Weak topologies for unbounded nets in CAT($0$) spaces

TL;DR

This paper investigates weak topologies in CAT(0) spaces that enable weak convergence of unbounded nets, generalizing Hilbert space concepts and exploring their relation to local compactness.

Contribution

It introduces and analyzes two weak topologies for unbounded nets in CAT(0) spaces, extending the understanding of weak convergence beyond bounded sequences.

Findings

Two weak topologies generalize the Hilbert space weak topology.

These topologies coincide with the strong topology only in locally compact CAT(0) spaces.

The study advances the theory of convergence in non-positively curved metric spaces.

Abstract

Weak topologies that yield weak convergence for bounded sequences and nets in CAT($0$) spaces have been studied in the past. We are here concerned with weak topologies that yield weak convergence of unbounded sequences and nets. We analyze two such topologies that generalize the weak topology on Hilbert spaces and that agree with the strong topology on a CAT($0$) space if and only if the space is locally compact.