Absorbing set
In functional analysis and related areas of mathematics an absorbing set in a vector space is a setwhich can be "inflated" or "scaled up" to eventually always include any given point of the vector space.Alternative terms are radial or absorbent set.Every neighborhood of the origin in every topological vector space is an absorbing subset.is a balanced set then this list can be extended to include: Ifto be an absorbing set, or to be a neighborhood of the origin in a topology) then this list can be extended to include: Ifthen this list can be extended to include: A set absorbing a point A set is said to absorb a pointThis notion of one set absorbing another is also used in other definitions: A subset of a topological vector spaceis called bounded if it is absorbed by every neighborhood of the origin.A set is called bornivorous if it absorbs every bounded subset.containing the origin is the one and only singleton subset that absorbs itself.is the unit circle (centered at the originIn contrast, every neighborhood of the origin absorbs every bounded subset ofif it satisfies any of the following equivalent conditions (here ordered so that each condition is an easy consequence of the previous one, starting with the definition): Ifto be absorbing) then it suffices to check any of the above conditions for all non-zerobe a linear map between vector spaces and letis a topological vector space (TVS) then any neighborhood of the origin inThis fact is one of the primary motivations for defining the property "absorbing inis a disk (a convex and balanced subset) thenis a non-convex balanced set that is not absorbing inabsorbing then the same is true of the symmetric setinto a seminormed space that carries its canonical pseduometrizable topology.as a limit point) forms a neighborhood basis of absorbing disks at the origin for this locally convex topology.is a topological vector space and if this convex absorbing subsetdoes not contain any non-trivial vector subspace thenEvery absorbing set contains the origin.is an absorbing disk in a vector spaceConsequently, if a topological vector spaceis a non-meager subset of itself (or equivalently for TVSs, if it is a Baire space) and ifnecessarily contains a non-empty open subset of
functional analysismathematicsvector spaceradialneighborhood of the origintopological vector spacereal numberscomplex numbersbalancedbalanced setbalanced hullbalanced coreabsolute valueEuclidean topologysingleton setboundedbornivorousunit circleneighborhoodbounded subsettrivial topologyvector topologyTVS-isomorphismEuclidean metricsubspace topologyclosedbarrelbarrelled spacealgebraic interiorconvexintervalquadrilateralunit ballnormed vector spaceseminormed vector spacesymmetric setabsolutely convex setMinkowski functionalseminormseminormed spacepseduometrizableneighborhood basislocally convexauxiliary normed spaceBanach spaceBanach disknon-meager subsetBaire spacetopological interiortotal setBornivorous setBounded set (topological vector space)Convex setLocally convex topological vector spaceRadial setStar domaincomplex planeBourbaki, NicolasÉléments de mathématiqueConway, JohnGraduate Texts in MathematicsSpringer-VerlagAmerican Mathematical SocietyDunford, NelsonSchwartz, Jacob T.Linear OperatorsWiley-InterscienceGrothendieck, AlexanderHogbe-Nlend, HenriLecture Notes in MathematicsKöthe, GottfriedCambridge University PressRudin, WalterMcGraw-Hill Science/Engineering/MathSchaefer, Helmut H.Schechter, EricTrèves, FrançoisWilansky, AlberttopicsglossaryBanachFréchetHilbertHölderNuclearOrliczSchwartzSobolevTopological vectorBarrelledCompleteReflexiveSeparableHahn–BanachRiesz representationClosed graphUniform boundedness principleKrein–MilmanMin–maxGelfand–NaimarkBanach–AlaogluAdjointCompactHilbert–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 polynomial