Proposition 6

The hypotheses and notations being those of def. 7, suppose in addition that E and F have the same finite dimension. Then the following conditions are equivalent: a) $d_\Phi$ is injective ; b) $d_\Phi$ is surjective ; c) $s_\Phi$ is injective ; d) $s_\Phi$ is surjective ; e) $\Phi$ is non-degenerate.

Indeed, since E, F, $E^*$ and $F^*$ have the same finite dimension,

a) and b) are equivalent, as are c) and d) (chap. II, § 3, no 4). Since $s_\Phi$ and $d_\Phi$ have the same rank (prop. 5), a) and c) are equivalent. Since e) is equivalent to the relation $E^0 = F^0 = \{0\}$, it is equivalent to the conjunction of a) and c), whence the equivalence of the stated conditions.