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 ${\displaystyle {\mathcal {D}}}$ of test functions. The space ${\displaystyle {\mathcal {D}}}$ itself consists of smooth functions of compact support.

Let ${\displaystyle X}$ and ${\displaystyle Y}$ be open sets in ${\displaystyle \mathbb {R} ^{n}}$. Every distribution ${\displaystyle k\in {\mathcal {D}}'(X\times Y)}$ defines a continuous linear map ${\displaystyle K\colon {\mathcal {D}}(Y)\to {\mathcal {D}}'(X)}$ such that

for every ${\displaystyle u\in {\mathcal {D}}(X),v\in {\mathcal {D}}(Y)}$. Conversely, for every such continuous linear map ${\displaystyle K}$, there exists one and only one distribution ${\displaystyle k\in {\mathcal {D}}'(X\times Y)}$ such that (1) holds. The distribution ${\displaystyle k}$ is the kernel of the map ${\displaystyle K}$.

Given a distribution ${\displaystyle k\in {\mathcal {D}}'(X\times Y)}$, one can always write the linear map ${\displaystyle K}$ informally as

so that

The traditional kernel functions ${\displaystyle K(x,y)}$ 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 ${\displaystyle {\mathcal {D}}}$ to its dual space ${\displaystyle {\mathcal {D}}'}$ 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 ${\displaystyle {\mathcal {D}}}$ arises by pairing the image distribution with a test function.

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

concentrated at the diagonal of the underlined Euclidean space, in terms of the Dirac delta function ${\displaystyle \delta }$. 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 ${\displaystyle K}$ a continuous kernel, only compact operators are created on a space such as the continuous functions on ${\displaystyle [0,1]}$. The operator ${\displaystyle I}$ is far from compact, and its kernel is intuitively speaking approximated by functions on ${\displaystyle [0,1]\times [0,1]}$ with a spike along the diagonal ${\displaystyle x=y}$ and vanishing elsewhere.