  We prove that the spaces of smooth and ultradifferentiable vectors associated
with a representation of a real Lie group on a Fr\'{e}chet space $E$ are
quasinormable if $E$ is so. A similar result is shown to hold for the linear
topological invariant $(\Omega)$. In the ultradifferentiable case, our results
particularly apply to spaces of Gevrey vectors of Beurling type. As an
application, we study the quasinormability and the property $(\Omega)$ for a
broad class of Fr\'{e}chet spaces of smooth and ultradifferentiable functions
on Lie groups globally defined via families of weight functions.
