Characterization of convex $μ$-compact sets

Details Characterization of convex $μ$-compact sets Characterization of convex $μ$-compact sets

2010-04-21

arxiv.org/abs/1004.3792v1

Russian Mathematical Surveys, 2008, 63:5

Mathematics

Functional Analysis

7 pages

Scientific paper

The class of $\mu$-compact sets can be considered as a natural extension of the class of compact metrizable subsets of locally convex spaces, to which the particular results well known for compact sets can be generalized. This class contains all compact sets as well as many noncompact sets widely used in applications. 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 be characterized by continuity of the operation of convex closure of a function (= the double Fenchel transform) with respect to monotonic pointwise converging sequences of continuous bounded and of lower semicontinuous lower bounded functions.

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