Corollary 2

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

The equivalence of (i) and (ii) follows from Banach’s th. (I, p. 17, th. 1).

In view of the equivalence of a) and c) in th. 1, condition (ii) says that $u$ is a strict morphism and that its image is dense in F for $\sigma(F, F')$. The equivalence of (ii) and (iii) then follows from the equivalence of a) and d) in th. 1 and from prop. 5 of IV, p. 6.

If $u : E \to F$ is a strict morphism of Fréchet spaces, the transpose ${}^t u$ is not necessarily a strict morphism from $F'_b$ into $E'_b$ (IV, p. 62, exerc. 3). However, we have the following partial result :