Dense strong continuity of mappings and the Radon-Nikodym property. (English) Zbl 0557.46016

Christensen, Jens Peter Reus; Kenderov, Petar Stojanov

Math. Scand. 54, 70-78 (1984).

Let \(F: X\to E\) \((F: Y\to E^*)\) be a weak-continuous (weak*-continuous) mapping from the Baire space X into the Banach space E (dual Banach space \(E^*)\). When do there exist a dense \(G_{\delta}\) subset of X at the points of which F is norm-continuous? It turns out that this is the case when E \((E^*)\) has the Radon-Nikodym property. The same holds true for multivalued mappings provided one uses a suitable notion of norm continuity of set-valued maps. An application is given to the theory of weak Asplund spaces. In particular, without using renorming theorems, it is proved that closed linear subspaces of a weakly compactly generated Banach space are weak Asplund.

MSC:

46B22	Radon-Nikodým, Kreĭn-Milman and related properties
46B10	Duality and reflexivity in normed linear and Banach spaces

Keywords:

Baire space; Radon-Nikodym property; multivalued mappings; norm continuity of set-valued maps; weak Asplund spaces; without using renorming theorems; closed linear subspaces of a weakly compactly generated Banach; space are weak Asplund; closed linear subspaces of a weakly compactly generated Banach space are weak Asplund