Bounded holomorphic functions attaining their norms in the bidual

TL;DR

This paper proves that, under certain conditions on a Banach space, the set of holomorphic functions whose Aron-Berner extensions attain their norms is dense, extending known results and providing counterexamples for related theorems.

Contribution

It establishes the density of norm-attaining holomorphic functions in specific Banach space settings and introduces a Lindenstrauss type theorem for polynomials.

Findings

Density of norm-attaining functions in $A_u(X)$ under certain hypotheses

Extension of results to dual spaces and spaces with property $(eta)$

Counterexamples for the Bishop-Phelps theorem in analytic and polynomial contexts

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.