Definition 6

Two elements $x \in E$ and $y \in F$ are said to be orthogonal with respect to $\Phi$ if $\Phi(x, y) = 0$. Two subsets $E' \subset E$ and $F' \subset F$ are said to be orthogonal if, whatever $x \in E'$ and $y \in F'$, $x$ and $y$ are orthogonal. The set of elements of $E$ (resp. F) orthogonal to a given submodule N of F (resp. M of E) is a submodule of E (resp. F), called the totally orthogonal submodule (or simply orthogonal submodule) to N (resp. M), and denoted by $N^0$ (resp. $M^0$).

Let H and $H'$ be two submodules of E or F. We have $H \subset (H^0)^0$ (denoted by $H^{00}$); if $H \subset H'$, we have ${H'}^0 \supset H^0$. It follows that we have $H^0 \supset (H^{00})^0$ and $H^0 \subset (H^0)^{00}$; on putting

we therefore have $H^0 = H^{000}$.

For the mapping $\Phi$ to be degenerate (No. 1, Def. 3), it is necessary and sufficient that at least one of the two submodules $E^0, F^0$ be $\neq \{0\}$. It is clear that $\Phi(x, y)$ does not change when an element of $F^0$ (resp. $E^0$) is added to $x$ (resp. $y$), and $\Phi$ therefore defines by passing to the quotient a bilinear mapping (or sesquilinear mapping) on $(E/F^0) \times (F/E^0)$; this is plainly non-degenerate; it is called the non-degenerate bilinear mapping (or sesquilinear mapping) associated with $\Phi$.

Let $(E_i)_{i \in I}$ be a family of left A-modules, $(F_i)_{i \in I}$ a family of right B-modules (resp. left B-modules), and $\Phi_i$ a bilinear (resp. right J-sesquilinear) mapping of $E_i \times F_i$ into G. Let $E$ (resp. $F$) denote the direct sum module of the $E_i$ (resp. $F_i$). It is immediately seen that the mapping $\Phi$ of $E \times F$ into G defined by

(the sum having a meaning since all but a finite number of its terms are zero) is bilinear (resp. right J-sesquilinear). It is called the direct sum of the mappings $\Phi_i$. It is clear that $E_i$ is orthogonal to $F_j$ with respect to $\Phi$ for $i \neq j$. Conversely, let $\Phi$ be a bilinear or sesquilinear mapping of $E \times F$ into G, and suppose that E is the direct sum of submodules $(E_i)_{i \in I}$ and F the direct sum of submodules $(F_i)_{i \in I}$ such that $E_i$ is orthogonal to $F_j$ for $i \neq j$; then $\Phi$ is the direct sum of its restrictions to the products $E_i \times F_i$ ($i \in I$).

For $\Phi$ to be non-degenerate, it is necessary and sufficient that each of the $\Phi_i$ be so; under these conditions, the submodule orthogonal to $E_i$ is $\sum_{j \neq i} F_j$.