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

Abstract

Let E = F' where F is a complex Banach space and let π1 : E'' = E ⊕ F⊥ → E be the canonical projection. Let P(nE) be the space of the complex valued continuous n-homogeneous polynomials defined in E. We characterize the elements P ∈ P(nE) whose Aron–Berner extension coincides with P ◦ π1 . The case of weakly continuous polynomials is considered. Finally we also study the same problem for holomorphic functions of bounded type.