Continuous linear representation of topological group over topological field

Definition

Suppose $G$ is a topological group, $K$ is a topological field, and $V$ is a topological vector space over $K$. A continuous linear representation of $G$ over $V$ (and hence over $K$) is a linear representation $\alpha :G\to GL(V)$, such that, when viewed as a map $G\times V\to V$, it is jointly continuous. In other words, it is continuous from $G\times V$ (endowed with the product topology) to $V$.