The Aron-Berner extension for polynomials defined in the dual of a Banach space
It is known that for any continuous \(n\)-homogeneous polynomial \(P:E \to \mathbb K\) on a Banach space \(E,\) there is a canonical norm-preserving extension of \(P\) to \(\widetilde{P}:E^{\prime\prime} \to \mathbb K\) (see, e.g., \textit{P. D. Berner} and the reviewer [Bull. Soc. Math. Fr. 106, 3--24 (1978; Zbl 0378.46043)], and \textit{A. M. Davie} and \textit{T. W. Gamelin} [Proc. Am. Math. Soc. 106, 351--356 (1989; Zbl 0683.46037)]). The question of interest here is how to determine whether a particular continuous \(n\)-homogeneous polynomial \(Q:E^{\prime\prime} \to \mathbb K\) is equal to some \(\widetilde{P},\) in the particular case where \(E = F'\) is a dual space. The authors show that this occurs if and only if \(dP(x)\) is weak-star continuous for every \(x \in E,\) which in turn holds if and only if \(dP(x) \in F\) for every \(x \in E.\) In addition, they show that if \(P\) is weakly continuous on bounded subsets of \(E\) and if \(dP(x) \in F\) for every \(x \in E,\) then in fact \(P\) is weak-star continuous on bounded subsets of \(E.\) The paper concludes with a discussion of the analogous problem for entire functions of bounded type on \(E.\) Remark: Related work has been done by \textit{I. Zalduendo} [Trans. Am. Math. Soc. 320, 747--763 (1990; Zbl 0709.46027)] and by \textit{C. Boyd, Y. S. Choi} and the reviewer [J. Aust. Math. Soc. 70, 387--400 (2001; Zbl 1036.46032)].
scientific article; zbMATH DE number 2113006