Topological tensor product

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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.

Motivation

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One of the original motivations for topological tensor products   is the fact that tensor products of the spaces of smooth real-valued 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  [1] We only get an isomorphism after constructing the topological tensor product; i.e.,

 

This article first details the construction in the Banach space case. The space   is not a Banach space and further cases are discussed at the end.

Tensor products of Hilbert spaces

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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 AB, 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 aibj form an orthonormal basis of AB.

Cross norms and tensor products of Banach spaces

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We shall use the notation from (Ryan 2002) 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 crossnorm (or cross norm)   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 reasonable crossnorm is also used for the definition above.

There is a cross norm   called the projective cross norm, given by   where  

It turns out that the projective cross norm agrees with the largest cross norm ((Ryan 2002), pp. 15-16).

There is a cross norm   called the injective cross norm, given by   where   Here   and   denote the topological duals of   and   respectively.

Note hereby that the injective cross norm is only in some reasonable sense the "smallest".

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

When   and   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   so the Hilbert space tensor product in the section above would be  

A uniform crossnorm   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   and   are two Banach spaces and   is a uniform cross norm then   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   on   and on the completed tensor product   for an element   in   (or  ) is denoted by  

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   and   are arbitrary Banach spaces and   is an arbitrary uniform cross norm then  

Tensor products of locally convex topological vector spaces

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The topologies of locally convex topological vector spaces   and   are given by families of seminorms. For each choice of seminorm on   and on   we can define the corresponding family of cross norms on the algebraic tensor product   and by choosing one cross norm from each family we get some cross norms on   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   are called the projective and injective tensor products, and denoted by   and   There is a natural map from   to  

If   or   is a nuclear space then the natural map from   to   is an isomorphism. Roughly speaking, this means that if   or   is nuclear, then there is only one sensible tensor product of   and  . This property characterizes nuclear spaces.

See also

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References

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  1. ^ "What is an example of a smooth function in C∞(R2) which is not contained in C∞(R)⊗C∞(R)".