Proposition 1

Let E and F be two 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 the following condition be satisfied:

(MS) For every semi-norm $p \in S(E)$, which is null on the kernel of u, there exists q in S(F) such that $p \leq q \circ u$.

Let N be the kernel and M the image of u; we introduce the canonical decomposition of u, let

The continuous semi-norms on E which are null on N, are the semi-norms $p_1 \circ \pi$ where $p_1$ ranges over S(E/N); similarly S(M) consists of the semi-norms $q_1$ for which there exists $q \in S(F)$ with $q_1 \leq q/F$. Finally, u is a strict morphism if and only if the bijective continuous linear mapping $\tilde{u}$ has a continuous inverse; this also means that every semi-norm in S(E/N) is of the form $q_1 \circ \tilde{u}$ with $q_1$ in S(M). Prop. 1 follows immediately from these remarks.