Some New Generalized Difference Spaces of Nonabsolute Type Derived from the Spaces ℓ (p) and ℓ (∞)

We introduce the sequence space ℓ (p) (λ)(B) of none absolute type which is a p-normed space and BK space in the cases 0 < p < 1 and 1 ⩽ p ⩽ ∞, respectively, and prove that ℓ (p) (λ)(B) and ℓ (p) are linearly isomorphic for 0 < p ⩽ ∞. Furthermore, we give some inclusion relations concerning the space ℓ (p) (λ)(B) and we construct the basis for the space ℓ (p) (λ)(B), where 1 ⩽ p < ∞. Furthermore, we determine the alpha-, beta- and gamma-duals of the space ℓ (p) (λ)(B) for 1 ⩽ p ⩽ ∞. Finally, we investigate some geometric properties concerning Banach-Saks type p and give Gurarii's modulus of convexity for the normed space ℓ (p) (λ)(B).