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On dualities between function spaces

Authors:	Ivan Singer

Institution:	(1) Institute of Mathematics, P.O. Box 1-764, 70700 Bucharest, Romania

Abstract:	By 6], the dualities between $\bar R^X$ and $\bar R^W$ , whereX andW are two sets and $\bar R = \left { - \infty , + \infty } \right]$ (i.e., the mappings $\Delta :\bar R^X \to \bar R^W$ satisfying $\mathop {\left( {\inf f_i } \right)^\Delta }\limits_{i \in I} = \mathop {\left( {\sup f_i } \right)^\Delta }\limits_{i \in I}$ for all $\left\{ {f_i } \right\}_{i \in I} \subseteq \bar R^X$ and all index setsI), can be represented with the aid of functions $G:X \times W \times \bar R \to \bar R$ . Here we show that they can be also represented with the aid of functions $e:X \times W \times R \to \bar R$ , whereR = (–, +). As an application, we show that every duality $\Delta :\bar R^X \to \bar R^W$ is completely determined by a suitable duality between 2^{X ×R} and 2^{W ×R} (i.e., a mapping 2^{X ×R} 2^{W ×R} satisfying $\Gamma (\mathop \cup \limits_{i \in I} M_i ) = \mathop \cap \limits_{i \in I} \Gamma \left( {M_i } \right)$ for all {M _i}_iI $\subseteq$ 2^{X ×R} and all index setsI), applied to the epigraphs of the functions $f \in \bar R^X$ .

Keywords:	Dualities between complete lattices of functions dualities between complete lattices of subsets
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