Equivalent complete norms and positivity
type: Item
instance of: scholarly article
title: Equivalent complete norms and positivity
zbMATH Open document ID: 1182.46006
author: Wolfgang Arendt
author: Robin Nittka
published in: Archiv der Mathematik
publication date: 2009-06-16T00:00:00Z
review text: In the first part of the article, automatic continuity of positive operators is characterized. Namely, it is proved that, if \((E, E_+)\) and \((F, F_+)\) are ordered Banach spaces, then the following (i) and (ii) are equivalent. {\parindent=7mm \begin{itemize}\item[(i)] At least one of the following is true: (1) \(E\) is finite-dimensional, (2) \(F_+ = \{0\}\), (3) \(\text{codim}(E_+ - E_+)\) is finite and \(F_+\) is proper. \item[(ii)] Every positive operator from \(E\) to \(F\) is continuous. \end{itemize}} As a consequence of this theorem, the authors obtain that, if \(E\) is an infinite-dimensional ordered Banach space, then \(E_+\) is proper, and \(E_+ - E_+\) has finite codimension if and only if each complete norm for which \(E_+\) is closed is equivalent to the given one. If \(E\) is an infinite-dimensional Banach space, one can construct infinitely many mutually non-equivalent complete norms on \(E\) for which all the corresponding spaces are isometrically isomorphic to the original one. Moreover, the cardinality of such a set of norms can be chosen to be as large as the dimension of \(E\). Different routes, set theoretic and analytic are taken to prove this. The set theoretical approach allows the authors to construct infinitely many complete norms such that the resulting Banach spaces are mutually non-isomorphic. The paper answers an old question of Laugwitz.
Mathematics Subject Classification ID: 46B40
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 46B03
Mathematics Subject Classification ID: 03E75
Mathematics Subject Classification ID: 46B26
zbMATH DE Number: 5565273
zbMATH Keywords: equivalent norms
zbMATH Keywords: positivity
zbMATH Keywords: discontinuous functionals
zbMATH Keywords: automatic continuity
zbMATH Keywords: cardinality of Hamel bases
reviewed by: Şafak Alpay
MaRDI profile type: Publication
full work available at URL: s00013-009-3190-6
OpenAlex ID: W2033057101
DOI: 10.1007/S00013-009-3190-6
description: scientific article; zbMATH DE number 5565273