  We define Sobolev norms of arbitrary real order for a Banach representation
$(\pi, E)$ of a Lie group, with regard to a single differential operator
$D=d\pi(R^2+\Delta)$. Here, $\Delta$ is a Laplace element in the universal
enveloping algebra, and $R>0$ depends explicitly on the growth rate of the
representation. In particular, we obtain a spectral gap for $D$ on the space of
smooth vectors of $E$. The main tool is a novel factorization of the delta
distribution on a Lie group.
