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  "body": "\n## Summary\n\nA Hilbert–Schmidt integral operator is an operator $f \\mapsto \\int K(\\cdot, y) f(y) \\, dy$ on a space $L^2$ whose kernel $K$ is square-integrable in both variables together.\nSuch operators are bounded by the $L^2$ norm of the kernel and are [[compact|M2NEP1K5AE]].\nThey form the [[Schatten class|153N5DW55Z]] of exponent $2$ on $L^2$, and they are the natural setting for the $L^2$ theory of integral equations.\n\n## Explanations\n\n### L1 human\n\nThe kernel $K(x, y)$ is like an infinite matrix, with rows indexed by $x$ and columns indexed by $y$.\nThe Hilbert–Schmidt condition asks that the sum of the squares of all entries be finite, here as a double integral.\nThis makes the operator behave much like a matrix.\nFor example, $K(x, y) = \\min(x, y)$ on $[0, 1]$ is bounded, so its square has a finite integral, and its operator is Hilbert–Schmidt.\n",
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