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 E^m, 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 R^m carries the Euclidean norm.^[3]

[edit] References

1. ^ M. D. Kirszbraun. Uber die zusammenziehenden und Lipschitzchen 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.