Banach Space - Tensor Product

Tensor Product

Let X and Y be two K-vector spaces. The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T′: X × Y → Z′ is any bilinear function into a K-vector space Z′, then only one linear function f: Z → Z′ with exists.

There are various norms that can be placed on the tensor product of the underlying vector spaces, amongst others the projective cross norm and injective cross norm. In general, the tensor product of complete spaces is not complete again.