Smooth norms in dense subspaces of $$\ell _p(\Gamma )$$ and operator ranges

Abstract

AbstractFor $$1\le p<\infty $$ 1 ≤ p < ∞ , we prove that the dense subspace $$\mathcal {Y}_p$$ Y p of $$\ell _p(\Gamma )$$ ℓ p ( Γ ) comprising all elements y such that $$y \in \ell _q(\Gamma )$$ y ∈ ℓ q ( Γ ) for some $$q \in (0,p)$$ q ∈ ( 0 , p ) admits a $$C^{\infty }$$ C ∞ -smooth norm which locally depends on finitely many coordinates. Moreover, such a norm can be chosen as to approximate the $$\left\| \cdot \right\| _p$$ · p -norm. This provides examples of dense subspaces of $$\ell _p(\Gamma )$$ ℓ p ( Γ ) with a smooth norm which have the maximal possible linear dimension and are not obtained as the linear span of a biorthogonal system. Moreover, when $$p>1$$ p > 1 or $$\Gamma $$ Γ is countable, such subspaces additionally contain dense operator ranges; on the other hand, no non-separable operator range in $$\ell _1(\Gamma )$$ ℓ 1 ( Γ ) admits a $$C^1$$ C 1 -smooth norm.