Density of polynomials whose Aron--Berner extensions attain their norms

Abstract

We establish a polynomial analogue of Lindenstrauss'theorem in full generality. More precisely, we prove that, for every real or complex Banach space X and every positive integer n, the n-homogeneous scalar-valued polynomials whose Aron-Berner extensions to the bidual X** attain their norms form a dense subset of the space of continuous n-homogeneous polynomials on X. This resolves a problem posed by Aron, Garcia and Maestre in 2003, extending their result for quadratic polynomials to arbitrary degrees without imposing any structural assumptions on the underlying Banach space. We further establish the corresponding theorem for vector-valued polynomials with arbitrary Banach space codomains. Our approach relies on a new perturbation scheme based on averaged linear contractions, which provides simultaneous control of the multilinear variables and yields norm attainment through iterated limits in the bidual.