Topological Modules/Topological tensor products

Tensor product of Hilbert spacesEdit

Proposition (tensor product of orthonormal bases is orthonormal basis of tensor product):

Let   be Hilbert spaces, and suppose that   is an orthonormal basis of   and   is an orthonormal basis of  . Then   is an orthonormal basis of  .

Proof: Let any element

 

of   be given; by definition, each element of   may be approximated by such elements. Let  . Then by definition of an orthonormal basis, we find   for   and   for   and then   resp.   such that

  and  .

Then note that by the triangle inequality,

 .

Now fix  . Then by the triangle inequality,

 

In total, we obtain that

 

(assuming that the given sum approximates   well enough) which is arbitrarily small, so that the span of tensors of the form   is dense in  . Now we claim that the basis is orthonormal. Indeed, suppose that  . Then

 .

Similarly, the above expression evaluates to   when   and  . Hence,   does constitute an orthonormal basis of  .