Schwartz kernel theorem

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

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