Proposition 1

Let E be a separated locally convex space, F a locally convex space and $u$ a continuous linear mapping of E into F. The following conditions are equivalent:

(i) The mapping $u$ is a strict morphism, its kernel is of finite dimension and its image is closed in F ;

(ii) There exists a closed neighbourhood V of 0 in E such that the restriction of $u$ to V is proper (TG, I, p. 72).

(i) $=\Rightarrow$ (ii) : Suppose that $u$ satisfies condition (i). Since the kernel of $u$ is of finite dimension, it has a topological supplement $E_1$ in E (III, p. 55, prop. 1), and there exists a compact neighbourhood C of 0 in Ker($u$). Identify E with $E_1\times$ Ker($u$); the set $V = E_1\times C$ is then a closed neighbourhood of 0 in E. The restriction of $u$ to V is composed of the projection of $E_1\times C$ onto $E_1$ which is proper (TG, I, p. 77, cor. 5) and of the restriction $u_1$ of $u$ to $E_1$. Now $u_1$ is a homeomorphism of $E_1$ onto a closed subspace of F, hence is proper (TG, I, p. 72, prop. 2). The composite of two proper mappings is proper (TG, I, p. 73, prop. 5, a)), hence the restriction of $u$ to V is proper.

(ii) $=\Rightarrow$ (i) : Let V be a closed neighbourhood of 0 in E such that the restriction $v$ of $u$ to V is proper. The set $V\cap$ Ker($u$) $=\overset{-1}{v}(\{0\})$ is then compact (TG, I, p. 75, th. 1); consequently, the vector space Ker($u$) is locally compact, hence of finite dimension (EVT, I, p. 15, th. 3). Let $E_1$ be a topological supplement of Ker($u$) in E (prop. 1 of III, p. 55); put $V_1= E_1\cap V$. The set $V_1$ is closed in V. The mapping $u|V_1$ is proper (TG, I, p. 74, cor. 1) and injective, hence is a homeomorphism of $V_1$ onto a closed subset of F (TG, I, p. 72, prop. 2). By Lemma 1, the restriction of $u$ to $E_1$ is a homeomorphism of $E_1$ onto a closed subspace of F, hence $u$ is a strict morphism with closed image.