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

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
Full Text: