that extends f and has the same Lipschitz constant as f.
Note that this result in particular applies to
Euclidean spacesEn and Em, 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 H1 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 with the
maximum norm and carries the Euclidean norm.[4] More generally, the theorem fails for equipped with any norm () (Schwartz 1969, p. 20).[2]
Explicit formulas
For an -valued function the extension is provided by where is the Lipschitz constant of on U.[5]
In general, an extension can also be written for -valued functions as where and conv(g) is the lower convex envelope of g.[6]
History
The theorem was proved by
Mojżesz David Kirszbraun, and later it was reproved by
Frederick Valentine,[7] who first proved it for the Euclidean plane.[8] Sometimes this theorem is also called Kirszbraun–Valentine theorem.