Skip to main content
SpringerLink
Log in

Analytic sets of Banach spaces

Conjuntos analíticos de espacios de banach

  • Published:
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas Aims and scope Submit manuscript

Abstract

We state and comment a number of open problems on the descriptive complexity of some natural classes of Banach spaces. We characterize the separable dual spaces on which the set of equivalent dual norms is analytic.

Resumen

Se exponen y comentan problemas abiertos sobre complejidad descriptiva en algunas clases naturales de espacios de Banach. Se caracterizan los espacios con dual separable para los que el conjunto de normas duales equivalentes es analítico

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Albiac, F. and Kalton, N. J., (2006). Topics in Banach space theory, Graduate Texts in Mathematics 233, Springer-Verlag.

  2. Alspach, D., (2000). The dual of the Bourgain-Delbaen spaces, Israel J. Math., 117, 1, 239–259. DOI: 10.1007/ BF02773572

    Article  MATH  MathSciNet  Google Scholar 

  3. Argyros, S. and Dodos, P., (2007). Genericity and amalgamation of classes of Banach spaces, Adv. Math., 209, 2, 666–748. DOI: 10.1016/j.aim.2006.05.013

    Article  MATH  MathSciNet  Google Scholar 

  4. Borel-Mathurin, L., (2009). Personnal communication.

  5. Bossard, B., (1996). Coanalytic families of norms on a separable Banach space, Illinois J. Math., 40, 2, 162–181.

    MATH  MathSciNet  Google Scholar 

  6. Bossard, B., (2002). A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces, Fund. Math., 172, 2, 117–152. DOI: 10.4064/fm172-2-3

    Article  MATH  MathSciNet  Google Scholar 

  7. Bossard, B.; Godefroy, G. and Kaufman, R., (1996). Hurewicz’s theorems and renormings of Banach spaces, J. Funct. Anal., 140, 1, 142–150. DOI: 10.1006/jfan.1996.0102

    Article  MATH  MathSciNet  Google Scholar 

  8. Cowell, M. and Kalton, N. J., Almost isometric embeddings into spaces with unconditional bases, Preprint, to appear.

  9. Dodos, P., Definability under duality, Houston J. Math., to appear.

  10. Dodos, P. and Ferenczi, V., (2007). Some strongly bounded classes of Banach spaces, Fund. Math., 193, 2, 171–179. DOI: 10.4064/fm193-2-5

    Article  MATH  MathSciNet  Google Scholar 

  11. Finet, C., (1984). Une classe d’espaces de Banach à prédual unique, Q. J. Math., 35, 4, 403–414. DOI: 10.1093/ qmath/35.4.403

    Article  MATH  MathSciNet  Google Scholar 

  12. Godefroy, G., (1984). Quelques remarques sur l’unicité des préduaux, Q. J. Math., 35, 147–152.

    Article  MATH  MathSciNet  Google Scholar 

  13. Godefroy, G., (1989). Existence and uniqueness of isometric preduals, a survey, Proceedings of the Iowa workshop on Banach spaces, Contemp. Math., 85, 131–194.

    MathSciNet  Google Scholar 

  14. Godefroy, G., (2007). Isometric embeddings and universal spaces, Extracta Math., 22, 2, 179–189.

    MATH  MathSciNet  Google Scholar 

  15. Godefroy, G., Descriptive set theory and the geometry of Banach spaces, in Perspectives in Mathematical Sciences, N. S. N. Sastry Ed., World Scientific, to appear.

  16. Godefroy, G. and Kalton, N. J., (1989). The ball topology and its applications, Proceedings of the Iowa workshop on Banach spaces, Contemp. Math., 85, 195–238.

    MathSciNet  Google Scholar 

  17. Godefroy, G. and Kalton, N. J., (2006). Universal spaces for strictly convex Banach spaces, RACSAM, 100, 137–146.

    MATH  Google Scholar 

  18. Godefroy, G.; Kalton, N. J. and Lancien, G., (2001). Szlenk indices and uniform homeomorphisms, Trans. Amer. Math. Soc., 353, 10, 3895–3918.

