Let $k$ be a global field, and let $G = \mathbf G(\mathbb A_k)$ for a connected, reductive group $\mathbf G$ over $k$. In these notes by Jayce Getz and Heekyoung Hahn, a unitary representation of $G$ is a Hilbert space $V$ together with a continuous homomorphism $\pi: G \rightarrow \operatorname{GL}(V)$ whose image is contained in the group $U(V)$ of unitary operators on $V$.

What is the topology on $\operatorname{GL}(V)$ (which I assume is the group of bounded linear operators on $V$) being considered here? Is it the induced topology coming from the norm topology?

I am trying to compare this definition with one given by Gerald Folland in *A Course in Abstract Harmonic Analysis*, which requires that for each $v \in V$ the map $g \mapsto \pi(g)v$ be continuous $G \rightarrow V$, where $V$ is taken in the norm topology. Are these two definitions of unitary representations different?

This matters because one later defines the Fell topology on the unitary dual $\hat{G}$ of $G$, and I want to know which representations are actually in $\hat{G}$.