Characterization of convex $μ$-compact sets - Mathematics > Functional AnalysisReport as inadecuate

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Abstract: The class of $\mu$-compact sets can be considered as a natural extension ofthe class of compact metrizable subsets of locally convex spaces, to which theparticular results well known for compact sets can be generalized. This classcontains all compact sets as well as many noncompact sets widely used inapplications. In this paper we give a characterization of a convex$\mu$-compact set in terms of properties of functions defined on this set.Namely, we prove that the class of convex $\mu$-compact sets can becharacterized by continuity of the operation of convex closure of a function =the double Fenchel transform with respect to monotonic pointwise convergingsequences of continuous bounded and of lower semicontinuous lower boundedfunctions.

Author: M.E.Shirokov


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