Weak topologies in complete metric spaces

Author:
Bijan Ahmadi Kakavandi

Journal:
Proc. Amer. Math. Soc. **141** (2013), 1029-1039

MSC (2010):
Primary 53C23; Secondary 54A10

DOI:
https://doi.org/10.1090/S0002-9939-2012-11743-5

Published electronically:
July 27, 2012

MathSciNet review:
3003694

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider some open questions concerning -convergence in complete metric spaces (i.e. Hadamard spaces). Suppose is a Hadamard space such that the sets are convex for each . We introduce a so-called half-space topology such that convergence in this topology is equivalent to -convergence for any sequence in . For a major class of Hadamard spaces, our results answer positively open questions nos. 1, 2 and 3 in [W. A. Kirk and B. Panyanak, *A concept of convergence in geodesic spaces*, Nonlinear Anal., 68 (2008) 3689-3696]. Moreover, we give a new characterization of -convergence and a new topology that we call the weak topology via a concept of a dual metric space. The relations between these topologies and the topology which is induced by the distance function have been studied. The paper concludes with some examples.

**1.**Bijan Ahmadi Kakavandi and Massoud Amini,*Duality and subdifferential for convex functions on complete 𝐶𝐴𝑇(0) metric spaces*, Nonlinear Anal.**73**(2010), no. 10, 3450–3455. MR**2680038**, https://doi.org/10.1016/j.na.2010.07.033**2.**I. D. Berg and I. G. Nikolaev,*Quasilinearization and curvature of Aleksandrov spaces*, Geom. Dedicata**133**(2008), 195–218. MR**2390077**, https://doi.org/10.1007/s10711-008-9243-3**3.**Martin R. Bridson and André Haefliger,*Metric spaces of non-positive curvature*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR**1744486****4.**Dmitri Burago, Yuri Burago, and Sergei Ivanov,*A course in metric geometry*, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR**1835418****5.**Rafa Espínola and Aurora Fernández-León,*𝐶𝐴𝑇(𝑘)-spaces, weak convergence and fixed points*, J. Math. Anal. Appl.**353**(2009), no. 1, 410–427. MR**2508878**, https://doi.org/10.1016/j.jmaa.2008.12.015**6.**Misha Gromov,*Metric structures for Riemannian and non-Riemannian spaces*, Progress in Mathematics, vol. 152, Birkhäuser Boston, Inc., Boston, MA, 1999. Based on the 1981 French original [ MR0682063 (85e:53051)]; With appendices by M. Katz, P. Pansu and S. Semmes; Translated from the French by Sean Michael Bates. MR**1699320****7.**Jürgen Jost,*Nonpositive curvature: geometric and analytic aspects*, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1997. MR**1451625****8.**W. A. Kirk and B. Panyanak,*A concept of convergence in geodesic spaces*, Nonlinear Anal.**68**(2008), no. 12, 3689–3696. MR**2416076**, https://doi.org/10.1016/j.na.2007.04.011**9.**Teck Cheong Lim,*Remarks on some fixed point theorems*, Proc. Amer. Math. Soc.**60**(1976), 179–182 (1977). MR**0423139**, https://doi.org/10.1090/S0002-9939-1976-0423139-X**10.**Zdzisław Opial,*Weak convergence of the sequence of successive approximations for nonexpansive mappings*, Bull. Amer. Math. Soc.**73**(1967), 591–597. MR**0211301**, https://doi.org/10.1090/S0002-9904-1967-11761-0**11.**E. N. Sosov,*On analogues of weak convergence in a special metric space*, Izv. Vyssh. Uchebn. Zaved. Mat.**5**(2004), 84–89 (Russian); English transl., Russian Math. (Iz. VUZ)**48**(2004), no. 5, 79–83. MR**2101682****12.**Karl-Theodor Sturm,*Probability measures on metric spaces of nonpositive curvature*, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002) Contemp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, pp. 357–390. MR**2039961**, https://doi.org/10.1090/conm/338/06080

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
53C23,
54A10

Retrieve articles in all journals with MSC (2010): 53C23, 54A10

Additional Information

**Bijan Ahmadi Kakavandi**

Affiliation:
Department of Mathematics, Shahid Beheshti University G. C., P.O. Box 1983963113, Tehran, Iran

Email:
b{\textunderscore}ahmadi@sbu.ac.ir

DOI:
https://doi.org/10.1090/S0002-9939-2012-11743-5

Keywords:
$CAT(0)$-space,
$Δ$-convergence,
$w$-convergence,
half-space topology,
weak topology

Received by editor(s):
July 31, 2011

Published electronically:
July 27, 2012

Communicated by:
Alexander N. Dranishnikov

Article copyright:
© Copyright 2012
American Mathematical Society