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 "a" and "b" run through orthonormal bases of "A" and "B", then the vectors "a"⊗"b" 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 formula_8 and formula_9 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 formula_8 and formula_9 are Banach spaces the algebraic tensor product of formula_8 and formula_9 means the tensor product of formula_8 and formula_9 as vector spaces and is denoted by formula_16 The algebraic tensor product formula_17 consists of all finite sums formula_18 where formula_19 is a natural number depending on formula_20 and formula_21 and formula_22 for formula_23 When formula_8 and formula_9 are Banach spaces, a ' (or ') formula_26 on the algebraic tensor product formula_17 is a norm satisfying the  conditions formula_28 formula_29 Here formula_30 and formula_31 are elements of the topological dual spaces of formula_8 and formula_33 respectively, and formula_34 is the dual norm of formula_35 The term is also used for the definition above. There is a cross norm formula_36 called the projective cross norm, given by formula_37 where formula_38 It turns out that the projective cross norm agrees with the largest cross norm (, proposition 2.1). There is a cross norm formula_39 called the injective cross norm, given by formula_40 where formula_38 Here formula_42 and formula_43 denote the topological duals of formula_8 and formula_33 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_46 and formula_47 When formula_8 and formula_9 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 formula_50 so the Hilbert space tensor product in the section above would be formula_51 A formula_52 is an assignment to each pair formula_53 of Banach spaces of a reasonable crossnorm on  formula_54 so that if formula_55 are arbitrary Banach spaces then for all (continuous linear) operators formula_56 and formula_57 the operator formula_58 is continuous and formula_59 If formula_8 and formula_9 are two Banach spaces and formula_52 is a uniform cross norm then formula_52 defines a reasonable cross norm on the algebraic tensor product formula_16 The normed linear space obtained by equipping formula_17 with that norm is denoted by formula_66 The completion of formula_67 which is a Banach space, is denoted by formula_68 The value of the norm given by formula_52 on formula_17 and on the completed tensor product formula_71 for an  element formula_20 in formula_71 (or formula_74) is denoted by formula_75 A uniform crossnorm formula_52 is said to be if, for every pair formula_53 of Banach spaces and every formula_78 formula_79 A uniform crossnorm formula_52 is if, for every pair formula_53 of Banach spaces and every formula_78 formula_83 A is defined to be a finitely generated uniform crossnorm. The projective cross norm formula_36 and the injective cross norm formula_39 defined above are tensor norms and they are called the projective tensor norm and the injective tensor norm, respectively. If formula_8 and formula_9 are arbitrary Banach spaces and formula_52 is an arbitrary  uniform cross norm then formula_89 Tensor products of locally convex topological vector spaces. The topologies of locally convex topological vector spaces formula_8 and formula_9 are given by families of seminorms. For each choice of seminorm on formula_8 and on formula_9 we can define the corresponding family of cross norms on the algebraic tensor product formula_94 and by choosing one cross norm from each family we get some cross norms on formula_94 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_96 are called the projective and injective tensor products, and denoted by formula_97 and formula_98 There is a natural map from formula_97 to formula_98 If formula_8 or formula_9 is a nuclear space then the natural map from formula_97 to formula_104 is an isomorphism. Roughly speaking, this means that if formula_8 or formula_9 is nuclear, then there is only one sensible tensor product of formula_8 and formula_9. This property characterizes nuclear spaces.