# Tauberian Conditions for Almost Convergence in a Geodesic Metric Space

## TL;DR

This paper extends Tauberian conditions for convergence from Banach spaces to Hadamard spaces, analyzing the relations between different types of convergence and demonstrating that almost periodic sequences are almost convergent in Hadamard spaces.

## Contribution

It introduces new Tauberian conditions for almost convergence in Hadamard spaces and connects these with mean and almost convergence concepts.

## Key findings

- Extended Tauberian conditions to Hadamard spaces
- Established equivalence between almost convergence and mean convergence
- Proved that almost periodic sequences are almost convergent in Hadamard spaces

## Abstract

In the present paper, after recalling the Karcher mean in Hadamard spaces, we study the relation between convergence, almost convergence and mean convergence (respect to the defined mean) of a sequence in Hadamard spaces. These results extend Tauberian conditions from Banach spaces to Hadamard spaces. Also, we show that every almost periodic sequence in Hadamard spaces is almost convergent.