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 G\^ateaux smooth norm. (Here, $\ell_{\infty}^{c} (\omega_1)$ denotes the Banach space of real-valued, bounded, and countably supported functions on $\omega_1$.)