Weak compactness of sublevel sets

In this paper we provide a short proof of the fact that if X is a Banach space and f:X \to \mathbb{R} \cup \{\infty \} is a proper function such that f-x^* attains its minimum for every x^* \in X^*, then all the sublevels of f are relatively weakly compact. This result has many applications.

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2017-08, Vol.145 (8), p.3377-3379
Main Author: MOORS, WARREN B.
Format: Article
Language:English
Subjects:
B. ANALYSIS
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Summary:In this paper we provide a short proof of the fact that if X is a Banach space and f:X \to \mathbb{R} \cup \{\infty \} is a proper function such that f-x^* attains its minimum for every x^* \in X^*, then all the sublevels of f are relatively weakly compact. This result has many applications.
ISSN:0002-9939
1088-6826
DOI:10.1090/proc/13466