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  "body": "\n## Statement\n\nLet $X$ be a real separable [[reflexive|NN8RWPC8WG]] [[Banach space|06GZ956657]] with norm $\\norm{\\cdot}$, and let $X^*$ be its [[continuous dual space|4A82S5FVRT]].\nWrite $\\langle g, u \\rangle$ for the value of $g \\in X^*$ at $u \\in X$.\nLet $T \\colon X \\to X^*$ be a map with these three properties.\n\n1. $T$ is [[monotone|Y1M03JQKKR]]: $\\langle T(u) - T(v), u - v \\rangle \\ge 0$ for all $u, v \\in X$.\n2. $T$ is hemicontinuous: for all $u, v, w \\in X$, the map $t \\mapsto \\langle T(u + t v), w \\rangle$ is continuous on $[0, 1]$.\n3. $T$ is coercive: $\\langle T(u), u \\rangle / \\norm{u} \\to +\\infty$ as $\\norm{u} \\to \\infty$.\n\nThen for every $g \\in X^*$ the equation $T(u) = g$ has a solution $u \\in X$.\nIf moreover $\\langle T(u) - T(v), u - v \\rangle \u003e 0$ for all $u \\ne v$, then the solution is unique.\n\n## Remarks\n\nA standard proof uses the Galerkin method.\nOn an increasing sequence of finite-dimensional subspaces whose union is dense, the Brouwer fixed point theorem gives approximate solutions.\nCoercivity bounds them, reflexivity gives a weakly convergent subsequence, and Minty's trick passes to the limit.\nThe trick: for a monotone hemicontinuous $T$, if $\\langle g - T(v), u - v \\rangle \\ge 0$ for all $v \\in X$, then $T(u) = g$.\nFor $X = \\R$ the theorem says that a nondecreasing continuous function $T$ with $T(u) \\to \\pm\\infty$ as $u \\to \\pm\\infty$ takes every real value.\nCoercivity cannot be dropped: $T(u) = \\arctan u$ on $\\R$ is monotone and continuous but never takes the value $2$.\n\n## Explanations\n\n### L1 human\n\nThe theorem solves equations $T(u) = g$ where $T$ is a nonlinear operator that \"never goes backwards\" and grows fast enough at infinity.\nOn the real line this is the intermediate value theorem for an increasing function that runs from $-\\infty$ to $+\\infty$.\nThe theorem carries this simple picture to infinite-dimensional spaces of functions.\n\n### L2 human\n\nThe theorem is the nonlinear analogue of the Lax–Milgram theorem.\nA bounded linear operator given by a coercive bilinear form is a special case of a coercive monotone operator.\nIts main use is in quasilinear elliptic equations such as $-\\operatorname{div}(\\abs{\\nabla u}^{p-2} \\nabla u) = f$, whose weak form defines a monotone coercive operator from the Sobolev space $W_0^{1,p}$ to its dual for $1 \u003c p \u003c \\infty$.\n",
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