Editing Kirszbraun theorem (section)

{{short description|Mathematical theorem related to real and functional analysis}}

In [[mathematics]], specifically [[real analysis]] and [[functional analysis]], the '''Kirszbraun theorem''' states that if {{mvar|U}} is a [[subset]] of some [[Hilbert space]] {{mvar|H{{sub|1}}}}, and {{mvar|H{{sub|2}}}} is another Hilbert space, and 

:<math> f: U \rightarrow H_2</math>

is a [[Lipschitz continuity|Lipschitz-continuous]] map, then there is a Lipschitz-continuous map 

:<math>F: H_1 \rightarrow H_2</math>

that extends {{mvar|f}} and has the same Lipschitz constant as {{mvar|f}}.

Note that this result in particular applies to [[Euclidean space]]s {{math|'''E'''{{sup|''n''}}}} and {{math|'''E'''{{sup|''m''}}}}, and it was in this form that Kirszbraun originally formulated and proved the theorem.<ref>{{cite journal |first=M. D. |last=Kirszbraun |title=Über die zusammenziehende und Lipschitzsche Transformationen |journal=Fundamenta Mathematicae |volume=22 |pages=77–108 |year=1934 |doi=10.4064/fm-22-1-77-108 |doi-access=free }}</ref> The version for Hilbert spaces can for example be found in (Schwartz 1969, p.&nbsp;21).<ref name="Schwartz1969">{{cite book |author-link=Jack Schwartz |first=J. T. |last=Schwartz |title=Nonlinear functional analysis |publisher=Gordon and Breach Science |location=New York |year=1969 }}</ref> If {{mvar|H{{sub|1}}}} 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 to need some form of the axiom of choice;  the [[Boolean prime ideal theorem]] is known to be sufficient.<ref>{{cite journal |first=D. H. |last=Fremlin |year=2011 |title=Kirszbraun's theorem |journal=Preprint |url=https://www1.essex.ac.uk/maths/people/fremlin/n11706.pdf }}</ref>

The proof of the theorem uses geometric features of Hilbert spaces; the corresponding statement for [[Banach space]]s 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 <math>\mathbb{R}^n</math> with the [[Uniform norm|maximum norm]] and <math>\mathbb{R}^m</math> carries the Euclidean norm.<ref>{{cite book |first=H. |last=Federer |title=Geometric Measure Theory |url=https://archive.org/details/geometricmeasure00fede_0 |url-access=registration |publisher=Springer |location=Berlin |year=1969 |page=[https://archive.org/details/geometricmeasure00fede_0/page/202 202] }}</ref> More generally, the theorem fails for <math> \mathbb{R}^m </math> equipped with any <math> \ell_p</math> norm (<math> p \neq 2</math>) (Schwartz 1969, p.&nbsp;20).<ref name="Schwartz1969" />