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](https://services.math.duke.edu/~hahn/Chapter3.pdf) 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 $\hat{G}$.



Definition of unitary representation of $\mathbf G(\mathbb A_k)$
It is absolutely essential that the space of (bounded/continuous) operators be given the "strong" operator topology (strictly weaker than the norm topology), and the map $G\times V\to V$ to be jointly continuous.


This is not a pathology: even in very simple cases, such as $G=\mathbb R$ acting on $V=L^2(\mathbb R)$, that joint continuity fails when operators are given the uniform norm topology, since there is no sufficient uniform bound on change in $g\in G$ so that $g\cdot v$ is close to $v$ for all $v\in V$. E.g., tent functions with ever-narrowing support illustrate this.


