Lemma 3

Let $c$ be a real number such that $c >((u))$. There exists a continuous linear mapping $h: E\rightarrow F$, of rank 1 and norm $< c$ such that the kernel of $u+h$ contains that of $u$ and is of dimension $n+ 1$.

Let $p$ denote the canonical mapping of E onto $E/$ Ker($u$). There exists $a\in E$ such that $\|p(a)\|= 1$ and $\|u(a)\|< c$ (formula (6) of III, p. 61). By the Hahn–Banach theorem (EVT, II, p. 24, cor. 2), there exists a continuous linear form $f$ on the normed space $E/$ Ker($u$) such that $f(p(a)) = 1$ and $\|f\|= 1$. The linear mapping $h:x\mapsto -(f\circ p)(x)u(a)$ of E into F is continuous, of rank 1, and one has $\|h\|\leqslant \|f\| \|p\| \|u(a)\|< c$. The kernel of the mapping $u+h$ contains that of $u$ and $a$; since $a\not \in$ Ker($u$), its dimension is therefore at least $n+ 1$. On the other hand, the linear mapping $u$ induces, by passing to subspaces, a linear mapping of Ker($u+h$) into Im($h$) with kernel Ker($u$)$\cap$ Ker($u+h$), so that dim(Ker($u+h$))$\leqslant$ dim(Ker($u$)) $+ 1$. The conclusion follows.