Skip to main content
SpringerLink
Log in

Topics in Weak Topologies on Banach Spaces

  • Chapter
  • First Online:
Book cover Banach Space Theory

Part of the book series: CMS Books in Mathematics ((CMSBM))

  • 4258 Accesses

Abstract

In this chapter we study the weak and weak* topologies of Banach spaces in more detail. We discuss several types of compacta (Eberlein, uniform Eberlein, scattered, Corson, and more), weakly Lindelöf determined spaces and properties of tightness in weak topologies. We discuss some applications in the structural properties of some Banach spaces.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. I. Aharoni and J. Lindenstrauss, Uniform equivalence between Banach spaces, Bull. Amer. Math. Soc. 84 (1978), 281–283.

    Article  MATH  MathSciNet  Google Scholar 

  2. K. Alster and R. Pol, On function spaces of compact subspaces of σ-products of the real line, Fund. Math. 107 (1980), 135–143.

    MathSciNet  Google Scholar 

  3. D. Amir and J. Lindenstrauss, The structure of weakly compact sets in Banach spaces, Ann. Math. 88 (1968), 35–44.

    Article  MathSciNet  Google Scholar 

  4. S. Argyros and V. Farmaki, On the structure of weakly compact subsets of Hilbert spaces and applications to the geometry of Banach spaces, Trans. Amer. Math. Soc. 289 (1985), 409–427.

    Article  MATH  MathSciNet  Google Scholar 

  5. S. Argyros and S. Mercourakis, On weakly Lindelöf Banach spaces, Rocky Mount. J. Math. 23 (1993), 395–446.

    Article  MATH  MathSciNet  Google Scholar 

  6. S. Argyros and S. Mercourakis, A note on the structure of WUR Banach spaces, Comment. Math. Univ. Carolinae 46 (2005), 399–408.

    MATH  MathSciNet  Google Scholar 

  7. S. Argyros, S. Mercourakis, and S. Negrepontis, Functional analytic properties of Corson compact spaces, Studia Math. 89 (1988), 197–229.

    MATH  MathSciNet  Google Scholar 

  8. D.P. Baturov, On subspaces of function spaces, Vestnik Moskov. Univ. Ser. Mat. no. 4 (1987), 66–69. English transl. in Moscow Univ. Math. Bull. 42 (1987). MR 89a:54018.

    Google Scholar 

  9. M. Bell and W. Marciszewski, On scattered Eberlein compact spaces, Israel J. Math. 158 (2007), 217–224.

    Article  MATH  MathSciNet  Google Scholar 

  10. S. Bellenot, R. Haydon, and E.W. Odell, Quasireflexive and tree spaces constructed in the spirit of R.C. James, Cont. Math. 85 (1989), 19–43.

    MathSciNet  Google Scholar 

  11. Y. Benyamini, M.E. Rudin, and M. Wage, Continuous images of weakly compact subsets of Banach spaces, Pacific J. Math. 70 (1977), 309–324.

    MATH  MathSciNet  Google Scholar 

  12. Y. Benyamini and T. Starbird, Embedding weakly compact sets into Hilbert space, Israel J. Math. 23 (1976), 137–141.

    Article  MATH  MathSciNet  Google Scholar 

  13. C. Bessaga and A. Pełczyński, Spaces of continuous functions IV, Studia Math. 19 (1960), 53–62.

    MATH  MathSciNet  Google Scholar 

  14. H.H. Corson, The weak topology of a Banach space, Trans. Amer. Math. Soc. 101 (1961), 1–15.

    Article  MATH  MathSciNet  Google Scholar 

  15. H.H. Corson and J. Lindenstrauss, On weakly compact subsets of Banach spaces, Proc. Amer. Math. Soc. 17 (1966), 407–412.

