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. [M. D. Kirszbraun. "Über die zusammenziehende und Lipschitzsche Transformationen." Fund. Math., (22):77–108, 1934.] The version for Hilbert spaces can for example be found in (Schwartz 1969) [J.T. Schwartz. "Nonlinear functional analysis". Gordon and Breach Science Publishers, New York, 1969.]

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. [H. Federer. "Geometric Measure Theory." Springer, Berlin 1969. Page 202.]

The theorem was proved by D. Kirszbraun, and later it was reproved by Valentine [F. A. Valentine, “A Lipschitz Condition Preserving Extension for a Vector Function,” American Journal of Mathematics, Vol. 67, No. 1 (Jan., 1945), pp. 83-93. ] , who first proved it for the Euclidean plane [F. A. Valentine, “On the extension of a vector function so as to preserve a Lipschitz condition,”Bulletin of the American Mathematical Society, vol. 49, pp. 100–108, 1943. ] . Sometimes this theorem is also called Kirszbraun–Valentine theorem.

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