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  "body": "\n## Definition\n\nLet $H$ and $K$ be [[Hilbert spaces|HM2HHE3J51]] and let $A \\colon H \\to K$ be a [[bounded linear operator|03TM135G0N]].\nThe operator $A$ is a *Hilbert–Schmidt operator* if there is an [[orthonormal basis|0S9GGTR0EX]] $(e_i)_{i \\in I}$ of $H$ such that\n$$\\sum_{i \\in I} \\norm{A e_i}^2 \u003c \\infty.$$\nThe *Hilbert–Schmidt norm* of $A$ is $\\norm{A}_{\\mathrm{HS}} = \\big( \\sum_{i \\in I} \\norm{A e_i}^2 \\big)^{1/2}$.\n\n## Remarks\n\nThe condition and the value of the sum do not depend on the orthonormal basis.\nIndeed, for an orthonormal basis $(f_j)_{j \\in J}$ of $K$, Parseval's identity gives $\\sum_i \\norm{A e_i}^2 = \\sum_i \\sum_j \\abs{\\ip{A e_i}{f_j}}^2 = \\sum_j \\norm{A^* f_j}^2$, and the right side does not involve $(e_i)$.\n\nEvery Hilbert–Schmidt operator is [[compact|M2NEP1K5AE]] and satisfies $\\norm{A} \\le \\norm{A}_{\\mathrm{HS}}$.\nA compact operator is Hilbert–Schmidt if and only if its [[singular values|QSJKPKPM7H]] $\\sigma_n$ satisfy $\\sum_n \\sigma_n^2 \u003c \\infty$, and then $\\norm{A}_{\\mathrm{HS}}^2 = \\sum_n \\sigma_n^2$.\nThe Hilbert–Schmidt operators form the Schatten class $S_2$; every [[trace class operator|C37R11WREF]] is Hilbert–Schmidt.\nOn $L^2$ spaces of $\\sigma$-finite measure spaces, they are the [[integral operators with square-integrable kernels|CB5G212Q03]], and $\\norm{A}_{\\mathrm{HS}}$ is the $L^2$ norm of the kernel.\n\nExample: on $\\ell^2$, the operator $A(x_1, x_2, \\dots) = (x_1, x_2/2, x_3/3, \\dots)$ is Hilbert–Schmidt.\nFor the standard basis, $\\norm{A e_n} = 1/n$ and $\\sum_n 1/n^2 = \\pi^2/6$.\n\nNon-example: on $\\ell^2$, the operator $B(x_1, x_2, \\dots) = (x_1, x_2/\\sqrt{2}, x_3/\\sqrt{3}, \\dots)$ is compact but not Hilbert–Schmidt, since $\\sum_n \\norm{B e_n}^2 = \\sum_n 1/n$ diverges.\nThe identity of an infinite-dimensional Hilbert space is not Hilbert–Schmidt either.\n\n## Explanations\n\n### L1 human\n\nAn operator is Hilbert–Schmidt when the total squared size of the images of the basis vectors is finite.\n",
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