Exercise 18, § 4

a) Let E, F be two normed spaces, $u$ a strict injective morphism from E into F ; then there is a number $m > 0$ such that $\|u(x)\| \geq m \|x\|$ for all $x \in E$. Show that if $w$ is a continuous linear mapping from E into F such that $\|w\| < m$, then $v = u + w$ is a strict injective morphism from E into F. Moreover, for all $y_0 \in u(E)$ such that $\|y_0\| = 1$, there exists $y \in v(E)$ such that $\|y - y_0\| \leq \|w\|/m$; for every $y \in v(E)$ with $\|y\| = 1$, there exists $y_0 \in u(E)$ such that $\|y - y_0\| \leq \|w\|/(m - \|w\|)$. b) Deduce from a) that, if in addition, E and F are Banach spaces, and if there exists a closed vector subspace G of F which is the complement of the closed subspace $u(E)$, then G is also the complement of $v(E)$ whenever $\|w\|$ is small enough (argue as in exerc. 17, c)).