Schwartz kernel theorem

In mathematics, the Schwartz kernel theorem is a foundational result in the theory of generalized functions, published by Laurent Schwartz in 1952. It states, in broad terms, that the generalized functions introduced by Schwartz (Schwartz distributions) have a two-variable theory that includes all reasonable bilinear forms on the space formula_1 of test functions. The space formula_1 itself consists of smooth functions of compact support.
Statement of the theorem.
Let formula_3 and formula_4 be open sets in formula_5.
Every distribution formula_6 defines a
continuous linear map formula_7 such that
for every formula_8.
Conversely, for every such continuous linear map formula_9
there exists one and only one distribution formula_6 such that () holds.
The distribution formula_11 is the kernel of the map formula_9.
Note.
Given a distribution formula_6 one can always write the linear map K informally as
so that 
Integral kernels.
The traditional kernel functions "K"("x", "y") of two variables of the theory of integral operators having been expanded in scope to include their generalized function analogues, which are allowed to be more singular in a serious way, a large class of operators from "D" to its dual space "D′" of distributions can be constructed. The point of the theorem is to assert that the extended class of operators can be characterised abstractly, as containing all operators subject to a minimum continuity condition. A bilinear form on "D" arises by pairing the image distribution with a test function.
A simple example is that the natural embedding [.] of the test function space D into D’ - sending every test function f into the corresponding distribution [f] - corresponds to the delta distribution
δ("x" − "y") 
concentrated at the diagonal of the underlined Euclidean space, in terms of the Dirac delta function δ. While this is at most an observation, it shows how the distribution theory adds to the scope. Integral operators are not so 'singular'; another way to put it is that for "K" a continuous kernel, only compact operators are created on a space such as the continuous functions on [0,1]. The operator "I" is far from compact, and its kernel is intuitively speaking approximated by functions on [0,1] × [0,1] with a spike along the diagonal "x" = "y" and vanishing elsewhere.
This result implies that the formation of distributions has a major property of 'closure' within the traditional domain of functional analysis. It was interpreted (comment of Jean Dieudonné) as a strong verification of the suitability of the Schwartz theory of distributions to mathematical analysis more widely seen. In his "Éléments d'analyse" volume 7, p. 3 he notes that the theorem includes differential operators on the same footing as integral operators, and concludes that it is perhaps the most important modern result of functional analysis. He goes on immediately to qualify that statement, saying that the setting is too 'vast' for differential operators, because of the property of monotonicity with respect to the support of a function, which is evident for differentiation. Even monotonicity with respect to singular support is not characteristic of the general case; its consideration leads in the direction of the contemporary theory of pseudo-differential operators.
Smooth manifolds.
Dieudonné proves a version of the Schwartz result valid for smooth manifolds, and additional supporting results, in sections 23.9 to 23.12 of that book.
Generalization to nuclear spaces.
Much of the theory of nuclear spaces was developed by Alexander Grothendieck while investigating the Schwartz kernel theorem and published in . We have the following generalization of the theorem. 
Schwartz kernel theorem: Suppose that "X" is nuclear, "Y" is locally convex, and "v" is a continuous bilinear form on formula_16. Then "v" originates from a space of the form formula_17 where formula_18 and formula_19 are suitable equicontinuous subsets of formula_20 and formula_21. Equivalently, "v" is of the form, 
where formula_24 and each of formula_25 and formula_26 are equicontinuous. Furthermore, these sequences can be taken to be null sequences (i.e. converging to 0) in formula_27 and formula_28, respectively.