On a weak topology for Hadamard spaces and its applications

TL;DR

This paper develops a new weak topology for Hadamard spaces, explores its properties, introduces a dual space concept, and applies these ideas to analyze geodesic segments and extend classical functional analysis results.

Contribution

It introduces a weak topology for Hadamard spaces, defines a dual space, and applies these concepts to geodesic analysis and functional analysis extensions.

Findings

Weak topology coincides with standard in Hilbert spaces

Existence of a geodesic segment as steepest descent direction

Extension of classical results to Hadamard spaces

Abstract

We investigate if an existing notion of weak sequential convergence in a Hadamard space can be induced by a topology. We provide an answer on what we call weakly proper Hadamard spaces. A notion of dual space is proposed and it is shown that our weak topology and dual space coincide with the standard ones in the case of a Hilbert space. Moreover we introduce the space of geodesic segments and a corresponding weak topology, and we show that this space is homeomorphic to its underlying Hadamard space. As an application of it we show the existence of a geodesic segment that acts as direction of steepest descent for a geodesically differentiable function whose geodesic derivative satisfies certain properties. Finally we extend several results from classical functional analysis to the setting of Hadamard spaces, and we compare our topology with other existing notions of weak topologies.