Norm continuity of weakly continuous mappings into Banach spaces

Abstract

Let T be the class of Banach spaces E for which every weakly continuous mapping from an α-favorable space to E is norm continuous at the points of a dense subset. We show that: • T contains all weakly Lindelöf Banach spaces; • l ∞ ∉ T , which brings clarity to a concern expressed by Haydon ([R. Haydon, Baire trees, bad norms and the Namioka property, Mathematika 42 (1995) 30–42], pp. 30–31) about the need of additional set-theoretical assumptions for this conclusion. Also, ( l ∞ / c 0 ) ∉ T . • T is stable under weak homeomorphisms; • E ∈ T iff every quasi-continuous mapping from a complete metric space to ( E , weak ) is densely norm continuous; • E ∈ T iff every quasi-continuous mapping from a complete metric space to ( E , weak ) is weakly continuous at some point.