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  "body": "\n## Definition\n\nLet $a \u003c b$.\nA *square-integrable kernel* on $[a, b]$ is a [[kernel|GSMBXT7FPK]] $K \\colon [a, b] \\times [a, b] \\to \\C$ that is Lebesgue [[measurable|YYNCJSR0PJ]] and satisfies\n$$\\norm{K}_{2} = \\left( \\int_a^b \\int_a^b \\abs{K(x, s)}^2 \\, ds \\, dx \\right)^{1/2} \u003c \\infty,$$\nwhere the integral is the [[Lebesgue integral|DWJF7ABPR1]].\n\n## Remarks\n\nExample: $K(x, s) = \\abs{x - s}^{-1/4}$ for $x \\ne s$ on $[0, 1]$.\nHere $\\abs{K}^2 = \\abs{x - s}^{-1/2}$, and $\\int_0^1 \\abs{x - s}^{-1/2} \\, ds \\le 2 \\int_0^{1} t^{-1/2} \\, dt = 4$ for each $x$, so $\\norm{K}_2^2 \\le 4$.\n\nNon-example: $K(x, s) = \\abs{x - s}^{-1/2}$ on $[0, 1]$.\nFor $0 \u003c x \u003c 1$, $\\int_0^1 \\abs{x - s}^{-1} \\, ds = \\infty$, so the double integral of $\\abs{K}^2$ is infinite.\n\nIn other words, $K$ lies in the [[Lp space|SASP14MA1Z]] $L^2([a, b] \\times [a, b])$.\nEvery continuous kernel on $[a, b]$ is square-integrable.\nFor $\\varphi \\in L^2(a, b)$, the function $(K \\varphi)(x) = \\int_a^b K(x, s) \\varphi(s) \\, ds$ lies in $L^2(a, b)$ and $\\norm{K \\varphi}_2 \\le \\norm{K}_2 \\norm{\\varphi}_2$, by the Cauchy–Schwarz inequality.\nThe same definition is used on a measurable subset of $\\R^m$.\n",
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