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Generic continuity of minimal set-valued mappings. (English) Zbl 0912.46017
Summary: We study classes of Banach spaces where every set-valued mapping from a complete metric space into subsets of the Banach space which satisfies certain minimal properties, is single-valued and norm upper semicontinuous at the points of a dense $$G_\delta$$ subset of its domain. Characterizations of these classes are developed and permanence properties are established. Sufficiency conditions for membership of these classes are defined in terms of fragmentability and $$\sigma$$-fragmentability of the weak topology. A characterization of non-membership is used to show that $$\ell_\infty(\mathbb{N})$$ is not a member of our classes of generic continuity spaces.

##### MSC:
 46B22 Radon-Nikodým, Kreĭn-Milman and related properties 46B20 Geometry and structure of normed linear spaces 58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds 47H04 Set-valued operators