Approximation of norms on Banach spaces

Relatively recently it was proved that if $Γ$ is an arbitrary set, then any equivalent norm on $c_0(Γ)$ 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 $d(w,1,Γ)$, and certain symmetric Nakano spaces and Orlicz spaces. We also show that, given an arbitrary ordinal number $α$, there exists a scattered compact space $K$ having Cantor-Bendixson height at least $α$, such that every equivalent norm on $C(K)$ can be approximated as above.

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