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  "body": "\n## Statement\n\nLet $a \u003c b$ and let $K \\in L^2([a, b] \\times [a, b])$ be a [[square-integrable kernel|WWMWX7AD4V]] that is [[Hermitian|2ZB3NFZ94T]], that is, $K(y, x) = \\overline{K(x, y)}$ for almost all $x, y$.\nAssume that $K$ is not $0$ almost everywhere.\nLet $T_K$ be its [[Hilbert–Schmidt integral operator|CB5G212Q03]] on $L^2[a, b]$.\nThen $T_K$ has an [[eigenfunction|4NSACSWRJZ]] with a real eigenvalue $\\mu$ such that $\\abs{\\mu} = \\norm{T_K} \u003e 0$.\nIn particular $K$ has at least one [[characteristic value|DC7HMHVREP]], namely $1/\\mu$.\n\n## Remarks\n\nSince $T_K$ is self-adjoint, $\\norm{T_K} = \\sup_{\\norm{\\varphi} = 1} \\abs{\\ip{T_K \\varphi}{\\varphi}}$, and $\\norm{T_K} \u003e 0$ because a kernel whose operator is $0$ is $0$ almost everywhere.\nTake unit vectors $\\varphi_k$ with $\\ip{T_K \\varphi_k}{\\varphi_k} \\to \\mu$, where $\\mu = \\pm \\norm{T_K}$.\nThen $\\norm{T_K \\varphi_k - \\mu \\varphi_k}^2 \\le 2\\mu^2 - 2\\mu \\ip{T_K \\varphi_k}{\\varphi_k} \\to 0$.\nSince $T_K$ is [[compact|M2NEP1K5AE]], a subsequence of $T_K \\varphi_k$ converges, so $\\varphi_k$ converges to some unit vector $\\varphi$, and $T_K \\varphi = \\mu \\varphi$.\n\nExample: for $K(x, y) = \\min(x, y)$ on $[0, 1]$, the largest eigenvalue is $4 / \\pi^2$, with eigenfunction $\\sin(\\pi x / 2)$, and $\\norm{T_K} = 4 / \\pi^2$.\n\nSymmetry is necessary.\nThe Volterra kernel, equal to $1$ for $y \\le x$ and $0$ otherwise, is square-integrable and not zero.\nBut its operator $\\varphi \\mapsto \\int_a^x \\varphi$ has no eigenfunction at all, so this kernel has no characteristic value.\nIn finite dimension, the statement is the fact that a nonzero Hermitian matrix has an eigenvalue whose absolute value is its operator norm.\n\n## Explanations\n\n### L1 human\n\nA symmetric kernel that is not zero always has at least one natural mode: a function that the integral operator only stretches.\nThe stretching factor of the best mode equals the largest amount by which the operator can stretch any function.\nWithout symmetry this can fail completely.\n\n### L2 human\n\nThis is the first step of the Hilbert–Schmidt theory.\nApplying it again on the orthogonal complement of the eigenfunctions already found produces the whole sequence of eigenvalues and eigenfunctions, and leads to the [[Hilbert–Schmidt expansion theorem|3CXAZ1G5RJ]].\nThe variational form $\\abs{\\mu} = \\max \\abs{\\ip{T_K \\varphi}{\\varphi}}$ is the starting point of the min-max characterization of the further characteristic values.\n",
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