  In this note we show that the representation of the additive group of the
Hilbert space $L^2([0,1],\R)$ on $L^2([0,1],\C)$ given by the multiplication
operators $\pi(f) := e^{if}$ is continuous but its space of smooth vectors is
trivial. This example shows that a continuous unitary representation of an
infinite dimensional Lie group need not be smooth.
