Proposition 4

Let $X_i$ ($1 \leq i \leq n$) and $Y$ be normed vector spaces over a non-discrete valued division ring $K$, and let $X = \prod_{i=1}^n X_i$. Then the set of all multilinear mappings of $X$ into $Y$ is closed in the space $\mathscr{F}_s(X; Y)$.

This set consists of all $u \in F(X; Y)$ which satisfy all the relations

($1 \leq i \leq n,\ x_i,\ x_i',\ x_i''$ arbitrary elements of $X_i$, $\lambda$ an arbitrary element of $K$); since both sides of the relations (1) are continuous functions of $u$ on $\mathscr{F}_s(X; Y)$ (§ 1, no. 2, Remark 6), the result follows (Chapter I, § 8, no. 1, Proposition 2).