Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-24T23:04:21.971Z Has data issue: false hasContentIssue false
  • English
  • Français

Limited Sets and Bibasic Sequences

Published online by Cambridge University Press:  20 November 2018

Ioana Ghenciu
Ioana Ghenciu*
Affiliation:
University of Wisconsin - River Falls, Department of Mathematics, River Falls, WI 54022-5001, USA. e-mail: ioana.ghenciu@uwrf.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Bibasic sequences are used to study relative weak compactness and relative norm compactness of limited sets.

Keywords

46B2046B2828B05limited setsL-setsbibasic sequencesthe Dunford-Pettis property
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 1 , 01 March 2015 , pp. 71 - 79
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Bator, E., Lewis, P., and Ochoa, J., Evaluation maps, restriction maps, and compactness. Colloq. Math. 78 (1998), 1–17. Google Scholar
[2] Bessaga, C. and Pelczynski, A., On bases and unconditional convergence of series in Banach spaces. Studia Math. 17 (1958), 151–174. Google Scholar
[3] Bombal, F., On V. sets and Pelczynski's property V Glasgow Math. J. 32 (1990), no. 1, 109–120. http://dx.doi.org/10.1017/S0017089500009113 Google Scholar
[4] Bourgain, J. and Diestel, J., Limited operators and strict cosingularity. Math. Nachr. 119 (1984), 55–58. http://dx.doi.org/10.1002/mana.19841190105 Google Scholar
[5] Castillo, J. M. and González, M., New results on the Dunford-Pettis property. Bull. London Math. Soc. 27 (1995), no. 6, 599–605. http://dx.doi.org/10.1112/blms/27.6.599 Google Scholar
[6] Cembranos, P. and Mendoza, J., Banach spaces of vector-valued functions. Lecture Notes in Mathematics, 1676, Springer-Verlag, Berlin, 1997.Google Scholar
[7] Collins, H. S. and Ruess, W., Weak compactness in the space of compact operators of vector-valued functions. Pacific J. Math. 106 (1983), no. 1, 45–71. http://dx.doi.org/10.2140/pjm.1983.106.45 Google Scholar
[8] Davis, W. J., Dean, D.W., and Lin, B. L., Bibasic sequences and norming basic sequences. Trans. Amer. Math. Soc. 176 (1973), 89–102. http://dx.doi.org/10.1090/S0002-9947-1973-0313763-9 Google Scholar
[9] Diestel, J., Sequences and series in Banach spaces. Graduate Texts in Mathematics, 92, Springer-Verlag, Berlin, 1984.Google Scholar
[10] Diestel, J., A survey of results related to the Dunford-Pettis property. In: Proceedings of the conference on integration, topology, and geometry in linear spaces (Univ. North Carolina, Chapel Hill, N.C., 1979), Contemp. Math. 2, American Mathematical Society, Providence, RI, 1980, pp. 15–60..Google Scholar
[11] Diestel, J. and Uhl, J. J., Jr., Vector measures. Mathematical Surveys, 15, American Mathematical Society, Providence, RI, 1977.Google Scholar
[12] Emmanuele, G., Banach spaces in which Dunford-Pettis sets are relatively compact. Arch. Math. (Basel) 58 (1992), no. 5, 477–485. http://dx.doi.org/10.1007/BF01190118 Google Scholar
[13] Ghenciu, I. and Lewis, P., Almost weakly compact operators. Bull. Pol. Acad. Sci. Math. 54 (2006), no. 3–4. 237–256. http://dx.doi.org/10.4064/ba54-3-6 Google Scholar
[14] Ghenciu, I. and Lewis, P., The Dunford-Pettis property, the Gelfand-Phillips property, and L-sets. Colloq. Math. 106 (2006), no. 2, 311–324. http://dx.doi.org/10.4064/cm106-2-11 Google Scholar
[15] Haydon, R., A non-reflexive Grothendieck space that does not contain `1 Israel J. Math. 40 (1981), no. 1, 65–73. http://dx.doi.org/10.1007/BF02761818 Google Scholar
[16] Haydon, R., Levy, M., and Odell, E., On sequences without weak convergent convex block subsequences. Proc. Amer. Soc. 100 (1987), no. 1, 94–98. Google Scholar
[17] Lewis, P., Dunford-Pettis sets. Proc. Amer. Math. Soc. 129 (2001), no. 11, 3297–3302. http://dx.doi.org/10.1090/S0002-9939-01-05963-9 Google Scholar
[18] Pełczynski, A. and Semadeni, Z., Spaces of continuous functions. III. Spaces C(Ω)for without perfect subsets. Studia Math. 18 (1959), 211–222. Google Scholar
[19] Semadeni, Z., Banach spaces of continuous functions. Vol. 1, Monografie Matematyczne, Tom 55, PWN–Polish Scientific Publishers, Warsaw, 1971.Google Scholar
[20] Singer, I., Bases in Banach spaces. II. Springer-Verlag, Berlin-New York, 1981.Google Scholar
[21] Schlumprecht, T., Limited sets in Banach spaces. Dissertation, Munich, 1987.Google Scholar
[22] Ülger, A., Continuous linear operators on C(K; X) and pointwise weakly precompact subsets of C(K; X). Math. Proc. Cambridge Philos. Soc. 111 (1992), no. 1, 143–150. http://dx.doi.org/10.1017/S0305004100075228 Google Scholar