Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T15:58:10.833Z Has data issue: false hasContentIssue false
  • English
  • Français

Strongly Extreme Points and Approximation Properties

Published online by Cambridge University Press:  20 November 2018

Trond A. Abrahamsen ,
Petr Hájek ,
Olav Nygaard  and
Stanimir L. Troyanski
Trond A. Abrahamsen
Affiliation:
Department of Mathematics, University of Agder, Postboks 422, 4604 Kristiansand, Norway, e-mail: trond.a.abrahamsen@uia.no , olav.nygaard@uia.no
Petr Hájek
Affiliation:
Mathematical Institute, Czech Academy of Science, Žitná 25, 115 67 Praha 1, Czech Republic and Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Zikova, 4, 160 00, Prague, Czech Republic, e-mail : hajek@math.cas.cz
Olav Nygaard
Affiliation:
Department of Mathematics, University of Agder, Postboks 422, 4604 Kristiansand, Norway, e-mail: trond.a.abrahamsen@uia.no , olav.nygaard@uia.no
Stanimir L. Troyanski
Affiliation:
Institute of Mathematics and Informatics, Bulgarian Academy of Science, bl.8, acad. G. Bonchev str. 1113 Sofia, Bulgaria and Departamento de Matématicas, Universidad de Murcia, Campus de Espinardo, 30100 Espinardo (Murcia), Spain, e-mail : stroya@um.es
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that if $x$ is a strongly extreme point of a bounded closed convex subset of a Banach space and the identity has a geometrically and topologically good enough local approximation at $x$, then $x$ is already a denting point. It turns out that such an approximation of the identity exists at any strongly extreme point of the unit ball of a Banach space with the unconditional compact approximation property. We also prove that every Banach space with a Schauder basis can be equivalently renormed to satisfy the suõcient conditions mentioned.

Keywords

46B2046B04denting pointstrongly extreme pointunconditional compact approximation property
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 61 , Issue 3 , 01 September 2018 , pp. 449 - 457
Copyright
Copyright © Canadian Mathematical Society 2018

References

[1] Abrahamsen, T. A., Häjek, P., Nygaard, O., Talponen, J., and Troyanski, S., Diameter 2 properties and convexity. Studia Math. 232(2016), no. 3, 227–242.Google Scholar
[2] Casazza, P. G. and Kaiton, N. J., Notes on approximation properties in separable Banach Spaces. In: Geometry of Banach Spaces, Proc. Conf. Strobl 1989, London Mathematical Society Lecture Note Series, 158, Cambridge University Press, Cambridge, 1990, pp. 49–63.Google Scholar
[3] Deville, R., Godefroy, G., and Zizler, V., Smoothness and renormings in Banach Spaces. Pitman Monographs and Surveys in Pure and Applied Mathematics, 64, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1993.Google Scholar
[4] Godefroy, G. and Kaiton, N. J., Approximating sequences and bidual projections. Quart. J. Math. Oxford Ser. 48(1997), no. 190, 179–202. http://dx.doi.Org/10.1093/qmath/48.2.179Google Scholar
[5] Godefroy, G., Kaiton, N. J., and Li, D., On subspaces of L1 which embed into l1 . J. Reine Angew. Math. 471(1996), 43–75.Google Scholar
[6] Godefroy, G., Kaiton, N. J., and Saphar, P. D., Unconditional Ideals in Banach Spaces. Studia Math. 104(1993), 13–59.Google Scholar
[7] Johnson, W. B. and J. Lindenstrauss, Handbook of the geometry of Banach Spaces. Vol. I, North-Holland, Amsterdam, 2001. http://dx.doi.org/10.1016/S1874-5849(01)80003-6Google Scholar
[8] Kunen, K. and Rosenthal, H., Martingaleproofs of some geometrical results in Banach Space theory. Pacific J. Math. 100(1982), no. 1, 153–175. http://dx.doi.Org/10.2140/pjm.1982.100.1 53Google Scholar
[9] Lin, B.-L., Lin, P.-K., and Troyanski, S. L., Characterizations of dentingpoints. Proc. Amer. Math. Soc. 102(1988), 526–528. http://dx.doi.org/10.1090/S0002-9939-1988-0928972-1Google Scholar