2.11. Tensor products 31 is given by formal symbols v ⊗ w, v ∈ V , w ∈ W , by the subspace spanned by the elements (v1 + v2) ⊗ w − v1 ⊗ w − v2 ⊗ w, v ⊗ (w1 + w2) − v ⊗ w1 − v ⊗ w2, av ⊗ w − a(v ⊗ w), v ⊗ aw − a(v ⊗ w), where v ∈ V, w ∈ W, a ∈ k. Exercise 2.11.2. Show that V ⊗ W can be equivalently defined as the quotient of the free abelian group V • W generated by v ⊗ w, v ∈ V, w ∈ W by the subgroup generated by (v1 + v2) ⊗ w − v1 ⊗ w − v2 ⊗ w, v ⊗ (w1 + w2) − v ⊗ w1 − v ⊗ w2, av ⊗ w − v ⊗ aw, where v ∈ V, w ∈ W, a ∈ k. The elements v ⊗ w ∈ V ⊗ W , for v ∈ V, w ∈ W are called pure tensors. Note that in general, there are elements of V ⊗ W which are not pure tensors. This allows one to define the tensor product of any number of vector spaces, V1 ⊗· · · ⊗ Vn. Note that this tensor product is associa- tive, in the sense that (V1 ⊗ V2) ⊗ V3 can be naturally identified with V1 ⊗ (V2 ⊗ V3). In particular, people often consider tensor products of the form V n = V ⊗ · · · ⊗ V (n times) for a given vector space V , and, more generally, E := V n ⊗ (V ∗ ) m . This space is called the space of tensors of type (m, n) on V . For instance, tensors of type (0, 1) are vectors, tensors of type (1, 0) — linear functionals (covectors), tensors of type (1, 1) — linear operators, of type (2, 0) — bilinear forms, tensors of type (2, 1) — algebra structures, etc. If V is finite dimensional with basis ei, i = 1,...,N, and ei is the dual basis of V ∗ , then a basis of E is the set of vectors ei 1 ⊗ · · · ⊗ ei n ⊗ ej1 ⊗ · · · ⊗ ejm,

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2011 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.