Let $\pi$ be a continuous irreducible representation of $G:=\mathrm{SL}(2,\mathbb{R})$ in a Banach space $H$, and $\pi^1$ the representation of $\mathcal{C}_c(G)$ induced by $\pi$. Suppose $\mathcal{S}$ is an $\mathrm{L}^1$dense subspace of $\mathcal{C}_c(G)$. Why does $\mathcal{S}$invariance imply $\mathcal{C}_c(G)$invariance for every closed subspace of $H$? In his book on $\mathrm{SL}(2,\mathbb{R})$, Serge Lang makes use of this in the proof of the fact that $K$finite vectors are dense, $K$ denoting $\mathrm{SO}(2)$ as usual (Theorem 3, p. 24). The answer is clear when $\pi$ is a bounded (in particular, when $H$ is a Hilbert space and $\pi$ is unitary), since in this case $\pi^1$ has a continuous extension to $\mathrm{L}^1(G)$.
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