Continuity of linear maps on \(\mathcal L_1\)-spaces
type: Item
instance of: scholarly article
title: Continuity of linear maps on \(\mathcal L_1\)-spaces
zbMATH Open document ID: 1232.46007
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-10-24T00:00:00Z
review text: Let \(L(X,Y)\) be the space of all linear (but not necessarily continuous) maps between two Banach spaces \(X\) and \(Y\). Let \(L_D(X,Y)\) be those \(T \in L(X,Y)\) whose restriction to some dense linear subspace of \(X\) is continuous. Inspired by a question of Bogachev, Kirchheim and Schachermayer the authors prove that a separable Banach space \(X\) is an \(\mathcal{L}_1\)-space if and only if \(L(X,c_0) = L_D(X,c_0)\). The proof is based on their characterization of \(\mathcal{L}_1\)-spaces: A separable Banach space \(X\) is an \(\mathcal{L}_1\)-space if and only if \(\mathrm{Ext}(X,H) = 0\) for every subspace \(H\) of \(c_0\). The condition \(\mathrm{Ext}(X,H) = 0\) means that whenever \(H\) is a subspace of a Banach space \(E\) in such a way that \(E/H\) is isomorphic to \(X\) then \(H\) must be complemented in \(E\).
reviewed by: Vegard Lima
Mathematics Subject Classification ID: 46B03
Mathematics Subject Classification ID: 46M18
zbMATH DE Number: 5962582
zbMATH Keywords: extension of operators
zbMATH Keywords: linear non-continuous maps
zbMATH Keywords: Banach spaces
author: Jesús M. F. Castillo
author: M. González
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.06.019
OpenAlex ID: W2061421303
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DOI: 10.1016/J.JMAA.2011.06.019
description: scientific article; zbMATH DE number 5962582