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  "body": "\n## Summary\n\nA Sobolev space $W^{k,p}(U)$ consists of the functions in the [[Lp space|SASP14MA1Z]] $L^p(U)$ whose [[weak derivatives|C3EC148VZD]] up to order $k$ also lie in $L^p(U)$.\nWith the sum of the $L^p$ norms of the derivatives it is a [[Banach space|06GZ956657]], and for $p = 2$ it is a [[Hilbert space|HM2HHE3J51]], written $H^k(U)$.\nSobolev spaces are the main setting of the modern theory of partial differential equations, because weak solutions are found in them by functional analysis.\nEmbedding theorems then translate the integrability of derivatives into integrability or continuity of the function itself.\n\n## Explanations\n\n### L1 human\n\nA Sobolev space measures a function together with its derivatives, but in an averaged sense rather than point by point.\nA function belongs to $W^{1,2}$ when the function and its slope have finite energy $\\int u^2 + \\abs{\\nabla u}^2$.\nCorners are allowed, but jumps are not.\nFor example, $\\abs{x}$ on $(-1, 1)$ lies in $W^{1,p}$ for every $p$, while the sign function on $(-1, 1)$ lies in no $W^{1,p}$.\n",
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