Topological Tensor Product - Cross Norms and Tensor Products of Banach Spaces

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 . The algebraic tensor product consists of all finite sums

where is a natural number depending on and and for .

When A and B are Banach spaces, a cross norm p on the algebraic tensor product is a norm satisfying the conditions

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 called the projective cross norm, given by

where .

There is a smallest cross norm called the injective cross norm, given by

where . Here A′ and B′ mean the topological duals of A and B, respectively.

The completions of the algebraic tensor product in these two norms are called the projective and injective tensor products, and are denoted by and .

The norm used for the 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 of Banach spaces of a reasonable crossnorm on so that if, are arbitrary Banach spaces then for all (continuous linear) operators and the operator is continuous and . If A and B are two Banach spaces and α is a uniform cross norm then α defines a reasonable cross norm on the algebraic tensor product . The normed linear space obtained by equipping with that norm is denoted by . The completion of, which is a Banach space, is denoted by . The value of the norm given by α on and on the completed tensor product for an element x in (or ) is denoted by or .

A uniform crossnorm is said to be finitely generated if, for every pair of Banach spaces and every ,

A uniform crossnorm is cofinitely generated if, for every pair of Banach spaces and every ,

A tensor norm is defined to be a finitely generated uniform crossnorm. The projective cross norm and the injective cross norm defined above are tensor norms and they are called the projective tensor norm and the injective tensor norm, respectively.

If A and B are arbitrary Banach spaces and α is an arbitrary uniform cross norm then