A note on the structure of WUR Banach spaces

Spiros A. Argyros; Sophocles Mercourakis

Commentationes Mathematicae Universitatis Carolinae (2005)

  • Volume: 46, Issue: 3, page 399-408
  • ISSN: 0010-2628

Abstract

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We present an example of a Banach space E admitting an equivalent weakly uniformly rotund norm and such that there is no Φ : E → c 0 ( Γ ) , for any set Γ , linear, one-to-one and bounded. This answers a problem posed by Fabian, Godefroy, Hájek and Zizler. The space E is actually the dual space Y * of a space Y which is a subspace of a WCG space.

How to cite

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Argyros, Spiros A., and Mercourakis, Sophocles. "A note on the structure of WUR Banach spaces." Commentationes Mathematicae Universitatis Carolinae 46.3 (2005): 399-408. <http://eudml.org/doc/249543>.

@article{Argyros2005,
abstract = {We present an example of a Banach space $E$ admitting an equivalent weakly uniformly rotund norm and such that there is no $\Phi :E\rightarrow c_0(\Gamma )$, for any set $\Gamma $, linear, one-to-one and bounded. This answers a problem posed by Fabian, Godefroy, Hájek and Zizler. The space $E$ is actually the dual space $Y^*$ of a space $Y$ which is a subspace of a WCG space.},
author = {Argyros, Spiros A., Mercourakis, Sophocles},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {WCG Banach space; weakly uniformly rotund norms; tree; weakly compactly generated (WCG) Banach space; weakly uniformly rotund norm; Reznichenko trees; Hilbert generated Banach space},
language = {eng},
number = {3},
pages = {399-408},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A note on the structure of WUR Banach spaces},
url = {http://eudml.org/doc/249543},
volume = {46},
year = {2005},
}

TY - JOUR
AU - Argyros, Spiros A.
AU - Mercourakis, Sophocles
TI - A note on the structure of WUR Banach spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2005
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 46
IS - 3
SP - 399
EP - 408
AB - We present an example of a Banach space $E$ admitting an equivalent weakly uniformly rotund norm and such that there is no $\Phi :E\rightarrow c_0(\Gamma )$, for any set $\Gamma $, linear, one-to-one and bounded. This answers a problem posed by Fabian, Godefroy, Hájek and Zizler. The space $E$ is actually the dual space $Y^*$ of a space $Y$ which is a subspace of a WCG space.
LA - eng
KW - WCG Banach space; weakly uniformly rotund norms; tree; weakly compactly generated (WCG) Banach space; weakly uniformly rotund norm; Reznichenko trees; Hilbert generated Banach space
UR - http://eudml.org/doc/249543
ER -

References

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  2. Argyros S.A., Mercourakis S., Examples concerning heredity problems of WCG Banach spaces, Proc. Amer. Math. Soc. 133 (2005), 773-785. (2005) Zbl1064.46015MR2113927
  3. Benyamini Y., Rudin M.E., Wage M., Continuous images of weakly compact subsets of Banach spaces, Pacific J. Math. 70 (1977), 309-324. (1977) Zbl0374.46011MR0625889
  4. Deville R., Godefroy G., Zizler V., Smoothness and Renormings in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Math. 64, Longman Scientific & Technical, Harlow, 1993. Zbl0782.46019MR1211634
  5. Fabian M., Gateaux Differentiability of Convex Functions and Topology, John Wiley & Sons, New York, 1997. Zbl0883.46011MR1461271
  6. Fabian M., Hájek P., Zizler V., On uniform Eberlein compacta and uniformly Gateaux smooth norms, Serdica Math. J. 23 (1997), 351-362. (1997) MR1660997
  7. Fabian M., Habala P., Hájek P., Montesinos V., Pelant J., Zizler V., Functional Analysis and Infinite-Dimensional Geometry, CMS Books in Mathematics, Springer, New York, 2001. MR1831176
  8. Fabian M., Godefroy G., Hájek P., Zizler V., Hilbert generated spaces, J. Funct. Anal. 200 (2003), 301-323. (2003) Zbl1039.46015MR1979014
  9. Fabian M., Montesinos V., Zizler V., Markushevich bases and Eberlein compacts, preprint. 
  10. James R.C., A separable somewhat reflexive Banach space with non-separable dual, Bull. Amer. Math. Soc. 80 (1974), 738-743. (1974) MR0417763
  11. Johnson W.B., Lindenstrauss J., Some remarks on weakly compactly generated Banach space, Israel J. Math. 17 (1974), 219-230. (1974) MR0417760
  12. Leiderman A.G., Sokolov G.A., Adequate families of sets and Corson compacts, Comment. Math. Univ. Carolinae 25 (1984), 233-246. (1984) Zbl0586.54022MR0768810
  13. Mercourakis S., On weakly countably determined Banach spaces, Trans. Amer. Math. Soc. 300 (1987), 307-327. (1987) Zbl0621.46018MR0871678
  14. Zizler V., Nonseparable Banach spaces, Handbook of the Geometry of Banach Spaces, Vol. 2, edited by W.B. Johnson and J. Lindenstrauss, North-Holland, Amsterdam, 2003. Zbl1041.46009MR1999613

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