Proposition 5

— Let $E_i (1 \leq i \leq n)$ and F be topological vector spaces on a non-discrete valued field K. In order that a multilinear mapping $f$ of $\prod_{i=1}^n E_i$ in F should be continuous in the product space $\prod_{i=1}^n E_i$ it is sufficient for it to be continuous at $(0, 0, ..., 0)$.

Let $(a_1, a_2, ..., a_n)$ be an arbitrary point of $\prod_{i=1}^n E_i$; we must show that for every neighbourhood W of 0 in F there exist neighbourhoods $V_i$ of 0 in $E_i$ ($1 \leq i \leq n$) such that the relations $z_i \in V_i$ imply

Now, we can write

where $H$ varies over the $2^n - 1$ subsets of the set of integers $\{ 1, 2, ..., n \}$, excluding the set $\{ 1, 2, ..., n \}$ itself, and where $u_H = f(y_1, y_2, ..., y_n)$, with $y_i = a_i$ if $i \in H$ and $y_i = z_i$ if $i \notin H$. There exist $2^n - 1$ balanced neighbourhoods $W_H$ of 0 in $F$ such that $\sum_H W_H \subset W$; on the other hand as $f$ is continuous at $(0, 0, ..., 0)$ by hypothesis, there exists in each $E_i$ a neighbourhood $U_i$ of 0 ($1 \leq i \leq n$) such that the $n$ relations $x_i \in U_i$ imply that $f(x_1, ..., x_n) \in \bigcap_H W_H$. As $U_i$ is absorbent, there exists $\lambda_i \neq 0$ in $K$ such that $\lambda_i a_i \in U_i$. Let $\lambda$ be an element of $K$ such that $|\lambda| \geq \prod_{i \in H} |\lambda_i|^{-1}$ for each subset $H$; we show that the neighbourhoods $V_i = \lambda^{-n} U_i$, fulfill the required condition. We can write $u_H = \mu f(x_1, ..., x_n)$ where $x_i \in U_i$ for $1 \leq i \leq n$ and $\mu = \lambda^{-np} (\prod_{i \in H} \lambda_i^{-1})$, $p$ being the number of integers of $\{ 1, 2, ..., n \}$ not in $H$. From the above $|\mu| \leq 1$, hence $u_H \in \mu W_H \subset W_H$ since $W_H$ is balanced. The proposition is established.