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(Ω) is a space of scalar-valued functions on a non-empty set Ω 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 δΩ→ Y and a topological isomorphism T(Ω)→ Yb' such that T(f)∘ δ= f for all f(Ω).