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  "body": "\n## Proof\n\nWe check the definition directly: the error term is $0$.\n\n1. Let $V$, $W$ and $A$ be as in the statement, and let $x \\in V$. The domain of $A$ is $V$, which is open in $V$.\n2. Let $h \\in V$ with $h \\ne 0$. Since $A$ is a [[linear map|0RJF668P1T]], $A(x + h) = A x + A h$.\n3. So $A(x + h) - A(x) - A h = 0$, and the quotient $\\|A(x + h) - A(x) - A h\\| / \\|h\\|$ equals $0$ for each $h \\ne 0$.\n4. Hence this quotient tends to $0$ as $h \\to 0$.\n5. The map $A$ is linear and bounded by hypothesis. By the definition of the [[Fréchet derivative|08EVNYZF42]], $A$ is Fréchet differentiable at $x$ with $DA(x) = A$.\n6. Since $x \\in V$ was arbitrary, this holds at each point of $V$.\n\n## Idea\n\nA linear map is its own best linear approximation. The error $A(x + h) - A x - A h$ is exactly $0$ by linearity, so the quotient in the defining limit is $0$ for each $h$.\n",
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