Weak topologies in metric spaces

In this talk I will talk about uniformly convex metric spaces and give some properties of those spaces. The main focus is on a notion of weak sequential convergence and on the so-called co-convex topology. Those two notions of weak convergence agree with the usual weak topology on Banach spaces. However, they might not agree in general: On CAT(0)-spaces one can show that convergence in the co-convex topology is weaker than the weak sequential convergence. I will construct a CAT(0)-space whose co-convex topology is not Hausdorff showing that the two topology are distinct.

In the end generalized barycenters of measures will be introduced. With such a notion one can proof a variant of the Banach-Saks property, i.e. every bounded sequence has a subsequence weakly sequentially converging to some point such that the sequence of (generalized) barycenters of Cesàro means strongly converges to the same limit point. Here the Cesàro means of a sequence are defined as the partial sums over the delta measures supported on the sequence.