Corollary 1

Let E and F be normed spaces, and let $u,v$ be non-zero continuous linear mappings of E into F whose kernels have the same finite dimension. One then has

Let $n$ be the common dimension of the kernels of $u$ and $v$. Let A be the set of continuous linear mappings of E into F whose kernel is of dimension $\geqslant n+ 1$. Since $u\not = 0$, one has dim(E) $> n$ and the set A contains 0. By proposition 9, $((u))$ and $((v))$ are respectively the distances from $u$ and from $v$ to the set A in $\mathscr{L}(E; F)$. It then suffices to apply the formula $|d(u,A)-d(v,A)|\leqslant \|u-v\|($cf. TG, IX, p. 13).