# Tensor products

If and are finite dimensional vector spaces then the Cartesian product is naturally a vector space called the direct sum of and and denoted . The tensor product is a more complicated object. To define it we start by defining for any set the free vector space over , . This is the set of all maps from to which are zero except at a finite number of points. We define the vector space structure by adding and scalar multiplying maps. Each gives rise to a function which is one at and zero elsewhere. We therefore have a map . By construction the span of the image of is all of .

The special property of the free vector space over is the following.

Proposition D.1   Let be any map from into a vector space then there is a unique linear map such that .

Proof. The general element of is

for . We define

Given two vector spaces and we can define . This is an infinite dimensional vector space. We shall denote by . Consider the subspace defined as the span of all elements of the form

and

for any real numbers and and vectors and . Let us denote

and define a map

by

We have

Proposition D.2   The map is bilinear.

Proof. We check the first factor only

From Proposition D.1 we know that any map , where is a vector space extends to a map . Standard linear algebra tells us that we can take the quotient to get a map if . The map is defined by . For example if and then defines a linear map from .

Let be a basis of and be a basis of . Consider the set of vectors in . We wish to show that they form a basis. First we check that they span the space . As the elements of are finite linear combinations of elements of the form it suffices to show that these are all in the span of the vectors . But this follows from the bilinearity. If and then

To show that they are linearly independent assume that

Let and be the dual bases of and . That is and . Then apply the map defined by and to this equation to obtain . So we have proved.

Proposition D.3   If and are finite dimensional vector spaces then

We can iterate tensor products. If and and are vector spaces we can form and . These different vector spaces are in fact isomorphic via the map

We use this map to identify these two spaces and ignore the brackets. We write for the triple tensor product. More generally we can form finitely many tensor products.

We also need to know about tensor products of maps. If is linear and is linear then we can define a map

by . This is a bilinear map so factors to a map which we denote by . It is defined by

We have seen that any bilinear map gives rise to a linear map . It is easy to show that this is an isomorphism. More generally if for any collection of vector spaces we denote by Mult the space of all multilinear maps from we have

Proposition D.4   If are vectors spaces then there is a natural isomorphism

defined by