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 \(X=\mathcal {F}(\Omega )\) $X=\mathcal{F}(\mathrm{\Omega })$ is a space of scalar-valued functions on a non-empty set \(\Omega \) $\mathrm{\Omega }$ 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 \(\delta :\Omega \rightarrow Y\) $\delta :\mathrm{\Omega }\to Y$ and a topological isomorphism \(T:\mathcal {F}(\Omega )\rightarrow Y_{b}'\) $T:\mathcal{F}(\mathrm{\Omega })\to Y_b^{\prime }$ such that \(T(f)\circ \delta = f\) $T(f)\circ \delta =f$ for all \(f\in \mathcal {F}(\Omega )\) $f\in \mathcal{F}(\mathrm{\Omega })$ .