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arXiv · 1804.05660

Approximation of norms on Banach spaces

Richard J. Smith·Stanimir Troyanski
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Abstract

Relatively recently it was proved that if $\Gamma$ is an arbitrary set, then any equivalent norm on $c_0(\Gamma)$ can be approximated uniformly on bounded sets by polyhedral norms and $C^\infty$ smooth norms, with arbitrary precision. We extend this result to more classes of spaces having uncountable symmetric bases, such as preduals of the `discrete' Lorentz spaces $d(w,1,\Gamma)$, and certain symmetric Nakano spaces and Orlicz spaces. We also show that, given an arbitrary ordinal number $\alpha$, there exists a scattered compact space $K$ having Cantor-Bendixson height at least $\alpha$, such that every equivalent norm on $C(K)$ can be approximated as above.

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BibTeXRIS

Richard J. Smith, Stanimir Troyanski. 2018-04-16. Approximation of norms on Banach spaces. https://doi.org/10.1016/j.jfa.2018.07.007

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