Proposition 13

The Mackey topology $\tau(F, E)$ is the direct sum of the Mackey topologies $\tau(F_i, E_i)$. The topology $\tau(E, F)$ is the product of the topologies $\tau(E_i, F_i)$.

The assertion on $\tau(F, E)$ follows from lemma 3 (ii) and the following property : for a closed, convex and balanced subset of $F^* = \prod_{i \in I} F_i^*$ to be compact for $\sigma(F^*, F)$, it is necessary and sufficient that its projection on each $F_i^*$ is compact for $\sigma(F_i^*, F_i)$.

To prove the assertion on $\tau(E, F)$, assume first that the topologies $\sigma(F_i, E_i)$ are Hausdorff, it is enough (lemma 3 (i)) to prove that every subset $A$ of $F$ which is convex, balanced and compact for $\sigma(F, E)$ is contained in a set of the form $\sum_{i \in J} A_i$ where $J \subset I$ is finite and where $A_i$ is convex, balanced and compact for $\sigma(F_i, E_i)$. But such a subset is bounded for $\sigma(F, E)$. By the proof of prop. 12, there exists a finite subset $J$ of $I$ such that $A \subset \sum_{i \in J} F_i$, and it is enough to take for $A_i$ the projection of $A$ on $F_i$.

In the general case, with the same notations as in the proof of prop. 12, we have $\tau(E_i, F_i) = \tau(E_i, F_i/N_i)$ and $\tau(E, F) = \tau(E, F/N)$ (IV, p. 2) and since $F/N$ is the topological direct sum of the $F_i/N_i$, we have reduced to the preceding case.

Q.E.D.

For the remainder of this paragraph, we assume that $(E_i)_{i \in I}$ is a family of locally convex spaces. Let $S$ denote the topological direct sum of the $E_i$ and $P$, their product. We define a linear mapping $\theta : S' \to \prod_{i \in I} E_i'$, said to be canonical, by

(where $S'$ denotes the dual of $S$, and $E_i'$ that of $E_i$).