Definition 11

Given A-modules $E_i, F_i (i = 1, \ldots, m)$ and, for each $i$, a sesquilinear form $\Phi_i$ for $J$ on $E_i \times F_i$, the sesquilinear form $\Phi$ for $J$ on $(\bigotimes_i E_i) \times (\bigotimes_i F_i)$ characterized by (35) is called the tensor product of the sesquilinear forms $\Phi_i$.

In the case where the $E_i$ and the $F_i$ are equal to one and the same module $E$, and where the $\Phi_i$ are equal to one and the same form $\Psi$, we say that $\Phi$ is the extension of $\Psi$ to $\bigotimes^m E$.

With the notations of Def. 11, let us study the mappings associated with $\Phi$. From formula (24) (No. 6) and from (35) one obtains the relation

We have therefore:

(36) $s_\Phi = j_s \circ (s_{\Phi_1} \otimes \cdots \otimes s_{\Phi_m}), \quad d_\Phi = j_d \circ (d_{\Phi_1} \otimes \cdots \otimes d_{\Phi_m})$

where $j_s$ (resp. $j_d$) denotes the canonical mapping of $\bigotimes_i F_i^*$ into $(\bigotimes_i F_i)^*$ (resp. of $\bigotimes_i E_i^*$ into $(\bigotimes_i E_i)^*$) (Chap. III, § 1, Nos. 4 and 7).