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A selection theorem for weak upper semi-continuous set-valued mappings

Let Φ be a set-valued mapping from a Baire space T into non-empty closed subsets of a Banach space X, which is upper semi-continuous with respect to the weak topology on X. In this paper, we give a condition on T which is sufficient to ensure that Φ admits a selection which is norm continuous at each point of a dense and Gδ subset of T. We also derive a variation of James' characterisation of weak compactness, which we use in conjunction with our selection theorem, to deduce some differentiability results for continuous convex functions defined on dual Banach spaces.