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  "body": "\n## Problem\n\nLet $G_1$ be a group and let $G_2$ be a group with a metric $d$.\nGiven $\\varepsilon \u003e 0$, does there exist $\\delta \u003e 0$ with the following property?\nIf a map $h \\colon G_1 \\to G_2$ satisfies $d(h(xy), h(x) h(y)) \u003c \\delta$ for all $x, y \\in G_1$, then there is a [[group homomorphism|0JYTG5K9Y4]] $H \\colon G_1 \\to G_2$ with $d(h(x), H(x)) \u003c \\varepsilon$ for all $x \\in G_1$.\n\n## Remarks\n\nThe question asks when an approximate homomorphism lies close to an exact one.\nUlam posed it in a talk at the University of Wisconsin in 1940.\nD. H. Hyers answered it in 1941 for additive maps between Banach spaces, with $\\delta = \\varepsilon$.\nThis answer is [[Hyers' theorem|VFRJ0P2P15]].\nThe answer depends on the groups: for some pairs of groups the property fails, so the problem is a family of questions rather than one.\nQuestions of this kind for other functional equations form the theory of [[Hyers–Ulam stability|TPYXQTYND6]].\n\n## Explanations\n\n### L1 human\n\nSuppose a map respects the group law only up to a small error.\nUlam asked whether such a map must be close to a map that respects the law exactly.\nFor additive maps between Banach spaces the answer is yes, and this started a large field.\n",
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      "first_stated": "1940",
      "posed_by": "Stanisław Ulam"
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