Smooth norms in dense subspaces of Banach spaces

Abstract

In the first part of our paper, we show that ℓ∞ has a dense linear subspace which admits an equivalent real analytic norm. As a corollary, every separable Banach space, as well as ℓ1(c), also has a dense linear subspace which admits an analytic renorming. By contrast, no dense subspace of c0(ω1) admits an analytic norm. In the second part, we prove (solving in particular an open problem of Guirao, Montesinos, and Zizler in [7]) that every Banach space with a long unconditional Schauder basis contains a dense subspace that admits a C∞-smooth norm. Finally, we prove that there is a proper dense subspace of ℓ∞c(ω1) that admits no Gâteaux smooth norm. (Here, ℓ∞c(ω1) denotes the Banach space of real-valued, bounded, and countably supported functions on ω1.)