Abstract
The properties of the space ℒ(X′x) 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 × (X′, X), equipped with topology of uniform convergence on bounded subsets of X′ are studied. It is shown that completeness and separability of a space X are hereditary for ℒ(X′x). Criteria for the compactness of subsets of ℒ(X′x) and conditions for the metrizability of compacta in ℒ(X′x) are given. The topological isomorphism between ℒ(X′x) 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.
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Translated from Matematicheskie Zametki, Vol. 22, No. 2, pp. 203–213, August, 1977.
The author thanks S. S. Kutateladze for useful discussions regarding this article.
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Tolstonogov, A.A. Support functions of convex compacta. Mathematical Notes of the Academy of Sciences of the USSR 22, 604–609 (1977). https://doi.org/10.1007/BF01780968
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DOI: https://doi.org/10.1007/BF01780968