Smooth norms in dense subspaces of Banach spaces

Abstract

In the first part of our paper, we show that ${\ell }_{\infty }$ has a dense linear subspace which admits an equivalent real analytic norm. As a corollary, every separable Banach space, as well as ${\ell }_1(\mathfrak{c})$, also has a dense linear subspace which admits an analytic renorming. By contrast, no dense subspace of $c_0({\omega }_1)$ admits an analytic norm. In the second part, we prove (solving in particular an open problem of Guirao, Montesinos, and Zizler) that every Banach space with a long unconditional Schauder basis contains a dense subspace that admits a $C^{\infty }$-smooth norm. Finally, we prove that there is a proper dense subspace of ${\ell }_{\infty }^c({\omega }_1)$ that admits no Gâteaux smooth norm. (Here, ${\ell }_{\infty }^c({\omega }_1)$ denotes the Banach space of real-valued, bounded, and countably supported functions on ${\omega }_1$.)

18 pp