A Local Hahn-Banach Theorem and Its Applications

TL;DR

This paper introduces a local version of the Hahn-Banach Theorem, applies it to Banach lattice duals, and simplifies the measure-free characterization of uniform integrability.

Contribution

It establishes a local Hahn-Banach Theorem and applies it to Banach lattices and measure theory, providing new insights and simplified proofs.

Findings

Established a local Hahn-Banach Theorem.

Applied the theorem to study uo-duals of Banach lattices.

Provided a simplified measure-free characterization of uniform integrability.

Abstract

An important consequence of the Hahn-Banach Theorem says that on any locally convex Hausdorff topological space $X$, there are sufficiently many continuous linear functionals to separate points of $X$. In the paper, we establish a `local' version of this theorem. The result is applied to study the uo-dual of a Banach lattice that was recently introduced in [3]. We also provide a simplified approach to the measure-free characterization of uniform integrability established in [8].