Smooth functions on \(c_0\)
type: Item
instance of: scholarly article
title: Smooth functions on \(c_0\)
zbMATH Open document ID: 0940.46023
DOI: 10.1007/BF02897057
author: Petr Hájek
published in: Israel Journal of Mathematics
publication date: 2000-02-16T00:00:00Z
review text: It is proved that if the derivative of a Fréchet differentiable real function on the Banach space \(c_0\) is locally uniformly continuous then it is locally compact (i.e. maps a neighborhood to a relatively compact subset of \(\ell^1\)). The intuitive meaning of this statement is, that every \(C^2\)-function on \(c_0\) depends locally only on ``almost'' finitely many coordinates. See \textit{H. Torunczyk}, Stud. Math. 46, 43-51 (1973; Zbl 0251.46022) and \textit{R. Haydon}, C. R. Acad. Sci., Paris, Sér. I 315, No. 11, 1175-1178 (1992; Zbl 0788.46008) for the importance these functions on \(c_0\) play for smoothness properties of Banach spaces. Among the corollaries given in this paper answers to questions posed by J. A.~Jaramillo and by S.~Bates are given.
Mathematics Subject Classification ID: 46G05
zbMATH DE Number: 1251441
zbMATH Keywords: \(c_0\)
zbMATH Keywords: locally compact mapping
zbMATH Keywords: uniform continuity
zbMATH Keywords: Fréchet differentiability
zbMATH Keywords: smoothness properties of Banach spaces
reviewed by: Andreas Kriegl
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02897057
OpenAlex ID: W4232518494
description: scientific article; zbMATH DE number 1251441