Kirszbraun theorem

From Wikipedia, the free encyclopedia

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.[1] The version for Hilbert spaces can for example be found in (Schwartz 1969)[2]

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 Rn with the maximum norm and Rm carries the Euclidean norm.[3]

The theorem was proved by D. Kirszbraun, and later it was reproved by Valentine [4], who first proved it for the Euclidean plane [5]. Sometimes this theorem is also called Kirszbraun–Valentine theorem.

[edit] References

  1. ^ M. D. Kirszbraun. Über die zusammenziehende und Lipschitzsche Transformationen. Fund. Math., (22):77–108, 1934.
  2. ^ J.T. Schwartz. Nonlinear functional analysis. Gordon and Breach Science Publishers, New York, 1969.
  3. ^ H. Federer. Geometric Measure Theory. Springer, Berlin 1969. Page 202.
  4. ^ 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.
  5. ^ 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.