    Article  MATH  MathSciNet  Google Scholar 

  16. W.J. Davis and J. Lindenstrauss, On total non-norming subspaces, Proc. Amer. Math. Soc. 31 (1972), 109–111.

    Article  MATH  MathSciNet  Google Scholar 

  17. R. Deville and G. Godefroy, Some applications of projectional resolutions of identity, Proc. London Math. Soc. 67 (1993), 183–199.

    Article  MATH  MathSciNet  Google Scholar 

  18. R. Deville, G. Godefroy, and V. Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs 64, London, Logman, 1993.

    Google Scholar 

  19. N. Dunford and J.T. Schwartz, Linear operators, Part I, Interscience, New York, 1958.

    Google Scholar 

  20. G.A. Edgar and R.F. Wheeler, Topological properties of Banach spaces, Pacific J. Math. 115 (1984), 317–350.

    MATH  MathSciNet  Google Scholar 

  21. M. Fabian, Differentiability of convex functions and topology: Weak Asplund spaces, Wiley, 1997.

    Google Scholar 

  22. M. Fabian, Overclasses of the class of Radon–Nikodým compact spaces, Methods in Banach Spaces, Editons J.M.F. Castillo and W.B. Johnson, London Math. Soc. Lect. Notes No. 337, 2006, 197–214.

    MathSciNet  Google Scholar 

  23. M. Fabian, G. Godefroy, and V. Zizler, The structure of uniformly Gâteaux smooth Banach spaces, Israel J. Math. 124 (2001), 243–252.

    Article  MATH  MathSciNet  Google Scholar 

  24. G. Godefroy, Espace de Banach, existence et unicité de certains préduaux, Ann. Inst. Fourier (Grenoble) 28 (1978), 87–105.

    MATH  MathSciNet  Google Scholar 

  25. G. Godefroy, N.J. Kalton, and G. Lancien, Lipschitz isomorphisms and subspaces of c 0 (ℕ), Geom. Funct. Anal. 10 (2000), 798–820.

    Article  MATH  MathSciNet  Google Scholar 

  26. S.P. Gul’ko, On properties of subsets of Σ-products, Dokl. Akad. Nauk SSSR 237 (1977), 505–507.

    MathSciNet  Google Scholar 

  27. J. Hagler and F.E. Sullivan, Smoothness and weak star sequential compactness, Proc. Amer. Math. Soc. 78 (1980), 497–503.

    MATH  MathSciNet  Google Scholar 

  28. P. Hájek, V. Montesinos, J. Vanderwerff, and V. Zizler, Biorthogonal systems in Banach spaces, CMS Books in Mathematics, Canadian Mathematical Society, Springer, 2007.

    Google Scholar 

  29. R. Haydon, Trees in renorming theory, Proc. London Math. Soc. 78 (1999), 541–584.

    Article  MATH  MathSciNet  Google Scholar 

  30. P. Holický, M. Šmídek, and L. Zajíček, Convex functions with non Borel set of Gâteaux differentiability points, Comment. Math. Univ. Carolinae 39 (1998), 469–482.

    MATH  Google Scholar 

  31. M. Jiménez and J.P. Moreno, Renorming Banach spaces with the Mazur intersection property, J. Funct. Analysis 144 (1997), 486–504.

    Article  MATH  Google Scholar 

  32. W.B. Johnson and J. Lindenstrauss, Some remarks on weakly compactly generated Banach spaces, Israel J. Math. 17 (1974), 219–230.

    Article  MATH  MathSciNet  Google Scholar 

  33. O. Kalenda, Valdivia compacta and subspaces of \(C(K)\) spaces, Extracta Math. 14 (1999), 355–371.

    MATH  MathSciNet  Google Scholar 

  34. O. Kalenda, An example concerning Valdivia compact spaces, Serdica Math. J. 25(2) (1999), 131–140.

    MATH  MathSciNet  Google Scholar 

  35. P. Koszmider, Banach spaces of continuous functionals with few operators, Math. Ann. 330 (2004), 151–183.

    Article  MATH  MathSciNet  Google Scholar 

  36. D. Kutzarova and S. Troyanski, Reflexive Banach spaces without equivalent norms which are uniformly convex or uniformly differentiable in every direction, Studia Math. 72 (1982), 91–95.

