obeying the rulesIf U, V, and W are VECTOR SPACES [I.3 §2.3] over some field, then a bilinear map from U × V to W is a map obeying the rules
(λu + μu′,υ) = λ (u,υ) + μ (u′,υ)
and
(u, λυ + μυ′) = λ (u, υ) + μ (u, υ′).
That is, it is linear in each variable separately.
Many important maps, such as INNER PRODUCTS [III.37], are bilinear. The tensor product U ⊗ V of two vector spaces U and V is a way of capturing the idea of the “most general” bilinear map that we can define on U × V. To get an idea of what this might mean, let us try to imagine a “completely arbitrary” bilinear map from U × V to a “completely arbitrary” vector space W, and let us use the notation u ⊗ v instead of (u, υ). Now because our linear map is perfectly general, all we know about it is what we can deduce from the fact that it is bilinear. For example, we know that u ⊗ υ1 + u ⊗υ2 = u ⊗ (υ1 + υ2). This example might suggest that all elements of U ⊗ V are of the form u ⊗ υ, but that is certainly not the case: for instance, in general there is no way of simplifying an expression such as ul ⊗ υ1 + u2 ⊗ υ2. (This reflects the fact that the set of values taken by a bilinear map from U × V to W is not in general a subspace of W.)
Thus, a typical element of U ⊗ V is a linear combination of elements of the form u ⊗ υ, with the rule that different linear combinations give the same element of U ⊗ V whenever they are forced to by the bilinearity property: for instance, (u1 + 2u2) ⊗ (υ1 – υ2) will always be equal to
ul ⊗ υ1 + 2u2 ⊗ υl - ul ⊗ υ2 - 2u2 ⊗ υ2.
A more formal way of expressing the above ideas is to say that U ⊗ V has a universal property. (See GEOMETRIC AND COMBINATORIAL GROUP THEORY [IV.10] for some other examples of universal properties. See also CATEGORIES [III.8].) The property in question is the following: given any bilinear map from U × V to a space W, we can find a linear map α from U ⊗ V to W such that (u, υ) = α(u ⊗ υ) for every u and υ. That is, every bilinear map defined on U × V is naturally associated with a linear map defined on U ⊗ V. (This linear map takes u ⊗ υ to (u, υ): the identifications made in the definition of the tensor product ensure that we can extend this to linear combinations of such elements in a consistent way.)
It is not hard to show that if U and V are finite dimensional, with bases u1, . . . , um and υ1, . . . , υn, then the vectors ui ⊗ υj form a basis for U ⊗ V. Other important properties of the tensor product are that it is commutative and associative, in the sense that U ⊗ V is naturally isomorphic to V ⊗ U and U ⊗ (V ⊗ W) is naturally isomorphic to (U ⊗ V) ⊗ W.
We have been discussing tensor products of vector spaces, but the definition can easily be generalized to any algebraic structure for which some notion of bilinearity makes sense, such as a MODULE [III.81 §3] or a C*-ALGEBRA [IV.15 §3]. Sometimes the tensor product of two structures is not what you would immediately expect. For instance, let ℤ n be the set of integers mod n, and consider both ℤ n and ℚ as modules over ℤ. Then their tensor product is zero. This reflects the fact that every bilinear map from ℤ n × ℚ must be the zero map.
Tensor products occur in many mathematical contexts. For a good example, see QUANTUM GROUPS [III.75].