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 (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.
Beltita et al. (Tue,) studied this question.