    MATH  MathSciNet  Google Scholar 

  37. H.E. Lacey, Separable quotients of Banach spaces, An. Acad. Brasil Ci. 44 (1972), 185–189.

    MathSciNet  Google Scholar 

  38. H.E. Lacey, The isometric theory of classical Banach spaces, Springer, 1974.

    Google Scholar 

  39. W. Marciszewski, On Banach spaces \(C(K)\) isomorphic to \(c_0(\varGamma)\), Studia Math. 156 (2003), 295–302.

    Article  MATH  MathSciNet  Google Scholar 

  40. E. Michael and M.E. Rudin, A note on Eberlein compacts, Pacific J. Math. 72 (1972), 487–495.

    MathSciNet  Google Scholar 

  41. A. Moltó, V. Montesinos, J. Orihuela, and S. Troyanski, Weakly uniformly rotund Banach spaces, Comment. Math. Univ. Carolinae 39 (1998), 749–753.

    MATH  Google Scholar 

  42. I. Namioka, Separate continuity and joint continuity, Pacific J. Math. 51 (1974), 515–531.

    MATH  MathSciNet  Google Scholar 

  43. I. Namioka, Fragmentability in Banach spaces: Interaction of topologies, Lecture Notes Paseky, 1999.

    Google Scholar 

  44. I.P. Natanson, Theory of functions of real variable, Moscow, 1950.

    Google Scholar 

  45. S. Negrepontis, Banach spaces and topology, Handbook of Set-Theoretic Topology, Editors K. Kunen and J.E. Vaughan, Elsevier Science Publishers BV, 1984.

    Google Scholar 

  46. J. Orihuela, On weakly Lindelöf Banach spaces, Progress in Funct. Anal. K.D. Bierstedt, Editors J. Bonet, J. Horváth, and M. Maestre, Elsevier Science Publishers B.V., 1992.

    Google Scholar 

  47. J. Orihuela, W. Schachermayer, and M. Valdivia, Every Radon–Nikodým Corson compact is an Eberlein compact, Studia Math. 98 (1991), 157–174.

    MATH  MathSciNet  Google Scholar 

  48. J. Pelant, P. Holický, and O. Kalenda, \(C(K)\) spaces which cannot be uniformly embedded into \(c_0(\varGamma)\), Fund. Math. 192 (2006), 245–254.

    Article  MATH  MathSciNet  Google Scholar 

  49. R.R. Phelps, Convex functions, monotone operators and differentiability, Lecture Notes in Mathematics 1364, Springer, 1989.

    Google Scholar 

  50. G. Plebanek, Banach spaces of continuous functions with few operators, Math. Annalen 328(1) (2004), 151–183.

    Google Scholar 

  51. R. Pol, Concerning function spaces on separable compact spaces, Bull. de L’Acad. Polon. Serie des Sci., Math., Astr., et Phys. 25 (1977), 993–997.

    MATH  MathSciNet  Google Scholar 

  52. R. Pol, On a question of H.H. Corson and some related problems, Fund. Math. 109 (1980), 143–154.

    MATH  MathSciNet  Google Scholar 

  53. D. Preiss and P. Simon, A weakly pseudocompact subspace of a Banach space is weakly compact, Comment. Math. Univ. Carolinae 15 (1974), 603–610.

    MATH  MathSciNet  Google Scholar 

  54. V. Pták, On a theorem of W.F. Eberlein, Studia Math. 14 (1954), 276–287.

    MathSciNet  Google Scholar 

  55. E.A. Rezniczenko, Normality and collectionwise normality of function spaces, Vestnik. Mosk. Univ. Ser. Mat. (1990), 56–58. English trans. Moscow Univ. Math. Bull. 45, no. 6 (1990), 25–26. MR 92b:46003.

