Geodesic
On a weak topology for Hadamard spaces
Sbornik. Mathematics, Tome 214 (2023) no. 10, pp. 1373-1389 Cet article a éte moissonné depuis la source Math-Net.Ru

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We investigate whether the existing notion of weak sequential convergence in Hadamard spaces can be induced by a topology. We provide an affirmative answer in what we call weakly proper Hadamard spaces. Several results from functional analysis are extended to the setting of Hadamard spaces. Our weak topology coincides with the usual one in the case of a Hilbert space. Finally, we compare our topology with other existing notions of weak topologies. Bibliography: 24 titles.
Keywords: weak convergence, weak topology
Mots-clés : Hadamard space. 
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A. Bёrdёllima. On a weak topology for Hadamard spaces. Sbornik. Mathematics, Tome 214 (2023) no. 10, pp. 1373-1389. http://geodesic.mathdoc.fr/item/SM_2023_214_10_a1/

[1] A. D. Alexandrov, “A theorem on triangles in a metric space and some of its applications”, Tr. Mat. Inst. Steklova, 38, Acad. Sci. USSR, Moscow, 1951, 5–23 (Russian) | MR | Zbl

[2] S. Baratella and Ng Siu-Ah, “A nonstandard proof of the Eberlein-Šmulian theorem”, Proc. Amer. Math. Soc., 131:10 (2003), 3177–3180 | DOI | MR | Zbl

[3] A. Bërdëllima, Investigations in Hadamard spaces, Ph.D. thesis, Georg-August-Univ., Göttingen, 2020, 131 pp. https://d-nb.info/1240476078/34

[4] I. D. Berg and I. G. Nikolaev, “Quasilinearization and curvature of Aleksandrov spaces”, Geom. Dedicata, 133 (2008), 195–218 | DOI | MR | Zbl

[5] M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren Math. Wiss., 319, Springer-Verlag, Berlin, 1999, xxii+643 pp. | DOI | MR | Zbl

[6] D. Burago, Yu. Burago and S. Ivanov, A course in metric geometry, Grad. Stud. Math., 33, Amer. Math. Soc., Providence, RI, 2001, xiv+415 pp. | DOI | MR | Zbl

[7] M. Bačák, Convex analysis and optimization in Hadamard spaces, De Gruyter Ser. Nonlinear Anal. Appl., 22, De Gruyter, Berlin, 2014, viii+185 pp. | DOI | MR | Zbl

[8] M. Bačák, Old and new challenges in Hadamard spaces, arXiv: 1807.01355

[9] R. Espínola and A. Fernández-León, “$\mathrm{CAT}(\kappa)$-spaces, weak convergence and fixed points”, J. Math. Anal. Appl., 353:1 (2009), 410–427 | DOI | MR | Zbl

[10] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progr. Math., 152, Transl. from the French, Birkhäuser Boston, Inc., Boston, MA, 1999, xx+585 pp. | MR | Zbl

[11] M. L. Gromov, “$\mathrm{CAT}(\kappa)$-spaces: construction and concentration”, J. Math. Sci. (N.Y.), 119:2 (2004), 178–200 | DOI | MR | Zbl

[12] J. Jost, “Equilibrium maps between metric spaces”, Calc. Var. Partial Differential Equations, 2 (1994), 173–204 | DOI | MR | Zbl

[13] B. A. Kakavandi, “Weak topologies in complete $CAT(0)$ metric spaces”, Proc. Amer. Math. Soc., 141:3 (2013), 1029–1039 | DOI | MR | Zbl

[14] M. Kell, “Uniformly convex metric spaces”, Anal. Geom. Metr. Spaces, 2:1 (2014), 359–380 | DOI | MR | Zbl

[15] J. L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto–New York–London, 1955, xiv+298 pp. | MR | Zbl

[16] Teck-Cheong Lim, “Remarks on some fixed point theorems”, Proc. Amer. Math. Soc., 60 (1976), 179–182 | DOI | MR | Zbl

[17] A. Lytchak and A. Petrunin, “Weak topology on $\mathrm{CAT}(0)$ spaces”, Israel J. Math., 255:2 (2023), 763–781 | DOI | MR | Zbl

[18] B. Mendelson, Introduction to topology, 3rd ed., Allyn and Bacon, Inc., Boston, MA, 1975, ix+206 pp. | MR | Zbl

[19] Ph. Miller, A. Bërdëllima and M. Wardetzky, Weak topologies for unbounded nets in $\mathrm{CAT}(0)$ spaces, arXiv: 2204.01291

[20] N. Monod, “Superrigidity for irreducible lattices and geometric splitting”, J. Amer. Math. Soc., 19:4 (2006), 781–814 | DOI | MR | Zbl

[21] J. R. Munkres, Topology, 2nd ed., Prentice Hail, Inc., Upper Saddle River, NJ, 2000, xvi+537 pp. | MR | Zbl

[22] S. Alexander, V. Kapovitch and A. Petrunin, An invitation to Alexandrov geometry. $\mathrm{CAT}(0)$ spaces, SpringerBriefs Math., Springer, Cham, 2019, xii+88 pp. | DOI | MR | Zbl

[23] E. N. Sosov, “Analogs of weak convergence in a special metric space”, Russian Math. (Iz. VUZ), 48:5 (2004), 79–83 | MR | Zbl

[24] R. Whitley, “An elementary proof of the Eberlein-Šmulian theorem”, Math. Ann., 172:2 (1967), 116–118 | DOI | MR | Zbl