Kirszbraun theorem

Kirszbraun theorem 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, und {Anzeigestil f:Urightarrow H_{2}} is a Lipschitz-continuous map, then there is a Lipschitz-continuous map {Anzeigestil F:H_{1}rightarrow H_{2}} 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, p. 21).[2] If H1 is a separable space (im Speziellen, 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.[3] 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 {Anzeigestil mathbb {R} ^{n}} with the maximum norm and {Anzeigestil mathbb {R} ^{m}} carries the Euclidean norm.[4] Allgemeiner, the theorem fails for {Anzeigestil mathbb {R} ^{m}} equipped with any {displaystyle ell _{p}} Norm ( {Anzeigestil pneq 2} ) (Schwartz 1969, p. 20).[2] Inhalt 1 Explicit formulas 2 Geschichte 3 Verweise 4 External links Explicit formulas For an {Anzeigestil mathbb {R} } -valued function the extension is provided by {Anzeigestil {tilde {f}}(x):=inf _{uin U}{groß (}f(u)+{Text{Lip}}(f)cdot d(x,u){groß )},} wo {Anzeigestil {Text{Lip}}(f)} is f's Lipschitz constant on U.[5] Im Algemeinen, an extension can also be written for {Anzeigestil mathbb {R} ^{m}} -valued functions as {Anzeigestil {tilde {f}}(x):=nabla _{j}({textrm {conv}}(g(x,j))(x,0)} wo {Anzeigestil g(x,j):=inf _{uin U}links{+{frac {{Text{Lip}}(f)}{2}}|x-u|^{2}Rechts}+{frac {{Text{Lip}}(f)}{2}}|x|^{2}+{Text{Lip}}(f)|j|^{2}} and conv(g) is the lower convex envelope of g.[6] History The theorem was proved by Mojżesz David Kirszbraun, and later it was reproved by Frederick Valentine,[7] who first proved it for the Euclidean plane.[8] Sometimes this theorem is also called Kirszbraun–Valentine theorem.

References ^ Kirszbraun, M. D. (1934). "Über die zusammenziehende und Lipschitzsche Transformationen". Grundlagen der Mathematik. 22: 77–108. doi:10.4064/fm-22-1-77-108. ^ Nach oben springen: a b Schwartz, J. T. (1969). Nonlinear functional analysis. New York: Gordon and Breach Science. ^ Fremlin, D. H. (2011). "Kirszbraun's theorem" (Pdf). Preprint. ^ Federer, H. (1969). Geometric Measure Theory. Berlin: Springer. p. 202. ^ McShane, E. J. (1934). "Extension of range of functions". Bulletin der American Mathematical Society. 40 (12): 837–842. ISSN 0002-9904. ^ Azagra, Daniel; Le Gruyer, Erwan; Mudarra, Carlos (2021). "Kirszbraun's Theorem via an Explicit Formula". Canadian Mathematical Bulletin. 64 (1): 142–153. doi:10.4153/S0008439520000314. ISSN 0008-4395. ^ Valentine, F. EIN. (1945). "A Lipschitz Condition Preserving Extension for a Vector Function". Amerikanisches Journal für Mathematik. 67 (1): 83–93. doi:10.2307/2371917. JSTOR 2371917. ^ Valentine, F. EIN. (1943). "On the extension of a vector function so as to preserve a Lipschitz condition". Bulletin der American Mathematical Society. 49 (2): 100–108. doi:10.1090/s0002-9904-1943-07859-7. HERR 0008251. External links Kirszbraun theorem at Encyclopedia of Mathematics. verbergen vte Funktionsanalyse (Themen – Glossar) Leerzeichen BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraisch/topologisch)lokal konvexreflexivseparable Theoreme Hahn-BanachRiesz-Darstellunggeschlossener Graphgleichmäßiges BeschränktheitsprinzipKakutani-FixpunktKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operatoren adjointboundcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebren Banach-AlgebraC*-AlgebraSpektrum einer C*-AlgebraOperator-Algebravon Gruppenalgebra einer lokalvariant-kompakten Gruppe SubraumproblemMahlersche Vermutung Anwendungen Hardy-RaumSpektraltheorie gewöhnlicher DifferentialgleichungenWärmekernindexsatzVariationsrechnungFunktionsrechnungIntegraloperatorJones-PolynomTopologische QuantenfeldtheorieNichtkommutative GeometrieRiemann-HypotheseVerteilung (oder verallgemeinerte Funktionen) Fortgeschrittene Themen Approximation PropertyBalanced SetChoquet-TheorieSchwache TopologieBanach-Mazur-AbstandTomita-Takesaki-Theorie Kategorien: Lipschitz mapsMetric geometryTheorems in real analysisTheorems in functional analysisHilbert space