On linearisation and existence of preduals

Abstract

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 $X=\mathcal{F}(\Omega )$ is a space of scalar-valued functions on a non-empty set $\Omega$ 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 $\delta :\Omega \to Y$ and a topological isomorphism $T:\mathcal{F}(\Omega )\to Y_b^{\prime }$ such that $T(f)\circ \delta =f$ for all $f\in \mathcal{F}(\Omega )$.

The former version arXiv:2307.09167v1 of this paper is split into two parts. This is the first part. arXiv admin note: text overlap with arXiv:2307.09167