Abstract

We prove that if XX is a complete locally convex space and f ⁣:X→R∪{+∞}f\colon X\to \mathbb{R}\cup \{+\infty \} is a function such that f−x∗f-x^\ast attains its minimum for every x∗∈Ux^\ast \in U, where UU is an open set with respect to the Mackey topology in X∗X^\ast, then for every γ∈R\gamma \in \mathbb{R} and x∗∈Ux^\ast \in U the set {x∈X:f(x)−⟨x∗,x⟩≤γ}\{ 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.

Author information

Contact details are reproduced from the original publication and may be historical.

Pedro Pérez-Aros

Instituto de Ciencias de la Ingeniería, Universidad de O’Higgins, Libertador Bernardo O'Higgins 611, Rancagua, Chile

pedro.perez@uoh.cl

Keywords

  • Convex functions
  • conjugate functions
  • inf-convolution
  • epi-pointed functions
  • weak compactness
  • inf-compact functions

Mathematics Subject Classification

46A25, 46A04, 46A50

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Suggested citation

P. Pérez-Aros, L. Thibault. “Weak Compactness of Sublevel Sets in Complete Locally Convex Spaces.” Journal of Convex Analysis 26 (2019), No. 3, 739–751. https://doi.org/10.68381/jca26039

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.