On a weak topology for Hadamard spaces

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Abstract

We investigate whether existing notions of weak sequential convergence in Hadamard spaces can be induced by a topology. We provide an affirmative answer on what we call weakly proper Hadamard spaces. Several results from classical functional analysis are extended to the setting of 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. We introduce the space of geodesic segments together with a corresponding weak topology, and we show that this space is homeomorphic to its underlying Hadamard space. As an application we show the existence of a geodesic segment that acts as the direction of steepest descent for a geodesically differentiable function satisfying certain properties. Finally we compare our topology with other existing notions of weak topologies.

About the authors

Arian Bёrdёllima

Institut für Mathematik, Technische Universität Berlin

Email: berdellima@gmail.com
Doctor of Science, Scientific Employee

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