On linearisation and uniqueness of preduals

TL;DR

This paper investigates the conditions under which spaces of scalar-valued functions admit unique preduals, focusing on strong linearisations and their extension to vector-valued cases, with implications for the structure of locally convex spaces.

Contribution

It provides sufficient conditions for lifting strong linearisations from scalar to vector-valued functions and characterizes spaces with unique preduals within certain classes.

Findings

Sufficient conditions for lifting strong linearisations.

Characterization of spaces with strongly unique preduals.

Extension of previous results on linearisations.

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 $\mathcal{F}(\Omega)$ of scalar-valued functions on a non-empty set $\Omega$ is said to admit a strong linearisation if there are 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 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 $\mathcal{F}(\Omega)$ with (strongly) unique predual in certain classes of locally convex Hausdorff spaces.