Abstract
We show that when the duality map is norm-to-weak upper semi-continuous at some point of a dual space, the pre-duality map shares this property. We show that if x is a point of very smoothness of a Banach space X, it fails in general to be a point of very smoothness of the bidual X **. This cannot happen however if the bidual X ** is a Grothendieck space.
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Godefroy, G., Indumathi, V. Norm-to-Weak Upper Semi-Continuity of the Duality and Pre-Duality Mappings. Set-Valued Analysis 10, 317–330 (2002). https://doi.org/10.1023/A:1020662918653
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DOI: https://doi.org/10.1023/A:1020662918653