Proposition 1

Let $\Phi_1$ (resp. $\Phi_2$) be a K-bilinear form on $E_0 \times E'_0$, invariant under $j$ and $j'$. The mapping which to $\Phi_1$ (resp. $\Phi_2$) associates the mapping $\Phi$ of $E \times E'$ into $A$ defined by

(resp. $\Phi(x, x') = - \Phi_2(x, j'(x')) + i \Phi_2(x, x')$)

$(x \in E, x' \in E)$, is an isomorphism of the K-vector space of K-bilinear forms on $E_0 \times E'_0$ invariant under $j$ and $j'$ onto the K-vector space of sesquilinear forms on $E \times E'$. Suppose moreover that $E = E'$; for $\Phi$ to be hermitian, it is necessary and sufficient that $\Phi_1$ be symmetric (resp. that $\Phi_2$ be antisymmetric) (cf. Exercise 4).

Indeed every mapping $\Phi$ of $E \times E'$ into $A$ can be written, in one and only one way, in the form $\Phi = \Phi_1 + i \Phi_2$, where $\Phi_1$ and $\Phi_2$ are mappings of $E \times E'$ into $K$. For the partial mapping $x \to \Phi(x, x')$ to be A-linear, it is necessary and sufficient, by the formula (3) (resp. (4)) of No. 2, that $\Phi_1$ (resp. $\Phi_2$) be K-linear in $x$ and that one have

(resp. $\Phi(x, x') = \Phi_2(j(x), x') + i \Phi_2(x, x')$).

Analogously, for $\overline{\Phi}(x, x')$ to be A-linear in $x'$, it is necessary and sufficient that $\Phi_1$ (resp. $\Phi_2$) be K-linear in $x'$ and that one have

(resp. $\Phi(x, x') = - \Phi_2(x, j'(x')) + i \Phi_2(x, x')$).

It follows immediately that, for $\Phi$ to be sesquilinear, it is necessary and sufficient that it be written in one or the other of the forms (9) and (11) (resp. (10) and (12)) with $\Phi_1$ (resp. $\Phi_2$) a K-bilinear form invariant under $j$ and $j'$.

It follows from this that, for a sesquilinear form $\Phi = \Phi_1 + i \Phi_2$ to be zero, it is necessary and sufficient that $\Phi_1$ (resp. $\Phi_2$) be zero. Now, if $E = E'$, one has $\Phi(y, x) = \Phi_1(y, x) + i \Phi_2(y, x)$ and $\overline{\Phi(x, y)} = \Phi_1(x, y) - i \Phi_2(x, y)$; for these two expressions to be equal, in other words for $\Phi$ to be hermitian, it is necessary and sufficient therefore that $\Phi_1$ be symmetric (resp. that $\Phi_2$ be antisymmetric).