Archiv der Mathematik

, Volume 92, Issue 5, pp 414–427

Equivalent complete norms and positivity


    • Institute of Applied AnalysisUniversity of Ulm
  • Robin Nittka
    • Institute of Applied AnalysisUniversity of Ulm

DOI: 10.1007/s00013-009-3190-6

Cite this article as:
Arendt, W. & Nittka, R. Arch. Math. (2009) 92: 414. doi:10.1007/s00013-009-3190-6


In the first part of the article we characterize automatic continuity of positive operators. As a corollary we consider complete norms for which a given cone E+ in an infinite dimensional Banach space E is closed and we obtain the following result: every two such norms are equivalent if and only if \(E_+ \cap (-E_+) = \{0\} \, {\rm and} \, E_+ - E_+\) has finite codimension.

Without preservation of an order structure, on an infinite dimensional Banach space one can always construct infinitely many mutually non-equivalent complete norms. We use different techniques to prove this. The most striking is a set theoretic approach which allows us to construct infinitely many complete norms such that the resulting Banach spaces are mutually non-isomorphic.

Mathematics Subject Classification (2000).



Equivalent normspositivitydiscontinuous functionalsautomatic continuitycardinality of Hamel bases

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009