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].
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