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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $U \\subset \\R^n$ be a bounded open set with $C^1$ boundary.\n\n1. Let $1 \\le p \u003c n$ and $p^* = \\frac{np}{n - p}$. For every $q$ with $1 \\le q \u003c p^*$, the inclusion of the [[Sobolev space|M4CEZ5K9HP]] $W^{1,p}(U)$ into the [[Lp space|SASP14MA1Z]] $L^q(U)$ is compact.\n2. Let $n \\le p \\le \\infty$. Then the inclusion of $W^{1,p}(U)$ into $L^p(U)$ is compact.\n\nHere compact means that every bounded sequence in $W^{1,p}(U)$ has a subsequence that converges in $L^q(U)$, respectively in $L^p(U)$.\n\n## Remarks\n\nThe inclusion is bounded by the [[Sobolev inequality|B03CA692K3]].\nFor compactness, one mollifies: the mollified functions $u^\\varepsilon$ form an equicontinuous and bounded family for each fixed $\\varepsilon$, so the [[Arzelà–Ascoli theorem|0381S8Z5P0]] applies to them.\nThe error $\\norm{u^\\varepsilon - u}_{L^1}$ is at most $\\varepsilon \\norm{\\nabla u}_{L^1}$, uniformly on bounded sets, and interpolation with the bound in $L^{p^*}$ gives the same uniform control in $L^q$ for $q \u003c p^*$.\nSo the inclusion is a [[compact operator|M2NEP1K5AE]].\n\nThe hypotheses are needed.\nThe inclusion into $L^{p^*}(U)$ itself is not compact: the rescaled functions $\\lambda^{(n-p)/p} \\varphi(\\lambda x)$ for a fixed bump $\\varphi$ are bounded in $W^{1,p}$ but concentrate at a point as $\\lambda \\to \\infty$.\nOn an unbounded set, such as $U = \\R^n$, translates $\\varphi(x - k e_1)$ of a bump are bounded in $W^{1,p}$ but have no convergent subsequence in $L^p$.\n\n## Explanations\n\n### L1 human\n\nA bounded family of functions with bounded slopes cannot oscillate wildly.\nThe theorem turns this into a compactness statement: from any such sequence one can extract a subsequence that converges in the mean.\nIt is the infinite-dimensional analogue of the fact that a bounded sequence of numbers has a convergent subsequence.\n\n### L2 human\n\nCompactness is how one passes to the limit in approximations of nonlinear equations: a weak limit in $W^{1,p}$ becomes a strong limit in $L^q$, so nonlinear terms in $u$ converge.\nIt also implies that the inverse of an elliptic operator such as $-\\Delta$ with Dirichlet conditions is compact on $L^2(U)$, which gives a discrete spectrum of eigenvalues.\nThe Poincaré inequality for functions with zero mean is proved from it by contradiction.\n",
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