Corollary

*Let M, N be two supplementary submodules in an A-module E and $p : E \to M, q : E \to N$ the corresponding projectors; then $^tp + ^tq : M^* \oplus N^* \to E^*$ is an isomorphism and $^tp$ (resp. $^tq$) is an isomorphism of $M^*$ (resp. $N^*$) onto the submodule of $E^*$ orthogonal to $N$ (resp. $M$). Moreover, if $i : M \to E$ and $j : N \to E$ are the canonical injections, $^tp \circ ^ti$ and $^tq \circ ^tj$ are the projectors $E^* \to ^tp(M^*)$, $E^* \to ^tq(N^*)$ corresponding to the decomposition of $E^*$ as the direct sum of $^tp(M^*)$ and $^tq(N^*)$.

$p \circ i = 1_M, \quad q \circ j = 1_N, \quad p \circ j = q \circ i = 0, \quad i \circ p + j \circ q = 1_E$, whence, by transposition (no. 5, formulae (16), (17) and (18)), $^ti \circ ^tp = 1_{M^*}, \ ^tj \circ ^tq = 1_{N^*}, \ ^tj \circ ^tp = ^ti \circ ^tq = 0. \ ^tp \circ ^ti + ^tq \circ ^tj = 1_{E^*}$ and the proposition follows from § 1, no. 6, Corollary 2 to Proposition 6.

Under the hypotheses of the Corollary, $M^*$ (resp. $N^*$) is often identified with the orthogonal $^tp(M^*)$ (resp. $^tq(N^*)$) of $N$ (resp. $M$) in $E^*$, thus identifying every linear form $u$ on $M$ (resp. $N$) with the linear form on $E$ extending $u$ and which is zero on $N$ (resp. $M$).

When an A-module $E$ admits a basis $(e_t)_{t \in T}$, it has been seen that giving this basis defines canonically an isomorphism $u : A_s^{(T)} \to E$. By virtue of Proposition 10 and no. 3, Proposition 5, the dual of $A_s^{(T)}$ is canonically identified with the product $A_d^T$; consider the contragredient isomorphism $\tilde{u} : A_d^T \to E^*$. If, for all $t \in T, f_t$ is the element of $A_d^T$ all of whose projections are zero with the exception of that of index $t$, which is equal to 1, and if we write $e_t^* = \tilde{u}(f_t)$, the elements $e_t^*$ of $E^*$ are, by (19) and (22), characterized by the relations

(23)

It amounts to the same to say that, for all $x = \sum_{t \in T} \xi_t e_t \in E, \ e_t^*(x) = \xi_t$; also $e_t^*$ is called the coordinate form of index $t$ on $E$. It follows from (23) that $(e_t^*)$ is a free system in $E^*$.

In particular, if $T$ is finite, the $e_t^*$ form a basis of $E^*$, the $f_t$ then forming the canonical basis of $A_d^T$. Hence: