## Associativity of the tensor product

All vector spaces are by default over or .

We consider three vector spaces , and . Let be a tensor product of and , and a tensor product of and .

The aim is first to show that , together with the mapping:

is a tensor product of , and .

**Proof**:

We note that as defined above is trilinear.

Let be a vector space and a trilinear mapping . We must show that there exists a unique linear mapping such that .

For any , the mapping is bilinear; hence there is a unique linear mapping such that for all , we have .

It is easy to check that the mapping is bilinear; hence there is a unique linear mapping such that for all , we have .

Then for any , with :

Hence is such that .

We must now show that if is such that , then .