Weak compactness of sublevel sets in complete locally convex spaces

Abstract

© 2019 Heldermann Verlag. All rights reserved.In this work we prove that if X is a complete locally convex space and {equation presented} is a function such that f -x∗attains its minimum for every x∗∈ U, where U is an open set with respect to the Mackey topology in X∗, then for every γ ∈ R and x∗∈ U the set {equation presented} is relatively weakly compact. This result corresponds to an extension of Theorem 2.4 in [J. Saint Raymond, Mediterr. J. Math. 10 (2013), no. 2, 927-940]. Directional James compactness theorems are also derived.