Corollary 1

Under the hypothesis of th. 1, the following conditions are equivalent : (i) u is a strict injective morphism ; (ii) ${}^t u$ is a strict surjective morphism for the weak topologies. (iii) ${}^t u$ is surjective.

The implication (i) $\Rightarrow$ (ii) follows immediately from the equivalence of conditions a), d) and e) of th. 1 and from IV, p. 6, prop. 5. It is clear that (ii) implies (iii). Finally, we prove that (iii) implies (i) : if ${}^t u$ is surjective $u$ is a strict morphism by the equivalence of a) and e) in th. 1 ; that $u$ is injective follows from prop. 5 of IV, p. 6.