Closure and Lower Closure Theorems under Weak Convergence
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*.
KeywordsWeak Convergence Lower Semicontinuous Measure Zero Measurable Subset Equivalence Theorem
Unable to display preview. Download preview PDF.