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 of test functions itself consists of smooth functions of compact support.

Statement of the theorem

. Every distribution defines a continuous linear map such that

such that (, in reference to the kernel of an integral transform informally as

so that

1. .

Integral kernels

The traditional kernel functions of two variables of the theory of integral operators to its dual space arises by pairing the image distribution with a test function.

to its corresponding distribution - corresponds to the delta distribution

concentrated at the diagonal of the underlined Euclidean space, in terms of the Dirac delta function a continuous kernel, only compact operators 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 volume7, 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. Then v. Equivalently, v is of the form,

, respectively.

See also

References

Bibliography

• .

External links