Let $G$ be a real Lie group, and $\pi: G \rightarrow \operatorname{GL}(V)$ a continuous representation of $G$ on a Hilbert space $V$.
If $X \in \mathfrak g$, "differentiation of a vector $v \in V$ in the direction $X$" is defined to be
$$\pi(X)v = \lim\limits_{h \to 0} \frac{1}{h} [\pi(\exp(hX))v - v]$$
if this limit exists. If $V$ is finite dimensional, say $V = \mathbb R^{m}$ as real vector spaces, then $\alpha: \mathbb R \rightarrow \mathbb R^m, \alpha(t) = \pi( \exp(tX))v$ defines a smooth map $\mathbb R \rightarrow \mathbb R^m$. Then $\pi(X)v$ is the evaluation of the derivative of a smooth map at $t = 0$.
If $V$ is infinite dimensional, to what extent can $\pi(X)v$ be realized as a derivative (beyond the obvious limit definition)? Can one make use of the notion of an "infinite dimensional" manifold here? Of course we are not going to be able to reduce to charts as in the finite dimensional case, since the limit defining $\pi(X)v$ need not be defined for a given $v$.