Proposition 10

Let $(E_i)_{i \in I}$ be a family of A-modules and for all $i \in I$ let $j_i : E_i \to E = \bigoplus_{i \in I} E_i$ be the canonical injection. Then the product mapping $x^* \mapsto ({}^t j_i(x^*))$

is an isomorphism of the dual $E^*$ of $E$ onto the product $\prod_{i \in I} E_i^*$.

This is a particular case of § 1, no. 6, Corollary 1 to Proposition 6, applied to the case where $\prod_{\lambda} F_{\lambda} = A_s$.

If, by means of the canonical injections $j_i$, the $E_i$ are identified with submodules of their direct sum $E$ and if, by means of the product mapping $x^* \mapsto (tj_i(x^*))$, $E^*$ is identified with $\prod_{i \in I} E_i^*$, it can then be said that $\prod_{i \in I} E_i^*$ is the dual of $\bigoplus_{i \in I} E_i$, the canonical bilinear form being given by

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