Let $X$ and $Y$ be locally convex topological vector spaces, say over $\mathbb{C}$. To set the stage a bit, I'll say that the topology on $X$ is given by a separating family of semi-norms $(p_i)_{i \in I}$ and, similarly, the topology on $Y$ is given by a family $(q_j)_{j \in J}$.
Now, suppose that $V \subset X$ is a dense subspace of $X$ and $T : V \to Y$ is a continuous linear map. Does there exist, then, a (unique) continuous extension $\overline T : X \to Y$ of $T$?
The answer is yes for Banach spaces, and this is an incredibly useful fact. The norm seems pretty crucial to the proof though. Any easy counterexamples?