Under certain hypotheses on the Banach space , we prove that the set of analytic functions in (the algebra of all holomorphic and uniformly continuous functions in the ball of ) whose Aron–Berner extensions attain their norms is dense in . This Lindenstrauss type result also holds for functions with values in a dual space or in a Banach space with the so-called property (). We show that the Bishop–Phelps theorem does not hold for for a certain Banach space , while our Lindenstrauss theorem does. In order to obtain our results, we first handle their polynomial cases.
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Daniel Carando, Martin Mazzitelli, Bounded Holomorphic Functions Attaining their Norms in the Bidual. Publ. Res. Inst. Math. Sci. 51 (2015), no. 3, pp. 489–512DOI 10.4171/PRIMS/162