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 .
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 .