Proposition 1

— Let E and F be two hilbertian spaces. For every mapping $u \in \mathscr{L}(E; F)$, there exists a unique mapping $u^* \in \mathscr{L}(F; E)$ such that

for all $x \in E$ and all $y \in F$. The mapping $u \mapsto u^*$ from $\mathscr{L}(E, F)$ into $\mathscr{L}(F; E)$ is bijective, isometric and semi-linear (with respect to the automorphism $\xi \mapsto \overline{\xi}$ of K).

Let $\mathscr{S}(E, F)$ be the space of all continuous sesquilinear forms on $E \times F$, endowed with the norm

We define the space $\mathscr{S}(F, E)$ similarly. We defined (V, p. 16, cor. 2) a Banach space isomorphism from $\mathscr{L}(E; F)$ onto $\mathscr{S}(F, E)$, denoted by $u \mapsto \Phi_u$ and characterized by

In an analogous way we define an isomorphism from $\mathscr{L}(F, E)$ onto $\mathscr{S}(E, F)$. Finally we define a mapping $\Phi \mapsto \Phi^*$ from $\mathscr{S}(F, E)$ onto $\mathscr{S}(E, F)$ by

This mapping is bijective, semi-linear and isometric. But formula (1) translates as $\Phi_{u^*} = (\Phi_u)^*$, hence the proposition.