Exercise 15, § 7

Let E, F be two vector spaces over a field K and $u : E \to F$ a linear mapping.

(a) The following conditions are equivalent: (1) $\operatorname{Ker}(u)$ is finite-dimensional; (2) there exists a linear mapping $v$ of F into E such that $v \circ u = l_E + w$, where $w$ is an endomorphism of E of finite rank; (3) for every vector space G over K and every linear mapping $f : G \to E$, the relation $\operatorname{rg}(u \circ f) < +\infty$ implies $\operatorname{rg}(f) < +\infty$.

(b) The following conditions are equivalent: (1) $\operatorname{Coker}(u)$ is finite-dimensional; (2) there exists a linear mapping $v$ of F into E such that $u \circ v = l_F + w$, where $w$ is an endomorphism of F of finite rank; (3) for every vector space G over K and every linear mapping $g : F \to G$, the relation $\operatorname{rg}(g \circ u) < +\infty$ implies $\operatorname{rg}(g) < +\infty$.