Proposition 8

— Let $(F_i, G_i)_{i \in I}$ be a family of pairs of spaces in duality. Let $F = \prod_{i \in I} F_i$ be the product space of the $F_i$ and $G = \bigoplus_{i \in I} G_i$ be the direct sum of the $G_i$. If, for all $x = (x_i) \in F$ and all $y = (y_i) \in G$, we write $\langle x, y \rangle = \sum_{i \in I} \langle x_i, y_i \rangle$ (a sum which has only finitely many non-zero terms) then the topology $\sigma(F, G)$ (relative to the bilinear form $(x, y) \mapsto \langle x, y \rangle$) is the product of the topologies $\sigma(F_i, G_i)$.

For, given a topology $\mathscr{T}$ on $F$; in order that, for all $y \in G$, the linear form $x \mapsto \langle x, y \rangle$ should be continuous for $\mathscr{T}$, it is necessary and sufficient, by the definition of $\langle x, y \rangle$, that each of the mappings $x \mapsto \langle \operatorname{pr}_i x, y_i \rangle$ should be continuous for $\mathscr{T}$, where $i$ is arbitrary in $I$ and $y_i$ in $G_i$; but this means that each of the mappings $\operatorname{pr}_i$ of $F$ in $F_i$ is continuous for $\mathscr{T}$ and for $\sigma(F_i, G_i)$ (I, p. 10, cor. 1); this completes the demonstration.