Proposition 6

With the same hypotheses on $E_i (1 \leq i \leq n)$ and on $F$ as in prop. 5, in order that a set $\mathscr{E}$ of multilinear maps of $\prod_{i=1}^n E_i$ in $F$ be equicontinuous it is sufficient that the set be equicontinuous at $(0, 0, ..., 0)$.

For, in the demonstration of prop. 5 the $U_i (1 \leq i \leq n)$ can be taken such that the relation $x_i \in U_i (1 \leq i \leq n)$ imply $f(x_1, ..., x_n) \in \bigcap_H W_H$ for every mapping $f \in \mathscr{E}$.