Definition 2

Let E and F be locally convex spaces, and let $u$ be a continuous linear mapping of E into F. One says that $u$ is a direct morphism if $u$ is a strict morphism whose kernel is a direct vector subspace of E and whose image is a direct vector subspace of F.

Let $u\in \mathscr{L}(E; F)$. In order that the kernel and image of $u$ be direct vector subspaces of E and F respectively, it is necessary and sufficient that there exist decompositions $E = E_1\oplus E_2$ and $F = F_1\oplus F_2$ into topological direct sums such that $u$ be represented by a matrix $\begin{pmatrix} u_1 & 0 \\ 0 & 0 \end{pmatrix}$ where $u_1\in \mathscr{L}(E_1; F_1)$ is bijective. Since the kernel of $u$ is then $E_2$, since the canonical linear mapping of $E_1$ onto $E/E_2$ is an isomorphism, and since the image of $u$ is $F_1$, we see that $u$ is a strict morphism if and only if $u_1$ is an isomorphism of $E_1$ onto $F_1$. Let us denote by $v$ the element of $\mathscr{L}(F; E)$ represented by the matrix $\begin{pmatrix} u_1^{-1} & 0 \\ 0 & 0 \end{pmatrix}$ . Then the linear mapping $u\circ v$ is the projector in F with kernel $F_2$ and image $F_1$, the linear mapping $v\circ u$ is the projector in E with kernel $E_2$ and image $E_1$, and one has $u\circ v\circ u=u$.

Conversely, one has the following result: