Remark 2

The preceding definitions and propositions (except for those concerning right and left derivatives) extend easily to the case where one replaces $\mathbf{R}$ by an arbitrary commutative non-discrete topological field $K$, and the topological vector spaces (resp. topological algebras) over $\mathbf{R}$ by topological vector spaces (resp. topological algebras) over $K$. In def. 1 and props. 1, 2 and 3 it is enough to replace I by a neighbourhood of $x_0$ in $K$; in prop. 4 one must assume further that the map $y \mapsto y'$ is defined and continuous on a neighbourhood of $f(x_0)$ in E. Prop. 5 generalizes in the following manner: let $K'$ be a non-discrete subfield of the topological field $K$, let E be a topological vector space over $K$; let $f$ be a function defined on a neighbourhood $V \subset K'$ of $x_0 \in K'$, with values in $K$ (considered as a topological vector space over $K'$), differentiable at $x_0$, and let g be a function defined on a neighbourhood of $f(x_0) \in K$, with values in E, and differentiable at the point $f(x_0)$; then the map $g \circ f$ is differentiable at $x_0$ and has derivative $g'(f(x_0))f'(x_0)$ there (E being then considered as a topological vector space over $K'$).

With the same notation, let $f$ be a function defined on a neighbourhood V of $a \in K$, with values in E, and differentiable at the point $a$; if $a \in K'$, then the restriction of $f$ to $V \cap K'$ is differentiable at $a$, and has derivative $f'(a)$ there. These considerations apply above all, in practice, to the case where $K = \mathbf{C}$ and $K' = \mathbf{R}$.

Finally, prop. 6 extends to the case where one replaces I by a neighbourhood of $x_0 \in K$, and $f$ by a homeomorphism of I onto a neighbourhood $J = f(I)$ of $y_0 = f(x_0)$ in $K$.