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
D
{\displaystyle {\mathcal {D}}}
of test functions. The space
D
{\displaystyle {\mathcal {D}}}
itself consists of smooth functions of compact support.
Contents
Statement of the theorem
Let
X
{\displaystyle X}
and
Y
{\displaystyle Y}
be open sets in
R
n
{\displaystyle \mathbb {R} ^{n}}
. Every distribution
k
∈
D
′
(
X
×
Y
)
{\displaystyle k\in {\mathcal {D}}'(X\times Y)}
defines a
continuous linear map
K
:
D
(
Y
)
→
D
′
(
X
)
{\displaystyle K\colon {\mathcal {D}}(Y)\to {\mathcal {D}}'(X)}
such that
for every
u
∈
D
(
X
)
,
v
∈
D
(
Y
)
{\displaystyle u\in {\mathcal {D}}(X),v\in {\mathcal {D}}(Y)}
. Conversely, for every such continuous linear map
K
{\displaystyle K}
, there exists one and only one distribution
k
∈
D
′
(
X
×
Y
)
{\displaystyle k\in {\mathcal {D}}'(X\times Y)}
such that (1) holds. The distribution
k
{\displaystyle k}
is called the kernel of the map
K
{\displaystyle K}
, in reference to the kernel of an integral transform. In line with this, one sometimes writes the linear map
K
{\displaystyle K}
informally as
K
v
=
∫
Y
k
(
⋅
,
y
)
v
(
y
)
d
y
{\displaystyle Kv=\int _{Y}k(\cdot ,y)v(y)dy}
so that
⟨
K
v
,
u
⟩
=
∫
X
∫
Y
k
(
x
,
y
)
v
(
y
)
u
(
x
)
d
y
d
x
{\displaystyle \langle Kv,u\rangle =\int _{X}\int _{Y}k(x,y)v(y)u(x)dydx}
.
Integral kernels
The traditional kernel functions
K
(
x
,
y
)
{\displaystyle 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
{\displaystyle {\mathcal {D}}}
to its dual space
D
′
{\displaystyle {\mathcal {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
{\displaystyle {\mathcal {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
{\displaystyle {\mathcal {D}}}
into
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 Grothendieck 1955. 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
×
Y
{\displaystyle X\times Y}
. Then v originates from a space of the form
X
A
′
′
⊗
^
ϵ
Y
B
′
′
{\displaystyle X_{A^{\prime }}^{\prime }{\widehat {\otimes }}_{\epsilon }Y_{B^{\prime }}^{\prime }}
where
A
′
{\displaystyle A^{\prime }}
and
B
′
{\displaystyle B^{\prime }}