Topological tensor product

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.
One of the original motivations for topological tensor products formula_1 is the fact that tensor products of the spaces of smooth functions on formula_2 do not behave as expected. There is an injection
but this is not an isomorphism. For example, the function formula_4 cannot be expressed as a finite linear combination of smooth functions in formula_5 We only get an isomorphism after constructing the topological tensor product; i.e.,
This article first details the construction in the Banach space case. formula_7 is not a Banach space and further cases are discussed at the end.
Tensor products 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".
Cross norms and tensor products of Banach spaces.
We shall use the notation from 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 formula_8. The algebraic tensor product formula_8 consists of all finite sums
where formula_11 is a natural number depending on formula_12 and formula_13 and formula_14 for
formula_15.
When "A" and "B" are Banach spaces, a cross norm "p" on the algebraic tensor product formula_8 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 cross norm formula_19 called the projective cross norm, given by
where formula_21.
It turns out that the projective cross norm agrees with the largest cross norm (, proposition 2.1).
There is a cross norm formula_22 called the injective cross norm, given by
where formula_21. Here "A"′ and "B"′ mean the topological duals of "A" and "B", 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 formula_25 and formula_26
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 formula_27
A uniform crossnorm α is an assignment to each pair formula_28 of Banach spaces of a reasonable crossnorm on formula_29 so that if formula_30 are arbitrary Banach spaces then for all (continuous linear) operators formula_31 and formula_32 the operator formula_33 is continuous and formula_34 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 formula_35 The normed linear space obtained by equipping formula_8 with that norm is denoted by formula_37 The completion of formula_38 which is a Banach space, is denoted by formula_39 The value of the norm given by α on formula_8 and on the completed tensor product formula_41 for an element "x" in formula_41 (or formula_43) is denoted by formula_44 or formula_45
A uniform crossnorm formula_46 is said to be finitely generated if, for every pair formula_28 of Banach spaces and every formula_48,
A uniform crossnorm formula_46 is cofinitely generated if, for every pair formula_28 of Banach spaces and every formula_48,
A tensor norm is defined to be a finitely generated uniform crossnorm. The projective cross norm formula_19 and the injective cross norm formula_22 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
Tensor products of locally convex topological vector spaces.
The topologies of locally convex topological vector spaces formula_57 and formula_58 are given by families of seminorms. For each choice of seminorm on formula_57 and on formula_58 we can define the corresponding family of cross norms on the algebraic tensor product formula_61 and by choosing one cross norm from each family we get some cross norms on formula_61 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 formula_63 are called the projective and injective tensor products, and denoted by formula_64 and formula_65 There is a natural map from formula_64 to formula_65
If formula_57 or formula_58 is a nuclear space then the natural map from formula_64 to formula_71 is an isomorphism. Roughly speaking, this means that if formula_57 or formula_58 is nuclear, then there is only one sensible tensor product of formula_57 and formula_58.
This property characterizes nuclear spaces.