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arXiv · 1509.00369

Smooth and polyhedral approximation in Banach spaces

Victor Bible·Richard J. Smith
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Abstract

We show that norms on certain Banach spaces $X$ can be approximated uniformly, and with arbitrary precision, on bounded subsets of $X$ by $C^{\infty}$ smooth norms and polyhedral norms. In particular, we show that this holds for any equivalent norm on $c_0(Γ)$, where $Γ$ is an arbitrary set. We also give a necessary condition for the existence of a polyhedral norm on a weakly compactly generated Banach space, which extends a well-known result of Fonf.

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Victor Bible, Richard J. Smith. 2015-09-20. Smooth and polyhedral approximation in Banach spaces. https://doi.org/10.1016/j.jmaa.2015.11.018

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