Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Topological tensor product
Summary
In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products (see Tensor product of Hilbert spaces), but for general Banach spaces or locally convex topological vector spaces the theory is notoriously subtle.
One of the original motivations for topological tensor products is the fact that tensor products of the spaces of smooth functions on do not behave as expected. There is an injection
but this is not an isomorphism. For example, the function cannot be expressed as a finite linear combination of smooth functions in We only get an isomorphism after constructing the topological tensor product; i.e.,
This article first details the construction in the Banach space case. is not a Banach space and further cases are discussed at the end.
Tensor product of Hilbert spaces
The algebraic tensor product of two Hilbert spaces A and B has a natural positive definite sesquilinear form (scalar product) induced by the sesquilinear forms of A and B. So in particular it has a natural positive definite quadratic form, and the corresponding completion is a Hilbert space A ⊗ B, called the (Hilbert space) tensor product of A and B.
If the vectors ai and bj run through orthonormal bases of A and B, then the vectors ai⊗bj form an orthonormal basis of A ⊗ B.
We shall use the notation from in this section. The obvious way to define the tensor product of two Banach spaces and is to copy the method for Hilbert spaces: define a norm on the algebraic tensor product, then take the completion in this norm. The problem is that there is more than one natural way to define a norm on the tensor product.
If and are Banach spaces the algebraic tensor product of and means the tensor product of and as vector spaces and is denoted by The algebraic tensor product consists of all finite sums
where is a natural number depending on and and for
When and are Banach spaces, a (or ) on the algebraic tensor product is a norm satisfying the conditions
Here and are elements of the topological dual spaces of and respectively, and is the dual norm of The term is also used for the definition above.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.