Analytic and $C^k$ approximations of norms in separable Banach spaces
Volume 120 / 1996
Studia Mathematica 120 (1996), 61-74
DOI: 10.4064/sm-120-1-61-74
Abstract
We prove that in separable Hilbert spaces, in $ℓ_{p}(ℕ)$ for p an even integer, and in $L_{p}[0,1]$ for p an even integer, every equivalent norm can be approximated uniformly on bounded sets by analytic norms. In $ℓ_{p}(ℕ)$ and in $L_{p}[0,1]$ 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 norms).
