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Abstract

We survey the current state of research in renormings of C(K) spaces.

Resumen

En este artículo analizamos el estado actual de la investigaciíon en teoría del renormamiento de espacios C(K).

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References

  1. Amir, D. and Lindenstrauss, J., (1968). The structure of weakly compact subsets in Banach spaces, Ann. of Math., 88, 35–46.

    Article  MathSciNet  Google Scholar 

  2. Argyros, S. A.; Arvanitakis, A. and Mercourakis, S., (2009). Reznichenko families of trees and their applications, J. Math. Anal. Appl., 350, 2, 792–810. DOI: 10.1016/j.jmaa.2008.07.028

    Article  MATH  MathSciNet  Google Scholar 

  3. Argyros, S. A.; Dodos, P. and Kanellopoulos, V., (2008). A classification of separable Rosenthal compacta and its applications, Dissertationes Math., 449, 52 pp. DOI: 10.4064/dm449-0-1

    Google Scholar 

  4. Argyros, S. A. and Mercourakis, S., (1993). On weakly Lindelöf Banach spaces, Rocky Mountain J. Math., 23, 395–446. DOI: 10.1216/rmjm/1181072569

    Article  MATH  MathSciNet  Google Scholar 

  5. Arvanitakis, A., (2002). Some remarks on Radon-Nikodým compact spaces, Fund. Math., 172, 41–60.

    Article  MATH  MathSciNet  Google Scholar 

  6. Asplund, E., (1968). Fréchet differentiability of convex functions, Acta Math., 121, 1, 31–47. DOI: 10.1007/BF02391908

    Article  MATH  MathSciNet  Google Scholar 

  7. Bessaga, C. and Pe⦊czyński, A., (1975). Selected topics in infinite-dimensional topology, Polish Scientific Publishers, Warsaw.

    MATH  Google Scholar 

  8. Bourgain, J.; Fremlin, D. H. and Talagrand, M., (1978). Pointwise compact sets of Baire-measurable functions, Amer. J. Math., 100, 4, 845–886. DOI:

    Article  MATH  MathSciNet  Google Scholar 

  9. Burke, D. K. and Hansell, R. W., (1996). Perfect maps and relatively discrete collections, in Papers on General Topology and Applications (Amsterdam, 1994), New York Acad. Sci., New York, 54–56. DOI: 10.1111/j.1749-6632.1996.tb36796.x

    Google Scholar 

  10. Burke, M. R.; Kubiś, W. and Todorĉević, S., (2006). Kadec norms on spaces of continuous functions, Serdica Math. J., 32, 227–258.

    MATH  MathSciNet  Google Scholar 

  11. Deville, R. and Godefroy, G., (1993). Some applications of projective resolutions of identity, Proc. Lond. Math. Soc., 67, 183–199. DOI: 10.1112/plms/s3-67.1.183

    Article  MATH  MathSciNet  Google Scholar 

  12. Deville, R.; Godefroy, G. and Zizler, V., (1993). Smoothness and Renormings in Banach Spaces, Longman, Harlow.

    MATH  Google Scholar 

  13. Fabian, M., (1997). Gâteaux Differentiability of Convex Functions and Topology, John Wiley and Sons, Inc., New York.

    MATH  Google Scholar 

  14. Fabian, M.; Godefroy, G. and Zizler, V., (2001). The structure of uniformly Gâteaux smooth Banach spaces, Israel J. Math., 124, 1, 243–252. DOI:

    Article  MATH  MathSciNet  Google Scholar 

  15. Fabian, M.; Godefroy, G.; Montesinos, V. and Zizler, V., (2004). Inner characterizations of weakly compactly generated Banach spaces and their relatives, J. Math. Anal. Appl., 297, 2, 419–455. DOI: 10.1016/j.jmaa.2004.02.015

    Article  MATH  MathSciNet  Google Scholar 

  16. Fetter, H. and Gamboa De Buen, B., (1997). The James Forest, London Math. Soc. Lecture Note Ser., 236, CUP.

  17. Fonf, V. P., (1980). A property of spaces of continuous functions on segments of ordinals, Sibirskii Mat. Zh. 21, 230–232 (English translation in Sib. Math. J., 6, (1980).)

