Approximation of norms on Banach spaces

Relatively recently it was proved that if Γ is an arbitrary set, then any equivalent norm on $c_0(\mathrm{\Gamma })$ 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 $d(w,1,\mathrm{\Gamma })$, 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.