9.9.1 Multilinear Maps
The constructions of the previous section extend from two factors to any finite number of them.
Definition 9.9.1 (Multilinear map). Let and be vector spaces over . A map is multilinear (or -linear) if it is linear in each variable when the others are held fixed. When , is a multilinear form.
Theorem 9.9.2 (Tensor product of several spaces). Let be finite-dimensional, and let be a basis of . There is a vector space together with a multilinear map such that for every vector space and every multilinear map there is a unique linear map with . It is unique up to unique isomorphism, the pure tensors with form a basis of it, and .
Proof. This is the proof of Theorem 9.8.6 with more indices. Let be the vector space with basis the symbols (). If , set which is multilinear. Given , the linear map with satisfies , by expanding using multilinearity; and it is the only linear map that can, since its values on a basis are forced. Uniqueness up to unique isomorphism is proved exactly as in Theorem 9.8.6(iii). ∎
As before, the space can also be built as a quotient of the vector space with basis all symbols , by the relations expressing multilinearity. Linear maps induce a linear map with , since the right side is multilinear in the ; exactly as in Proposition 9.8.7, this construction is compatible with composition and identities.
Proposition 9.9.3 (Associativity). For finite-dimensional vector spaces and , there is an isomorphism with . In particular .
Proof. The map is multilinear: it is linear in because is linear in and is linear in its first argument, and similarly for each . By Theorem 9.9.2 it induces a linear map with the stated values. Choose bases of all the spaces. Then sends the basis tensor to . The tensors form a basis of , and similarly for the ’s, so by Theorem 9.8.6 these images form a basis of the target. So sends a basis to a basis. ∎
Thus nothing new happens with more than two factors, and we may drop parentheses in iterated tensor products.
9.9.2 Tensor Powers and the Tensor Algebra
Definition 9.9.4 (Tensor powers). For a vector space and , the th tensor power of is By convention, and . If , then . For a linear map we write .
By Proposition 9.9.3, . Composing this isomorphism with the bilinear map gives a bilinear multiplication , which is concatenation of tensors. When or it is multiplication by a scalar.
Definition 9.9.5 (Tensor algebra). The tensor algebra of is the direct sum whose elements are finite sums with , with the multiplication that extends the products bilinearly.
Note. is a ring: the multiplication distributes over addition because it is bilinear, it has the identity , and it is associative because and are trilinear in and agree on pure tensors, where both are concatenation. It is also a vector space over whose multiplication is bilinear; a ring with this extra structure is called a -algebra. The tensor algebra is graded: it is the direct sum of the subspaces , and . It is not commutative once : for linearly independent , the tensors and are distinct elements of a basis of . As a vector space, is infinite-dimensional unless .
The tensor algebra is the free -algebra on : every linear map into a -algebra extends uniquely to a -linear ring homomorphism with , namely . This formula defines on each , because the right side is multilinear in the ; is multiplicative because and are bilinear in and agree on pure tensors; and is unique because is spanned by and the products of elements of .
Proposition 9.9.6 (Multilinear forms). For finite-dimensional vector spaces there are isomorphisms under which corresponds to the form . In particular is the space of -linear forms on .
Proof. The second isomorphism is the universal property: is a linear bijection from multilinear forms to linear functionals on . For the first, argue as in Proposition 9.8.13(i): the form depends multilinearly on , so it induces a linear map , and this map sends the tensor product of dual basis vectors to the functional that is on and on the other basis tensors. So it sends a basis to the dual basis. ∎