Open mapping theorem (functional analysis)
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem[1] (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.The proof here uses the Baire category theorem, and completeness of bothThe proof is based on the following lemmas, which are also somewhat of independent interest.between topological vector spaces is said to be nearly open if, for each neighborhoodThe next lemma may be thought of as a weak version of the open mapping theorem.be a continuous linear map between normed spaces.In general, a continuous bijection between topological spaces is not necessarily a homeomorphism.The open mapping theorem, when it applies, implies the bijectivity is enough: Corollary (Bounded inverse theorem) — [8] A continuous bijective linear operator between Banach spaces (or Fréchet spaces) has continuous inverse.is continuous and bijective and thus is a homeomorphism by the bounded inverse theorem; in particular, it is an open mapping.As a quotient map for topological groups is open,Here is a formulation of the open mapping theorem in terms of the transpose of an operator.Hence, the above result is a variant of a special case of the closed range theorem.in the dense subspace and the sum converging in norm.The open mapping theorem may not hold for normed spaces that are not complete.Consider the space X of sequences x : N → R with only finitely many non-zero terms equipped with the supremum norm.The map T : X → X defined by is bounded, linear and invertible, but T−1 is unbounded.This does not contradict the bounded inverse theorem since X is not complete, and thus is not a Banach space.To see this, one need simply note that the sequence is an element ofThe open mapping theorem has several important consequences: The open mapping theorem does not imply that a continuous surjective linear operator admits a continuous linear section.If one drops the requirement that a section be linear, a surjective continuous linear operator between Banach spaces admits a continuous section; this is the Bartle–Graves theorem.is not essential to the proof, but completeness is: the theorem remains true in the case whenbe a continuous linear operator from a complete pseudometrizable TVSis a topological vector space (TVS) homomorphism if the induced mapOn the other hand, a more general formulation, which implies the first, can be given: Open mapping theorem[15] — Letbe a surjective linear map from a complete pseudometrizable TVSis (a closed linear operator and thus also) an open mapping.[18] Many authors use a different definition of "nearly/almost open map" that requires that the closure ofA bijective linear map is nearly open if and only if its inverse is continuous.is a continuous linear bijection from a complete Pseudometrizable topological vector space (TVS) onto a Hausdorff TVS that is a Baire space, thenWebbed spaces are a class of topological vector spaces for which the open mapping theorem and the closed graph theorem hold.
functional analysisOpen mapping theoremStefan BanachJuliusz Schauderboundedcontinuous linear operatorBanach spacessurjectiveopen mapFréchet spacesBaire category theoremcompletenessnormed vector spacenearly opennon-meagertransposeHahn–Banach theoremclosed range theoremTerence Taoclosed graph theoremsequencessupremum normcompletespace
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ℓp spacebijectiveinverse operatorsequenceF-spacespseudometrizable TVSnonmeagerF-spacekernelquotient spaceclosedisomorphismtopological vector spacesTVS homomorphismlinear mapBaire spacelocally convexbarrelled spaceclosed linear operatoralmost open maplocally convex TVSbarrelled TVSPseudometrizabletopological vector spacehomeomorphismWebbed spacesAlmost open linear mapBounded inverse theoremClosed graphClosed graph theorem (functional analysis)Open mapping theorem (complex analysis)Surjection of Fréchet spacesUrsescu theoremWebbed spaceTao, TerenceBorwein, J. M.Springer-VerlagBanach, StefanBourbaki, NicolasÉléments de mathématiqueConway, JohnGraduate Texts in MathematicsDieudonné, JeanGrothendieck, AlexanderKöthe, GottfriedCambridge University PressRudin, WalterMcGraw-Hill Science/Engineering/MathSchaefer, Helmut H.Trèves, FrançoisWilansky, AlbertPlanetMathtopicsglossaryBanachFréchetHilbertHölderNuclearOrliczSchwartzSobolevTopological vectorBarrelledReflexiveSeparableHahn–BanachRiesz representationUniform boundedness principleKrein–MilmanMin–maxGelfand–NaimarkBanach–AlaogluAdjointCompactHilbert–SchmidtNormalTrace classUnboundedUnitaryBanach algebraC*-algebraSpectrum of a C*-algebraOperator algebraGroup algebra of a locally compact groupVon Neumann algebraInvariant subspace problemMahler's conjectureHardy spaceSpectral theory of ordinary differential equationsHeat kernelIndex theoremCalculus of variationsFunctional calculusIntegral linear operatorJones polynomialTopological quantum field theoryNoncommutative geometryRiemann hypothesisDistributionGeneralized functions