On linearisation and existence of preduals

We study the problem of existence of preduals of locally convex Hausdorff spaces. We derive necessary and sufficient conditions for the existence of a predual with certain properties of a bornological locally convex Hausdorff space X. Then we turn to the case that X=F () is a space of scalar-valued functions on a non-empty set and characterise those among them which admit a special predual, namely a strong linearisation, i. e. there are a locally convex Hausdorff space Y, a map Y and a topological isomorphism T () Y₁' such that T (f) = f for all f ().