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 $$V$$.