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

ei1 ⊗ · · · ⊗ ein ⊗

ej1

⊗ · · · ⊗

ejm

,