Proposition 15

— (i) The map $\phi$ is an isomorphism from the topological direct sum of the strong duals of the $E_i$ onto the strong dual of $P = \prod_{i \in I} E_i$.

(ii) For a subset $A$ of $P'$ to be equicontinuous, it is necessary and sufficient that it is contained in a finite sum $\sum_{i \in J} \phi(A_i)$, where $J \subset I$ is finite and where $A_i$ is equicontinuous in $E'_i$ for all $i \in J$.

(iii) The Mackey topology $\tau(P, P')$ is the product of the topologies $\tau(E_i, E'_i)$.

(iv) The topology $\beta(P, P')$ is the product of the topologies $\beta(E_i, E'_i)$.

It is immediate that $\phi$ is injective. A fundamental system of neighbourhoods of 0 in $P$ consists of sets of the form $V = \prod_{i \in J} V_i \times \prod_{i \in I - J} E_i$, where $J \subset I$ is finite and $V_i$ is a neighbourhood of 0 in $E_i$ for $i$ in $J$. The polar of $V$ in $P'$ is equal to $\sum_{i \in J} \phi(V_i^0)$.

This proves the surjectivity of $\phi$ and also assertion (ii).

Assertions (i) and (iv) follow from prop. 12 (IV, p. 12) and (iii) from prop. 13 (IV, p. 12).