    MATH  MathSciNet  Google Scholar 

  18. Fonf, V. P., (1990). Three characterizations of polyhedral Banach spaces, Ukrainian Math. J. 42, 1145–1148 (translated from Russian). DOI: 10.1007/BF01056615

    Article  MATH  MathSciNet  Google Scholar 

  19. Fonf, V. P.; Lindenstrauss, J. and Phelps, R. R., (2001). Infinite Dimensional Convexity, in Handbook of the Geometry of Banach Spaces, Vol. 1, W. B. Johnson and J. Lindenstrauss, eds., Elsevier Science, 599–670. DOI: 10.1016/S1874-5849(01)80017-6

  20. Fonf, V. P.; Pallares, A. J.; Smith, R. J. and Troyanski, S., (2008). Polyhedral norms on non-separable Banach spaces, J. Funct. Anal., 255, 2, 449–470. DOI: 10.1016/j.jfa.2008.03.001

    Article  MATH  MathSciNet  Google Scholar 

  21. Godefroy, G., (1980). Compacts de Rosenthal, Pacific J. Math., 91, 293–306.

    MATH  MathSciNet  Google Scholar 

  22. Gruenhage, G., (1984). Generalized metric spaces, in Handbook of set-theoretic topology, K. Kunen and J. E. Vaughan eds., North Holland, Amsterdam, 235–293.

    Google Scholar 

  23. Gruenhage, G., (1987). A note on Gul’ko compact spaces, Proc. Amer. Math. Soc., 100, 2, 371–376. DOI: 10.2307/2045974

    MATH  MathSciNet  Google Scholar 

  24. Hájek, P., (1995). Smooth norms that depend locally on finitely many coordinates, Proc. Amer. Math. Soc., 123, 12, 3817–3821. DOI: 10.2307/2161911

    Article  MATH  MathSciNet  Google Scholar 

  25. Hájek, P. and Haydon, R. G., (2007). Smooth norms and approximation in Banach spaces of the type C(K), Q. J. Math., 58, 2, 221–228. DOI: 10.1093/qmath/ham010

    Article  MATH  MathSciNet  Google Scholar 

  26. Hájek, P. and Procházka, A., C k-smooth approximations of LUR norms. http://arxiv.org/abs/0901.3623

  27. Há jek, P. and Procházka, A., (2009). A smooth version of Deville’s Lemma. Announced at “Function theory on infinite-dimensional spaces XI”, Madrid, 2009.

  28. Hájek, P. and Zizler, V., (2006). Functions locally dependent on finitely many coordinates, RACSAM, Rev. R. Acad. Cien. Serie A. Mat., 100, 147–154.

    MATH  Google Scholar 

  29. Hansell, R. W., (2001). Descriptive sets and the topology of nonseparable Banach spaces, Serdica Math. J. 27, 1–66.

    MATH  MathSciNet  Google Scholar 

  30. Haydon, R. G., (1990). A counterexample to several questions about scattered compact spaces, Bull. Lond. Math. Soc., 22, 261–268.

    Article  MATH  MathSciNet  Google Scholar 

  31. Haydon, R. G., (1992). Normes infiniment différentiables sur certains espaces de Banach, C. R. Acad. Sci. Paris, 315, 1175–1178.

    MATH  MathSciNet  Google Scholar 

  32. Haydon, R. G., (1995). Baire trees, bad norms and the Namioka property, Mathematika, 42, 1, 30–42. DOI: 10.1112/S0025579300011323

    Article  MATH  MathSciNet  Google Scholar 

  33. Haydon, R. G., (1996). Smooth functions and partitions of unity on certain Banach spaces, Q. J. Math. 47, 455–468. DOI: 10.1093/qjmath/47.188.455

    Article  MATH  MathSciNet  Google Scholar 

  34. Haydon, R. G., (1999). Trees in renorming theory, Proc. Lond. Math. Soc., 78, 3, 541–584. DOI: 10.1112/S0024611599001768

    Article  MATH  MathSciNet  Google Scholar 

  35. Haydon, R. G., (2008). Locally uniformly convex norms in Banach spaces and their duals, J. Funct. Anal., 254, 2023–2039. DOI: 10.1016/j.jfa.2007.11.005

    MATH  MathSciNet  Google Scholar 

  36. Haydon, R. G.; Jayne, J. E.; Namioka, I. and Rogers, C. A., (2000). Continuous functions on totally ordered spaces that are compact in their order topologies, J. Funct. Anal., 178, 23–63. DOI:

    Article  MATH  MathSciNet  Google Scholar 

  37. Haydon, R. G.; Moltó, A. and Orihuela, J., (2007). Spaces of functions with countably many discontinuities, Israel J. Math., 158, 1, 19–39. DOI: 10.1007/s11856-007-0002-1

    Article  MATH  MathSciNet  Google Scholar 

  38. Jayne, J. E.; Namioka, I. and Rogers, C. A., (1990). Norm fragmented weak* compact sets, Collect. Math., 41, 133–163.

