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?