Proposition 6

— Let $(E_i)_{i \in I}$ be a family of vector spaces, and for each $i \in I$, let $A_i$ be a non-empty subset of $E_i$. Then the set $A = \prod_{i \in I} A_i$ is convex in $E = \prod_{i \in I} E_i$, if, and only if, for all $i \in I$, the set $A_i$ is convex in $E_i$.

In fact, each projection $\mathrm{pr}_i$ is a linear mapping and we have $A_i = \mathrm{pr}_i A$ and $A = \bigcap_{i \in I} \mathrm{pr}_i^{-1}(A_i)$; the proposition follows from props. 2 and 5 above.