Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Andreas H. Hamel

Dept. of Mathematical Sciences, Yeshiva University, 2495 Amsterdam Avenue, New York, NY 10033, U.S.A.

hamel@yu.edu

Keywords

  • Extended real-valued functions
  • directional derivative
  • subdifferential
  • Fenchel conjugate
  • set-valued function
  • conlinear space
  • infimal convolution

Mathematics Subject Classification

49N15; 54C60, 90C46

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Suggested citation

A. H. Hamel, C. Schrage. “Notes on Extended Real- and Set-Valued Functions.” Journal of Convex Analysis 19 (2012), No. 2, 355–384. https://doi.org/10.68381/jca19020

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Published by Heldermann Verlag, 2012. Rights now held by Banach Press LLC.