Given a vector space of continuous scalar-valued functions, , on a set , the authors give an abstract construction that allows each function in to be linearised. Specifically, they show that there is a complete locally convex space and a map such that for each in there is in with . Furthermore, if is any other space with this property, then there is a continuous injection of onto a dense subspace of . When is a Fréchet space, this mapping is an isomorphism.
The pair also allows linearisation of certain spaces of vector-valued functions. Given a locally convex space , the authors use to denote the space of all continuous functions such that belongs to for all in . They proceed to prove that each function in factors linearly through and thus can be algebraically identified with the space of continuous linear operators from into .
The linearisation process is examined in the cases of the spaces of all continuous -homogeneous polynomials, all -homogeneous integral polynomials, all holomorphic functions, all bounded holomorphic functions, all holomorphic functions of bounded type and all integral holomorphic functions. In each of these individual cases, it is shown that this abstract linearisation process yields a predual which had previously been constructed in the literature. The example of shows that the existence of the linearising pair is strictly stronger than the existence of a predual.