Given an action $\alpha$ of $V$ a Lie group on $B$ a Fréchet space with seminorms $ \{ \| \cdot \|\_j \} $, let $B^\infty$ be the space of smooth vectors. Is this dense in $B$? Can I guarantee it is non-empty? Is there any requirements on $G$ or $\alpha$ or $B$? For the case I am (supposed to be) working on right now, we also have that $\alpha$ is isometric for all the seminorms and strongly continuous. 


Also, since I am almost illiterate on the subject of Lie groups, I would also be thankful for some easy to read references! Thank you so much.



are the smooth vectors of a Frechet space dense?
The answer is "yes" quite generally, and the following argument can be found in many texts (Knapp, Wallach, Varadarajan, ...): for a test function $f$ on the Lie group $G$, there is the "averaged" action on any quasi-complete, locally convex repn space $\pi,V$ for $G$, namely, $f\cdot v=\int\_G f(g)\,\pi(g)(v)\;dg$. These images are smooth vectors. The collection of all such is the Garding subspace. Taking $f$'s to run through an \_approximate\_identity\_ shows density.


(In fact, Dixmier and Malliavin showed that the Garding subspace is the whole space of smooth vectors.)


