Corollary

The hypotheses and notations being those of Definition 7, we suppose in addition that E is of finite dimension and that $\Phi$ is non degenerate. Then we have $\dim E = \dim F$ and, for every basis $(e_i)$ ($1 \leq i \leq \dim E$) of E, there exists a basis $(f_i)$ of F such that $\Phi(e_i, f_k) = \delta_{ik}$ ($i, k = 1, ..., \dim E$).

Indeed, since $\Phi$ is non degenerate, we have $E^0 = F^0 = \{0\}$, whence $\dim E = \dim F$ (prop. 4). It follows (prop. 6) that $d_\Phi$ is an isomorphism of F (resp. $F^j$) onto $E^*$; therefore, if $(e_i^*)$ is the dual basis of $(e_i)$, the elements $f_i = d_\Phi^{-1}(e_i^*)$ form a basis of F which, in view of formula (23) (resp. formula (24)), satisfies $\Phi(e_i, f_k) = \delta_{ik}$.

It is immediate that, in this corollary, one can exchange the roles of E and of F, replacing $d_\Phi$ by $s_\Phi$ in the proof.