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Topological Properties of the Set of Norm-Attaining Linear Functionals

Abstract If X is a separable non-reflexive Banach space, then the set NA of all norm-attaining elements of X * is not a w *-G δ subset of X *. However if the norm of X is locally uniformly rotund, then the set of norm attaining elements of norm one is w *-G δ. There exist separable spaces such that NA is a norm-Borel set of arbitrarily high class. If X is separable and non-reflexive, there exists an equivalent Gâteaux-smooth norm on X such that the set of all Gâteaux-derivatives is not norm-Borel.