Exercise 20, § 4

a) Let E, F be two Banach spaces, $u$ a strict morphism from E into F, such that $u^{-1}(0)$ is finite dimensional. Show that for every continuous linear mapping $w$ from E into F with small enough norm, $v = u + w$ is a strict morphism from E into F and $\dim v^{-1}(0) \leq \dim u^{-1}(0)$. (Write E as the topological direct sum of $u^{-1}(0)$ and a closed subspace and use exerc. 18 of IV, p. 66.) b) Let E, F be two Banach spaces, $u$ a continuous linear mapping from E into F such that $u(E)$ has finite codimension in F. Then $u(E)$ is closed and $u$ is a strict morphism (I, p. 28, exerc. 4). Show that for every continuous linear mapping $w$ from E into F, of small enough norm, $v = u + w$ is a strict morphism from E into F and

(consider $^tv = ^tu + ^tw$, and use a) and IV, p. 30, cor. 3).