Proposition 4

Let $E_i$ ($1 \leq i \leq n$) and $F$ be topological vector spaces over $K$; we suppose that, for every $i$, the topology of $E_i$ is defined by a directed set of semi-norms $\Gamma_i$, and that the topology of $F$ is defined by a set of semi-norms $\Gamma$. Then a set $H$, of multilinear mappings of $\prod_{i=1}^n E_i$ in F is equicontinuous if, and only if, for each semi-norm $q \in \Gamma$, and each index i, there exists a semi-norm $p_i \in \Gamma_i$, and a number $a > 0$, such that for each function $u \in \mathbf{H}$ and point $(x_i) \in \prod_{i=1}^n E_i$,

The condition is sufficient since it implies that H is equicontinuous at (0, 0, ..., 0) and therefore everywhere (I, p. 9, prop. 6).

We show that the condition is necessary. By hypothesis, for every semi-norm $q \in \Gamma$ and every number $\beta > 0$, we have $q(u(x_1, x_2, ..., x_n)) \leq \beta$ for every function $u \in \mathbf{H}$ provided that $p_i(x_i) \leq \alpha_i$ are true for each index $i$, $1 \leq i \leq n$, and certain appropriately chosen numbers $\alpha_i > 0$ and semi-norms $p_i \in \Gamma_i$. As K is non-discrete, we can also suppose that, for every $i$, we have $\alpha_i = |\lambda_i| < 1$ where $\lambda_i \in \mathbf{K}$. Then let $(x_1, x_2, ..., x_n)$ be any point of $\prod_{i=1}^n E_i$, and for each index $i$, let $m_i \in \mathbf{Z}$ be an integer such that $p_i(x_i) \leq |\lambda_i|^{m_i + 1}$; this can be written as $p_i(\lambda_i^{-m_i} x_i) \leq |\lambda_i|$ ($1 \leq i \leq n$), therefore, by hypothesis, we have

Suppose firstly that one of the $p_i(x_i)$ is zero, then we can take $m_i \in \mathbf{N}$ arbitrarily large, therefore $q(u(x_1, x_2, ..., x_n)) = 0$. If, on the contrary, all the $p_i(x_i)$ are $\neq 0$, take the integer $m_i$ such that $|\lambda_i|^{m_i + 2} < p_i(x_i) \leq |\lambda_i|^{m_i + 1}$ for each $i$; then we have $|\lambda_i|^{m_i} < |\lambda_i|^{-2} p_i(x_i)$, from which, by (6), the relation (5) follows with

Q.E.D.