If $G$ is a topological group (for example a Lie group) and $V$ is a real vector space of finite dimension it's defined the concept of regular representation of $G$ over $V$ in the following way:
$P : G \rightarrow GL(V) $ is a regular representation of $G$ on $V$ if is a representation of $G$ as group (i.e. $P$ is a group homomorphism) such that the induced map $G \times V \rightarrow V$ is continuous.
My question is: Is this equivalent to
$P$ is a continuous homomorphism of groups.
Note: On $V$ and $GL(V)$ there are the natural topologies.