Rigged Hilbert space

In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional analysis.Such spaces were introduced to study spectral theory.They bring together the 'bound state' (eigenvector) and 'continuous spectrum', in one place.Using this notion, a version of the spectral theorem for unbounded operators on Hilbert space can be formulated.[1] "Rigged Hilbert spaces are well known as the structure which provides a proper mathematical meaning to the Dirac formulation of quantum mechanics.on the real line R, but isn't square-integrable for the usual (Lebesgue) measure on R. To properly consider this function as an eigenfunction requires some way of stepping outside the strict confines of the Hilbert space theory.This was supplied by the apparatus of distributions, and a generalized eigenfunction theory was developed in the years after 1950.[3] A rigged Hilbert space is a pair (H, Φ) with H a Hilbert space, Φ a dense subspace, such that Φ is given a topological vector space structure for which the inclusion map[4][5] Identifying H with its dual space H*, the adjoint to i is the mapIn the case of complex Hilbert spaces, we use a Hermitian inner product; it will be complex linear in u (math convention) or v (physics convention), and conjugate-linear (complex anti-linear) in the other variable.Note that even though Φ is isomorphic to Φ* (via Riesz representation) if it happens that Φ is a Hilbert space in its own right, this isomorphism is not the same as the composition of the inclusion i with its adjoint i*The concept of rigged Hilbert space places this idea in an abstract functional-analytic framework.Formally, a rigged Hilbert space consists of a Hilbert space H, together with a subspace Φ which carries a finer topology, that is one for which the natural inclusionIt is no loss to assume that Φ is dense in H for the Hilbert norm.We consider the inclusion of dual spaces H* in Φ*.The latter, dual to Φ in its 'test function' topology, is realised as a space of distributions or generalised functions of some sort, and the linear functionals on the subspace Φ of typefor v in H are faithfully represented as distributions (because we assume Φ dense).Now by applying the Riesz representation theorem we can identify H* with H. Therefore, the definition of rigged Hilbert space is in terms of a sandwich:The most significant examples are those for which Φ is a nuclear space; this comment is an abstract expression of the idea that Φ consists of test functions and Φ* of the corresponding distributions.An example of a nuclear countably Hilbert space, respectively, rigging the Hilbert space of square-integrable functions.
mathematicsdistributionsquare-integrablefunctional analysisspectral theorybound stateeigenvectorcontinuous spectrumspectral theoremunbounded operatorsDirac formulation of quantum mechanicseigenfunctiondifferential operatorreal lineLebesgueHilbert spacedistributionstopological vector spaceinclusion mapIsrael GelfandRiesz representationfiner topologyno lossdual spaceslinear functionalsRiesz representation theoremnuclear spaceSchwartz spacetempered distributionssquare-integrable functionsSobolev spacesFourier inversion theoremMinlos, R. A.Encyclopedia of MathematicsEMS PressJ. DieudonnéGel'fand, I. M.Vilenkin, N. YatopicsglossaryBanachFréchetHilbertHölderNuclearOrliczSchwartzSobolevTopological vectorBarrelledCompleteLocally convexReflexiveSeparableHahn–BanachClosed graphUniform boundedness principleKrein–MilmanMin–maxGelfand–NaimarkBanach–AlaogluAdjointBoundedCompactHilbert–SchmidtNormalTrace classTransposeUnboundedUnitaryBanach 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 hypothesisGeneralized functionsApproximation propertyBalanced setChoquet theoryWeak topologyBanach–Mazur distanceTomita–Takesaki theory*-algebrasInvolution/*-algebraB*-algebraNoncommutative topologyProjection-valued measureSpectrumSpectral radiusOperator spaceGelfand–Mazur theoremGelfand–Naimark theoremGelfand representationPolar decompositionSingular value decompositionSpectral theory of normal C*-algebrasIsospectraloperatorHermitian/Self-adjointKrein–Rutman theoremNormal eigenvalueSpectral asymmetrySpectral gapDecomposition of a spectrumContinuousDirect integralDiscreteSpectral abscissaBorel functional calculusMin-max theoremPositive operator-valued measureRiesz projectorSpectral theory of compact operatorsAmenable Banach algebraApproximate identityBanach function algebraDisk algebraNuclear C*-algebraUniform algebraAlon–Boppana boundBauer–Fike theoremNumerical rangeSchur–Horn theoremDirac spectrumEssential spectrumPseudospectrumStructure spaceShilov boundaryAbstract index groupBanach algebra cohomologyCohen–Hewitt factorization theoremExtensions of symmetric operatorsFredholm theoryLimiting absorption principleSchröder–Bernstein theorems for operator algebrasSherman–Takeda theoremUnbounded operatorWiener algebraAlmost Mathieu operatorCorona theoremHearing the shape of a drumDirichlet eigenvalueKuznetsov trace formulaLax pairProto-value functionRamanujan graphRayleigh–Faber–Krahn inequalitySpectral geometrySpectral methodSturm–Liouville theorySuperstrong approximationTransfer operatorTransform theoryWeyl lawWiener–Khinchin theoremHilbert spacesInner productL-semi-inner productPrehilbert spaceOrthogonal complementOrthonormal basisBessel's inequalityCauchy–Schwarz inequalityHilbert projection theoremParseval's identityPolarization identityCompact operator on Hilbert spaceDensely definedSelf-adjointSesquilinear formCn(K) with K compact & n<∞Segal–Bargmann F