is generally sufficient): is the set of the functions from the Cartesian product to that have a finite number of nonzero values. The pointwise operations make a vector space. The function that maps to and the other elements of to is denoted The set is straightforwardly a basis of which is called the tensor product of the bases and The tensor product of two vectors is defined from their decomposition on the bases. More precisely, if are vectors decomposed on their respective bases, then the tensor product of and is If arranged into a rectangular array, the coordinate vector of is the outer product of the coordinate vectors of and . Therefore, the tensor product is a generalization of the outer product. It is straightforward to verify that the map is a bilinear map from to A limitation of this definition of the tensor product is that, if one changes bases, a different tensor product is defined. However, the decomposition on one basis of the elements of the other basis defines a canonical isomorphism between the two tensor products of vector spaces, which allows identifying them. Also, contrarily to the two following alternative definitions, this definition cannot be extended into a definition of the tensor product of modules over a ring. As a quotient space A construction of the tensor product that is basis independent can be obtained in the following way. Let and be two vector spaces over a field . One considers first a vector space that has the Cartesian product as a basis. That is, the basis elements of are the pairs with and To get such a vector space, one can define it as the vector space of the functions that have a finite number of nonzero values, and identifying with the function that takes the value on and otherwise. Let be the linear subspace of that is spanned by the relations that the tensor product must satisfy. More precisely is spanned by the elements of one of the forms where and Then, the tensor product is defined as the quotient space and the image of in this quotient is denoted 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.) Universal property In this section, the universal property satisfied by the tensor product is