, Volume 92, Issue 5, pp 414-427
Date: 24 Apr 2009

Equivalent complete norms and positivity

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Abstract.

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.

Dedicated to Professor Heinz König on the occasion of his 80th birthday
Received: 28 January 2009