Let $K$ be a finite extension of the $p$-adic field $\mathbb{Q}_p$ and let $G_K:=Gal(K/\mathbb{Q}_p)$ is the Galois group. Now if we consider the continuous representation of $G_K$ as $\rho: G_K \to GL(V)$, where $V$ is a vector space over $\mathbb{Q}_p$.
My question:
What topology is there in the codomain $GL(V)$ so that $\rho$ is continuous?
Offcourse if i give indiscrete topology, then it continuous. But what is the topology in the above case?