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  "body": "\n## Statement\n\nLet $V$ and $W$ be normed vector spaces over $\\mathbb{R}$, let $U \\subseteq V$ be open, and let $f \\colon U \\to W$. Let $x \\in U$, and let $A \\colon V \\to W$ be a bounded linear map.\nThen $A$ is the [[Fréchet derivative|08EVNYZF42]] of $f$ at $x$ if and only if\n$$f(x + h) = f(x) + A h + r(h), \\qquad \\text{where } r(h) = o(\\|h\\|) \\text{ as } h \\to 0.$$\nHere $r(h) = o(\\|h\\|)$ means that for each $\\varepsilon \u003e 0$ there is a $\\delta \u003e 0$ such that $\\|r(h)\\| \\le \\varepsilon \\|h\\|$ for each $h$ with $\\|h\\| \u003c \\delta$ and $x + h \\in U$.\n\n## Remarks\n\nThe remainder $r(h)$ is defined by the equation, so the statement says that the error of the linear approximation tends to $0$ faster than $\\|h\\|$.\nThe little-o form includes $h = 0$, where $r(0) = 0$. This is why it uses $\\le$ and does not divide by $\\|h\\|$.\nThis form is often the most convenient one for proofs, for example of the [[chain rule|0EAAS2EHDR]].\n\n## Explanations\n\n### L1 human\n\nThe equation splits $f(x + h)$ into three parts: the value at $x$, a linear part in $h$, and an error.\nThe derivative condition says only that the error is tiny compared with the step: halving the step more than halves the error.\n",
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