Eberlein theorem and norm continuity of pointwise continuous mappings into function spaces

Abstract

For a pseudocompact (strongly pseudocompact) space T we show that every strongly bounded (bounded) subset A of the space C(T) of all continuous functions on T has compact closure with respect to the pointwise convergence topology. This generalization of the Eberlein–Grothendieck theorem allows us to prove that, for any strongly pseudocompact spaces T, there exist many points of norm continuity for any pointwise continuous, C(T)-valued mapping h, defined on a Baire space X, which is homeomorphic to a dense Borel subset of a pseudocompact space. In particular, this is so, if X is pseudocompact. In the case when T is pseudocompact the same “norm-continuity phenomenon” has place for every strongly pseudocompact space X or, for every Baire space X which is homeomorphic to a Borel subset of some countably compact space.