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  "body": "\n## Statement\n\nLet $X$ and $Y$ be real [[Banach spaces|06GZ956657]], let $U \\subseteq X$ be open, and let $f \\colon U \\to \\R$ and $g \\colon U \\to Y$ be continuously [[Fréchet differentiable|06EH97V2A7]].\nLet $x_0 \\in U$ with $g(x_0) = 0$, and suppose that $x_0$ is a [[local extremum|0WFBH1Q3WC]] of the restriction of $f$ to $M = \\set{x \\in U : g(x) = 0}$.\nSuppose that the derivative $Dg(x_0) \\colon X \\to Y$ is surjective.\nThen there is a unique continuous linear functional $\\lambda \\in Y^*$, an element of the [[dual space|07JBSZE0V4]] of $Y$, such that\n$$Df(x_0) = \\lambda \\circ Dg(x_0).$$\n\n## Remarks\n\nLet $h \\in \\ker Dg(x_0)$.\nBy [[Lyusternik's theorem|3PGXH0T1K6]] there is a curve $t \\mapsto x_0 + th + r(t)$ in $M$ with $r(t) = o(t)$, and $t \\mapsto f(x_0 + th + r(t))$ has a local extremum at $t = 0$, so $Df(x_0) h = 0$.\nThus $Df(x_0)$ vanishes on $\\ker Dg(x_0)$, and $\\lambda(Dg(x_0) h) = Df(x_0) h$ is a well defined linear map on $Y$.\nIt is continuous by the [[open mapping theorem|03YEM7JEDN]], and it is unique because $Dg(x_0)$ is onto.\n\nFor $X = \\R^n$ and $Y = \\R^m$, the functional $\\lambda$ is a vector $(\\lambda_1, \\dots, \\lambda_m)$, and the result is the [[Lagrange multiplier theorem|0VKKP8X6Q6]].\n\nExample: an isoperimetric problem minimises $J(y) = \\int_a^b L(x, y, y') \\, dx$ subject to $K(y) = \\int_a^b G(x, y, y') \\, dx = \\ell$ and fixed ends.\nHere $Y = \\R$, and the result gives a number $\\lambda$ with $\\delta J(y_0; \\eta) = \\lambda \\, \\delta K(y_0; \\eta)$ for all admissible variations $\\eta$.\n\nSurjectivity is needed.\nFor $X = \\R^2$, $Y = \\R$, $f(x_1, x_2) = x_1$ and $g(x_1, x_2) = x_2^2 - x_1^3$, the point $0$ minimises $f$ on $M$, since $x_1 \\ge 0$ on $M$.\nBut $Dg(0) = 0$, so $\\lambda \\circ Dg(0) = 0 \\ne Df(0)$ for every $\\lambda$.\n\n## Explanations\n\n### L1 human\n\nAt a constrained extremum, the derivative of the objective is a combination of the derivatives of the constraints.\nIn infinite dimensions there may be infinitely many constraints, so the coefficients form a continuous linear functional on the constraint space instead of a list of numbers.\n\n### L2 human\n\nThis is the abstract form of the multiplier rule used for constrained problems in the calculus of variations and in optimal control.\nWith $Y = \\R$ it gives Euler's multiplier rule for isoperimetric problems, and with $Y$ a function space it handles pointwise constraints.\nThe onto condition on $Dg(x_0)$ is the infinite-dimensional constraint qualification; without it the multiplier may not exist, as the example shows.\n",
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