# Kirszbraun theorem

In mathematics, specifically real analysis and functional analysis, the **Kirszbraun theorem** states that if *U* is a subset of some Hilbert space *H*_{1}, and *H*_{2} is another Hilbert space, and

*f*:*U*→*H*_{2}

is a Lipschitz-continuous map, then there is a Lipschitz-continuous map

*F*:*H*_{1}→*H*_{2}

that extends *f* and has the same Lipschitz constant as *f*.

Note that this result in particular applies to Euclidean spaces **E**^{n} and **E**^{m}, and it was in this form that Kirszbraun originally formulated and proved the theorem.^{[1]} The version for Hilbert spaces can for example be found in (Schwartz 1969, p. 21).^{[2]} If *H*_{1} is a separable space (in particular, if it is a Euclidean space) the result is true in Zermelo–Fraenkel set theory; for the fully general case, it appears to need some form of the axiom of choice; the Boolean prime ideal theorem is known to be sufficient.^{[3]}

The proof of the theorem uses geometric features of Hilbert spaces; the corresponding statement for Banach spaces is not true in general, not even for finite-dimensional Banach spaces. It is for instance possible to construct counterexamples where the domain is a subset of **R**^{n} with the maximum norm and **R**^{m} carries the Euclidean norm.^{[4]} More generally, the theorem fails for equipped with any norm () (Schwartz 1969, p. 20).^{[2]}

For an **R**-valued function the extension is provided by where is f's Lipschitz constant on U.

## History

The theorem was proved by Mojżesz David Kirszbraun, and later it was reproved by Frederick Valentine,^{[5]} who first proved it for the Euclidean plane.^{[6]} Sometimes this theorem is also called **Kirszbraun–Valentine theorem**.

## References

- ↑ Kirszbraun, M. D. (1934). "Über die zusammenziehende und Lipschitzsche Transformationen".
*Fund. Math*.**22**: 77–108. - 1 2 Schwartz, J. T. (1969).
*Nonlinear functional analysis*. New York: Gordon and Breach Science. - ↑ Fremlin, D. H. (2011). "Kirszbraun's theorem" (PDF).
*Preprint*. - ↑ Federer, H. (1969).
*Geometric Measure Theory*. Berlin: Springer. p. 202. - ↑ Valentine, F. A. (1945). "A Lipschitz Condition Preserving Extension for a Vector Function".
*American Journal of Mathematics*.**67**(1): 83–93. doi:10.2307/2371917. - ↑ Valentine, F. A. (1943). "On the extension of a vector function so as to preserve a Lipschitz condition".
*Bulletin of the American Mathematical Society*.**49**: 100–108. doi:10.1090/s0002-9904-1943-07859-7. MR 0008251.