Metric functionals and weak convergence

TL;DR

This paper introduces a new concept of weak convergence in general metric spaces using metric functionals, and shows it aligns with classical weak convergence in normed linear spaces.

Contribution

The paper defines weak convergence via metric functionals in arbitrary metric spaces and proves its consistency with traditional weak convergence in normed spaces.

Findings

Weak convergence in metric spaces can be characterized by metric functionals.

The new notion coincides with classical weak convergence in normed linear spaces.

Provides a unified framework for weak convergence beyond linear spaces.

Abstract

We introduce a notion of weak convergence in arbitrary metric spaces. Metric functionals are key in our analysis: weak convergence of sequences in a given metric space is tested against all the metric functionals defined on said space. When restricted to bounded sequences in normed linear spaces, we prove that our notion of weak convergence agrees with the standard one.