Theorem 1

*Let E and F be two Fréchet spaces and u a continuous linear mapping from E into F. The following conditions are equivalent :

a) $u$ is a strict morphism. b) $u$ is a strict morphism for the weakened topologies. c) The image of $u$ is closed in F. d) $^t u$ is a strict morphism from $F'$ into $E'$ for the weak topologies. e) The image of $^t u$ is closed in $E'$ for the weak topology $\sigma(E', E)$. f) The image of $^t u$ is closed in $E'$ for the strong topology $\beta(E', E)$. g) $^t u$ is a strict morphism from $F'_c$ into $E'_c$ (the duals endowed with the topology of compact convergence).

The equivalence of a), b) and e) follows from prop. 3 of IV, p. 28 and the remark preceding it. That of a) and c) is precisely cor. 3 of I, p. 19. The remark of IV, p. 28, also shows that d) is equivalent to the fact that, the image of $u$ is closed for the weakened topology $\sigma(F, F')$ of F; the equivalence of c) and d) then follows from prop. 2 of IV, p. 4.

We now prove the equivalence of e) and f). It is enough to prove that f) implies e). Suppose that the image of $^t u$ is closed for $\beta(E', E)$ in $E'$. On account of the Banach-Dieudonné theorem (IV, p. 25, cor. 2), it is enough to prove that for every convex balanced neighbourhood U of 0 in E, the intersection $B = ^t u(F') \cap U$ is compact for $\sigma(E', E)$. The strong dual $E'_b$ of the Fréchet space E is complete (IV, p. 22, prop. 2), hence the closed subset B of $E'_b$ is complete, and so the normed space $E'_B$ is complete (III, p. 8, corollary). Let $(V_n)$ be a decreasing sequence forming a fundamental system of neighbourhoods of 0 in F. Then $F'$ is the union of sets $C = V_n^\circ$ which are compact for $\sigma(F', F)$, hence $E'_B = \bigcup_n B_n$, with $B_n = E'_B \cap ^t u(C_n)$. Since $E'_B$ is a Baire space, and each of the sets $B_n$ is convex balanced and closed, there exists a real number $r > 0$ and an integer $n$ such that $B \subset r.B_n$. Then we have $B = U^\circ \cap {}^t u(r.C_n)$; since the sets $U^\circ$ and $r.C_n$ are compact and ${}^t u$ is continuous for the weak topologies, $B$ is compact for $\sigma(E', E)$. This completes the proof of the equivalence of e) and f).

Finally the equivalence of g) and the preceding conditions follows from prop. 18 of GT, X, § 2, No. 10 and the following lemma.