described. As for every universal property, two objects that satisfy the property are related by a unique isomorphism. It follows that this is a (non-constructive) way to define the tensor product of two vector spaces. In this context, the preceding constructions of tensor products may be viewed as proofs of existence of the tensor product so defined. A consequence of this approach is that every property of the tensor product can be deduced from the universal property, and that, in practice, one may forget the method that has been used to prove its existence. The "universal-property definition" of the tensor product of two vector spaces is the following (recall that a bilinear map is a function that is separately linear in each of its arguments): The tensor product of two vector spaces and is a vector space denoted as together with a bilinear map from to such that, for every bilinear map there is a unique linear map such that (that is, for every and ). Linearly disjoint Like the universal property above, the following characterization may also be used to determine whether or not a given vector space and given bilinear map form a tensor product. For example, it follows immediately that if and are positive integers then and the bilinear map defined by sending to form a tensor product of and Often, this map will be denoted by so that denotes this bilinear map's value at As another example, suppose that is the vector space of all complex-valued functions on a set with addition and scalar multiplication defined pointwise (meaning that is the map and is the map ). Let and be any sets and for any and let denote the function defined by If and are vector subspaces then the vector subspace of together with the bilinear map form a tensor product of and Properties Dimension If and are vectors spaces of finite dimension, then is finite-dimensional, and its dimension is the product of the dimensions of and . This results from the fact that a basis of is formed by taking all tensor products of a basis element of and a basis element of . Associativity The tensor product is associative in the sense that, given three vector spaces there is a canonical isomorphism that maps to This allows omitting parentheses in the tensor product of more than two vector spaces or vectors. Commutativity as vector