Operator topologies

In the mathematical field of functional analysis there are several standard topologies which are given to the algebra B(X) of bounded linear operators on a Banach space X.The topologies listed below are all locally convex, which implies that they are defined by a family of seminorms.(In topology proper, these terms can suggest the opposite meaning, so strong and weak are replaced with, respectively, fine and coarse.)The diagram on the right is a summary of the relations, with the arrows pointing from strong to weak.On norm bounded sets of B(H), the weak (operator) and ultraweak topologies coincide.However, when H is separable, all the topologies above are metrizable when restricted to the unit ball (or to any norm-bounded subset).For example, the dual space of B(H) in the weak or strong operator topology is too small to have much analytic content.

Diagram of relations among topologies on the space $B(X)$ of bounded operators

mathematicalfunctional analysistopologiesbounded linear operatorsBanach spaceoperator normunit balluniform operator topologystrong operator topologyweak topologyweak operator topologyseminormsHilbert spacepredualnorm topologyweak (Banach space) topologyMackey topologyultrastrong topologyweak-* operator topologyBanach–Alaoglu theoremconvexfirst-countableBounded operatorContinuous linear operatorList of topologiesModes of convergenceNorm (mathematics)Topologies on spaces of linear mapsVague topologyWeak convergence (Hilbert space)AsplundBanachBanach latticeGrothendieck HilbertInner product spacePolarization identityPolynomiallyReflexiveL-semi-inner productStrictlyUniformlyUniformly smoothInjectiveProjectiveTensor productof Hilbert spacesBarrelledCompleteF-spaceFréchetLocally convexMinkowski functionalsMackeyMetrizableNormedQuasinormedStereotypeBanach–Mazur compactumDual spaceDual normUltraweakoperatorStrongUltrastrongUniform 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/CircledConvex 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<∞Duality