Integral linear operator

An integral bilinear form is a bilinear functional that belongs to the continuous dual space of formula_1, the injective tensor product of the locally convex topological vector spaces (TVSs) "X" and "Y". An integral linear operator is a continuous linear operator that arises in a canonical way from an integral bilinear form. 
These maps play an important role in the theory of nuclear spaces and nuclear maps.
Definition - Integral forms as the dual of the injective tensor product.
Let "X" and "Y" be locally convex TVSs, let formula_2 denote the projective tensor product, formula_3 denote its completion, let formula_4 denote the injective tensor product, and formula_1 denote its completion. 
Suppose that formula_6 denotes the TVS-embedding of formula_4 into its completion and let formula_8 be its transpose, which is a vector space-isomorphism. This identifies the continuous dual space of formula_4 as being identical to the continuous dual space of formula_1. 
Let formula_11 denote the identity map and formula_12 denote its transpose, which is a continuous injection. Recall that formula_13 is canonically identified with formula_14, the space of continuous bilinear maps on formula_15. In this way, the continuous dual space of formula_4 can be canonically identified as a vector subspace of formula_14, denoted by formula_18. The elements of formula_18 are called integral (bilinear) forms on formula_15. The following theorem justifies the word integral. 
Integral linear maps.
A continuous linear map formula_21 is called integral if its associated bilinear form is an integral bilinear form, where this form is defined by formula_22. It follows that an integral map formula_21 is of the form: 
for suitable weakly closed and equicontinuous subsets "S" and "T" of formula_25 and formula_26, respectively, and some positive Radon measure formula_27 of total mass ≤ 1. 
The above integral is the weak integral, so the equality holds if and only if for every formula_28, formula_29. 
Given a linear map formula_30, one can define a canonical bilinear form formula_31, called the associated bilinear form on formula_32, by formula_33. 
A continuous map formula_30 is called integral if its associated bilinear form is an integral bilinear form. An integral map formula_35 is of the form, for every formula_36 and formula_37:
for suitable weakly closed and equicontinuous aubsets formula_39 and formula_40 of formula_25 and formula_42, respectively, and some positive Radon measure formula_27 of total mass formula_44.
Relation to Hilbert spaces.
The following result shows that integral maps "factor through" Hilbert spaces.
Proposition: Suppose that formula_45 is an integral map between locally convex TVS with "Y" Hausdorff and complete. There exists a Hilbert space "H" and two continuous linear mappings formula_46 and formula_47 such that formula_48. 
Furthermore, every integral operator between two Hilbert spaces is nuclear. Thus a continuous linear operator between two Hilbert spaces is nuclear if and only if it is integral.
Sufficient conditions.
Every nuclear map is integral. An important partial converse is that every integral operator between two Hilbert spaces is nuclear.
Suppose that "A", "B", "C", and "D" are Hausdorff locally convex TVSs and that formula_49, formula_50, and formula_51 are all continuous linear operators. If formula_50 is an integral operator then so is the composition formula_53. 
If formula_45 is a continuous linear operator between two normed space then formula_45 is integral if and only if formula_56 is integral. 
Suppose that formula_45 is a continuous linear map between locally convex TVSs. 
If formula_45 is integral then so is its transpose formula_59. Now suppose that the transpose formula_59 of the continuous linear map formula_45 is integral. Then formula_45 is integral if the canonical injections formula_63 (defined by formula_64 value at ) and formula_65 are TVS-embeddings (which happens if, for instance, formula_66 and formula_67 are barreled or metrizable). 
Properties.
Suppose that "A", "B", "C", and "D" are Hausdorff locally convex TVSs with "B" and "D" complete. If formula_49, formula_50, and formula_51 are all integral linear maps then their composition formula_53 is nuclear. 
Thus, in particular, if is an infinite-dimensional Fréchet space then a continuous linear surjection formula_72 cannot be an integral operator.