Universal property of a tensor product
= Universal property of a tensor product
For <vector spaces> $V_1,\ldots,V_r$, every <multilinear map> $F:V_1\times\cdots\times V_r\to W$ factors uniquely through a <linear map> $\widetilde F:V_1\otimes\cdots\otimes V_r\to W$ sending a decomposable tensor to $F$ of its factors. Thus multilinear formulas define <linear maps> on <tensor products> without choosing a basis. <Tensor contraction> is an example, using the evaluation pairing between a vector and a <covector>.