information:classification
As a quotient space

A construction of the tensor product that is basis independent can be obtained in the following way.

Let V and W be two vector spaces over a field F.

One considers first a vector space L that has the Cartesian product V × W {\displaystyle V\times W} as a basis. That is, the basis elements of L are the pairs (v, w) {\displaystyle (v,w)} with v ∈ V {\displaystyle v\in V} and w ∈ W {\displaystyle w\in W}. To get such a vector space, one can define it as the vector space of the functions V × W → F {\displaystyle V\times W\to F} that have a finite number of nonzero values and identifying (v, w) {\displaystyle (v,w)} with the function that takes the value 1 on (v, w) {\displaystyle (v,w)} and 0 otherwise.

Let R be the linear subspace of L that is spanned by the relations that the tensor product must satisfy. More precisely, R is spanned by the elements of one of the forms:

where v, v 1, v 2 ∈ V {\displaystyle v,v_{1},v_{2}\in V}, w, w 1, w 2 ∈ W {\displaystyle w,w_{1},w_{2}\in W} and s ∈ F {\displaystyle s\in F}.

Then, the tensor product is defined as the quotient space:

and the image of (v, w) {\displaystyle (v,w)} in this quotient is denoted v ⊗ w {\displaystyle v\otimes w}.

It is straightforward to prove that the result of this construction satisfies the universal property considered below. (A very similar construction can be used to define the tensor product of modules.)