Graphics Reference
In-Depth Information
C.6.1. Theorem.
Let
V
i
, 1 £ i £ k, be vector spaces.
(1) (Existence) A tensor product (
A
,a) of the
V
i
exists. The image of a will actu-
ally span
A
.
(2) (Uniqueness) Given another tensor product (
B
,b), then there is a unique iso-
morphism m :
A
Æ
B
with b=m
a.
Proof.
To prove part (1), here is how one can define
A
and a when k = 2. It should
be obvious how to generalize the construction to handle the case of an arbitrary k.
Let
M
be the free vector space with basis (
v
1
,
v
2
), where
v
i
Œ
V
i
. Let
N
be the vector
subspace of
M
generated by all elements of
M
of the form
(
)
-
(
)
-¢
(
)
(
)
-
(
)
-
(
)
v
+¢
v
,
v
v
,
v
v
,
v
,
v
,
v
+¢
v
v
,
v
v
,
v
¢
,
1
1
2
1
2
1
2
1
2
2
1
2
1
2
(
)
-
(
)
(
)
-
(
)
r
vv
,
r
vv
,
,
v v
,
r
r
vv
,
,
12
12
1 2
12
where
v
i
,
v
i
¢Œ
V
i
and r Œ
R
. Define A =
M
/
N
and
a :
VV A
¥Æ
1
by
(
)
=
(
)
+
a
vv
,
vv
,
N
.
12
12
Given a map f :
V
1
¥
V
2
Æ
W
, define g
0
:
M
Æ
W
by
Ê
Á
ˆ
˜
=
Â
Â
(
)
(
)
g
a
vv
,
a
f
vv
,
.
0
vv
,
1
2
vv
,
1
2
12
12
(
vv
,
)δ
V V
(
vv
,
)δ
V V
12
1 2
12
1 2
One can show that g
0
sends
N
to
0
and hence induces a map g :
A
Æ
W
. Clearly,
f = g
a. The map g is unique because the image of a spans
A
. It follows that the pair
(
A
,a) is a tensor product for
V
1
and
V
2
. See [AusM63].
To prove the uniqueness of the tensor product, let (
B
,b) be another such. See
Figure C.4. Since b is multilinear, the universal factorization property of (
A
,a) implies
that there is a unique linear map m :
A
Æ
B
with b=m
a. Similarly, the universal fac-
torization property of (
B
,b) implies that there is a unique linear map m¢ :
B
Æ
A
with
a=m¢
b. Therefore, b=m
m¢
b and a=m¢
m
a. This implies that m is an isomorphism.
The theorem is proved.
Theorem C.6.1(2) shows that it is the universal factorization property of a tensor
product that is important and not the particular construction that is used. For that
reason one usually talks about “the” tensor product and uses a uniform notation.
a
...
V
1
¥ V
2
¥¥ V
k
A
m
b
Figure C.4.
The uniqueness of tensor
products.
m¢
B
