Theorem 1

Let E and F be separated locally convex spaces and $u$ a strict morphism of E into F, whose kernel is of finite dimension and whose image is closed. Let $h$ be a compact linear mapping of E into F. The linear mapping $u+h$ is a strict morphism, its kernel is of finite dimension and its image is closed.

According to prop. 1 of III, p. 71, there exists a closed neighbourhood V of 0 in E such that the restriction of $u$ to V is proper. Since $h$ is compact, there exists a closed neighbourhood W of 0 contained in V such that the set $h(W)$ is relatively compact. Put $C =h(W)$. The restriction of $u$ to W is proper (TG, I, p. 74, cor. 1). The mapping $\alpha :x\mapsto (u(x), h(x))$ from W into $F\times C$ is proper because the mapping pr$_1\circ \alpha =u|W$ from W into F is proper (TG, I, p. 73, prop. 5, d)). The mapping $\beta : (y, z)\mapsto (y+z, z)$ from $F\times C$ into $F\times C$ is a homeomorphism, and the mapping pr$_1$ from $F\times C$ into F is proper (TG, I, p. 77, cor. 5). The composed mapping pr$_1\circ \beta \circ \alpha$ from W into F is therefore proper (TG, I, p. 73, prop. 5, a)). But this mapping is none other than the restriction of $u+h$ to W. According to the prop. 1 of III, p. 71$,u+h$ is a strict morphism, its kernel is of finite dimension and its image is closed.