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  "body": "\n## Statement\n\nLet $X \\subseteq \\R^m$ and $Y \\subseteq \\R^n$ be open sets.\nLet $\\mathcal{D}(X)$ be the space of test functions on $X$, and let $\\mathcal{D}'(X)$ be the space of distributions on $X$.\nFor $\\varphi \\in \\mathcal{D}(X)$ and $\\psi \\in \\mathcal{D}(Y)$, let $(\\varphi \\otimes \\psi)(x, y) = \\varphi(x) \\psi(y)$.\n\n1. Each $K \\in \\mathcal{D}'(X \\times Y)$ defines a linear map $A_K \\colon \\mathcal{D}(Y) \\to \\mathcal{D}'(X)$ by $\\langle A_K \\psi, \\varphi \\rangle = \\langle K, \\varphi \\otimes \\psi \\rangle$. It is sequentially continuous: if $\\psi_j \\to 0$ in $\\mathcal{D}(Y)$, then $\\langle A_K \\psi_j, \\varphi \\rangle \\to 0$ for every $\\varphi \\in \\mathcal{D}(X)$.\n2. Conversely, for each linear map $A \\colon \\mathcal{D}(Y) \\to \\mathcal{D}'(X)$ that is sequentially continuous in this sense, there is exactly one $K \\in \\mathcal{D}'(X \\times Y)$ with $A = A_K$.\n\nThe distribution $K$ is the *Schwartz kernel* of $A$.\n\n## Remarks\n\nThe hard part is the existence of $K$ in part 2.\nOne proof expands test functions in Fourier series on large cubes and controls the coefficients.\nAnother uses that the spaces of test functions are [[nuclear spaces|XC6XPMC2FQ]], so that continuous bilinear forms on $\\mathcal{D}(X) \\times \\mathcal{D}(Y)$ extend to $\\mathcal{D}(X \\times Y)$.\n\nExample: the identity map of $\\mathcal{D}(\\R)$, viewed as a map into $\\mathcal{D}'(\\R)$, has kernel $\\delta(x - y)$.\nIndeed $\\langle \\delta(x - y), \\varphi \\otimes \\psi \\rangle = \\int \\varphi(x) \\psi(x)\\, dx$.\nThe operator $\\psi \\mapsto \\psi'$ has kernel $\\delta'(x - y)$, the derivative in $x$ of the [[Dirac delta|0KPHENZ5S7]] on the diagonal.\n\n## Explanations\n\n### L1 human\n\nA matrix is the same as a linear map between spaces of columns.\nThe kernel theorem says that the same holds for very general linear maps on functions: each one is \"integration against a kernel\" $K(x, y)$.\nThe price is that $K$ may be a distribution rather than a function, like the delta function for the identity map.\n\n### L2 human\n\nThe theorem lets one describe operators such as differential operators, pseudodifferential operators and Fourier integral operators through their kernels.\nProperties of the operator then become properties of $K$, for example the singular support or the wave front set of $K$.\nFunction kernels are not enough: the identity operator of $L^2(\\R)$ is not an integral operator with a function kernel, so one really needs distributions.\n",
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