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.
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José G. Llavona, Luiza A. Moraes, The Aron–Berner Extension for Polynomials Deﬁned in the Dual of a Banach Space. Publ. Res. Inst. Math. Sci. 40 (2004), no. 1, pp. 221–230