Proposition 3

Let E and F be two locally convex spaces, and u a continuous linear mapping from E into F. We assume that E is Hausdorff and that F is metrizable. For u to be a strict morphism, it is necessary and sufficient that the image of $^t u$ be closed in $E'$ for the weak topology $\sigma(E', E)$.

The necessity follows from prop. 2.

Conversely, suppose that the image of $^t u$ is closed for $\sigma(E', E)$ and introduce the canonical decomposition of $u$ as in the proof of prop. 1. By the above remarks, the inverse mapping $\tilde{u}^{-1}$ of $\tilde{u}$ is continuous for the weakened topologies. But the subspace $M = u(E)$ of F is metrizable, hence bornological (III, p. 12, prop. 2); consequently (IV, p. 7, prop. 7, (ii)), $\tilde{u}^{-1}$ is continuous, hence $u$ is a strict morphism.