Remark 6

Let $(G_i)_{i\in I}$ be a family of separated topological vector spaces, and for every $i\in I$, let $u_i$ be a compact linear mapping from a topological vector space E into $G_i$. If the set I is finite, or if E is a seminormed space, the linear mapping $x\mapsto (u_i(x))_{i\in I}$ from E into $\prod_{i\in I}G_i$ is compact (remark 2 and TG, I, p. 63, th. 3).