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Weak compactness of sublevel sets


Author: Warren B. Moors
Journal: Proc. Amer. Math. Soc. 145 (2017), 3377-3379
MSC (2010): Primary 46B20, 46B22
DOI: https://doi.org/10.1090/proc/13466
Published electronically: February 24, 2017
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Abstract: 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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Warren B. Moors
Affiliation: Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand
Email: moors@math.auckland.ac.nz

DOI: https://doi.org/10.1090/proc/13466
Keywords: Weakly compact sets, James' theorem, sublevel sets
Received by editor(s): June 20, 2016
Received by editor(s) in revised form: June 26, 2016, July 17, 2016, and August 29, 2016
Published electronically: February 24, 2017
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2017 American Mathematical Society