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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 $g \\colon U \\to Y$ be continuously [[Fréchet differentiable|06EH97V2A7]].\nLet $x_0 \\in U$ with $g(x_0) = 0$, and suppose that the derivative $Dg(x_0) \\colon X \\to Y$ is surjective.\nLet $M = \\set{x \\in U : g(x) = 0}$.\nThen for every $h$ in the [[kernel|0S23E6YE83]] of $Dg(x_0)$ there are $\\varepsilon \u003e 0$ and a map $r \\colon (-\\varepsilon, \\varepsilon) \\to X$ with\n$$\\lim_{t \\to 0} \\frac{\\norm{r(t)}}{\\abs{t}} = 0 \\quad \\text{and} \\quad x_0 + t h + r(t) \\in M \\text{ for all } \\abs{t} \u003c \\varepsilon.$$\nConsequently, the set of tangent vectors to $M$ at $x_0$ is exactly $\\ker Dg(x_0)$.\n\n## Remarks\n\nThe converse inclusion is easy: if $x_0 + th + r(t) \\in M$ with $r(t) = o(t)$, then differentiating $g(x_0 + th + r(t)) = 0$ at $t = 0$ gives $Dg(x_0) h = 0$.\nThe main inclusion is proved by a Newton-type iteration that corrects $x_0 + th$ back onto $M$.\nThe [[open mapping theorem|03YEM7JEDN]] gives a constant $C$ such that each $v \\in Y$ equals $Dg(x_0) w$ for some $w$ with $\\norm{w} \\le C \\norm{v}$, and this makes the corrections converge.\nWhen $\\ker Dg(x_0)$ has a closed complement, for example when $X$ is a Hilbert space, the result also follows from the implicit function theorem.\n\nSurjectivity cannot be dropped.\nFor $g \\colon \\R^2 \\to \\R$, $g(x_1, x_2) = x_1^2 + x_2^2$, one has $M = \\set{0}$ and $Dg(0) = 0$, so $\\ker Dg(0) = \\R^2$, while the only tangent vector to the single point $M$ is $0$.\n\n## Explanations\n\n### L1 human\n\nThe zero set of a nice map looks, near a regular point, like a flat space: the kernel of the derivative.\nThe theorem says that each direction in this kernel is really followed by a curve inside the zero set, up to an error that is small compared with the step.\nIt works in infinite dimensions, where no coordinates are available.\n\n### L2 human\n\nThis is the key step in the proof of [[Lagrange multipliers on Banach spaces|Q7QK3PF806]].\nIn the calculus of variations, $X$ is a function space and $g$ encodes constraints such as a fixed integral, so the theorem describes the admissible variations of a constrained problem.\nThe condition that $Dg(x_0)$ is onto plays the role of the constraint qualification.\n",
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