There are natural function spaces on Lie groups that are nevertheless not continuous (and, thus, are not representations in any usual, useful sense). For example, already on $G=\mathbb R$, the Frechet space $V$ of all continuous functions, and/or the Frechet space of bounded continuous functions, with the translation action of $G$, are not repn spaces, in the sense that $G\times V\rightarrow V$ is not continuous. The reason is the existence of not-uniformly-continuous continuous functions. For example, $f(x)=\sin(x^2)$.

There are natural function spaces on Lie groups that are nevertheless not continuous (and, thus, are not representations in any usual, useful sense). For example, already on $G=\mathbb R$, the Fréchet space $V$ of all continuous functions, and/or the Fréchet space of bounded continuous functions, with the translation action of $G$, are not repn spaces, in the sense that $G\times V\rightarrow V$ is not continuous. The reason is the existence of not-uniformly-continuous continuous functions. For example, $f(x)=\sin(x^2)$.