On some new characterizations of weakly compact sets in Banach spaces

Abstract

We show several characterizations of weakly compact sets in Banach spaces. Given a bounded closed convex set C of a Banach space X, the following statements are equivalent: (i) C is weakly compact; (ii) C can be affinely uniformly embedded into a reflexive Banach space; (iii) there exists an equivalent norm on X which has the w2R-property on C; (iv) there is a continuous and w *-lower semicontinuous seminorm p on the dual X * with p ≥ sup C such that p 2 is everywhere Fréchet differentiable in X * ; and as a consequence, the space X is a weakly compactly generated space if and only if there exists a continuous and w *-l.s.c. Fréchet smooth (not necessarily equivalent) norm on X * .

On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate

“…The uniform Fréchet differentiability of p 2 is equivalent to that p is uniformly Fréchet differentiable on S p . By [5], C is weakly compact. Since p is w * lower semicontinuous on X * , the Fréchet derivative dp(x * ) ∈ C for every x * ∈ S p [5].…”

“…By [5], C is weakly compact. Since p is w * lower semicontinuous on X * , the Fréchet derivative dp(x * ) ∈ C for every x * ∈ S p [5]. Let q be the Minkowski functional generated by C, and let X q = ∪ ∞ n=1 nC.…”

On the sum of simultaneously proximinal sets

“…Lemma 3.7. [3] Let X be a Banach space and C be a closed convex set of X. Then the following statements are equivalent.…”

Characterization of Reflexivity by Convex Functions

“…Journal of Function Spaces The 2 property of ℎ on gives that → 0 (16) for some 0 ∈ . This, combined with (14), means that ⟨ * , 0 ⟩ = sup * .…”

“…A sufficient and necessary condition for a Banach space to be reflexive is that it admits an equivalent 2 norm; that is, if a bounded sequence ( ) ⊂ satisfies (1), then ( ) is weakly convergent in . The localized versions of the two renorming theorems have been considered in [16].…”