
Journal of Convex Analysis 19 (2012), No. 2, 355384 Copyright Heldermann Verlag 2012 Notes on Extended Real and SetValued Functions Andreas H. Hamel Dept. of Mathematical Sciences, Yeshiva University, 2495 Amsterdam Avenue, New York, NY 10033, U.S.A. hamel@yu.edu Carola Schrage Institut für Mathematik, MartinLutherUniversität, TheodorLieserStraße 5, 06120 Halle, Germany carola.schrage@mathematik.unihalle.de 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 +∞, socalled residuations. New definitions and results for directional derivatives, subdifferentials and LegendreFenchel conjugates for extended realvalued functions are given which admit to include the proper as well as the improper case. For setvalued functions, scalar representation theorems and a new conjugation theory are established. The common denominator is that the appropriate image spaces for setvalued functions share fundamental structures with the extended real numbers: They are order complete, residuated monoids with a multiplication by nonnegative real numbers. Keywords: Extended realvalued functions, directional derivative, subdifferential, Fenchel conjugate, setvalued function, conlinear space, infimal convolution. MSC: 49N15; 54C60, 90C46 [ Fulltextpdf (228 KB)] for subscribers only. 