Assume that a Banach space X has density character Aleph[1] and that there is an Asplund space Y and a bounded linear operator T from Y onto a dense set in X. We show that such Y then can be chosen a reflexive space provided that X admits a projectional resolution of identity in each equivalent norm on X. We prove that the dual unit ball B[x*] in its weak star topology is a Corson compact provided that the norm of X is Gateaux differentiable and B[x*] is a Valdivia compact.
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