Weak Compactness of Sublevel Sets in Complete Locally Convex Spaces

We prove that if $X$ is a complete locally convex space and $f\colon X\to \mathbb{R}\cup \{+\infty \}$ is a function such that $f-x^\ast$ attains its minimum for every $x^\ast \in U$, where $U$ is an open set with respect to the Mackey topology in $X^\ast$, then for every $\gamma \in \mathbb{R}$ and $x^\ast \in U$ the set $\{ x\in X: f(x)- \langle x^\ast, x \rangle \leq \gamma\}$ 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.