On some geometric parameters in Banach spaces

Abstract

Let X be a normed linear space and S ( X ) = { x ∈ X : ‖ x ‖ = 1 } be the unit sphere of X. Let δ ( ϵ ) : [ 0 , 2 ] → [ 0 , 1 ] , ρ X ( ϵ ) : [ 0 , + ∞ ) → [ 0 , + ∞ ) , and J ( X ) = sup { ‖ x + y ‖ ∧ ‖ x − y ‖ } , x and y ∈ S ( X ) be the modulus of convexity, the modulus of smoothness, and the modulus of squareness of X, respectively. Let E ( X ) = sup { ‖ x + y ‖ 2 + ‖ x − y ‖ 2 : x , y ∈ S ( X ) } . In this paper we proved some sufficient conditions on δ ( ϵ ) , ρ X ( ϵ ) , J ( X ) , E ( X ) , and w ( X ) = sup { λ > 0 : λ ⋅ lim inf n → ∞ ‖ x n + x ‖ ⩽ lim inf n → ∞ ‖ x n − x ‖ } , where the supremum is taken over all the weakly null sequence x n in X and all the elements x of X for the uniform normal structure.