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arXiv · 2307.09167

On linearisation and uniqueness of preduals

Karsten Kruse
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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.

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Karsten Kruse. 2023-07-18. On linearisation and uniqueness of preduals. https://doi.org/10.1002/mana.202400355

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