Tensor product

In mathematics, the tensor product ${\displaystyle V\otimes W}$ of two vector spaces V and W (over the same field) is itself a vector space, together with an operation of bilinear composition, denoted by ${\displaystyle \otimes }$, from ordered pairs in the Cartesian product ${\displaystyle V\times W}$ into ${\displaystyle V\otimes W}$, in a way that generalizes the outer product. The tensor product of V and W is the vector space generated by the symbols ${\displaystyle v\otimes w}$, with ${\displaystyle v\in V}$ and ${\displaystyle w\in W}$, in which the relations of bilinearity are imposed for the product operation ${\displaystyle \otimes }$, and no other relations are assumed to hold. The tensor product space is thus the "freest" (or most general) such vector space, in the sense of having the least constraints.

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