On linearisation and uniqueness of preduals

Abstract

We study strong linearisations and the uniqueness of preduals of locally convex Hausdorff spaces of scalar-valued functions. Strong linearisations are special preduals. A locally convex Hausdorff space $\mathcal{F}(\hat{I}\text{c◯})$ of scalar-valued functions on a non-empty set $\hat{I}\text{c◯}$ is said to admit a strong linearisation if there are a locally convex Hausdorff space $Y$, a map $\hat{I}´:\hat{I}\text{c◯}\to Y$ and a topological isomorphism $T:\mathcal{F}(\hat{I}\text{c◯})\to Y_b^{\prime }$ such that $T(f)\circ \hat{I}´=f$ for all $f\in \mathcal{F}(\hat{I}\text{c◯})$. We give sufficient conditions that allow us to lift strong linearisations from the scalar-valued to the vector-valued case, covering many previous results on linearisations, and use them to characterise the bornological spaces $\mathcal{F}(\hat{I}\text{c◯})$ with (strongly) unique predual in certain classes of locally convex Hausdorff spaces.

The former version arXiv:2307.09167v1 of this paper is split into two parts. This is the second part