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Bounded holomorphic functions attaining their norms in the bidual

Abstract

Under certain hypotheses on the Banach space XX, we prove that the set of analytic functions in Au(X)\mathcal{A}_u(X) (the algebra of all holomorphic and uniformly continuous functions in the ball of XX) whose Aron-Berner extensions attain their norms, is dense in Au(X)\mathcal{A}_u(X). The result holds also for functions with values in a dual space or in a Banach space with the so-called property (β)(\beta). For this, we establish first a Lindenstrauss type theorem for continuous polynomials. We also present some counterexamples for the Bishop-Phelps theorem in the analytic and polynomial cases where our results apply.Comment: Accepted in Publ. Res. Inst. Math. Sc

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