Some Geometric Properties of a Generalized Cesàro Sequence Space

Every Banach space X with the Banach-Saks property is reflexive, but the converse is not true (see [4, 5]). Kakutani [6] proved that any uniformly convex Banach space X has the Banach-Saks property. Moreover, he also proved that if X is a reflexive Banach space and θ ∈ (0, 2) such that for every sequence (xn) in S(X) weakly convergent to zero, there exist n1, n2 ∈ N satisfying the Banach-Saks property. For a sequence (xn) ⊂ X, we define A(xn) = lim n→∞ inf{‖xi + xj‖ : i, j ≥ n, i 6= j}.