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  "body": "\n## Summary\n\nA Hilbert–Schmidt operator is a bounded operator between [[Hilbert spaces|HM2HHE3J51]] whose images of an orthonormal basis have a finite sum of squared norms.\nThese operators are [[compact|M2NEP1K5AE]], and they form a Hilbert space themselves, with the Hilbert–Schmidt norm.\nOn $L^2$ spaces they are exactly the [[integral operators with square-integrable kernels|CB5G212Q03]].\nThis makes them the natural class for the Fredholm theory of integral equations.\n\n## Explanations\n\n### L1 human\n\nThink of an operator as an infinite matrix.\nIt is Hilbert–Schmidt when the sum of the squares of the absolute values of all the entries is finite.\nThis is the infinite version of the Frobenius norm of a matrix.\nFor example, the diagonal operator on sequences that multiplies the $n$-th entry by $1/n$ is Hilbert–Schmidt, since $\\sum 1/n^2$ is finite.\n",
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