Let $G$ be a locally compact topological group, and $V$ be a locally convex topological vector space over $\mathbb{C}$. An abstract representation of $G$ is a homomorphism $\rho\colon G\rightarrow \text{Aut}(V)$. We call $\rho$ a topological representation if additionally, the map $G\times V\rightarrow V$ which takes $(g,x)\mapsto \rho_g(X)$ is continuous with respect to the product topology on $G\times V$.
Following the book "Fourier Analysis on Number Fields" by Ramakrishnan and Valenza, I've seen some nice results which characterize when an abstract representation is topological. However, I haven't been able to find (or come up with) any examples of abstract representations which are not topological! Are there any good examples of these?