TITLE: are the smooth vectors of a Frechet space dense?
QUESTION [5 upvotes]: 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.

REPLY [3 votes]: In full generality, the answer is "no". If you have a unitary irreducible representation of a $p$-adic Lie group on a $p$-adic Banach space, then the locally constant vectors are usually not dense. There may even be no nonzero locally algebraic vector. The locally constant vectors are dense however if you have an $\ell$-adic Lie group acting on a $p$-adic space, with $\ell \neq p$ (this is a theorem of Vigneras).