Proposition 11

— (i) The weakened topology on $E/M$ is the quotient of that on $E$; the weakened topology on $M$ is induced by that of $E$.

(ii) The Mackey topology on E/M is the quotient of that on E; the Mackey topology on M is finer than the topology induced by τ(E, E').

Assertion (i) follows from prop. 7 of II, p. 48.

The canonical injection $i : M \to E$ is continuous for the weakened topologies, hence for the Mackey topologies $\tau(M, M')$ and $\tau(E, E')$ (IV, p. 7, prop. 7). Similarly, the canonical projection $p : E \to E/M$ is continuous for the Mackey topologies. We see immediately that the quotient topology on $E/M$ obtained from $\tau(E, E')$ is compatible with the duality between $E/M$ and $(E/M)'$, hence is coarser than the Mackey topology on $E/M$, by Mackey's theorem (IV, p. 2, th. 1). This proves (ii).