The Aron-Berner extension for polynomials defined in the dual of a Banach space
type: Item
instance of: scholarly article
title: The Aron-Berner extension for polynomials defined in the dual of a Banach space
zbMATH Open document ID: 1074.46028
author: Jose G. Llavona
author: Luiza A. Moraes
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2004-11-01T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 46G25
Mathematics Subject Classification ID: 46G20
zbMATH DE Number: 2113006
zbMATH Keywords: Aron-Berner extension
zbMATH Keywords: weakly continuous polynomials
reviewed by: Richard M. Aron
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1145475970
OpenAlex ID: W2043234762
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DOI: 10.2977/PRIMS/1145475970
description: scientific article; zbMATH DE number 2113006