2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/208142In 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 $π(f) := e^{if}$ 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.5 pagesRepresentation TheoryFunctional Analysis22E65; 22E45text