Densely defined operator

In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function.In a topological sense, it is a linear operator that is defined "almost everywhere".Densely defined operators often arise in functional analysis as operations that one would like to apply to a larger class of objects than those for which they a priori "make sense".[clarification needed] A closed operator that is used in practice is often densely defined.A densely defined linear operatorfrom one topological vector space,is a linear operator that is defined on a dense linear subspaceand takes values inwhen the context makes it clear thatmight not be the set-theoretic domain ofof all real-valued, continuous functions defined on the unit interval; letdenote the subspace consisting of all continuously differentiable functions.Equipwith the supremum norminto a real Banach space.The differentiation operatoris a densely defined operator fromto itself, defined on the dense subspaceThe operatoris an example of an unbounded linear operator, sinceThis unboundedness causes problems if one wishes to somehow continuously extend the differentiation operatorThe Paley–Wiener integral, on the other hand, is an example of a continuous extension of a densely defined operator.In any abstract Wiener spacethere is a natural continuous linear operator (in fact it is the inclusion, and is an isometry) fromgoes to the equivalence classis dense inSince the above inclusion is continuous, there is a unique continuous linear extensionThis extension is the Paley–Wiener map.
mathematicsoperator theoryfunctiontopologicallinear operatorfunctional analysisa prioriclosed operatortopological vector spacedomainreal-valuedcontinuous functionscontinuously differentiable functionssupremum normBanach spacedifferentiation operatorunbounded linear operatorPaley–Wiener integralcontinuous extensionabstract Wiener spaceadjointcontinuous linear operatorisometryequivalence classBlumberg theoremClosed graph theorem (functional analysis)Linear extension (linear algebra)Partial functionHilbert spacesInner productL-semi-inner productHilbert spacePrehilbert spaceOrthogonal complementOrthonormal basisBessel's inequalityCauchy–Schwarz inequalityRiesz representationHilbert projection theoremParseval's identityPolarization identityCompact operator on Hilbert spaceHilbert–SchmidtNormalSelf-adjointSesquilinear formTrace classUnitaryCn(K) with K compact & n<∞Segal–Bargmann FAsplundBanachBanach latticeGrothendieck HilbertInner product spacePolynomiallyReflexiveStrictlyUniformlyUniformly smoothInjectiveProjectiveTensor productof Hilbert spacesBarrelledCompleteF-spaceFréchetLocally convexMinkowski functionalsMackeyMetrizableNormedQuasinormedStereotypeBanach–Mazur compactumDual spaceDual normOperatorUltraweakStrongUltrastrongUniform convergenceLinear operatorsBilinearsesquilinearBoundedClosedCompacton Hilbert spacesContinuousDensely definedkernelFunctionalspositivePseudo-monotoneNuclearStrictly singularTransposeBanach algebrasC*-algebrasOperator spaceSpectrumC*-algebraradiusSpectral theoryof ODEsSpectral theoremPolar decompositionSingular value decompositionAnderson–KadecBanach–AlaogluBanach–MazurBanach–SaksBanach–Schauder (open mapping)Banach–Steinhaus (Uniform boundedness)Closed graphClosed rangeEberlein–ŠmulianFreudenthal spectralGelfand–MazurGelfand–NaimarkGoldstineHahn–Banachhyperplane separationKrein–MilmanMackey–ArensMazur's lemmaM. Riesz extensionRiesz's lemmaSchauder fixed-pointBanach manifoldbundleBochner spaceConvex seriesDifferentiation in Fréchet spacesDerivativesGateauxfunctionalholomorphicIntegralsBochnerDunfordGelfand–PettisregulatedPaley–WienerFunctional calculusMeasuresLebesgueProjection-valuedVectorWeaklyStronglyAbsolutely convexAbsorbingAffineBalanced/CircledConvexConvex cone (subset)Linear cone (subset)RadialRadially convex/Star-shapedSymmetricZonotopeAffine hullAlgebraic interior (core)Bounding pointsConvex hullExtreme pointInteriorLinear spanMinkowski additionAbsolute continuity AC b a ( Σ ) {\displaystyle ba(\Sigma )} c spaceBanach coordinate BKBesov B p , q s ( R ) {\displaystyle B_{p,q}^{s}(\mathbb {R} )} Birnbaum–OrliczBounded variation BVBs spaceContinuous C(K) with K compact HausdorffHardy HpMorrey–Campanato L λ , p ( Ω ) {\displaystyle L^{\lambda ,p}(\Omega )} Schwartz S ( R n ) {\displaystyle S\left(\mathbb {R} ^{n}\right)} Sequence spaceSobolev Wk,pSobolev inequalityTriebel–LizorkinWiener amalgam W ( X , L p ) {\displaystyle W(X,L^{p})} Differential operatorFinite element methodMathematical formulation of quantum mechanicsOrdinary Differential Equations (ODEs)Validated numericstopicsglossaryHölderOrliczSchwartzSobolevTopological vectorSeparableUniform boundedness principleMin–maxUnboundedBanach 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 variationsIntegral linear operatorJones polynomialTopological quantum field theoryNoncommutative geometryRiemann hypothesisDistributionGeneralized functionsApproximation propertyBalanced setChoquet theoryWeak topologyBanach–Mazur distanceTomita–Takesaki theory