Closure and Lower Closure Theorems under Weak Convergence

Abstract

If X is a normed linear space over the reals with norm ‖x‖, let X* be the dual of X, that is, the space of all linear bounded operators x* on X, the linear operation being denoted by (x*, x), or X* × X → R. A sequence [x _k ] of elements of X then is said to be convergent in X to x provided ‖x _k – x‖ → 0 as k → ∞. A sequence [x _k ] of elements of X is said to be weakly convergent in X to x provided (x*, x _k ) → (x*, x) as k → ∞ for all x* ∈ X*.