Let $G$ be a Lie group and $(\pi,V)$ be a continuous representation of $G$ on $V$ a Fréchet space. Let $dx$ denote the Haar measure on $G$.
The representation $\pi$ induces a representation of $M_c(G)$ - the space of bounded Borel measures on $G$ of compact support, defined as
$\pi(\mu)v = \int_G \pi(x)v \, d\mu(x). $
This is a representation, since $\pi(\mu_1)\pi(\mu_2)=\pi(\mu_1*\mu_2)$. Furthermore, it is continuous, since for a seminorm $\rho$, we have $\|\pi(\mu)v\|_\rho = \Big(\int_G |d\mu|\Big) \, \|v\|_\rho. $
I would like to know if continuity holds if, instead of the mentioned measures, we work with $C^\infty_c(G)$ - the space of smooth functions with compact support, given a Frechet topology. If known, also for distributions, Schwartz space, and tempered distributions would be nice :)
In particular, I wanted to prove that if we pick a Dirac sequence [i.e. $f_n\in C^\infty_c(G)$ sequence of $L^1$-normalized positive functions with support tending to $1_G$.], how can I prove that $\| \pi(f_n)v - v \| \to 0$. The measure approach doesn't prove it, apparently.
Thank you.