Proposition 1

Let U be an open subset of $\mathbf{R}^n$ and $k\in \mathbf{N}$. Let E, F and G be topological vector spaces and let $b: E\times F\rightarrow G$ be a continuous bilinear mapping. Let $f$ and $g$ be functions of class $C^k$ from U into E and F respectively. Then the mapping $h:x\mapsto b(f(x), g(x))$ is of class $C^k$ in U; moreover, for every $\alpha \in \mathbf{N}^n$ such that $|\alpha |\leqslant k$ and for every $x\in U$, we have

$\beta \leqslant \alpha$

The mapping $h$ is the composite of the mapping $(f, g) : U\rightarrow E\times F$, which is of class $C^k$, and of the mapping $b: E\times F\rightarrow G$ which is of class $C^{\infty}$. It is therefore of class $C^k$. The expression for its partial derivatives results from Leibniz's formula (FVR, I, p. 28, prop. 2, cf. A, III, p. 122, corollary).

We recall that a separated locally convex topological vector space is a Montel space if it is barrelled and if every bounded subset is relatively compact (EVT, IV, p. 18, def. 4).

We also recall that if E and F are locally convex spaces, and if E is bornological (EVT, III, p. 12, def. 1), a linear mapping $u: E\rightarrow F$ is continuous if and only if the image by $u$ of every bounded subset of E is bounded in F (EVT, III, p. 11, prop. 1, (iiibis), when F is semi-normed, the general case resulting formally from this, cf. EVT, II, p. 7, prop. 5, b)).