    MATH  MathSciNet  Google Scholar 

  39. Jiménez-Sevilla, M. and Moreno, J. P., (1997). Renorming Banach spaces with the Mazur Intersection Property, J. Funct. Anal., 144, 486–504. DOI: 10.1006/jfan.1996.3014

    Article  MATH  MathSciNet  Google Scholar 

  40. Lindenstrauss, J., (1972). Weakly compact sets-their topological properties and the Banach spaces they generate, in Symposia on Infinite Dimensional Topology, Ann. Math. Stud., 69, 235–273.

  41. Marciszewski, W., (2004). Rosenthal compacta, in Encyclopedia of General Topology, K. P. Hart, J. Nagata and J. E. Vaughan, eds., Elsevier, 142–144.

  42. Martínez, J. F.; Moltó, A.; Orihuela, J. and Troyanski, S., (2010). On locally uniformly rotund renormings in C(K) spaces, Canad. J. Math., 62, 595–613. DOI: 10.4153/CJM-2010-037-1

    Article  MATH  MathSciNet  Google Scholar 

  43. Moltó, A.; Orihuela, J.; Troyanski, S. and Valdivia, M., (1999). On weakly locally uniformly rotund Banach spaces, J. Funct. Anal., 163, 252–271. DOI: 10.1006/jfan.1998.3376

    Article  MATH  MathSciNet  Google Scholar 

  44. Moltó, A.; Orihuela, J.; Troyanski, S. and Valdivia, M., (2009). A Nonlinear Transfer Technique for Renorming, Lecture Notes in Mathematics 1951, Springer-Verlag, Berlin.

    Book  MATH  Google Scholar 

  45. Namioka, I., (1987). Radon-Nikodým compact spaces and fragmentability, Mathematika, 34, 2, 258–281. DOI: 10.1112/S0025579300013504

    Article  MATH  MathSciNet  Google Scholar 

  46. Namioka, I. and Phelps, R. R., (1975). Banach spaces which are Asplund spaces, Duke Math. J., 42, 4, 735–750. DOI: 10.1215/S0012-7094-75-04261-1

    Article  MATH  MathSciNet  Google Scholar 

  47. Negrepontis, S., (1984). Banach spaces and topology, in Handbook of set-theoretic topology K. Kunen and J. E. Vaughan, eds., North Holland, Amsterdam, 235–293.

    Google Scholar 

  48. Odell, E. and Rosenthal, H. P., (1975). A double-dual characterization of separable Banach spaces containing l 1, Israel J. Math., 20, 375–384.

    Article  MATH  MathSciNet  Google Scholar 

  49. Cncina, L. and Raja, M., (2004). Descriptive compact spaces and renorming, Studia Math., 165, 39–52. DOI: 10.4064/sm165-1-3

    Article  MathSciNet  Google Scholar 

  50. Orihuela, J.; Schachermayer, W. and Valdivia, M., (1991). Every Radon-Nikodym Corson compact space is Eberlein compact, Studia Math., 98, 157–174.

    MATH  MathSciNet  Google Scholar 

  51. Orihuela, J.; Smith, R. J. and Troyanski, S., Scattered spaces and strictly convex norms, in preparation.

  52. Raja, M., (1999). Locally uniformly rotund norms, Mathematika, 46, 2, 343–358. DOI: 10.1112/S00255 79300007816

    Article  MATH  MathSciNet  Google Scholar 

  53. Raja, M., (2002). On dual locally uniformly rotund norms, Israel J. Math., 129, 1, 77–91. DOI: 10.1007/BF02773154

    Article  MATH  MathSciNet  Google Scholar 

  54. Raja, M., (2003). Weak* locally uniformly rotund norms and descriptive compact spaces, J. Funct. Anal., 197, 1, 1–13. DOI: 10.1016/S0022-1236(02)00037-X

    Article  MATH  MathSciNet  Google Scholar 

  55. Ribarska, N. K., (1987). Internal characterization of fragmentable spaces, Mathematika, 34, 2, 243–257. DOI: 10.1112/S0025579300013498

