Real functions Functions of several real variables Theorem

Kirszbraun theorem

Builds on Lipschitz condition Inner product space

Statement

Let $n, m \ge 1$ , let $S \subseteq \R^n$ be any subset, and let $L \ge 0$ . Give $\R^n$ and $\R^m$ the Euclidean norm. Let $f \colon S \to \R^m$ satisfy the Lipschitz condition $\norm{f(x) - f(y)} \le L \norm{x - y}$ for all $x, y \in S$ . Then there is a map $F \colon \R^n \to \R^m$ with $F = f$ on $S$ and $\norm{F(x) - F(y)} \le L \norm{x - y}$ for all $x, y \in \R^n$ .

Remarks

The point is that the extension keeps the same Lipschitz constant $L$ . For $m = 1$ this follows from the McShane–Whitney extension. Extending each coordinate separately for $m \ge 2$ only gives the constant $\sqrt{m} \, L$ . The theorem holds more generally for maps from a subset of one Hilbert space to another Hilbert space. It can fail for norms that do not come from an inner product. One proof first extends $f$ to one extra point, which reduces to a statement about intersections of balls in Euclidean space, and then uses Zorn's lemma or a compactness argument.

Going deeper

For a student

The theorem is a basic tool in geometric measure theory, where Lipschitz maps are often defined on irregular sets. It lets one assume that a Lipschitz map is defined on all of $\R^n$ , and then apply results such as Rademacher's theorem. The key geometric step is about balls. Take two families of balls with the same radii, where the distances between centres in the second family are at most those in the first. If the first family has a common point, then so does the second.

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Tag	`2ZM39HSFTQ`
URI	`m2://2ZM39HSFTQ`
Kind	statement, theorem
Writing	W0 stub
MSC	26B35, 54C20, 46C05
Status	This copy of the site has no build data. Run `m2 status 2ZM39HSFTQ`.
About	Lipschitz condition
Uses	Lipschitz condition, Inner product space
Same as	Wikidata Q4454954 Wikipedia Kirszbraun theorem
Changed	2 November 2026

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