Corollary 4

Let $E_i$ ($1 \leq i \leq n$) and $F$ be locally convex spaces, and $f$ be a continuous multilinear map from $\prod_{i=1}^n E_i$ into $F$. If $B_i$ is a bounded subset of $E_i$, for $1 \leq i \leq n$, then $f(\prod_{i=1}^n B_i)$ is bounded in $F$.