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

Abstract

For $1\le p<\infty$, we prove that the dense subspace ${\mathcal{Y}}_p$ of ${\ell }_p(\Gamma )$ comprising all elements $y$ such that $y\in {\ell }_q(\Gamma )$ for some $q\in (0,p)$ admits a $C^{\infty }$-smooth norm which locally depends on finitely many coordinates. Moreover, such a norm can be chosen as to approximate the ${\Vert \cdot \Vert }_p$-norm. This provides examples of dense subspaces of ${\ell }_p(\Gamma )$ 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 $p>1$ or $\Gamma$ is countable, such subspaces additionally contain dense operator ranges; on the other hand, no non-separable operator range in ${\ell }_1(\Gamma )$ admits a $C^1$-smooth norm.

References in corpus (4)

• Dense lineability and spaceability in certain subsets of ${\ell }_{\infty }$
• Smooth norms in dense subspaces of Banach spaces
• Surjective homomorphisms from algebras of operators on long sequence spaces are automatically injective
• Smooth and polyhedral norms via fundamental biorthogonal systems