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arXiv · 1011.3179

Notes about extended real- and set-valued functions

Andreas H. Hamel·Carola Schrage
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Abstract

An order theoretic and algebraic framework for the extended real numbers is established which includes extensions of the usual difference to expressions involving $-\infty$ and/or $+\infty$, so-called residuations. Based on this, 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.

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Andreas H. Hamel, Carola Schrage. 2010-11-14. Notes about extended real- and set-valued functions. https://arxiv.org/abs/1011.3179

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