 . 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 Δ δnk and S snk by δnk ⎧ ⎨ ⎩ −1 n−k, if n − 1 ≤ k ≤ n, 0, otherwise, snk ⎧ ⎨ ⎩ 1, if 0 ≤ k ≤ n, 0, otherwise. 2.4 Malkowsky and Savas 12 , Choudhary and Mishra 13 , and Altay and Basar 14 have defined the sequence spaces Z u, v;X , p , and u, v; p , respectively. By using the matrix domain, the spaces Z u, v;X , p , and u, v; p may be redefined by Z u, v;X XG u,v , p p S, and u, v; p p G u,v , respectively. If λ ⊂ w is a sequence space and x xk ∈ λ, Sx -transform with 2.4 corresponds to nth partial sum of the series ∑ n xn and it is denoted by s sn . By using 2.4 and any infinite lower triangular matrix A, we can define two infinite lower triangular matrices A and  as follows: A AS and  ΔA. Let x xk be a sequence in λ. By considering the multiplication of infinite lower triangular matrices, we have A Sx Ax, that is, tn n ∑ v 0 anvsv n ∑ v 0 anvxv. 2.5 Also since  ΔA,we have Âx ΔA x, that is, tn − tn−1 n ∑ v 0 ânvxv. 2.6 Now let us write the following equality: