Topological tensor product

One of the original motivations for topological tensor productsis the fact that tensor products of the spaces of smooth real-valued functions oncannot be expressed as a finite linear combination of smooth functions in[1] We only get an isomorphism after constructing the topological tensor product; i.e., This article first details the construction in the Banach space case.is not a Banach space and further cases are discussed at the end.The algebraic tensor product of two Hilbert spaces A and B has a natural positive definite sesquilinear form (scalar product) induced by the sesquilinear forms of A and B.So in particular it has a natural positive definite quadratic form, and the corresponding completion is a Hilbert space A ⊗ B, called the (Hilbert space) tensor product of A and B.If the vectors ai and bj run through orthonormal bases of A and B, then the vectors ai⊗bj form an orthonormal basis of A ⊗ B.The obvious way to define the tensor product of two Banach spacesis to copy the method for Hilbert spaces: define a norm on the algebraic tensor product, then take the completion in this norm.The problem is that there is more than one natural way to define a norm on the tensor product.are Banach spaces the algebraic tensor product ofare Banach spaces, a crossnorm (or cross norm)are elements of the topological dual spaces ofIt turns out that the projective cross norm agrees with the largest cross norm ((Ryan 2002), pp.called the injective cross norm, given byNote hereby that the injective cross norm is only in some reasonable sense the "smallest".The completions of the algebraic tensor product in these two norms are called the projective and injective tensor products, and are denoted byare Hilbert spaces, the norm used for their Hilbert space tensor product is not equal to either of these norms in general.so the Hilbert space tensor product in the section above would beare arbitrary Banach spaces then for all (continuous linear) operatorsdefines a reasonable cross norm on the algebraic tensor productThe normed linear space obtained by equippingA tensor norm is defined to be a finitely generated uniform crossnorm.is an arbitrary uniform cross norm thenwe can define the corresponding family of cross norms on the algebraic tensor productThere are in general an enormous number of ways to do this.are called the projective and injective tensor products, and denoted byis a nuclear space then the natural map fromThis property characterizes nuclear spaces.
mathematicstopological vector spacesHilbert spacesnuclear spaceswell-behavedtensor productsTensor product of Hilbert spacesBanach spaceslocally convex topological vector spacesisomorphismsesquilinear formpositive definite quadratic formorthonormal basestensor producttopological dual spacesdual normInjective tensor productProjective tensor productseminormsnuclear spaceFréchet spaceFredholm kernelInductive tensor productProjective topologyGrothendieck, A.Memoirs of the American Mathematical SocietyFunctional analysistopicsglossaryBanachFréchetHilbertHölderNuclearOrliczSchwartzSobolevTopological vectorBarrelledCompleteLocally convexReflexiveSeparableHahn–BanachRiesz representationClosed 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 hypothesisDistributionGeneralized functionsApproximation propertyBalanced setChoquet theoryWeak topologyBanach–Mazur distanceTomita–Takesaki theoryAuxiliary normed spacesof Hilbert spacesTopologiesFredholm determinantHilbert–Schmidt operatorHypocontinuityIntegralbetween Banach spacesGrothendieck trace theoremSchwartz kernel theorem