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 $\mathcal A\_u (X)$ (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.