Strong operator topology

In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space H induced by the seminorms of the form, as x varies in H. Equivalently, it is the coarsest topology such that, for each fixed x in H, the evaluation map(taking values in H) is continuous in T. The equivalence of these two definitions can be seen by observing that a subbase for both topologies is given by the sets(where T0 is any bounded operator on H, x is any vector and ε is any positive real number).This language translates into convergence properties of Hilbert space operators.

functional analysismathematicslocally convextopologybounded operatorsHilbert spaceseminormscoarsest topologysubbasestrongerweak operator topologynorm topologymeasurable functional calculuscontinuous functional calculuslinear functionalsconvex setStrongly continuous semigroupTopologies on the set of operators on a Hilbert spaceRudin, WalterMcGraw-Hill Science/Engineering/MathSchaefer, Helmut H.Trèves, FrançoisBanach spaceAsplundBanachBanach latticeGrothendieck HilbertInner product spacePolarization identityPolynomiallyReflexiveL-semi-inner productStrictlyUniformlyUniformly smoothInjectiveProjectiveTensor productof Hilbert spacesBarrelledCompleteF-spaceFréchetMinkowski functionalsMackeyMetrizableNormedQuasinormedStereotypeBanach–Mazur compactumDual spaceDual normOperatorUltraweakStrongUltrastrongUniform convergenceLinear operatorsAdjointBilinearsesquilinearBoundedClosedCompacton Hilbert spacesContinuousDensely definedkernelHilbert–SchmidtFunctionalspositivePseudo-monotoneNormalNuclearSelf-adjointStrictly singularTrace classTransposeUnitaryOperator theoryBanach algebrasC*-algebrasOperator spaceSpectrumC*-algebraradiusSpectral theoryof ODEsSpectral theoremPolar decompositionSingular value decompositionAnderson–KadecBanach–AlaogluBanach–MazurBanach–SaksBanach–Schauder (open mapping)Banach–Steinhaus (Uniform boundedness)Bessel's inequalityCauchy–Schwarz inequalityClosed graphClosed rangeEberlein–ŠmulianFreudenthal spectralGelfand–MazurGelfand–NaimarkGoldstineHahn–Banachhyperplane separationKrein–MilmanMackey–ArensMazur's lemmaM. Riesz extensionParseval's identityRiesz's lemmaRiesz representationSchauder fixed-pointAbstract Wiener spaceBanach 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)} Segal–Bargmann FSequence 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 numericsHilbert spacesInner productPrehilbert spaceOrthogonal complementOrthonormal basisHilbert projection theoremCompact operator on Hilbert spaceSesquilinear formCn(K) with K compact & n<∞topicsglossaryHölderOrliczSchwartzSobolevTopological vectorSeparable