  Let $\mathcal{X}$ be a Banach space with a fundamental biorthogonal system
and let $\mathcal{Y}$ be the dense subspace spanned by the vectors of the
system. We prove that $\mathcal{Y}$ admits a $C^\infty$-smooth norm that
locally depends on finitely many coordinates (LFC, for short), as well as a
polyhedral norm that locally depends on finitely many coordinates. As a
consequence, we also prove that $\mathcal{Y}$ admits locally finite,
$\sigma$-uniformly discrete $C^\infty$-smooth and LFC partitions of unity and a
$C^1$-smooth LUR norm. This theorem substantially generalises several results
present in the literature and gives a complete picture concerning smoothness in
such dense subspaces. Our result covers, for instance, every WLD Banach space
(hence, all reflexive ones), $L_1(\mu)$ for every measure $\mu$,
$\ell_\infty(\Gamma)$ spaces for every set $\Gamma$, $C(K)$ spaces where $K$ is
a Valdivia compactum or a compact Abelian group, duals of Asplund spaces, or
preduals of Von Neumann algebras. Additionally, under Martin Maximum {\sf MM},
all Banach spaces of density $\omega_1$ are covered by our result.
