I have a question about the definition of continuous representation of topological group.
Let $G$ be a group and $\rho : G \rightarrow {\rm Aut}_k(V)$ be a representation of $G$ , where $k$ is a field
and $V$ is a topological space and finite dimensional vector space.
${\rm Map}(V,V)$ is the topological space by compact-open topology.
We consider ${\rm Aut}_k (V)$ as a topological subspace of ${\rm Map}(V,V)$.
$\rho$ is a continuous map under this setting.
[My question]
Two conditions $(1)$ and $(2)$ is equivalent $??$
$(1)$ $\rho$ is a continuous map of topological spaces.
$(2)$ For any element $g \in G$ , $\rho (g)$ is a continuous map.
I think $(2)$ is the definition of continuous representation.
$(1)$ may be the definition. Please give me opinions $!!$