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{{Short description|Theorem}}
In [[mathematics]], the '''Schwartz kernel theorem''' is a foundational result in the theory of [[generalized function]]s, published by [[Laurent Schwartz]] in 1952. It states, in broad terms, that the generalized functions introduced by Schwartz ([[Schwartz distribution]]s) have a two-variable theory that includes all reasonable [[bilinear form]]s on the space <math>\mathcal{D}</math> of [[test functions]]. The space <math>\mathcal{D}</math> itself consists of [[smooth function]]s of [[compact support]].

==Statement of the theorem==
Let <math>X</math> and <math>Y</math> be open sets in <math>\mathbb{R}^n</math>. Every distribution <math>k \in \mathcal{D}'(X \times Y)</math> defines a
continuous linear map <math>K \colon \mathcal{D}(Y) \to \mathcal{D}'(X)</math> such that

{{NumBlk|:|<math>\left \langle k, u \otimes v \right \rangle = \left \langle Kv , u \right \rangle</math>|{{EquationRef|1}}}}

for every <math>u \in \mathcal{D}(X), v \in \mathcal{D}(Y)</math>. Conversely, for every such continuous linear map <math>K</math>, there exists one and only one distribution <math>k \in \mathcal{D}'(X \times Y)</math> such that ({{EquationNote|1}}) holds. The distribution <math>k</math> is the [[Kernel (linear algebra)|kernel]] of the map <math>K</math>.

===Note===
Given a distribution <math>k \in \mathcal{D}'(X \times Y)</math>, one can always write the linear map <math>K</math> informally as
:<math>Kv = \int_{Y} k(\cdot,y) v(y) d y</math>
so that 
:<math>\langle Kv,u \rangle = \int_{X} \int_{Y} k(x,y) v(y) u(x) d y d x</math>.

==Integral kernels==

The traditional [[kernel function]]s <math>K(x,y)</math> of two variables of the theory of [[integral operator]]s 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 <math>\mathcal{D}</math> to its [[dual space]] <math>\mathcal{D}'</math> 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 <math>\mathcal{D}</math> arises by pairing the image distribution with a test function.

A simple example is that the natural embedding of the test function space <math>\mathcal{D}</math> into <math>\mathcal{D}'</math> - sending every test function <math>f</math> into the corresponding distribution <math>[f]</math> - corresponds to the delta distribution

:<math>\delta(x-y)</math>

concentrated at the diagonal of the underlined Euclidean space, in terms of the [[Dirac delta function]] <math>\delta</math>. 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 <math>K</math> a continuous kernel, only [[compact operator]]s are created on a space such as the continuous functions on <math>[0,1]</math>. The operator <math>I</math> is far from compact, and its kernel is intuitively speaking approximated by functions on <math>[0,1]\times[0,1]</math> with a spike along the diagonal <math>x=y</math> 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&nbsp;7, p.&nbsp;3 he notes that the theorem includes [[differential operator]]s 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 operator]]s.

== Smooth manifolds ==

[[Jean Dieudonné|Dieudonné]] proves a version of the Schwartz result valid for [[smooth manifold]]s, and additional supporting results, in sections 23.9 to 23.12 of that [[Treatise on Analysis|book]].

== Generalization to nuclear spaces ==
Much of the theory of [[nuclear space]]s was developed by [[Alexander Grothendieck]] while investigating the Schwartz kernel theorem and published in {{harvnb|Grothendieck|1955}}. We have the following generalization of the theorem. 

'''Schwartz kernel theorem''':{{sfn | Schaefer|Wolff| 1999 | p=172}} Suppose that ''X'' is [[Nuclear space|nuclear]], ''Y'' is locally convex, and ''v'' is a continuous bilinear form on <math>X \times Y</math>. Then ''v'' originates from a space of the form <math>X^{\prime}_{A^{\prime}} \widehat{\otimes}_{\epsilon} Y^{\prime}_{B^{\prime}}</math> where <math>A^{\prime}</math> and <math>B^{\prime}</math> are suitable equicontinuous subsets of <math>X^{\prime}</math> and <math>Y^{\prime}</math>. Equivalently, ''v'' is of the form, 
: <math>v(x, y) = \sum_{i=1}^{\infty} \lambda_i \left\langle x, x_i^{\prime} \right\rangle \left\langle y, y_i^{\prime} \right\rangle</math> for all <math>(x, y) \in X \times Y</math>
where <math>\left( \lambda_i \right) \in l^1</math> and each of <math>\{ x^{\prime}_1, x^{\prime}_2, \ldots \}</math> and <math>\{ y^{\prime}_1, y^{\prime}_2, \ldots \}</math> are equicontinuous. Furthermore, these sequences can be taken to be null sequences (i.e. converging to 0) in <math>X^{\prime}_{A^{\prime}}</math> and <math>Y^{\prime}_{B^{\prime}}</math>, respectively.

==See also==
* {{annotated link|Fredholm kernel}}
* {{annotated link|Injective tensor product}}
* {{annotated link|Nuclear operator}}
* {{annotated link|Nuclear space}}
* {{annotated link|Projective tensor product}}
* {{annotated link|Rigged Hilbert space}}
* {{annotated link|Trace class}}

== References ==
{{Reflist}}

==Bibliography==
* {{Grothendieck Produits Tensoriels Topologiques et Espaces Nucléaires}}
* {{cite book|mr=0717035|first=L.|last= Hörmander|authorlink=Lars Hörmander|title=The analysis of linear partial differential operators I|series= Grundl. Math. Wissenschaft. |volume= 256 |publisher= Springer  |year=1983|isbn=3-540-12104-8 |doi=10.1007/978-3-642-96750-4}}.
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer|Wolff| 1999 | p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Treves | 2006 | p=}} -->
* {{cite book | author=Wong, Yau-Chuen | title=Schwartz spaces, nuclear spaces, and tensor products | publisher=Springer-Verlag | publication-place=Berlin New York | year=1979 | isbn=3-540-09513-6 | oclc=5126158 }} <!-- {{sfn | Wong | 1979 | p=}} -->

==External links==

* {{springer|id=N/n067820|title=Nuclear bilinear form|author=G. L. Litvinov}}

{{Topological tensor products and nuclear spaces}}
{{Functional analysis}}

[[Category:Generalized functions]]
[[Category:Transforms]]
[[Category:Theorems in functional analysis]]
[[Category:Topological tensor products]]
[[Category:Schwartz distributions]]