Valdivia compacta and equivalent norms

Ondřej Kalenda

Studia Mathematica (2000)

  • Volume: 138, Issue: 2, page 179-191
  • ISSN: 0039-3223

Abstract

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We prove that the dual unit ball of a Banach space X is a Corson compactum provided that the dual unit ball with respect to every equivalent norm on X is a Valdivia compactum. As a corollary we show that the dual unit ball of a Banach space X of density ℵ 1 is Corson if (and only if) X has a projectional resolution of the identity with respect to every equivalent norm. These results answer questions asked by M. Fabian, G. Godefroy and V. Zizler and yield a converse to Amir-Lindenstrauss’ theorem.

How to cite

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Kalenda, Ondřej. "Valdivia compacta and equivalent norms." Studia Mathematica 138.2 (2000): 179-191. <http://eudml.org/doc/216697>.

@article{Kalenda2000,
abstract = {We prove that the dual unit ball of a Banach space X is a Corson compactum provided that the dual unit ball with respect to every equivalent norm on X is a Valdivia compactum. As a corollary we show that the dual unit ball of a Banach space X of density $ℵ_1$ is Corson if (and only if) X has a projectional resolution of the identity with respect to every equivalent norm. These results answer questions asked by M. Fabian, G. Godefroy and V. Zizler and yield a converse to Amir-Lindenstrauss’ theorem.},
author = {Kalenda, Ondřej},
journal = {Studia Mathematica},
keywords = {Corson compact space; Valdivia compact space; projectional resolution of the identity; countably 1-norming Markushevich basis; equivalent norm; Banach space; Corson compacta; Valdivia compacta; dual unit ball},
language = {eng},
number = {2},
pages = {179-191},
title = {Valdivia compacta and equivalent norms},
url = {http://eudml.org/doc/216697},
volume = {138},
year = {2000},
}

TY - JOUR
AU - Kalenda, Ondřej
TI - Valdivia compacta and equivalent norms
JO - Studia Mathematica
PY - 2000
VL - 138
IS - 2
SP - 179
EP - 191
AB - We prove that the dual unit ball of a Banach space X is a Corson compactum provided that the dual unit ball with respect to every equivalent norm on X is a Valdivia compactum. As a corollary we show that the dual unit ball of a Banach space X of density $ℵ_1$ is Corson if (and only if) X has a projectional resolution of the identity with respect to every equivalent norm. These results answer questions asked by M. Fabian, G. Godefroy and V. Zizler and yield a converse to Amir-Lindenstrauss’ theorem.
LA - eng
KW - Corson compact space; Valdivia compact space; projectional resolution of the identity; countably 1-norming Markushevich basis; equivalent norm; Banach space; Corson compacta; Valdivia compacta; dual unit ball
UR - http://eudml.org/doc/216697
ER -

References

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  1. [AL] D. Amir and J. Lindenstrauss, The structure of weakly compact sets in Banach spaces, Ann. of Math. 88 (1968), 35-46. Zbl0164.14903
  2. [AMN] S. Argyros, S. Mercourakis and S. Negrepontis, Functional-analytic properties of Corson-compact spaces, Studia Math. 89 (1988), 197-229. Zbl0656.46014
  3. [DG] R. Deville and G. Godefroy, Some applications of projective resolutions of identity, Proc. London Math. Soc. 67 (1993), 183-199. Zbl0798.46008
  4. [D] J. Diestel, Sequences and Series in Banach Spaces, Springer, Berlin, 1984. 
  5. [FGZ] M. Fabian, G. Godefroy and V. Zizler, A note on Asplund generated Banach spaces, Bull. Acad. Polon. Sci. 47 (1999), 221-230. Zbl0946.46016
  6. [HHZ] P. Habala, P. Hájek and V. Zizler, Introduction to Banach Spaces, lecture notes, Matfyzpress, Prague, 1996. 
  7. [K1] O. Kalenda, An example concerning Valdivia compact spaces, Serdica Math. J. 25 (1999), 131-140. Zbl0946.46020
  8. [K2] O. Kalenda, Continuous images and other topological properties of Valdivia compacta, Fund. Math. 162 (1999), 181-192. Zbl0989.54019
  9. [K3] O. Kalenda, A characterization of Valdivia compact spaces, Collect. Math. (to appear). Zbl0949.46004
  10. [K4] O. Kalenda, Valdivia compacta and subspaces of C(K) spaces, Extracta Math. (to appear). Zbl0983.46020
  11. [V1] M. Valdivia, Resolutions of the identity in certain Banach spaces, Collect. Math. 39 (1988), 127-140. Zbl0718.46006
  12. [V2] M. Valdivia, Projective resolutions of the identity in C(K) spaces, Arch. Math. (Basel) 54 (1990), 493-498. Zbl0707.46009
  13. [V3] M. Valdivia, Simultaneous resolutions of the identity operator in normed spaces, Collect. Math. 42 (1991), 265-285. Zbl0788.47024
  14. [V4] M. Valdivia, On certain compact topological spaces, Rev. Mat. Univ. Complut. Madrid 10 (1997), 81-84. Zbl0870.54025

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