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We prove that in separable Hilbert spaces, in $ℓ_{p}(ℕ)$ for p an even integer, and in for p an even integer, every equivalent norm can be approximated uniformly on bounded sets by analytic norms. In $ℓ_{p}(ℕ)$ and in for p ∉ ℕ (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...
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