Injective tensor product

In mathematics, the injective tensor product is a particular topological tensor product, a topological vector space (TVS) formed by equipping the tensor product of the underlying vector spaces of two TVSs with a compatible topology.It was introduced by Alexander Grothendieck and used by him to define nuclear spaces.Injective tensor products have applications outside of nuclear spaces: as described below, many constructions of TVSs, and in particular Banach spaces, as spaces of functions or sequences amount to injective tensor products of simpler spaces.be locally convex topological vector spaces overdenotes the weak-* topology.Although written in terms of complex TVSs, results described generally also apply to the real case.is isomorphic to the (vector space) tensor productdenote the respective dual spaces with the topology of bounded convergence.is a locally convex topological vector space, then, whose basic open sets are constructed as follows.to form a locally convex TVS topology on[1][clarification needed] This topology is called theis the underlying scalar field,as above, and the topological vector space consisting ofIts norm can be expressed in terms of the (continuous) duals ofDenoting the unit balls of the dual spacesunder the injective norm is isomorphic as a topological vector space toare two linear maps between locally convex spaces.-topology is the finest locally convex topology onthat makes continuous the canonical mapand called the projective tensor product of[5] The injective and projective topologies both figure in Grothendieck's definition of nuclear spaces.consists of exactly those continuous bilinear formsfor some closed, equicontinuous subsetsrunning through a norm bounded subset of the space of Radon measures ona Banach space, certain constructions related toin Banach space theory can be realized as injective tensor products.be the space of unconditionally summable sequences indenotes the Banach space of continuous functions onbe a complete, Hausdorff, locally convex topological vector space, and let
topological tensor producttopological vector spaceAlexander Grothendiecknuclear spacesBanach spacestopological vector spacesweak-* topologytensor productinduced fromequicontinuouscompletenormed spacesTVS-embeddingsProjective tensor productfinestinductive topologyRadon measureRadon measuresunconditionally summablesequence spacesBanach space of continuous functionsDifferentiable vector-valued functions from Euclidean spacespace of k {\displaystyle k} -times continuously differentiable Y {\displaystyle Y} -valued functionsSchwartz spacesSchaefer, Helmut H.Trèves, FrançoisGrothendieck, AlexanderTopological tensor productsAuxiliary normed spacesNuclear spaceof Hilbert spacesTopologiesInductive tensor productFredholm determinantFredholm kernelHilbert–Schmidt operatorHypocontinuityIntegralNuclearbetween Banach spacesTrace classGrothendieck trace theoremSchwartz kernel theorem