    Article  MATH  MathSciNet  Google Scholar 

  56. Ribarska, N. K., (1988). A Radon-Nikodým compact which is not a Gruenhage space, C. R. Acad. Bulgare Sci., 41, 9–11.

    MATH  MathSciNet  Google Scholar 

  57. Ribarska, N. K., (1992). The dual of a Gâteaux smooth space is weak star fragmentable, Proc. Amer. Math. Soc., 114, 4, 1003–1008. DOI: 10.2307/2159619

    MATH  MathSciNet  Google Scholar 

  58. Ribarska, N. K. and Babev, V. D., (2009). A stability property for locally uniformly rotund renorming, J. Math. Anal. Appl., 350, 2, 811–828. DOI: 10.1016/j.jmaa.2008.07.004

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  60. Rosenthal, H. P., (1977). Point-wise compact subsets of the first Baire class, Amer. J. Math., 99, 362–378. DOI: 10.2307/2373824

    Article  MATH  MathSciNet  Google Scholar 

  61. Smith, R. J., (2005). Trees, Talagrand Operators and Renorming Theory, DPhil thesis, University of Oxford, UK.

    Google Scholar 

  62. Smith, R. J., (2005). Bounded linear Talagrand operators on ordinal spaces, Q. J. Math., 56, 383–395. DOI: 10.1093/qmath/hah038

    Article  MATH  MathSciNet  Google Scholar 

  63. Smith, R. J., (2006). On trees and dual rotund norms, J. Funct. Anal., 231, 177–194. DOI: 10.1016/j.jfa.j.jfa.2005.04.011

    Article  MATH  MathSciNet  Google Scholar 

  64. Smith, R. J., (2007). Trees, Gâteaux smooth norms and a problem of Haydon, Bull. Lond. Math. Soc., 39, 112–120.

    Article  MATH  MathSciNet  Google Scholar 

  65. Smith, R. J., (2007). Trees, linear orders and Gâteaux smooth norms, J. Lond. Math. Soc., 76, 633–646.DOI: 10.1112/jlms/jdm085

    Article  MATH  MathSciNet  Google Scholar 

  66. Smith, R. J., (2009). Gruenhage compacta and strictly convex dual norms, J. Math. Anal. App., 350, 2, 745–757. DOI: 10.1016/j.jmaa.2008.07.017

    Article  MATH  Google Scholar 

  67. Smith, R. J. and Troyanski, S., (2009). Unconditional bases and strictly convex dual renormings, Bull. Lond. Math. Soc., 41, 5, 831–840. DOI: 10.1112/blms/bdp059

    Article  MATH  MathSciNet  Google Scholar 

  68. Stegall, C., (1991). The topology of certain spaces of measures, Topology Appl., 41, 73–112. DOI: 10.1016/ 0166-8641(91)90102-R

    Article  MATH  MathSciNet  Google Scholar 

  69. Talagrand, M., (1979). Espaces de Banach faiblement K-analytiques, Ann. of Math., 110, 407–438.

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  71. Todorĉević, S., (1984). Trees and linearly ordered sets, in Handbook of set-theoretic topology, K. Kunen and J. E. Vaughan eds., North Holland, Amsterdam, 235–293.

    Google Scholar 

  72. Todorĉević, S., (1999). Compact subsets of the first Baire class, J. Amer. Math. Soc., 12, 1179–1212.

    Article  MATH  MathSciNet  Google Scholar 

  73. Todorĉević, S., (2005). Representing trees as relatively compact subsets of the first Baire class, Bull. Cl. Sci. Math. Nat. Sci. Math., 30, 29–45.

    Google Scholar 

  74. Troyanski, S., (1971). On locally uniformly convex and differentiable norms in certain non-separable Banach spaces, Studia Math., 37, 173–180.

    MATH  MathSciNet  Google Scholar 

  75. Zizler, V., (2003). Nonseparable Banach spaces, in Handbook of the Geometry of Banach Spaces, Vol. 2, W. B. Johnson and J. Lindenstrauss eds., Elsevier Science, 1743–1816.

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Correspondence to Richard J. Smith.

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This article is dedicated to Professor Valdivia on the occasion of his 80th birthday.

Submitted by Vicente Montesinos

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Smith, R.J., Troyanski, S. Renormings of C(K) spaces. RACSAM 104, 375–412 (2010). https://doi.org/10.5052/RACSAM.2010.24

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