On linearisation, existence and uniqueness of preduals

Karsten Kruse

Published 2023-07-18Version 1

We study the problem of existence and uniqueness 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 is a locally convex Hausdorff space $Y$, a map $\delta\colon\Omega\to Y$ and a topological isomorphism $T\colon\mathcal{F}(\Omega)\to Y_{b}'$ such that $T(f)\circ \delta= f$ for all $f\in\mathcal{F}(\Omega)$. We also lift such a linearisation to the case of vector-valued functions, covering many previous results on linearisations, and use it to characterise the bornological spaces $\mathcal{F}(\Omega)$ with (strongly) unique predual in certain classes of locally convex Hausdorff spaces.