The Aron–Berner Extension for Polynomials Defined in the Dual of a Banach Space

Abstract

Let $E=F^{\prime }$ where $F$ is a complex Banach space and let ${\pi }_1:E^{\prime \prime }=E\oplus F^{\perp }\to E$ be the canonical projection. Let $P({}^{n}E)$ be the space of the complex valued continuous $n$-homogeneous polynomials defined in $E$. We characterize the elements $P\in P({}^{n}E)$ whose Aron–Berner extension coincides with $P\circ {\pi }_1$ . The case of weakly continuous polynomials is considered. Finally we also study the same problem for holomorphic functions of bounded type.