I'm currently reading *Traces of Hecke Operators* by Knightly and Li, while simultaneously revisiting the adelic/representation-theoretic point of view on automorphic forms.


In Knightly and Li, they give a familiar definition of a representation. That is, for a locally compact group $G$ and a normed vector space $V$, they note that a representation is a homomorphism
$$ \pi: G \longrightarrow \mathrm{GL}(V) $$
such that the map
$$\begin{align}
G \times V &\longrightarrow V \\
(g,v) &\mapsto \pi(g)v
\end{align}$$
is continuous. Sometimes I've seen this stated first in terms of the continuity of $g \mapsto \pi(g)v$.


But I note that both $G$ and $\mathrm{GL}(V)$ are topological groups, so it would make sense to consider the continuity of $\pi$ directly as a function $G \longrightarrow \mathrm{GL}(V)$. I don't know how the continuity of $G \to \mathrm{GL}(V)$ relates to the continuity of $G \times V \to V$.


Intuitively, I suspect that the continuity of $G \times V \to V$ implies continuity of the representation map $G \to \mathrm{GL}(V)$, but not the converse. Is this right?



With a linear representation, how does the continuity of $G \to \mathrm{GL}(V)$ relate to that of $G \times V \to V$?
For locally compact groups continuity of $\pi$, joint continuity of the action map $G\times V\to V$ and separate continuity in both variables are equivalent. See Theorem 2.3 of Karl H. Hofmann, Sidney A. Morris
The Structure of Compact Groups (edition 3).