    Article  MATH  MathSciNet  Google Scholar 

  19. Godefroy, G.; Kaufman, R. and Yahdi, M., (2001). The topological complexity of a natural class of norms on Banach spaces, Ann. Pure Appl. Logic, 111, 3–13. DOI: 10.1016/S0168-0072(01)00031-8

    Article  MathSciNet  Google Scholar 

  20. Hájek, P. and Lancien, G., (2007). Various slicing indices of Banach spaces, Mediterr. J. Math., 4, 2, 179–190. DOI: 10.1007/s00009-007-0111-4

    Article  MATH  MathSciNet  Google Scholar 

  21. Hájek, P.; Lancien, G. and Procházka, A., (2009).Weak* dentability indices of spaces C([0, α]), J.Math. Anal. Appl., 353, 1, 239–243. DOI: 10.1016/j.jmaa.2008.11.075

    Article  MATH  MathSciNet  Google Scholar 

  22. Huff, R., (1980). Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math., 10, 4, 743–749.

    Article  MATH  MathSciNet  Google Scholar 

  23. Johnson, W. B.; Lindenstrauss, J. and Schechtman, G., (1996). Banach spaces determined by their uniform structure, Geom. Funct. Anal., 3, 430–470.

    Article  MathSciNet  Google Scholar 

  24. Johnson, W. B. and Zippin, M., (1972). On subspaces of quotients of (ΣG n )t p and d(ΣG n)c 0, Israel J. Math., 13, 3–4, 311–316. DOI: 10.1007/BF02762805

    Article  MathSciNet  Google Scholar 

  25. Johnson, W. B. and Zheng, B., (2008). A characterization of subspaces and quotients of reflexive spaces with unconditional bases, Duke Math. J., 141, 3, 505–518. DOI: 10.1215/00127094-2007-003

    Article  MATH  MathSciNet  Google Scholar 

  26. Kalton, N. J., (2001). On subspaces of c 0 and extensions of operators into C(K) spaces, Q. J. Math., 52, 3, 313–328.

    Article  MATH  MathSciNet  Google Scholar 

  27. Kechris, A., (1995). Classical descriptive set theory, Graduate Texts in Mathematics 156, Springer-Verlag.

  28. Knaust, H.; Odell, E. and Schlumprecht, T., (1989). On asymptotic structure, Szlenk index and UKK properties in Banach spaces, Positivity, 3, 173–199.

    Article  MathSciNet  Google Scholar 

  29. Kwapien, S., (1972). Isomorphic characterization of inner product spaces by orthogonal series with vector valued coefficients, Studia Math., 44, 583–595.

    MATH  MathSciNet  Google Scholar 

  30. Lancien, G., (2006). A survey on the Szlenk index and some of its applications, RACSAM, 100, 209–235.

    MATH  MathSciNet  Google Scholar 

  31. Odell, E.; Schlumprecht, T. Zsak, A., (2007). Banach spaces of bounded Szlenk index, Studia Math., 183, 1, 63–97. DOI: 10.4064/sm183-1-4

    Article  MATH  MathSciNet  Google Scholar 

  32. Pelczynski, A., (1969). Universal bases, Studia Math., 32, 247–268.

    MATH  MathSciNet  Google Scholar 

  33. Rosenthal, H. P., (1983). A characterization of c 0 and some remarks concerning the Grothendieck property, in Longhorn Notes, Texas Funct. Anal. Seminar 1982–83 (Austin, Tx), 95–108.

  34. Samuel, C., (1983). Indices de Szlenk des C(K), Séminaire de géométrie des espaces de Banach, vol. I-II, Publi. Math. Univ. Paris 7, 81–91.

  35. Vanderwerff, J., (1992). Fréchet differentiable norms on spaces of countable dimension, Arch. Math. (Basel), 58, 5, 471–476. DOI: 10.1007/BF01190117

    MATH  MathSciNet  Google Scholar 

  36. Yahdi, M., (1998). The topological complexity of sets of convex differentiable functions, Rev. Mat. Complut., 11, 1, 79–91.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut de Mathématiques de Jussieu (IMJ), Bureau 4B3, 175, rue du Chevaleret, 75013, Paris.

    Gilles Godefroy

Authors
  1. Gilles Godefroy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gilles Godefroy.

Additional information

Dedicated to Professor Manuel Valdivia on the occasion of his 80th birthday

Submitted by José Orihuela

Rights and permissions

Reprints and permissions

About this article

Cite this article

Godefroy, G. Analytic sets of Banach spaces. RACSAM 104, 365–374 (2010). https://doi.org/10.5052/RACSAM.2010.23

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.5052/RACSAM.2010.23

Keywords

Mathematics Subject Classifications

Search

Navigation