Definition 1

A mapping $\Phi$ of the product $E \times F$ into $G$ is said to be bilinear if it satisfies the following conditions:

(1) $\Phi(x + x', y) = \Phi(x, y) + \Phi(x', y)$ for all $x \in E$, $x' \in E$, $y \in F$ ; (2) $\Phi(x, y + y') = \Phi(x, y) + \Phi(x, y')$ for all $x \in E$, $y \in F$, $y' \in F$ ; (3) $\Phi(ax, y) = a\Phi(x, y)$ for all $a \in A$, $x \in E$, $y \in F$ ; (4) $\Phi(x, yb) = \Phi(x, y)b$ for all $x \in E$, $y \in F$, $b \in B$.

The tensor product $E \otimes_{\mathbf{Z}} F$ is canonically endowed with a structure of $(A, B)$-bimodule characterized by $a(x \otimes y)b = ax \otimes yb$ (Chap. III, 2e éd., App. II, n° 3), and the giving of a bilinear mapping $\Phi$ of $E \times F$ into $G$ is equivalent to that of a mapping $\Psi$ of $E \otimes_{\mathbf{Z}} F$ into $G$ which is a homomorphism for the structures of $(A, B)$-bimodules and which satisfies $\Psi(x \otimes y) = \Phi(x, y)$ for all $x \in E$ and $y \in F$.

The conditions imposed on $\Phi$ by Definition 1 signify that the partial mappings $d_\Phi(y) : x \to \Phi(x, y)$ and $s_\Phi(x) : y \to \Phi(x, y)$ are respectively an A-linear mapping from $E$ into $G$ and a B-linear mapping from $F$ into $G$. Let us endow the commutative group $\mathscr{L}_A(E, G)$ (resp. $\mathscr{L}_B(F, G)$) with the structure of right B-module (resp. left A-module) defined by $ub(x) = u(x) . b$ ($u \in \mathscr{L}_A(E, G)$, $x \in E$, $b \in B$) (resp. $a \nu(y) = a . \nu(y)$ ($a \in A$, $\nu \in \mathscr{L}_B(F, G)$, $y \in F$)). Then conditions (1) to (4) are respectively equivalent to:

for all $x, x'$ in $E$, $y, y'$ in $F$, $a \in A$, $b \in B$; in other words, the mapping $d_\Phi$ from $F$ into $\mathscr{L}_A(E, G)$ is B-linear, and the mapping $s_\Phi$ from $E$ into $\mathscr{L}_B(F, G)$ is A-linear. We have, by definition