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  "body": "\n## Statement\n\nLet $a \u003c b$.\n\n1. Let $K \\colon [a, b] \\times [a, b] \\to \\C$ be a [[continuous kernel|ZM5G144PZG]].\nThen the operator $(T f)(x) = \\int_a^b K(x, y) f(y) \\, dy$ maps $C[a, b]$ to itself and is a [[compact operator|M2NEP1K5AE]] on $C[a, b]$ with the sup norm.\n2. Let $(X, \\mu)$ be a $\\sigma$-finite measure space and let $K \\in L^2(X \\times X, \\mu \\otimes \\mu)$.\nThen the [[Hilbert–Schmidt integral operator|CB5G212Q03]] $(T f)(x) = \\int_X K(x, y) f(y) \\, d\\mu(y)$ is a compact operator on $L^2(X, \\mu)$, and $\\norm{T} \\le \\norm{K}_{L^2(X \\times X)}$.\n\n## Remarks\n\nFor part 1, let $f$ run over the unit ball of $C[a, b]$.\nThen $\\abs{Tf(x)} \\le (b - a) \\max \\abs{K}$, and $\\abs{Tf(x) - Tf(x')} \\le (b - a) \\max_y \\abs{K(x, y) - K(x', y)}$, which is small when $x$ and $x'$ are close, by uniform continuity of $K$.\nSo the image of the unit ball is bounded and equicontinuous, and the [[Arzelà–Ascoli theorem|0381S8Z5P0]] shows that it is relatively compact.\n\nFor part 2, the Cauchy–Schwarz inequality gives $\\abs{Tf(x)}^2 \\le \\int_X \\abs{K(x, y)}^2 \\, d\\mu(y) \\cdot \\norm{f}^2$, and integrating in $x$ gives the norm bound.\n[[Approximating $K$ by degenerate kernels|FJKNJ14AD2]] in $L^2$ then writes $T$ as a norm limit of [[finite-rank operators|Z8FABSYQTS]], and such a limit is compact.\n\nExample: the Volterra operator $(Vf)(x) = \\int_0^x f(y) \\, dy$ on $L^2[0, 1]$ has the square-integrable kernel equal to $1$ for $y \\le x$ and $0$ otherwise, so it is compact.\n\nThe integrability of the kernel matters.\nOn $L^2(\\R)$, the convolution $(Tf)(x) = \\int_\\R e^{-\\abs{x - y}} f(y) \\, dy$ has a kernel that is not in $L^2(\\R^2)$.\nThis $T$ is not compact: it commutes with translations, so the images of the translates of a fixed nonzero function have no convergent subsequence.\n\n## Explanations\n\n### L1 human\n\nAn integral operator averages a function against a kernel.\nAveraging smooths things out, so the outputs of all functions of size at most one form a tame family, without wild oscillations.\nCompactness is the precise form of this: every sequence of outputs has a convergent subsequence.\n\n### L2 human\n\nCompactness is what makes integral equations of the second kind behave like finite linear systems.\nOnce the operator is known to be compact, [[Fredholm's theorem|XQ8DYKW4SH]] and the spectral theory of compact operators apply, and for symmetric kernels so does the [[Hilbert–Schmidt theorem|A1G25R99PY]].\n",
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