Smooth norms in dense subspaces of Banach spaces

TL;DR

This paper explores the existence of dense subspaces with smooth or analytic norms in various Banach spaces, revealing both possibilities and limitations through constructive and counterexample results.

Contribution

It demonstrates the existence of dense subspaces with analytic or smooth norms in certain Banach spaces and provides counterexamples showing limitations in others.

Findings

Dense subspace of $\, ext{ell}_ ext{infty}$ admits an analytic norm.

Every separable Banach space has a dense subspace with an analytic renorming.

There exists a dense subspace of $ ext{c}_0( ext{omega}_1)$ with no analytic norm.

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 $\omega_1$.)