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
l{D}
l{D}
Let
X
Y
Rn
k\inl{D}'(X x Y)
K\colonl{D}(Y)\tol{D}'(X)
for every
u\inl{D}(X),v\inl{D}(Y)
K
k\inl{D}'(X x Y)
k
K
Given a distribution
k\inl{D}'(X x Y)
Kv=\intYk( ⋅ ,y)v(y)dy
\langleKv,u\rangle=\intX\intYk(x,y)v(y)u(x)dydx
The traditional kernel functions
K(x,y)
l{D}
l{D}'
l{D}
A simple example is that the natural embedding of the test function space
l{D}
l{D}'
f
[f]
\delta(x-y)
\delta
K
[0,1]
I
[0,1] x [0,1]
x=y
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.
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.
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
X x Y
\prime | |
X | |
A\prime |
\widehat{ ⊗ }\epsilon
\prime | |
Y | |
B\prime |
A\prime
B\prime
X\prime
Y\prime
v(x,y)=
infty | |
\sum | |
i=1 |
λi\left\langlex,
\prime | |
x | |
i |
\right\rangle\left\langley,
\prime | |
y | |
i |
\right\rangle
(x,y)\inX x Y
\left(λi\right)\inl1
\{
\prime | |
x | |
1, |
\prime | |
x | |
2, |
\ldots\}
\{
\prime | |
y | |
1, |
\prime | |
y | |
2, |
\ldots\}
\prime | |
X | |
A\prime |
\prime | |
Y | |
B\prime |
. 0717035. Lars Hörmander. The analysis of linear partial differential operators I. Grundl. Math. Wissenschaft. . 256 . Springer . 1983. 3-540-12104-8 . 10.1007/978-3-642-96750-4. .