In the present paper, a CAT(0) space
is a geodesic space for which each geodesic triangle is at least as ‘thin’ as its comparison triangle in the Euclidean plane. A notion of convergence introduced independently by T.-C. Lim
[Proc. Am. Math. Soc. 60, 179–182 (1976; Zbl 0346.47046
)] and T. Kuczumow
[Ann. Univ. Mariae Curie-Skłodowska, Sect. A 32, 79–88 (1978; Zbl 0463.47035
)] is shown in CAT(0) spaces to be very similar to the usual weak convergence in Banach spaces. In particular, many Banach space results involving weak convergence have precise analogues in this setting. The paper ends with several open questions.