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On the Young-Fenchel transform for convex functions

Author: Gerald Beer
Journal: Proc. Amer. Math. Soc. 104 (1988), 1115-1123
MSC: Primary 49A50; Secondary 26E25, 46B99, 54B20
MathSciNet review: 937844
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Abstract: Let $ \Gamma (X)$ be the proper lower semicontinuous convex functions on a reflexive Banach space $ X$. We exhibit a simple Vietoris-type topology on $ \Gamma (X)$, compatible with Mosco convergence of sequences of functions, with respect to which the Young-Fenchel transform (conjugate operator) from $ \Gamma (X)$ to $ \Gamma ({X^*})$ is a homeomorphism. Our entirely geometric proof of the bicontinuity of the transform halves the length of Mosco's proof of sequential bicontinuity, and produces a stronger result for nonseparable spaces.

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Keywords: Convex function, conjugate convex function, Young-Fenchel transform, Mosco convergence, Mosco topology, hyperspace
Article copyright: © Copyright 1988 American Mathematical Society

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