  We establish the exponential law for suitably topologies on spaces of
vector-valued smooth functions on topological groups, where smoothness is
defined by using differentiability along continuous one-parameter subgroups. As
an application, we investigate the canonical correspondences between the
universal enveloping algebra, the invariant local operators, and the
convolution algebra of distributions supported at the unit element of any
finite-dimensional Lie group, when one passes from finite-dimensional Lie
groups to pre-Lie groups. The latter class includes for instance any locally
compact groups, Banach-Lie groups, additive groups underlying locally convex
vector spaces, and also mapping groups consisting of rapidly decreasing Lie
group-valued functions.
