Injective tensor product

In mathematics, the injective tensor product of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the . Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS with any need to extend definitions (such as "differentiable at a point") from real/complex-valued functions to -valued functions. Throughout let and be topological vector spaces and be a linear map. is a topological homomorphism or homomorphism, if it is linear, continuous, and is an open map, where has the subspace topology induced by If is a subspace of then both the quotient map and the canonical injection are homomorphisms. In particular, any linear map can be canonically decomposed as follows: where defines a bijection. The set of continuous linear maps (resp. continuous bilinear maps ) will be denoted by (resp. ) where if is the scalar field then we may instead write (resp. ). The set of separately continuous bilinear maps (that is, continuous in each variable when the other variable is fixed) will be denoted by where if is the scalar field then we may instead write We will denote the continuous dual space of by or and the algebraic dual space (which is the vector space of all linear functionals on whether continuous or not) by To increase the clarity of the exposition, we use the common convention of writing elements of with a prime following the symbol (for example, denotes an element of and not, say, a derivative and the variables and need not be related in any way). Topology of uniform convergence and Mackey topology denotes the coarsest topology on making every map in continuous and or denotes endowed with this topology.