Weak Compactness and Variational Characterization of the Convexity |
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Authors: | Jean Saint Raymond |
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Institution: | 1. Analyse Fonctionnelle, Institut de Mathématique de Jussieu, Université Pierre et Marie Curie, Bo?te 186 – 4 place Jussieu, F- 75252, Paris Cedex 05, France
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Abstract: | We prove that if X is a Banach space and ${f : X \rightarrow \mathbb{R} \cup \{+\infty\}}$ is a proper function such that f ? ? attains its minimum for every ? ε X *, then the sublevels of f are all relatively weakly compact in X. As a consequence we show that a Banach space X where there exists a function ${f : X \rightarrow \mathbb{R}}$ such that f ? ? attains its minimum for every ? ε X * is reflexive. We also prove that if ${f : X \rightarrow \mathbb{R} \cup \{+\infty\}}$ is a weakly lower semicontinuous function on the Banach space X and if for every continuous linear functional ? on X the set where the function f ? ? attains its minimum is convex and non-empty then f is convex. |
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