Remark 7

Soient $A$ a field, $E$ a vector space over $A$, $\Phi$ a sesquilinear form on $E$, $u$ an endomorphism of $E$.

a) For there to exist one and only one endomorphism $u^*$ of $E$ such that $\Phi(u(x), y) = \Phi(x, u^*(y))$ for $x, y$ in $E$, it is necessary and sufficient that $d_\Phi$ be injective and that $u(d_\Phi(E)) \subset d_\Phi(E)$.

b) Give an example where $E$ is of infinite dimension and $d_\Phi$ is injective, but where $u(d_\Phi(E))$ is not contained in $d_\Phi(E)$.