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On linearisation and uniqueness of preduals

Karsten Kruse

TL;DR

The paper develops a general framework for strong linearisations of locally convex function spaces and shows how to lift such linearisations from scalar-valued to vector-valued functions. It then uses this framework to characterize when spaces $\mathcal{F}(\Omega)$ possess (strongly) unique preduals within several natural classes $\mathcal{C}$, including complete barrelled, DF-, Fréchet, and completely normable spaces. A key technical advance is the vector-valued linearisation theorem, which yields a topological isomorphism $\mathcal{F}(\Omega,E)_{\sigma,b}\cong L_{b}(Y,E)$ whenever $E$ is complete, thereby enabling extension results and a precise equivalence theory for preduals across different linearisations. The results unify and extend numerous prior vector-valued linearisations and provide a coherent approach to uniqueness and equivalence of preduals in a broad locally convex setting, with implications for transfer of scalar results to vector-valued contexts and for extension problems. Overall, the work advances understanding of how strong linearisations govern duality structure in spaces of functions and clarifies when preduals are unique up to topological isomorphism within rich categories of locally convex spaces.

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 $\mathcal{F}(Ω)$ of scalar-valued functions on a non-empty set $Ω$ is said to admit a strong linearisation if there are a locally convex Hausdorff space $Y$, a map $δ\colonΩ\to Y$ and a topological isomorphism $T\colon\mathcal{F}(Ω)\to Y_{b}'$ such that $T(f)\circ δ= f$ for all $f\in\mathcal{F}(Ω)$. 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 $\mathcal{F}(Ω)$ with (strongly) unique predual in certain classes of locally convex Hausdorff spaces.

On linearisation and uniqueness of preduals

TL;DR

The paper develops a general framework for strong linearisations of locally convex function spaces and shows how to lift such linearisations from scalar-valued to vector-valued functions. It then uses this framework to characterize when spaces possess (strongly) unique preduals within several natural classes , including complete barrelled, DF-, Fréchet, and completely normable spaces. A key technical advance is the vector-valued linearisation theorem, which yields a topological isomorphism whenever is complete, thereby enabling extension results and a precise equivalence theory for preduals across different linearisations. The results unify and extend numerous prior vector-valued linearisations and provide a coherent approach to uniqueness and equivalence of preduals in a broad locally convex setting, with implications for transfer of scalar results to vector-valued contexts and for extension problems. Overall, the work advances understanding of how strong linearisations govern duality structure in spaces of functions and clarifies when preduals are unique up to topological isomorphism within rich categories of locally convex spaces.

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 of scalar-valued functions on a non-empty set is said to admit a strong linearisation if there are a locally convex Hausdorff space , a map and a topological isomorphism such that for all . 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 with (strongly) unique predual in certain classes of locally convex Hausdorff spaces.
Paper Structure (5 sections, 22 theorems, 39 equations)

This paper contains 5 sections, 22 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Notions and preliminaries
  3. Linearisation and uniqueness
  4. Linearisation of vector-valued functions
  5. Equivalence and uniqueness of preduals

Key Result

Proposition 2.3

Let $X$ be a dual space with quasi-barrelled predual $(Y,\varphi)$. Then the map is a topological isomorphism into, i.e. a topological isomorphism to its range.

Theorems & Definitions (57)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3: kruse2023a
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • proof
  • Example 2.8
  • ...and 47 more