Projective tensor product

In functional analysis, an area of mathematics, the projective tensor product of two locally convex topological vector spaces is a natural topological vector space structure on their tensor product.Namely, given locally convex topological vector spacesa locally convex topological vector space such that the canonical mapand called the projective tensor product ofbe locally convex topological vector spaces.is the unique locally convex topological vector space with underlying vector spaceis the balanced convex hull of the setis generated by the collection of such tensor products of the seminorms on[3] Throughout, all spaces are assumed to be locally convex.denotes the completion of the projective tensor product ofcan always be linearly embedded as a dense vector subspace of some complete locally convex TVS, which is generally denoted by, namely, the space of continuous bilinear forms[9] In a Hausdorff locally convex space[10] The following fundamental result in the theory of topological tensor products is due to Alexander Grothendieck.be metrizable locally convex TVSs and letis the sum of an absolutely convergent seriesThe next theorem shows that it is possible to make the representation of) be a balanced open neighborhood of the origin inbe a compact subset of the convex balanced hull ofdenote the families of all bounded subsets ofis the space of continuous bilinear formsthe topology of uniform convergence on sets inwhich is also called the topology of bi-bounded convergence.This is equivalent to the problem: Given a bounded subsetis a subset of the closed convex hull ofGrothendieck proved that these topologies are equal whenThey are also equal when both spaces are Fréchet with one of them being nuclear.be a locally convex topological vector space and letbe locally convex topological vector spaces withThen, denoting strong dual spaces with a subscripted
functional analysismathematicslocally convex topological vector spacestensor productstrongestuniversal propertyseminormsbalancedHausdorffFréchet spacesbarelledcoarsercomplemented subspacesBanach spacesAlexander Grothendieckabsolutely convergentnull sequencesstrong topologyDF-spacesreflexive spaceLebesgue spaceInductive tensor productInjective tensor productTensor product of Hilbert spacesTopological tensor productSchaefer, Helmut H.Trèves, FrançoisGrothendieck, AlexanderTopological tensor productsnuclear spacesAuxiliary normed spacesNuclear spaceof Hilbert spacesTopologiesFredholm determinantFredholm kernelHilbert–Schmidt operatorHypocontinuityIntegralNuclearbetween Banach spacesTrace classGrothendieck trace theoremSchwartz kernel theoremtopicsglossaryBanachFréchetHilbertHölderOrliczSchwartzSobolevTopological vectorBarrelledCompleteLocally convexReflexiveSeparableHahn–BanachRiesz representationClosed graphUniform boundedness principleKrein–MilmanMin–maxGelfand–NaimarkBanach–AlaogluAdjointBoundedCompactHilbert–SchmidtNormalTransposeUnboundedUnitaryBanach 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 hypothesisDistributionGeneralized functionsApproximation propertyBalanced setChoquet theoryWeak topologyBanach–Mazur distanceTomita–Takesaki theory