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  "body": "\n## Definition\n\nLet $X$ be a nonempty set and let $\\ell(X)$ be the vector space of all functions $X \\to \\R$.\nLet $\\mathcal{F}$ be a linear subspace of $\\ell(X)$ and let $\\mathcal{E} \\colon \\mathcal{F} \\times \\mathcal{F} \\to \\R$ be a symmetric bilinear form, with $\\mathcal{E}(u) = \\mathcal{E}(u, u)$.\nThe pair $(\\mathcal{E}, \\mathcal{F})$ is a *resistance form* on $X$ if:\n\n1. $\\mathcal{F}$ contains the constant functions, $\\mathcal{E}(u) \\ge 0$ for all $u \\in \\mathcal{F}$, and $\\mathcal{E}(u) = 0$ if and only if $u$ is constant.\n2. Let $u \\sim v$ mean that $u - v$ is constant. Then $\\mathcal{E}$ induces an inner product on the quotient space $\\mathcal{F} / {\\sim}$, and $\\mathcal{F} / {\\sim}$ with this inner product is a [[Hilbert space|HM2HHE3J51]].\n3. For all $x \\ne y$ in $X$ there is $u \\in \\mathcal{F}$ with $u(x) \\ne u(y)$.\n4. For all $x, y \\in X$,\n$$R(x, y) = \\sup \\set{ \\frac{(u(x) - u(y))^2}{\\mathcal{E}(u)} : u \\in \\mathcal{F}, \\ \\mathcal{E}(u) \u003e 0 } \u003c \\infty.$$\n5. (Markov property.) For every $u \\in \\mathcal{F}$, the function $\\bar u = \\min(\\max(u, 0), 1)$ lies in $\\mathcal{F}$ and $\\mathcal{E}(\\bar u) \\le \\mathcal{E}(u)$.\n\nThe number $R(x, y)$ is the *effective resistance* between $x$ and $y$, and $R$ is a metric on $X$, the *resistance metric*.\n\n## Remarks\n\nExample: let $X$ be the vertex set of a finite connected graph, with a conductance $c_{xy} = c_{yx} \u003e 0$ on each edge $\\set{x, y}$, let $\\mathcal{F} = \\ell(X)$, and let $\\mathcal{E}(u) = \\sum_{\\set{x, y}} c_{xy} (u(x) - u(y))^2$, the sum over edges.\nClause 1 holds because $\\mathcal{E}(u) = 0$ forces $u$ to be equal at the two ends of every edge, hence constant on the connected graph.\nClauses 2 and 4 hold because $\\mathcal{F} / {\\sim}$ is finite-dimensional and $\\mathcal{E}$ is positive definite on it.\nClause 3 holds with the indicator function of $\\set{x}$.\nClause 5 holds because $t \\mapsto \\min(\\max(t, 0), 1)$ does not increase distances, so no term $(u(x) - u(y))^2$ grows.\nHere $R(x, y)$ is the [[effective resistance|XAE5YJ3H68]] of the electrical network.\n\nNon-example: the same formula on a graph with two connected components fails clause 1.\nThe function equal to $0$ on one component and $1$ on the other has energy $0$ but is not constant.\n\nOn $X = [0, 1]$, the form $\\mathcal{E}(u, v) = \\int_0^1 u'(s) v'(s) \\, ds$ on the absolutely continuous functions $u$ with $u' \\in L^2$ is a resistance form with $R(x, y) = \\abs{x - y}$.\nOn a domain of $\\R^n$ with $n \\ge 2$, the Dirichlet energy $\\int \\abs{\\nabla u}^2$ does not give a resistance form.\nThere are smooth functions with arbitrarily small energy that equal $1$ at a given point $x$ and vanish at another point $y$, so clause 4 fails.\nThe definition and the main theory of resistance forms are due to Kigami.\n",
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