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 ${\mathcal{A}}_u(X)$. The result holds also for functions with values in a dual space or in a Banach space with the so-called property $(\beta )$. 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.

Accepted in Publ. Res. Inst. Math. Sci