Proposition 1

The set of vector functions defined on an interval $I \subset \mathbf{R}$, taking values in a given topological vector space $E$, and differentiable at the point $x_0$, is a vector space over $\mathbf{R}$, and the map $f \mapsto Df(x_0)$ is a linear mapping of this space into $E$.

In other words, if $f$ and $g$ are defined on $I$ and differentiable at the point $x_0$, then $f + g$ and $fa$ ($a$ an arbitrary scalar) are differentiable at $x_0$ and their derivatives there are $f'(x_0) + g'(x_0)$ and $f'(x_0)a$ respectively. This follows immediately from the continuity of $x + y$ and of $xa$ on $E \times E$ and $E$ respectively.