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  "body": "\n## Statement\n\nLet $E_1$ be a real [[normed vector space|03J5RAKB30]], let $E_2$ be a real [[Banach space|06GZ956657]], and let $\\delta \\ge 0$.\nLet $f \\colon E_1 \\to E_2$ be [[$\\delta$-additive|J6GPWAX3X2]], that is, $\\norm{f(x + y) - f(x) - f(y)} \\le \\delta$ for all $x, y \\in E_1$.\nThen:\n\n1. For each $x \\in E_1$ the limit $A(x) = \\lim_{n \\to \\infty} 2^{-n} f(2^n x)$ exists in $E_2$.\n2. The map $A \\colon E_1 \\to E_2$ is [[additive|KF719SZD5T]] and satisfies $\\norm{f(x) - A(x)} \\le \\delta$ for all $x \\in E_1$.\n3. $A$ is the only additive map $E_1 \\to E_2$ such that $f - A$ is bounded.\n4. If moreover, for each $x \\in E_1$, the map $t \\mapsto f(tx)$ from $\\R$ to $E_2$ is continuous, then $A$ is [[linear|0NRYHJPQG2]].\n\n## Remarks\n\nSetting $x = y$ gives $\\norm{f(2x) - 2f(x)} \\le \\delta$, so consecutive terms of the sequence $2^{-n} f(2^n x)$ differ by at most $2^{-n-1}\\delta$.\nThe sequence is therefore Cauchy, it converges by completeness of $E_2$, and summing the geometric bounds gives $\\norm{f(x) - A(x)} \\le \\delta$.\nAdditivity follows from $\\norm{2^{-n}(f(2^n(x + y)) - f(2^n x) - f(2^n y))} \\le 2^{-n}\\delta$.\nFor uniqueness, the difference $D$ of two such additive maps is additive and bounded, and $D(x) = D(nx)/n \\to 0$.\nExample: $f(x) = x + \\sin x$ on $\\R$ is $3$-additive, and the theorem gives $A(x) = x$ with $\\abs{f(x) - x} \\le 1$.\nCompleteness of $E_2$ is used to make the limit exist.\nD. H. Hyers proved the theorem in 1941, answering [[Ulam's stability problem|CBYB39FTFG]] for the Cauchy equation.\n\n## Explanations\n\n### L1 human\n\nIf a map between normed spaces adds vectors correctly up to a fixed error, then it is close to a map that adds them exactly.\nThe exact map is found by scaling up, applying $f$, and scaling back down, so the fixed error becomes negligible.\nThe distance between the two maps is at most the error itself.\n\n### L2 human\n\nThe theorem shows that the Cauchy equation $f(x + y) = f(x) + f(y)$ is [[Hyers–Ulam stable|TPYXQTYND6]] with constant $1$.\nThe method, now called the direct method, works for many other equations, such as the Jensen and quadratic equations, after a suitable rescaling.\nReplacing the constant $\\delta$ by a bound that grows with $\\norm{x}$ and $\\norm{y}$ leads to the [[Aoki–Rassias theorem|93WX6PZC8R]].\n",
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