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
,
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