Remark

— The duality between $F$ and $G$ is separating in $F$ (resp. in $G$) if and only if for all $i \in I$, the duality between $F_i$ and $G_i$ is separating in $F_i$ (resp. in $G_i$). If the duality between $F$ and $G$ is separating in $F$ (resp. $G$), then, in $F$ (resp. $G$), the subspace orthogonal to one $G_i$ (resp. $F_i$), canonically identified with a subspace of $G$ (resp. $F$) is the subspace of the product of the $F_\kappa$ where $\kappa \neq i$ (resp. the direct sum of the $G_\kappa$ such that $\kappa \neq i$).