Remark 1

The formulas (7) and (8) show that, if $x \in E$, in order that $\Phi(x, x') = 0$ for all $x' \in E'$, it is necessary and sufficient that $\Phi_1(x, x') = 0$ (resp. $\Phi_2(x, x') = 0$) for all $x' \in E'$. 2) The adjoint of an endomorphism $u$ of $E$ with respect to $\Phi$ ($§ 1$, n° 8) is the same as the adjoint of $u$ (considered as an endomorphism of $E_0$) with respect to $\Phi_1$ (resp. $\Phi_2$).