Journal of Convex Analysis 26 (2019), No. 3, 739--751
Copyright Heldermann Verlag 2019
Weak Compactness of Sublevel Sets in Complete Locally Convex Spaces
Pedro Pérez-Aros
Instituto de Ciencias de la Ingeniería, Universidad de O’Higgins, Libertador Bernardo O'Higgins 611, Rancagua, Chile
pedro.perez@uoh.cl
Lionel Thibault
Institut A. Grothendieck, Université de Montpellier, 34095 Montpellier 5, France
lionel.thibault@umontpellier.fr
[Abstract-pdf]
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.
Keywords: Convex functions, conjugate functions, inf-convolution, epi-pointed functions, weak compactness, inf-compact functions.
MSC: 46A25, 46A04, 46A50
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