Let $\pi : G \rightarrow U(H)$ be a strongly continuous unitary representation of a Lie group, $G$, on a Hilbert space, $H$. Let $H_\infty$ be the space of smooth vectors in $H$, those $v$ for which $g\mapsto \pi(g)v$ is smooth. $\pi$ induces a representation, $\pi'$, of the Lie algebra, $\frak{g}$, of $G$ on $H_\infty$ in the obvious way: $$\pi'(X)v=\frac{d}{dt}|_{t=0}\pi(e^{tX})v$$
Is each $\pi'(X)$ continuous?