    Google Scholar 

  56. H.P. Rosenthal, On injective Banach spaces and the spaces \(L^\infty(\mu)\) for finite measures μ, Acta Math. 124 (1970), 205–247.

    Article  MATH  MathSciNet  Google Scholar 

  57. H.P. Rosenthal, The heredity problem for weakly compactly generated Banach spaces, Comp. Math. 28 (1974), 83–111.

    MATH  Google Scholar 

  58. H.L. Royden, Real analysis, Third Edition, Macmillan, 1988.

    Google Scholar 

  59. W. Rudin, Continuous functions on compact spaces without perfect subsets, Proc. Amer. Math. Soc. 8 (1957), 39–42.

    Article  MATH  MathSciNet  Google Scholar 

  60. J. Rychtář, Pointwise uniformly rotund norms, Proc. Amer. Math Soc. 133 (2005), 2259–2266.

    Article  MATH  MathSciNet  Google Scholar 

  61. B.E. Shapirovskii, Special types of embeddings in Tychonoff cubes: Subspaces of Σ-products and cardinal invariants, Collection, Topology VII, Colloq. Math. Soc. Janos Bolyai 23, North-Holland, 1980, 1055–1086.

    Google Scholar 

  62. C. Stegall, The Radon–Nikodým property in conjugate Banach spaces II, Trans. Amer. Math. Soc. 264 (1981), 507–519.

    MATH  MathSciNet  Google Scholar 

  63. M. Talagrand, Espaces de Banach faiblement \({{\mathbb K}}\)-analytiques, Ann. Math. 119 (1979), 407–438.

    Article  MathSciNet  Google Scholar 

  64. M. Talagrand, Serabilité vague dans l’espace des measures sur un compact, Israel J. Math. 37 (1980), 171–180.

    Article  MATH  MathSciNet  Google Scholar 

  65. M. Talagrand, Renormages de quelques \(C(K)\), Israel J. Math. 54 (1986), 327–334.

    Article  MATH  MathSciNet  Google Scholar 

  66. W.-K. Tang, Uniformly differentiable bump functions, Arch. Math. 68 (1997), 55–59.

    Article  MATH  Google Scholar 

  67. S. Todorcevic, Topics in topology, Lecture Notes in Mathematics 1652, Springer, 1997.

    Google Scholar 

  68. M. Valdivia, Some more results on weak compactness, J. Funct. Anal. 24 (1977), 1–10.

    Article  MATH  MathSciNet  Google Scholar 

  69. M. Valdivia, Projective resolutions of identity in \(C(K)\) spaces, Arch. Math. 54 (1990), 493–498.

    Article  MATH  MathSciNet  Google Scholar 

  70. J. Vanderwerff, J.H.M. Whitfield, and V. Zizler, Markushevich bases and Corson compacta in duality, Canad. J. Math. 46 (1994), 200–211.

    Article  MATH  MathSciNet  Google Scholar 

  71. P. Wojtaszczyk, Banach spaces for analysts, Cambridge Studies in Advanced Mathematics 25, 1991.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute of the Academy of Sciences of the Czech Republic, Žitná 25, Praha 1, 11567, Prague, Czech Republic

    Marián Fabian & Petr Hájek

  2. Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 16627, Prague, Czech Republic

    Petr Habala

  3. Departamento de Matematica Aplicada, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022, Valencia, Spain

    Vicente Montesinos

  4. Department of Mathematical and Statistical Sciences, University of Alberta, Central Academic Building, Edmonton, T6G 2G1, Alberta, Canada

    Václav Zizler

Authors
  1. Marián Fabian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Petr Habala
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Petr Hájek
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Vicente Montesinos
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Václav Zizler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marián Fabian .

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V. (2011). Topics in Weak Topologies on Banach Spaces. In: Banach Space Theory. CMS Books in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7515-7_14

Download citation

Publish with us

Policies and ethics

Search

Navigation