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.
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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.
Cross norms and tensor products of Banach spaces
We shall use the notation from (Ryan 2002) in this section. The obvious way to define the tensor product of two Banach spaces A and B 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 A and B are Banach spaces the algebraic tensor product of A and B means the tensor product of A and B as vector spaces and is denoted by
where
When A and B are Banach spaces, a cross norm p on the algebraic tensor product
Here a′ and b′ are in the topological dual spaces of A and B, respectively, and p′ is the dual norm of p. The term reasonable crossnorm is also used for the definition above.
There is a largest cross norm
where
There is a smallest cross norm
where
The completions of the algebraic tensor product in these two norms are called the projective and injective tensor products, and are denoted by
When A and B are Hilbert spaces, the norm used for their Hilbert space tensor product is not equal to either of these norms in general. Some authors denote it by σ, so the Hilbert space tensor product in the section above would be
A uniform crossnorm α is an assignment to each pair
A uniform crossnorm
A uniform crossnorm
A tensor norm is defined to be a finitely generated uniform crossnorm. The projective cross norm
If A and B are arbitrary Banach spaces and α is an arbitrary uniform cross norm then
Tensor products of locally convex topological vector spaces
The topologies of locally convex topological vector spaces A and B are given by families of seminorms. For each choice of seminorm on A and on B we can define the corresponding family of cross norms on the algebraic tensor product A⊗B, and by choosing one cross norm from each family we get some cross norms on A⊗B, defining a topology. There are in general an enormous number of ways to do this. The two most important ways are to take all the projective cross norms, or all the injective cross norms. The completions of the resulting topologies on A⊗B are called the projective and injective tensor products, and denoted by A⊗γB and A⊗λB. There is a natural map from A⊗γB to A⊗λB.
If A or B is a nuclear space then the natural map from A⊗γB to A⊗λB is an isomorphism. Roughly speaking, this means that if A or B is nuclear, then there is only one sensible tensor product of A and B. This property characterizes nuclear spaces.