Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-25T05:29:29.014Z Has data issue: false hasContentIssue false
  • English
  • Français

Connection Between Strong and Norming Markushevich Bases in Non-Separable Banach Spaces

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  21 December 2009

G. Alexandrov  and
A. Plichko
G. Alexandrov
Affiliation:
Department of Mathematics and Informatics, University of Sofia, 5, J. Bourchier Blvd., 1164 Sofia, Bulgaria. E-mail: [email protected]
A. Plichko
Affiliation:
Department of Mathematics, Cracow University of Technology, ul. Warszawska 24, Cracow 31–155, Poland. E-mail: [email protected]
Get access

Abstract

It is shown that every Banach space with a norming Markushevich basis has a strong Markushevich basis. It is also shown that the converse is false.

MSC classification

Secondary: 46B26: Nonseparable Banach spaces 46B15: Summability and bases
Type
Research Article
Information
Mathematika , Volume 53 , Issue 2 , December 2006 , pp. 321 - 328
Copyright
Copyright © University College London 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[A1]Alexandrov, G. A., Locally uniformly convex equivalent norms in non-separable Banach spaces. Ph.D. Dissertation, Kharkiv (1980) (Russian).Google Scholar
[A2]Alexandrov, G. A., Strong M-basis and equivalent norms in non-separable Banach spaces (Russian). Godishnik Vissh. Uchebn. Zaved., Prilozhna Mat. 19 (1983), 31–44.Google Scholar
[AP]Alexandrov, G. A. and Plichko, A. N., On connection between strong M-bases and equivalent locally uniformly convex norms in Banach spaces (Russian). C. R. Acad. Bulgare Sci. 40 (1987), 15–16.Google Scholar
[AL]Argyros, S., Lambrou, M. and Longstaff, W. E., Atomic Boolean subspace lattices and applications to the theory of bases. Mem. Amer. Math. Soc. 91 (1991), 445.Google Scholar
[B]Banach, S., Théorie des opérations linéaires. Monografie Matematyczne (Warszawa-Lwów, 1932).Google Scholar
[G]Godefroy, G., Asplund spaces and decomposable nonseparable Banach spaces. Rocky Mountain J. Math. 25 (1995), 1013–1024.CrossRefGoogle Scholar
[JZ]John, K. and Zizler, V., Some remarks on non-separable Banach spaces with Markuševič basis. Comment. Math. Univ. Carol. 15 (1974), 679–691.Google Scholar
[K]Kalenda, O., M-bases in spaces of continuous functions on ordinals. Colloq. Math. 92 (2002), 179–187.CrossRefGoogle Scholar
[P1]Plichko, A. N., On projective resolutions of the identity operator and Markushevich bases (Russian). Dokl. Akad. Nauk. SSSR 263 (1982), 543–546. English transl.: Soviet Math. Dokl. 25 (1982), 386–389.Google Scholar
[P2]Plichko, A. N., Projective resolutions, Markushevich bases and equivalent norms (Russian). Matem. Zametki 34 (1983), 719–726. English transl.: Math. Notes 34 (1983), 851–855.Google Scholar
[P3]Plichko, A. N., On the bases and complements in non-separable Banach spaces. Sibirsk. Mat. Zh. 25 (1984), no. 4, 155–162 (Russian). English transl.: Sib. Mat. Zh. 25 (1984), 636–641.Google Scholar
[S]Semadeni, Z., Banach spaces of continuous functions, I. Monografie Matematyczne (Warszawa, 1971).Google Scholar
[Si]Singer, I., Bases in Banach spaces, II. Springer-Verlag (Berlin etc., 1981).CrossRefGoogle Scholar
[Sh]Sinha, D. P., On strong M-bases in Banach spaces with PRI. Collect. Math. 51 (2000), 277–284.Google Scholar
[R]Rosenthal, H. P., The heredity problem for weakly compactly generated Banach spaces. Compos. Math. 28 (1974), 83–111.Google Scholar
[T]Terenzi, P., Every separable Banach space has a bounded strong norming biorthogonal sequence which is also a Steinitz basis. Studia Math. 111 (1994), 207–222.CrossRefGoogle Scholar
[V]Valdivia, M., On certain classes of Markushevich bases. Arch. Math. (Basel) 62 (1994), 445–458.CrossRefGoogle Scholar
[VW]Vanderwer, J.., Whitfield, J. H. M. and Zizler, V., Markushevich bases and Corson compacta in duality. Canad. J. Math. 46 (1994), 200–211.CrossRefGoogle Scholar