= Algebraic dual of a direct sum
{title2=$(\bigoplus_iV_i)^*\cong\prod_iV_i^*$}
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.
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