On the diameter of the Banach-Mazur set

Gilles Godefroy

Czechoslovak Mathematical Journal (2010)

  • Volume: 60, Issue: 1, page 95-100
  • ISSN: 0011-4642

Abstract

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On every subspace of l ∞ ( ℕ ) which contains an uncountable ω -independent set, we construct equivalent norms whose Banach-Mazur distance is as large as required. Under Martin’s Maximum Axiom (MM), it follows that the Banach-Mazur diameter of the set of equivalent norms on every infinite-dimensional subspace of l ∞ ( ℕ ) is infinite. This provides a partial answer to a question asked by Johnson and Odell.

How to cite

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Godefroy, Gilles. "On the diameter of the Banach-Mazur set." Czechoslovak Mathematical Journal 60.1 (2010): 95-100. <http://eudml.org/doc/37991>.

@article{Godefroy2010,
abstract = {On every subspace of $l_\{\infty \}(\mathbb \{N\})$ which contains an uncountable $\omega $-independent set, we construct equivalent norms whose Banach-Mazur distance is as large as required. Under Martin’s Maximum Axiom (MM), it follows that the Banach-Mazur diameter of the set of equivalent norms on every infinite-dimensional subspace of $l_\{\infty \}(\mathbb \{N\})$ is infinite. This provides a partial answer to a question asked by Johnson and Odell.},
author = {Godefroy, Gilles},
journal = {Czechoslovak Mathematical Journal},
keywords = {Banach-Mazur diameter; elastic Banach spaces; Martin's Maximum axiom; Banach-Mazur diameter; elastic Banach space; Martin's maximum axiom},
language = {eng},
number = {1},
pages = {95-100},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the diameter of the Banach-Mazur set},
url = {http://eudml.org/doc/37991},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Godefroy, Gilles
TI - On the diameter of the Banach-Mazur set
JO - Czechoslovak Mathematical Journal
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 1
SP - 95
EP - 100
AB - On every subspace of $l_{\infty }(\mathbb {N})$ which contains an uncountable $\omega $-independent set, we construct equivalent norms whose Banach-Mazur distance is as large as required. Under Martin’s Maximum Axiom (MM), it follows that the Banach-Mazur diameter of the set of equivalent norms on every infinite-dimensional subspace of $l_{\infty }(\mathbb {N})$ is infinite. This provides a partial answer to a question asked by Johnson and Odell.
LA - eng
KW - Banach-Mazur diameter; elastic Banach spaces; Martin's Maximum axiom; Banach-Mazur diameter; elastic Banach space; Martin's maximum axiom
UR - http://eudml.org/doc/37991
ER -

References

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  1. Bačák, M., Hájek, P., 10.1016/j.jfa.2008.05.016, J. Funct. Anal. 255 (2008), 2090-2094. (2008) MR2462587DOI10.1016/j.jfa.2008.05.016
  2. Finet, C., Godefroy, G., 10.1090/conm/085/983383, Contemp. Math. 85 (1989), 87-110. (1989) Zbl0684.46016MR0983383DOI10.1090/conm/085/983383
  3. Foreman, M., Magidor, M., Shelah, S., 10.2307/1971415, Ann. Math. 127 (1988), 1-47. (1988) Zbl0645.03028MR0924672DOI10.2307/1971415
  4. Fremlin, D. H., Sersouri, A., 10.1093/qmath/39.3.323, Q. J. Math. 39 (1988), 323-331. (1988) Zbl0662.46018MR0957274DOI10.1093/qmath/39.3.323
  5. Godefroy, G., Louveau, A., 10.1007/BF02764903, Isr. J. Math. 67 (1989), 109-116. (1989) Zbl0712.46004MR1021365DOI10.1007/BF02764903
  6. Godefroy, G., Talagrand, M., 10.1007/BF02760538, Isr. J. Math. 41 (1982), 321-330. (1982) Zbl0498.46016MR0657864DOI10.1007/BF02760538
  7. Granero, A. S., Jimenez-Sevilla, M., Montesinos, A., Moreno, J. P., Plichko, A. N., 10.4064/sm157-2-1, Stud. Math. 157 (2003), 97-120. (2003) MR1980708DOI10.4064/sm157-2-1
  8. Hájek, P., Santalucia, V. Montesinos, Vanderwerff, J., Zízler, V., Biorthogonal Systems in Banach Spaces. CMS Books in Mathematics, Springer New York (2008). (2008) MR2359536
  9. Jimenéz-Sevilla, M., Moreno, J. P., 10.1006/jfan.1996.3014, J. Funct. Anal. 144 (1997), 486-504. (1997) MR1432595DOI10.1006/jfan.1996.3014
  10. Johnson, W. B., Odell, E., 10.4007/annals.2005.162.423, Ann. Math. 162 (2005), 423-437. (2005) Zbl1098.46011MR2178965DOI10.4007/annals.2005.162.423
  11. Kalton, N. J., 10.1090/conm/085/983390, Contemp. Math. 85 (1989), 319-323. (1989) Zbl0678.46011MR0983390DOI10.1090/conm/085/983390
  12. Negrepontis, S., Banach Spaces and Topology. Handbook of Set-Theoretic Topology, K. Kunen, J. E. Vaughan North-Holland Amsterdam (1984), 1045-1142. (1984) MR0776642
  13. Todorcevic, S., 10.1007/s00208-006-0762-7, Math. Ann. 335 (2006), 687-715. (2006) Zbl1112.46015MR2221128DOI10.1007/s00208-006-0762-7
  14. Todorcevic, S., 10.1007/s00208-006-0762-7, Math. Ann. 335 (2006), 687-715. (2006) Zbl1112.46015MR2221128DOI10.1007/s00208-006-0762-7

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