Algebraic dual of a direct sum ($({⨁}_i{V}_i{)}^{\ast }\cong {\prod }_i{V}_i^{\ast }$)

Restricting a linear functional to each summand gives a family of functionals, with no finite-support restriction. Conversely an arbitrary such family acts on a vector by the finite sum of its nonzero coordinates. These constructions are inverse and prove the displayed algebraic dual space identity. A group action on the summands induces the product action by their algebraic contragredients.