Proposition 2

Let $E$ and $F$ be two topological vector spaces over $\mathbf{R}$, and $u$ a continuous linear map from $E$ into $F$. If $f$ is a vector function defined on an interval $I \subset \mathbf{R}$, taking values in $E$, and differentiable at the point $x_0 \in I$, then the composite function $u \circ f$ has a derivative equal to $u(f'(x_0))$ at $x_0$.

Indeed, since $\frac{u(f(x)) - u(f(x_0))}{x - x_0} = u \left( \frac{f(x) - f(x_0)}{x - x_0} \right)$, this follows from the continuity of $u$.