Kirszbraun theorem

In mathematics, specifically real analysis and functional analysis, the Kirszbraun theorem states that if U is a subset of some Hilbert space H1, and H2 is another Hilbert space, and

f : U → H2

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

F: H1 → H2

that extends f and has the same Lipschitz constant as f.
Note that this result in particular applies to Euclidean spaces En and Em, and it was in this form that Kirszbraun originally formulated and proved the theorem. The version for Hilbert spaces can for example be found in (Schwartz 1969, p. 21). 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...