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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Conjugate convex functions and the epi-distance topology
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by Gerald Beer PDF
Proc. Amer. Math. Soc. 108 (1990), 117-126 Request permission

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
  • © Copyright 1990 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 108 (1990), 117-126
  • MSC: Primary 46A55; Secondary 46G99, 58C99
  • DOI: https://doi.org/10.1090/S0002-9939-1990-0982400-8
  • MathSciNet review: 982400