Journal Home Page

Cumulative Index

List of all Volumes

Complete Contents
of this Volume

Previous Article

Next Article

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

Lionel Thibault
Institut A. Grothendieck, Université de Montpellier, 34095 Montpellier 5, France


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

[ Fulltext-pdf  (133  KB)] for subscribers only.