Remark 2

Let $F$ be a locally convex space, $k$ an integer $\geq 0$ and $f : E^k \to F$ a multilinear mapping. For $f$ to be continuous, it is necessary and sufficient that the restriction of $f$ to $E_n^k$ is continuous for every $n$. We verify immediately that $E^k$ has the final locally convex topology for the family of linear mappings $v_n \times \cdots \times v_n : E_n \times \cdots \times E_n \to E \times \cdots \times E$ (II, p. 28, cor. 2 and p. 30, prop. 7) and that $u_n \times \cdots \times u_n$ is a compact injective linear mapping from $(E_n)^k$ into $(E_{n+1})^k$. It is now enough to apply lemma 1.