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  "body": "\n## Statement\n\nLet $E_1$ be a real [[normed vector space|03J5RAKB30]] and let $E_2$ be a real [[Banach space|06GZ956657]].\nLet $\\theta \\ge 0$ and $0 \\le p \u003c 1$.\nLet $f \\colon E_1 \\to E_2$ satisfy\n$$\\norm{f(x + y) - f(x) - f(y)} \\le \\theta \\left( \\norm{x}^p + \\norm{y}^p \\right) \\quad \\text{for all } x, y \\in E_1.$$\nThen there is a unique [[additive map|KF719SZD5T]] $A \\colon E_1 \\to E_2$ such that\n$$\\norm{f(x) - A(x)} \\le \\frac{2\\theta}{2 - 2^p} \\norm{x}^p \\quad \\text{for all } x \\in E_1,$$\nwhere $\\norm{0}^0$ is read as $1$.\nIt is given by $A(x) = \\lim_{n \\to \\infty} 2^{-n} f(2^n x)$.\nIf 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\nThe proof follows [[Hyers' theorem|VFRJ0P2P15]].\nSetting $x = y$ gives $\\norm{f(2x) - 2f(x)} \\le 2\\theta \\norm{x}^p$, so consecutive terms of $2^{-n} f(2^n x)$ differ by at most $\\theta \\, 2^{n(p-1)} \\norm{x}^p$.\nBecause $p \u003c 1$, these bounds form a convergent geometric series with sum $\\frac{2\\theta}{2 - 2^p}\\norm{x}^p$.\nFor $p = 0$ the hypothesis is a constant bound $2\\theta$, and the theorem gives back Hyers' theorem with $\\delta = 2\\theta$.\nExample: $f(x) = x + \\sqrt{\\abs{x}}$ on $\\R$ satisfies the hypothesis with $\\theta = 1$ and $p = 1/2$, because $0 \\le \\sqrt{\\abs{x}} + \\sqrt{\\abs{y}} - \\sqrt{\\abs{x + y}} \\le \\sqrt{\\abs{x}} + \\sqrt{\\abs{y}}$; here $A(x) = x$.\nThe hypothesis $p \u003c 1$ cannot be dropped: Z. Gajda showed that the statement fails for $p = 1$, already for maps $\\R \\to \\R$.\nT. Aoki proved the additive case in 1950, and Th. M. Rassias proved the linear case in 1978.\n\n## Explanations\n\n### L1 human\n\nHyers' theorem allows a fixed error in the rule $f(x + y) = f(x) + f(y)$.\nThis theorem allows an error that grows with the size of $x$ and $y$, but more slowly than linearly.\nThe conclusion is the same in spirit: $f$ is close to an exact additive map, with a distance that grows like $\\norm{x}^p$.\n\n### L2 human\n\nThe theorem started the study of stability with a control function, often called Hyers–Ulam–Rassias stability.\nThe same rescaling argument works for $p \u003e 1$ with $A(x) = \\lim_n 2^n f(2^{-n} x)$, and the excluded case $p = 1$ marks the boundary where the geometric series diverges.\nSimilar results hold for the Jensen, quadratic and many other [[functional equations|SRAN4EKWS0]].\n",
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