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 distribution
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
s) have a two-variable theory that includes all reasonable
bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear ...
s on the space
of
test functions
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
. The space
itself consists of
smooth functions of
compact support
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalle ...
.
Statement of the theorem
Let
and
be open sets in
.
Every distribution
defines a
continuous linear map
such that
for every
.
Conversely, for every such continuous linear map
there exists one and only one distribution
such that () holds.
The distribution
is the kernel of the map
.
Note
Given a distribution
one can always write the linear map K informally as
:
so that
:
.
Integral kernels
The traditional
kernel function In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by James Mercer in the early 20th century, in the context of solvi ...
s
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
to its
dual space 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
arises by pairing the image distribution with a test function.
A simple example is that the natural embedding of the test function space
into
- sending every test function
into the corresponding distribution