Analytic and C ^k approximations of norms in separable Banach spaces

We prove that in separable Hilbert spaces, in ℓp(Nℕ) for p an even integer, and in Lp[0,1] for p an even integer, every equivalent norm can be approximated uniformly on bounded sets by analytic norms. In ℓp(Nℕ) and in Lp[0,1] for p ∉ Nℕ (resp. for p an odd integer), every equivalent norm can be approximated uniformly on bounded sets by C ^[p]-smooth norms (resp. by C ^p-1-smooth norms).