Proposition 2

Let E and F be two Hausdorff locally convex spaces and u a continuous linear mapping from E into F. In order that u be a strict morphism, it is necessary and sufficient that its transpose $^t u : F' \to E'$ satisfy the following conditions:

a) The image of $^t u$ is closed in $E'$ for $\sigma(E', E)$.

b) Every equicontinuous subset of $E'$, contained in the image of $^t u$ is the image under $^t u$ of an equicontinuous subset of $F'$.

If this is so, we have $\operatorname{Ker} ^t u = (\operatorname{Im} u)^{\circ}$ and $\operatorname{Im} ^t u = (\operatorname{Ker} u)^{\circ}$ and there exist canonical isomorphisms from Coker $^t u$ onto the dual of Ker u and from Ker $^t u$ onto the dual of Coker u.

Let N be the kernel and I the image of u. By cor. 2 of II, p. 47, the kernel of $^t u$ is the orthogonal of I, and the closure of the image of $^t u$ for $\sigma(E', E)$ is the orthogonal $N^{\circ}$ of N. The conjunction of a) and b) is then equivalent to the following condition:

b') Every equicontinuous subset of $E'$ contained in $N^{\circ}$ is the image under $^t u$ of an equicontinuous subset of $F'$.

Since $N^{\circ}$ can be identified with the dual of E/N, prop. 9, (i) of IV, p. 8, shows that the equicontinuous subsets of $E'$ contained in $N^{\circ}$ are the sets which are contained in a set of the form $H_p$, where p is a continuous semi-norm on E, vanishing on N. The condition b') then says that, for every semi-norm $p \in S(E)$ which is null on N, there exists $q \in S(F)$ such that $H_p \subset ^t u(H_q)$. By Hahn-Banach theorem (II, p. 23, cor. 1 and 2, p. 63, th. 1 and cor. 1), we have $^t u(H_q) = H_{q \circ u}$, and the relations $H_p \subset H_{p'}$ and $p \leq p'$ are equivalent for all semi-norms $p$ and $p'$ in S(E). Consequently, the relation $H_p \subset ^t u(H_q)$ is equivalent to the relation $p \leq q \circ u$. By prop. 1, property b') implies that u is a strict morphism.

Suppose that u is a strict morphism. We have already seen that the kernel of $^t u$ is the orthogonal of I and the image of $^t u$ is the orthogonal of N. The cokernel of $u$ is the space $F/I$ and its dual can be identified with $I^\circ = \mathrm{Ker}\ ^t u$. Similarly, the dual of $N = \mathrm{Ker}\ u$ can be identified with $E'/N^\circ$ (IV, p. 8), i.e. with the cokernel of $^t u$ since $N^\circ$ is the image of $^t u$.