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  "body": "\n## Definition\n\nLet $U \\subseteq \\R^n$ be open, let $k \\ge 0$ be an integer and let $1 \\le p \\le \\infty$.\nThe *Sobolev space* $W^{k,p}(U)$ is the set of functions $u \\in L^p(U)$ with the following property.\nFor every multi-index $\\alpha$ with $\\abs{\\alpha} \\le k$, the [[weak derivative|C3EC148VZD]] $D^\\alpha u$ exists and lies in $L^p(U)$.\nIts norm is\n$$\\norm{u}_{W^{k,p}(U)} = \\Big( \\sum_{\\abs{\\alpha} \\le k} \\int_U \\abs{D^\\alpha u}^p \\, dx \\Big)^{1/p} \\text{ for } p \u003c \\infty, \\qquad \\norm{u}_{W^{k,\\infty}(U)} = \\sum_{\\abs{\\alpha} \\le k} \\operatorname*{ess\\,sup}_U \\abs{D^\\alpha u}.$$\nThe space $W^{k,p}_0(U)$ is the closure in $W^{k,p}(U)$ of the space $C_c^\\infty(U)$ of test functions.\n\n## Remarks\n\nHere $L^p(U)$ is the [[Lp space|SASP14MA1Z]], and functions equal almost everywhere are identified.\nEach $W^{k,p}(U)$ is a Banach space, and $H^k(U) = W^{k,2}(U)$ is a Hilbert space with inner product $\\sum_{\\abs{\\alpha} \\le k} \\int_U D^\\alpha u \\, D^\\alpha v \\, dx$.\nFor $k = 0$ one gets $W^{0,p}(U) = L^p(U)$.\nFor $1 \\le p \u003c \\infty$, the smooth functions in $W^{k,p}(U)$ are dense in it (Meyers–Serrin).\n\nExample: $u(x) = \\abs{x}$ lies in $W^{1,p}(-1, 1)$ for all $p$.\nIndeed $u$ is bounded, and its weak derivative $\\sgn x$ is bounded, so both lie in every $L^p(-1, 1)$.\n\nNon-example: $\\sgn x$ is bounded, so it lies in every $L^p(-1, 1)$, but it has no weak derivative on $(-1, 1)$.\nSo it is in no space $W^{1,p}(-1, 1)$.\n",
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