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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Conjugate convex functions and the epi-distance topology


Author: Gerald Beer
Journal: Proc. Amer. Math. Soc. 108 (1990), 117-126
MSC: Primary 46A55; Secondary 46G99, 58C99
MathSciNet review: 982400
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Abstract: Let $ \Gamma (X)$ denote the proper, lower semicontinuous, convex functions on a normed linear space, and let $ {\Gamma ^ * }({X^ * })$ denote the proper, weak*-lower semicontinuous, convex functions on the dual $ {X^ * }$ of $ X$. It is well-known that the Young-Fenchel transform (conjugate operator) is bicontinuous when $ X$ is reflexive and both $ \Gamma (X)$ and $ {\Gamma ^ * }({X^ * })$ are equipped with the topology of Mosco convergence. We show that without reflexivity, the transform is bicontinuous, provided we equip both $ \Gamma (X)$ and $ {\Gamma ^ * }({X^ * })$ with the (metrizable) epi-distance topology of Attouch and Wets. Convergence of a sequence of convex functions $ \left\langle {{f_n}} \right\rangle $ to $ f$ in this topology means uniform convergence on bounded subsets of the associated sequence of distance functional $ \left\langle {d( \cdot ,{\text{epi}}{f_n})} \right\rangle $ to $ d( \cdot ,{\text{epi}}f)$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1990-0982400-8
PII: S 0002-9939(1990)0982400-8
Keywords: Convex function, Young-Fenchel transform, conjugate convex function, polar, epi-distance topology, uniform convergence on bounded sets, Mosco convergence
Article copyright: © Copyright 1990 American Mathematical Society