On dualities between function spaces

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 Γ between 2^{X ×R} and 2^{W ×R} (i.e., a mapping Γ∶ 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} _iεI $\subseteq$ 2^{X ×R} and all index setsI), applied to the epigraphs of the functions$f \in \bar R^X$.