Proposition 4.2.3.label Let $\curl{E_j}_{1}^{n}$ and $F$ be locally convex spaces, and $T:\prods{n}{j=1}E_{j}\to F$ be $n$-linear map, then the following are equivalent:

1. (1)

   $T$ is continuous

2. (2)

   For every continuous seminorm $\braks{\cd }_{F }$ on $F$, there exists continuous seminorms $\curl{\braks{\cd }_{j }}_{1}^{n}$ on $\curl{E_j }_{1}^{n}$ such that for every $x\in\prods{n}{j=1}E_{j}$

   \begin{align*}\braks{Tx }_{F }\leq \prods{n}{j=1}\braks{x_j }_{E_j}\end{align*}