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Chapter 14
Tensor Products
In the preceding chapters, we have seen several ways to construct new vector
spaces from old ones. Two of the most important such constructions are the
B
direct sum <l = and the vector space ²<Á = ³ of all linear transformations
from to . In this chapter, we consider another very important construction,
<
=
known as the tensor product .
Universality
We begin by describing a general type of universality that will help motivate the
definition of tensor product. Our description is strongly related to the formal
notion of a universal pair in category theory, but we will be somewhat less
formal to avoid the need to formally define categorical concepts. Accordingly,
the terminology that we shall introduce is not standard, but does not contradict
any standard terminology.
Referring to Figure 14.1, consider a set and two functions and , with
(
domain .
(
f
A S
W
g
X
Figure 14.1
Suppose that there exists a function ¢: ¦ ? for which this diagram
commutes, that is,
~ k
This is sometimes expressed by saying that can be factored through . What
does this say about the relationship between the functions and ?
