Ranges of operators and derivatives
type: Item
instance of: scholarly article
title: Ranges of operators and derivatives
zbMATH Open document ID: 1202.46021
author: Petr Hájek
author: V. Montesinos
author: A. J. Guirao
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-03-29T00:00:00Z
full work available at URL: http://hdl.handle.net/10251/150314
review text: The authors develop a method of constructing differentiable and Lipschitz functions between Banach spaces. They show that their method provides a unified way to prove results of \textit{S.\,M.\thinspace Bates} [Isr.\ J.\ Math.\ 100, 209--220 (1997; Zbl 0898.46044)], \textit{D.\,Azagra} and \textit{R.\,Deville} [J.~Funct.\ Anal.\ 180, No.\,2, 328--346 (2001; Zbl 0983.46016)] and \textit{D.\,Azagra, M.\,Jiménez-Sevilla} and \textit{R.\,Deville} [Math.\ Proc.\ Camb.\ Philos.\ Soc.\ 134, No.\,1, 163--185 (2003; Zbl 1034.46039)]. They also obtain some new results under special set theoretic assumptions. Using results of Todorčević, they prove that, under Martin's Maximum axiom: {\parindent5mm \begin{itemize}\item[(1)] For every two Banach spaces \(X\) and \(Y\) of density \(\aleph_1\), there exists a \(C^1\) Fréchet smooth and Lipschitz mapping from \(X\) onto \(Y\). \item[(2)] For every Asplund Banach space \(X\) of density \(\aleph_1\), there exists a \(C^1\)-Fréchet smooth and Lipschitz function \(f:X\longrightarrow\mathbb{R}\) whose range of the derivative contains all \(B_{X^\ast}\). \end{itemize}} Under the weaker axiom MA\(_\aleph{}_1{}\): {\parindent5mm \begin{itemize}\item[(3)] There is no surjective \(C^1\)-smooth function with locally uniformly continuous derivative from \(c_0(\omega_1)\) onto \(\ell_2\). \end{itemize}} The authors notice that the last statement fails under CH. However, they do not know whether the results (1) and (2) above hold in ZFC. They even ask whether every two Banach spaces of density \(\mathfrak c\) are the \(C^\infty\)-smooth image of each other.
Mathematics Subject Classification ID: 46B26
Mathematics Subject Classification ID: 46T20
Mathematics Subject Classification ID: 03E50
zbMATH DE Number: 5687677
zbMATH Keywords: Fréchet smoothness
zbMATH Keywords: surjective mappings
zbMATH Keywords: Martin's axiom
reviewed by: Antonio Avilés
MaRDI profile type: Publication
OpenAlex ID: W2083974097
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DOI: 10.1016/J.JMAA.2009.12.010
description: scientific article; zbMATH DE number 5687677