Bounded holomorphic functions attaining their norms in the bidual

Abstract

Under certain hypotheses on the Banach space X, we prove that the set of analytic functions in Au(X) (the algebra of all holomorphic and uniformly continuous functions in the ball of X) whose Aron–Berner extensions attain their norms is dense in Au(X). This Lindenstrauss type result also holds for functions with values in a dual space or in a Banach space with the so-called property (β). We show that the Bishop–Phelps theorem does not hold for Au(c0, Z00) for a certain Banach space Z, while our Lindenstrauss theorem does. In order to obtain our results, we first handle their polynomial cases.