Approximation of norms on Banach spaces

Abstract

Relatively recently it was proved that if Γ is an arbitrary set, then any equivalent norm on c0(Γ) can be approximated uniformly on bounded sets by polyhedral norms and C∞ 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.