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 ${\mathcal{A}}_u(c_0,Z^{\prime \prime })$ for a certain Banach space $Z$, while our Lindenstrauss theorem does. In order to obtain our results, we first handle their polynomial cases.