A study on the dual of $C(X)$ with the topology of (strong) uniform convergence on a bornology

TL;DR

This paper investigates the dual space of continuous functions on a metric space under various topologies related to uniform convergence on bornologies, providing measure-theoretic decompositions and characterizations of these duals.

Contribution

It introduces a measure-theoretic decomposition of continuous linear functionals on $C(X)$ with specific topologies and characterizes bornologies for dual space equality.

Findings

Characterization of bornologies where duals coincide

Conditions for normability of the topology of uniform convergence on bounded sets

A topology on measures connected to the dual space under uniform convergence

Abstract

This article begins by deriving a measure-theoretic decomposition of continuous linear functionals on $C(X)$, the space of all real-valued continuous functions on a metric space $(X, d)$, equipped with the topology $\tau_\mathcal{B}$ of uniform convergence on a bornology $\mathcal{B}$. We characterize the bornologies for which $(C(X), \tau_{\mathcal{B}})^*=(C(X), \tau_{\mathcal{B}}^s)^*$, where $\tau_{\mathcal{B}}^s$ represents the topology of strong uniform convergence on $\mathcal{B}$. Furthermore, we examine the normability of $\tau_{ucb}$, the topology of uniform convergence on bounded subsets, on $(C(X), \tau_{\mathcal{B}})^*$, and explore its relationship with the operator norm topology. Finally, we derive a topology on measures that shares a connection with $(C(X), \tau_{\mathcal{B}})^*$ when endowed with $\tau_{ucb}$.