On weak convergence in metric spaces

This note gives an exposition of various extensions of the notion of weak convergence to metric spaces. They are motivated by applications, such as existence of xed points of non-expansive maps, and analysis of the defect of compactness relative to gauge groups in Banach spaces, where weak convergence is generally less useful than respectively, asymptotic centers in [14] and polar convergence in a preliminary version of [35]). The note compares notions of convergence of weak type found in literature, in particular the notion of -convergence introduced by Lim in [25], polar convergence introduced by the authors, and the modes of convergence of weak type introduced by Jost [20], Sosov [36] and Monod [28] in Hadamard spaces. Some applications of polar convergence, such as the existence of xed points for nonexpansive maps and a suitable variant of the Brezis-Lieb Lemma are produced.