Proposition 5

Under the hypotheses of Proposition 4, the set $\mathscr{L}(X_1, \ldots, X_n; Y)$ of continuous multilinear mappings of $X$ into $Y$ is closed in $\mathscr{F}(X; Y)$ with respect to the topology of bounded convergence; it is complete with respect to the uniformity of bounded convergence if $Y$ is complete.

For if $\mathscr{G}$ is the set of all bounded subsets of $X$, $\mathscr{L}(X_1, \ldots, X_n; Y)$ is the intersection of the set of all multilinear mappings of $X$ into $Y$ and the set $\mathscr{B}_\mathscr{G}(X; Y)$ (Chapter IX, § 3, no. 5, Theorem 1); the result thus follows from Proposition 4 and Proposition 2, Corollary 1.

For the remainder of this subsection, $K$ denotes a non-discrete valued field.

Then $\mathscr{L}(X_1, \ldots, X_n; Y)$ is a vector subspace of $\mathscr{F}(X; Y)$. Let $B$ be the unit ball in $X$, the set of all $(x_i)_{1 \leq i \leq n}$ such that $\sup_{1 \leq i \leq n} \|x_i\| \leq 1$.

Then the mapping $u \to u|B$ of $\mathscr{L}(X_1, \ldots, X_n; Y)$ into $\mathscr{B}(B; Y)$ is injective; moreover, the inverse image, under this mapping, of the uniformity of uniform convergence on $\mathscr{B}(B; Y)$ is the uniformity of bounded convergence on $\mathscr{L}(X_1, \ldots, X_n; Y)$. For every bounded subset of $X$ is contained in a set of the form $\mu B$ (for some $\mu \in K^*$), and if $u$ is an element of $\mathscr{L}(X_1, \ldots, X_n; Y)$, to say that $\|u(z)\| \leq a$ for all $z \in \mu B$ is equivalent to saying that $\|u(z)\| \leq a/|\mu|^n$ for all $z \in B$. It is easily verified that the number

is a norm on $\mathscr{L}(X_1, \ldots, X_n; Y)$ and defines the uniformity of bounded convergence on this set, and clearly we have

Unless the contrary is expressly stated, whenever $\mathscr{L}(X_1, \ldots, X_n; Y)$ is considered as a normed space, it is the norm defined above which is in question.