Support functions of convex compacta

The properties of the space $\mathscr L(X'_\varkappa)$ of all sublinear functionals, defined on a space $X'$ (topologically adjoint to a Hausdorff locally convex barrelled space $X$) and continuous in the Arens topology $\varkappa(X',X)$, equipped with topology of uniform convergence on bounded subsets of $X$prime are studied. It is shown that completeness and separability of a space $X$ are hereditary for $\mathscr L(X'_\varkappa)$. Criteria for the compactness of subsets of $\mathscr L(X'_\varkappa)$ and conditions for the metrizability of compacta in $\mathscr L(X'_\varkappa)$ are given. The topological isomorphism between $\mathscr L(X'_\varkappa)$ and the space of all nonempty convex compacta in $X$ with the Vietoris topology is established. The results obtained here are applied for the study of the properties of multiple-valued integrals.