Norm-to-weak upper semi-continuity of the pre-duality map

Abstract

For a Banach space X we study the norm-weak upper semi-continuity of the pre-duality map. We give an example to show that these points in general do not continue to be points of norm-weak upper semi-continuity for the pre-duality map on the third dual. We give several geometric conditions on X that ensures the existence of these points independent of the predual and also that they continue to be points of norm-weak upper semi-continuity in all the duals of odd order. We also consider these questions in the space of operators.