Difference and functional equations Stability of functional equations Problem

Ulam's stability problem

Builds on Group homomorphism Metric space

Problem

Let $G_1$ be a group and let $G_2$ be a group with a metric $d$ . Given $\varepsilon > 0$ , does there exist $\delta > 0$ with the following property? If a map $h \colon G_1 \to G_2$ satisfies $d(h(xy), h(x) h(y)) < \delta$ for all $x, y \in G_1$ , then there is a group homomorphism $H \colon G_1 \to G_2$ with $d(h(x), H(x)) < \varepsilon$ for all $x \in G_1$ .

Remarks

The question asks when an approximate homomorphism lies close to an exact one. Ulam posed it in a talk at the University of Wisconsin in 1940. D. H. Hyers answered it in 1941 for additive maps between Banach spaces, with $\delta = \varepsilon$ . This answer is Hyers' theorem. The answer depends on the groups: for some pairs of groups the property fails, so the problem is a family of questions rather than one. Questions of this kind for other functional equations form the theory of Hyers–Ulam stability.

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Tag	`CBYB39FTFG`
URI	`m2://CBYB39FTFG`
Kind	problem
Writing	W0 stub
MSC	39B82, 39B52
Status	This copy of the site has no build data. Run `m2 status CBYB39FTFG`.
About	Hyers–Ulam stability
Uses	Group homomorphism, Metric space

Ulam's stability problem

Problem

Remarks

Reading path

Further reading

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