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  "body": "\n## Statement\n\nLet $X$ and $Y$ be [[Banach spaces|06GZ956657]], let $D \\subseteq X$ be open, and let $F \\colon D \\to Y$ be [[Fréchet differentiable|06EH97V2A7]] with derivative $F'(x)$ at $x$.\nLet $x_0 \\in D$ be a point where $F'(x_0)$ is invertible with bounded inverse.\nAssume that there are constants $\\eta \u003e 0$ and $L \u003e 0$ such that:\n\n1. $\\norm{F'(x_0)^{-1} F(x_0)} \\le \\eta$;\n2. $\\norm{F'(x_0)^{-1} (F'(x) - F'(y))} \\le L \\norm{x - y}$ for all $x, y \\in D$, a [[Lipschitz condition|0AW1E6YTJY]];\n3. $h = L \\eta \\le \\tfrac12$;\n4. the closed ball $\\bar B(x_0, t^*)$ lies in $D$, where $t^* = \\dfrac{1 - \\sqrt{1 - 2h}}{L}$.\n\nThen the [[Newton iterates|07EXVNF9VP]] $x_{k+1} = x_k - F'(x_k)^{-1} F(x_k)$ are well defined, they lie in $\\bar B(x_0, t^*)$, and they converge to a point $x^* \\in \\bar B(x_0, t^*)$ with $F(x^*) = 0$.\n\n## Remarks\n\nThe proof compares the iterates with Newton's method for the real quadratic $p(t) = \\tfrac{L}{2} t^2 - t + \\eta$, started at $t_0 = 0$.\nThose scalar iterates increase to the smallest root $t^*$ of $p$, and an induction shows $\\norm{x_{k+1} - x_k} \\le t_{k+1} - t_k$, so $(x_k)$ is a Cauchy sequence.\nThe quadratic $p$ has a real root exactly when $h \\le \\tfrac12$, which explains the condition on $h$.\nIf $h \u003c \\tfrac12$, the convergence is quadratic; if $h = \\tfrac12$, it may be only linear.\nExample: for $F(x) = x^2 - 2$ on $\\R$ and $x_0 = 3/2$, one has $\\eta = 1/12$ and $L = 2/3$ on $D = \\R$, so $h = 1/18 \\le 1/2$, and Newton's method converges to $\\sqrt 2$.\n\n## Explanations\n\n### L1 human\n\nNewton's method solves an equation by repeatedly replacing it with its linear approximation.\nIt can fail if the start is poor.\nThe Newton–Kantorovich theorem gives a test that uses only data at the starting point.\nIf the first Newton step is small compared with how fast the derivative can change, then the method converges.\nThe theorem also says in which ball the solution lies.\n\n### L2 human\n\nThe theorem works in any Banach space, so it applies to nonlinear integral and differential equations, not only to systems in $\\R^n$.\nIt is also an existence theorem: it proves that a solution exists near $x_0$ without knowing it in advance.\nThis is the basis of computer-assisted proofs, where $\\eta$ and $L$ are bounded rigorously by interval arithmetic.\nCompared with the Banach fixed point theorem, it needs a derivative but gives quadratic convergence when $h \u003c \\tfrac12$.\n",
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