Approximation of norms on Banach spaces

Abstract 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 ∞ 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.