ABSTRACT
We denote by X a real Banach space1 with norm ‖·‖ and by X∗ its topological dual, i.e., the vector space of all linear continuous functionals from X to R, which, endowed with the dual norm ‖x∗‖ = sup‖x‖≤1 |(x, x∗)|, for x∗ ∈ X∗, is a real Banach space too. As usual, if x ∈ X and x∗ ∈ X∗, (x, x∗) denotes x∗(x). Let Fin (X∗) be the class of all finite subsets in X∗ and let F ∈ Fin (X∗). The function ‖ · ‖F : X → R, defined by
‖x‖F = max{|(x, x∗)|; x∗ ∈ F}
for each x ∈ X , is a seminorm on X . The family of seminorms {‖ · ‖F ; F ∈ Fin (X∗)} defines the so-called
to
weak topology. Equipped with this topology, X is a separated locally convex topological vector space, denoted by Xw.
