Let $G$ be a real Lie group, and $V$ a Hilbert space. Wikipedia defines a unitary representation of $G$ on $V$ to be a homomorphism of $G$ into the group of unitary operators of $V$ such that for each $v \in V$, the map $g \mapsto \pi(g)v$ is continuous, where $V$ is given its norm topology.
On the other hand, many places for example here (page 23, proposition 5.5(2)) seem to define a Hilbert space representation as one for which the product mapping $G \times V \rightarrow V, (g,v) \mapsto \pi(g)v$ is continuous.
Are these inequivalent definitions? Or does each $g \mapsto \pi(g)v$ being continuous imply the continuity of $(g,v) \mapsto \pi(g)v$? Maybe by some application of the uniform boundedness principle.
Hint: Yes, it is indeed equivalent. First notice it is enough to show that $G \times V \to V, (g,v) \mapsto \pi(g)v$ is continuous at $(e,0)$ and that every Lie group is locally compact. Choose a compact neighbourhood $C \subseteq G$ from $e$. Then $\{π(g) : g ∈ C\}$ is a family of operators that suffices the conditions of the uniform boundedness principle. Can you take it from there?