Corollary 2

Let E and F be Banach spaces, and let $u$ and $v$ be non-zero continuous linear mappings of E into F whose images have the same finite codimension in F. Then

The morphism $u$ is strict (III, p. 52, lemma 6). The kernel of the continuous linear mapping $^tu$ is the orthogonal (Im($u$))$^{\circ}$ of Im($u$) (EVT, IV, p. 27, prop. 2), hence dim(Ker($^tu$)) $=$ codim$_F$(Im($u$)), and analogously dim(Ker($^tv$)) $=$ codim$_F$(Im($v$)). The kernels of $^tu$ and $^tv$ therefore have the same finite dimension. Since

(EVT, IV, p. 7, prop. 8 and prop. 8 of III, p. 63), the assertion follows from corollary 1, applied to $^tu$ and $^tv$.