  Let $G$ be a locally convex Lie group and $\pi:G \to \mathrm{U}(\mathcal{H})$
be a continuous unitary representation. $\pi$ is called smooth if the space of
$\pi$-smooth vectors $\mathcal{H}^\infty\subset \mathcal{H}$ is dense. In this
article we show that under certain conditions, concerning in particular the
structure of the Lie algebra $\mathfrak{g}$ of $G$, a continuous unitary
representation of $G$ is automatically smooth. As an application, this yields a
dense space of smooth vectors for continuous positive energy representations of
oscillator groups, double extensions of loop groups and the Virasoro group.
Moreover we show the existence of a dense space of analytic vectors for the
class of semibounded representations of Banach-Lie groups. Here $\pi$ is called
semibounded, if $\pi$ is smooth and there exists a non-empty open subset
$U\subset\mathfrak{g}$ such that the operators $i\mathrm{d}\pi(x)$ from the
derived representation are uniformly bounded from above for $x\in U$.
