Proposition 14

(i) The mapping $\theta$ is an isomorphism from the strong (resp. weak) dual of $S = \bigoplus_{i \in I} E_i$ onto the product of the strong (resp. weak) duals of the $E_i$:

(ii) For a subset $A$ of $S'$ to be equicontinuous, it is necessary and sufficient that the projection of $\theta(A)$ onto $E_i'$ be equicontinuous for all $i \in I$.

(iii) The Mackey topology $\tau(S, S')$ is the direct sum of the Mackey topologies $\tau(E_i, E_i')$.

(iv) The topology $\beta(S, S')$ is the direct sum of the topologies $\beta(E_i, E_i')$.

That $\theta$ is bijective follows immediately from the definition of a topological direct sum (II, p. 30, prop. 6). Assertion (i) then follows from prop. 12 of IV, p. 12, for the strong topologies, and from prop. 8 of II, p. 50, for the weak topologies. Similarly (iii) follows from prop. 13 (IV, p. 12) and (iv) from prop. 12 (IV, p. 12).

To prove (ii), let $A$ be a subset of $S'$. Put

let $q_i$ denote the restriction of $q$ to $E_i$, whence

where $A_i$ denotes the projection of $\theta(A)$ on $E_i'$. For $A$ to be equicontinuous, it is necessary and sufficient that $q$ is finite (that is, that each $q_i$ is finite) and continuous. In view of the characterization of continuous semi-norms on a topological direct sum (II, p. 27, prop. 5), this is the same as assuming that each $q_i$ is continuous, or in fact, that each set $A_i$ is equicontinuous.

Let $\phi$ be the linear mapping, said to be canonical, from $\bigoplus_{i \in I} E_i'$ into the dual $P'$ of $P = \prod_{i \in I} E_i$, defined by the formula

for $x = (x_i)$ in $P$ and $x' = (x'_i)$ in $\bigoplus_{i \in I} E'_i$.