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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 $!!$

神宮寺春姫
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    (2) is not good enough : it doesn't take the topology of $G$ into account; moreover for many purposes $V$ will be normed and finite dimensional so all linear maps are continuous and so (2) wouldn't make sense. – Maxime Ramzi Aug 04 '19 at 11:15
  • "Let $G$ be a group": should be "Let $G$ be a topological group"? – YCor Aug 04 '19 at 23:51
  • This is covered by your question https://math.stackexchange.com/q/3162758. – Paul Frost Aug 05 '19 at 11:40

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