# nt.number theory – With a linear representation, how does the continuity of \$G to mathrm{GL}(V)\$ relate to that of \$G times V to V\$?

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?