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Kirszbraun theorem
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{{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. 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. 20).<ref name="Schwartz1969" /> == Explicit formulas == For an <math>\mathbb{R}</math>-valued function the extension is provided by <math>\tilde f(x):=\inf_{u\in U}\big(f(u)+\text{Lip}(f)\cdot d(x,u)\big),</math> where <math>\text{Lip}(f)</math> is the Lipschitz constant of <math>f</math> on {{mvar|U}}.<ref>{{Cite journal |last=McShane |first=E. J. |date=1934 |title=Extension of range of functions |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-40/issue-12/Extension-of-range-of-functions/bams/1183497871.full |journal=Bulletin of the American Mathematical Society |volume=40 |issue=12 |pages=837–842 |doi=10.1090/S0002-9904-1934-05978-0 |issn=0002-9904|doi-access=free }}</ref> In general, an extension can also be written for <math>\mathbb{R}^{m}</math>-valued functions as <math>\tilde f(x):= \nabla_{y}(\textrm{conv}(g(x,y))(x,0)</math> where <math>g(x,y):=\inf_{u\in U}\left\{\langle f(u),y \rangle +\frac{\text{Lip}(f)}{2}\|x-u\|^{2}\right\}+\frac{\text{Lip}(f)}{2} \|x\|^{2}+\text{Lip}(f)\|y\|^{2}</math> and conv(''g'') is the lower convex envelope of ''g''.<ref>{{Cite journal |last1=Azagra |first1=Daniel |last2=Le Gruyer |first2=Erwan |last3=Mudarra |first3=Carlos |date=2021 |title=Kirszbraun's Theorem via an Explicit Formula |journal=[[Canadian Mathematical Bulletin]] |language=en |volume=64 |issue=1 |pages=142–153 |doi=10.4153/S0008439520000314 | doi-access=free |issn=0008-4395|arxiv=1810.10288 }}</ref> ==History== The theorem was proved by [[Mojżesz David Kirszbraun]], and later it was reproved by [[Frederick Valentine]],<ref>{{cite journal |first=F. A. |last=Valentine |title=A Lipschitz Condition Preserving Extension for a Vector Function |journal=[[American Journal of Mathematics]] |volume=67 |issue=1 |year=1945 |pages=83–93 |doi=10.2307/2371917 |jstor=2371917 }}</ref> who first proved it for the Euclidean plane.<ref>{{cite journal |first=F. A. |last=Valentine |title=On the extension of a vector function so as to preserve a Lipschitz condition |journal=Bulletin of the American Mathematical Society |volume=49 |pages=100–108 |year=1943 |issue=2 |mr=0008251 |doi=10.1090/s0002-9904-1943-07859-7|doi-access=free }}</ref> Sometimes this theorem is also called '''Kirszbraun–Valentine theorem'''. ==References== {{Reflist}} ==External links== * [https://www.encyclopediaofmath.org/index.php/Kirszbraun_theorem ''Kirszbraun theorem''] at [[Encyclopedia of Mathematics]]. {{Functional analysis}} {{DEFAULTSORT:Kirszbraun Theorem}} [[Category:Lipschitz maps]] [[Category:Metric geometry]] [[Category:Theorems in real analysis]] [[Category:Theorems in functional analysis]] [[Category:Hilbert spaces]]
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