Let $G=\mathrm{SL}(2,\mathbb{R})$ and $K=\mathrm{SO}(2)$. Suppose $\pi$ is a continuous irreducible representation of $G$ in a Banach space $H$. Can one always find a nonzero $v\in H$ such that $\langle \pi(k)v : k\in K\rangle$ is finitedimensional? Serge Lang uses this implicitly in the proof of Theorem 3 on p. 24 of his book on $\mathrm{SL}(2,\mathbb{R}).$ (I know that the answer is affirmative when $H$ is a Hilbert space and $\pi$ is unitary on $K$.)

You must combine lemma 5 and lemma 1 in Serge Lang$(2,\mathbb{R})$: lemma 5 (which holds for Banach spaces) tells you how to construct $K$finite vectors. Now you must rule out the case where $\pi(f)v=0$ for every $v\in H$ and $f\in \oplus_{m,n} S_{n,m}$ (notations as in Lang). But the latter direct sum is dense in $C_c(G)$, by lemma 1. And using an approximation of identity in $C_c(G)$, you see that $\pi(f)v=0$ for every $f\in C_c(G)$, forces $v=0$. 

