Weak Compactness of Sublevel Sets in Complete Locally Convex Spaces

We prove that if X is a complete locally convex space and f : X -> R boolean OR {+infinity} is a function such that f - x* attains its minimum for every x* is an element of U, where U is an open set with respect to the Mackey topology in X* , then for every gamma is an element of R and x* is an element of U the set {x is an element of X: f (x) - (x*, x) <= gamma} is relatively weakly compact. This result corresponds to an extension of Theorem 2.4 in a recent paper of J. Saint Raymond [Mediterr. J. Math. 10(2) (2013) 927-940]. Directional James compactness theorems are also derived.