Extensions of smooth mappings into biduals and weak continuity
type: Item
instance of: scholarly article
title: Extensions of smooth mappings into biduals and weak continuity
zbMATH Open document ID: 1280.46051
author: Yun Sung Choi
author: Petr Hájek
author: Han Ju Lee
published in: Advances in Mathematics
publication date: 2013-02-11T00:00:00Z
review text: Results on extensions of multilinear mappings and polynomials between Banach spaces to their biduals have appeared numerous times in the literature, probably starting with the results of \textit{R. Arens} [Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)], where bilinear mappings are extended. The classical Aron-Berner extension results [\textit{R. M. Aron} and \textit{P. D. Berner}, Bull. Soc. Math. Fr. 106, 3--24 (1978; Zbl 0378.46043)] build extension operators for the space of multilinear forms and then use Taylor expansions to extend holomorphic functions of bounded type from a Banach space to its bidual. The present paper presents a number of extension results in the context of uniformly differentiable mappings between real Banach spaces -- although many of the results can be carried on, as the authors mention, to the complex Banach space case or, even, to the locally convex setting. They cover the classical cases mentioned above and many other previous results in the literature. For example, the following generalization of a result of Lindenstrauss is presented: Let \(X\) be an \({\mathcal L}_{\infty,\lambda}\)-space for \(\lambda\geq 1\), and let \(Y\) be a Banach space. Let \(T:B_{X}\rightarrow Y\) be a weakly compact, uniformly smooth (in the interior of \(B_X\)) operator. Then \(T\) can be extended, preserving its best smoothness properties, into a mapping from the \((1/\lambda)\)-multiple of the unit ball of any superspace. Moreover, \(T\) maps weakly Cauchy sequence into norm-convergent ones, giving a smooth version of the Dunford-Pettis property for the \({\mathcal L}_{\infty}\)-spaces that is a generalization of the work of Dunford, Pettis, Grothendieck, Lindenstrauss and Pełczyński, and Ryan, who proved the result for polynomials. It is also shown that a uniformly differentiable mapping \(T\), not necessarily weakly compact, still maps weakly Cauchy sequences from \(\lambda B_X\) into norm-convergent sequences in \(Y\), provided that \(Y^{**}\…
Mathematics Subject Classification ID: 46T20
Mathematics Subject Classification ID: 46B03
zbMATH DE Number: 6135756
zbMATH Keywords: extension to biduals
zbMATH Keywords: Dunford-Pettis property
zbMATH Keywords: smoothness
zbMATH Keywords: approximation by polynomials
zbMATH Keywords: Taylor polynomials
zbMATH Keywords: extensions of multilinear mappings
zbMATH Keywords: uniformly differentiable mappings
reviewed by: V. Montesinos
MaRDI profile type: Publication
full work available at URL: j.aim.2012.11.001
OpenAlex ID: W2093702823
Wikidata QID: Q57618268
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Extensions of smooth mappings into biduals and weak continuity (DOI)
DOI: 10.1016/J.AIM.2012.11.001