A Weak Topology on Metric Spaces

Abstract

We use metric functionals to construct a topology on metric spaces that induces $d$-weak convergence. We show that this topology is invariant under surjective isometries and characterize its convergence in terms of weak metrics. We prove that the topology is $T_1$ on every metric space. For normed linear spaces, we further show that it is Hausdorff and coarser than the classical weak topology. Finally, we construct a sequence in $C[0,1]$ that is unbounded in the sup-norm yet converges $d$-weakly, thereby correcting an assertion in an earlier work.

arXiv commentRevised version with new results on the separation properties of the metric-functional weak topology and its comparison with the classical weak topology on normed linear spaces. The presentation has also been improved