Linearization of functions

Abstract

Given a space F(U ) of functions f : U −→ C which are continuous, we construct another space F∗(U ) and a map e : U −→ F∗(U )linearizing all functions f ∈ F(U )(i.e. there are Lf ∈ F∗(U ) such that Lf ◦ e = f ). Such linearizations are stronger than mere preduals for F(U ), for example for F(U ) = 1, linearizations correspond to preduals of 1 which are isomorphic to c0. We also address the vector-valued case. A number of such linearizing constructions are to be found in the literarture, mostly for certain spaces of holomorphic functions. The procedure presented here generalizes all these special cases.