Journal of Convex Analysis 19 (2012), No. 2, 355--384
Copyright Heldermann Verlag 2012
Notes on Extended Real- and Set-Valued Functions
Andreas H. Hamel
Dept. of Mathematical Sciences, Yeshiva University, 2495 Amsterdam Avenue, New York, NY 10033, U.S.A.
Institut für Mathematik, Martin-Luther-Universität, Theodor-Lieser-Straße 5, 06120 Halle, Germany
An order theoretic and algebraic framework for the extended real numbers is established which includes extensions of the usual difference to expressions involving -∞ and/or +∞, so-called residuations. New definitions and results for directional derivatives, subdifferentials and Legendre--Fenchel conjugates for extended real-valued functions are given which admit to include the proper as well as the improper case. For set-valued functions, scalar representation theorems and a new conjugation theory are established. The common denominator is that the appropriate image spaces for set-valued functions share fundamental structures with the extended real numbers: They are order complete, residuated monoids with a multiplication by non-negative real numbers.
Keywords: Extended real-valued functions, directional derivative, subdifferential, Fenchel conjugate, set-valued function, conlinear space, infimal convolution.
MSC: 49N15; 54C60, 90C46
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