Linearization of functions (Q1434169)

Given a vector space of continuous scalar-valued functions, \({\mathcal F}(U)\), on a set \(U\), the authors give an abstract construction that allows each function in \({\mathcal F}(U)\) to be linearised. Specifically, they show that there is a complete locally convex space \({\mathcal F}_*(U)\) and a map \(e\colon {\mathcal F}_*(U)\to {\mathbb C}\) such that for each \(f\) in \({\mathcal F}(U)\) there is \(L_f\) in \({\mathcal F}_*(U)\) with \(f=L_f\circ e\). Furthermore, if \(Y\) is any other space with this property, then there is a continuous injection of \({\mathcal F}_*(U)\) onto a dense subspace of \(Y\). When \(Y\) is a Fréchet space, this mapping is an isomorphism. The pair \(({\mathcal F}_*(U),e)\) also allows linearisation of certain spaces of vector-valued functions. Given a locally convex space \(F\), the authors use \(\omega {\mathcal F}(U,F)\) to denote the space of all continuous functions \(g\colon U\to F\) such that \(\gamma\circ g\) belongs to \({\mathcal F}(U)\) for all \(\gamma\) in \(F'\). They proceed to prove that each function in \(\omega{\mathcal F}(U,F)\) factors linearly through \(e\) and thus \(\omega{\mathcal F}(U,F)\) can be algebraically identified with the space of continuous linear operators from \({\mathcal F}_*(U)\) into \(F\). The linearisation process is examined in the cases of the spaces of all continuous \(n\)-homogeneous polynomials, all \(n\)-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 \(\ell_1\) shows that the existence of the linearising pair \(({\mathcal F}_*(U),e)\) is strictly stronger than the existence of a predual.