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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|0VKD1TP7F3]] with a Borel [[measure|PR11BZQQPG]] $\\mu$, and let $1 \\le p \u003c \\infty$.\nLet $u \\colon X \\to \\R$ be a function, defined at every point.\nFor a Borel function $g \\colon X \\to [0, \\infty]$ and a [[rectifiable curve|0HATW2VWAZ]] $\\gamma \\colon [a, b] \\to X$, let $\\int_\\gamma g \\, ds$ be the integral of $g$ along $\\gamma$ with respect to arc length.\nThe function $g$ is an *upper gradient* of $u$ if\n$$\\abs{u(\\gamma(a)) - u(\\gamma(b))} \\le \\int_\\gamma g \\, ds$$\nfor every rectifiable curve $\\gamma \\colon [a, b] \\to X$.\nLet $\\widetilde{N}^{1,p}(X)$ be the set of all $u$ in the [[Lp space|SASP14MA1Z]] $L^p(X, \\mu)$ with an upper gradient in $L^p(X, \\mu)$.\nOn it put\n$$\\norm{u}_{\\widetilde{N}^{1,p}} = \\left( \\int_X \\abs{u}^p \\, d\\mu + \\inf_g \\int_X g^p \\, d\\mu \\right)^{1/p},$$\nthe infimum over all upper gradients $g$ of $u$.\nThe *Newtonian space* $N^{1,p}(X)$ is the quotient of $\\widetilde{N}^{1,p}(X)$ by the relation $u \\sim v$ if $\\norm{u - v}_{\\widetilde{N}^{1,p}} = 0$, with the induced norm.\n\n## Remarks\n\nExample: on $X = \\R$ with Lebesgue measure, $u(x) = \\max(0, 1 - \\abs{x})$ lies in $\\widetilde{N}^{1,p}(\\R)$.\nIndeed $u \\in L^p$, and $g = \\mathbf{1}_{[-1, 1]}$ is an upper gradient in $L^p$, since $\\abs{u(s) - u(t)} \\le \\abs{\\int_s^t g}$ for real $s, t$ and a curve from $\\gamma(a)$ to $\\gamma(b)$ covers at least the interval between them.\n\nNon-example: on $X = [0, 1]$ with Lebesgue measure, $u = \\mathbf{1}_{[1/2, 1]}$ is in $L^p$ but has no upper gradient in $L^p$.\nAn upper gradient $g$ would need $\\int_{1/2 - \\varepsilon}^{1/2 + \\varepsilon} g \\ge 1$ for every $\\varepsilon \u003e 0$, and these integrals tend to $0$ when $g$ is integrable.\n\nThe space $N^{1,p}(X)$ is a [[Banach space|06GZ956657]].\nIts elements are classes of functions defined at every point, and two functions in one class agree outside a set of $p$-capacity zero, not only almost everywhere.\nThe definition is often given with $p$-weak upper gradients, which may fail the curve inequality on a family of curves of $p$-modulus zero.\nThis gives the same space and the same norm.\nIf $X$ contains no nonconstant rectifiable curves, for example a Cantor set, then $g = 0$ is an upper gradient of every function and $N^{1,p}(X) = L^p(X, \\mu)$.\nFor an open set $\\Omega \\subset \\R^n$ with Lebesgue measure and $1 \u003c p \u003c \\infty$, $N^{1,p}(\\Omega)$ agrees with the [[Sobolev space|M4CEZ5K9HP]] $W^{1,p}(\\Omega)$, each class of $W^{1,p}$ being represented by suitable pointwise defined functions.\n",
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