Abstract
Under certain hypotheses on the Banach space X , we prove that the set of analytic functions in \mathcal A_u (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 \mathcal A_u (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 ( \beta ). We show that the Bishop–Phelps theorem does not hold for \mathcal A_u (c_0, Z'') for a certain Banach space Z , while our Lindenstrauss theorem does. In order to obtain our results, we first handle their polynomial cases.
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