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, and {displaystyle f:Urightarrow H_{2}} is a Lipschitz-continuous map, then there is a Lipschitz-continuous map {displaystyle 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 (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.[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 {displaystyle mathbb {R} ^{n}} with the maximum norm and {displaystyle mathbb {R} ^{m}} carries the Euclidean norm.[4] More generally, the theorem fails for {displaystyle mathbb {R} ^{m}} equipped with any {displaystyle ell _{p}} norm ( {displaystyle pneq 2} ) (Schwartz 1969, p. 20).[2] Contents 1 Explicit formulas 2 History 3 References 4 External links Explicit formulas For an {displaystyle mathbb {R} } -valued function the extension is provided by {displaystyle {tilde {f}}(x):=inf _{uin U}{big (}f(u)+{text{Lip}}(f)cdot d(x,u){big )},} where {displaystyle {text{Lip}}(f)} is f's Lipschitz constant on U.[5] In general, an extension can also be written for {displaystyle mathbb {R} ^{m}} -valued functions as {displaystyle {tilde {f}}(x):=nabla _{y}({textrm {conv}}(g(x,y))(x,0)} where {displaystyle g(x,y):=inf _{uin U}left{+{frac {{text{Lip}}(f)}{2}}|x-u|^{2}right}+{frac {{text{Lip}}(f)}{2}}|x|^{2}+{text{Lip}}(f)|y|^{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". Fundamenta Mathematicae. 22: 77–108. doi:10.4064/fm-22-1-77-108. ^ Jump up to: 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 of the 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. A. (1945). "A Lipschitz Condition Preserving Extension for a Vector Function". American Journal of Mathematics. 67 (1): 83–93. doi:10.2307/2371917. JSTOR 2371917. ^ Valentine, F. A. (1943). "On the extension of a vector function so as to preserve a Lipschitz condition". Bulletin of the American Mathematical Society. 49 (2): 100–108. doi:10.1090/s0002-9904-1943-07859-7. MR 0008251. External links Kirszbraun theorem at Encyclopedia of Mathematics. hide vte Functional analysis (topics – glossary) Spaces BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraic/topological)locally convexreflexiveseparable Theorems Hahn–BanachRiesz representationclosed graphuniform boundedness principleKakutani fixed-pointKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operators adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebras Banach algebraC*-algebraspectrum of a C*-algebraoperator algebragroup algebra of a locally compact groupvon Neumann algebra Open problems invariant subspace problemMahler's conjecture Applications Hardy spacespectral theory of ordinary differential equationsheat kernelindex theoremcalculus of variationsfunctional calculusintegral operatorJones polynomialtopological quantum field theorynoncommutative geometryRiemann hypothesisdistribution (or generalized functions) Advanced topics approximation propertybalanced setChoquet theoryweak topologyBanach–Mazur distanceTomita–Takesaki theory Categories: Lipschitz mapsMetric geometryTheorems in real analysisTheorems in functional analysisHilbert space