I am reading this paper and this came up:
So $G$ is a real Lie group, $\{X_i\}$ is a fixed basis for its Lie algebra $\frak{g}$. For any unitary $G$-representation $V$, the Sobolev norm $S_k$ is defined as $$S_k(v) = \sqrt{\sum_{\text{ord}(D) \leq k} \|Dv\|_2^2},$$ where the sum is over all monomials $D$ in the $X_i$'s of degree less than or equal $k$.
I cannot see how those monomials induce an operator on $V$. What's going on?
You have to pass to the subspace of smooth vectors, i.e. those vectors $v \in V$ for which the orbit map $G \to V$ defined by $g \mapsto g\cdot v$ is a smooth $V$-valued function. On these vectors you can differentiate the $G$-action to obtain an action of $\mathfrak g$, and from there, as Blazej suggests in comments, you can compose the $X_i$ actions to get an action of monomials. It is a theorem of Garding that the smooth vectors are dense, so you don't really lose anything by passing to the smooth vectors. (In fact, Ed Nelson proved that even the analytic vectors are dense in $V$.)
See Wikipedia for more on this.