Proposition 9.2.2. Let $\seqf{E_j}$ and $F$ be locally convex spaces, and $T: \prod_{j = 1}^{n} E_{j} \to F$ be $n$-linear map, then the following are equivalent:
$T$ is continuous.
For every continuous seminorm $[\cdot]_{F}$ on $F$, there exists continuous seminorms $\seqf{[\cdot]_j}$ on $\seqf{E_j}$, such that for every $x \in \prod_{j = 1}^{n} E_{j}$,
\[[Tx]_{F} \le \prod_{j = 1}^{n} [x_{j}]_{E_j}\]
Proof. $(1) \Rightarrow (2)$: By continuity of $T$, there exists continuous seminorms $\seqf{[\cdot]_j}$ on $\seqf{E_j}$ such that for any $x \in \prod_{j = 1}^{n} E_{j}$, $\max_{1 \le j \le n}[x_{j}]_{E_j}< 1$ implies that $[Tx]_{F} < 1$. In which case, the inequality follows from linearity.$\square$