Abstract

Dual weak barrelledness led us to prove that XX is a PP-space if and only if every pointwise eventually zero sequence in Cp(X)C_{p}(X) is summable, and other better known characterizations. Novel ones recall utility functions from economics and Arkhangel'skii's (strict) τ\tau-continuity. Mackey ℵ0\aleph_0-barrelled duality leads us to prove that XX is discrete if and only if every bounded σ\sigma-compact set in Cp(X)C_{p}(X) is relatively compact. We relax the σ\sigma-compact hypothesis of Velichko and the σ\sigma-countably compact hypothesis of Tkachuk/Shakhmatov to prove : X is a P-space if and only if Cp(X)C_{p}(X) is σ\sigma-relatively sequentially complete.

Author information

Contact details are reproduced from the original publication and may be historical.

Juan Carlos Ferrando

Centro de Investigación Operativa, Universidad Miguel Hernandez, 03202 Elche, Spain

jc.ferrando@umh.es

Jerzy Kąkol

Faculty of Mathematics and Informatics, A. Mickiewicz University, Matejki 48-49, 60-769 Poznań, Poland

kakol@amu.edu.pl

Stephen A. Saxon

Dept. of Mathematics, University of Florida, P.O.Box 118105, Gainesville, FL 32611, U.S.A.

stephen_saxon@yahoo.com

Keywords

  • P-spaces
  • relatively compact
  • weak barrelledness

Mathematics Subject Classification

54C35, 46A08

References

  1. A. V. Arkhangel’skii: Topological Function Spaces, Kluwer, Dordrecht (1992).https://doi.org/10.1007/978-94-011-2598-7Google Scholar
  2. H. Buchwalter, J. Schmets: Sur quelques propriétés de l’espace Cs (T), J. Math. Pures Appl., IX Sér. 52 (1973) 337–352.Google Scholar
  3. J. C. Candeal, E. Indurain, G. B. Mehta: Some utility theorems on inductive limits of preordered topological spaces, Bull. Aust. Math. Soc. 52 (1995) 235–246.https://doi.org/10.1017/S0004972700014660Google Scholar
  4. J. C. Ferrando, J. Kąkol: A note on spaces Cp (X) K-analytic-framed in RX, Bull. Aust. Math. Soc. 78 (2008) 141–146.https://doi.org/10.1017/S0004972708000567Google Scholar
  5. J. C. Ferrando, J. Kąkol, S. A. Saxon: The dual of the locally convex space Cp (X), Funct. Approximatio, Comment. Math. 50(2) (2014) 389–399.https://doi.org/10.7169/facm/2014.50.2.11Google Scholar
  6. J. C. Ferrando, L. M. Sánchez Ruiz: On sequential barrelledness, Arch. Math. 57 (1991) 597–605.https://doi.org/10.1007/BF01199065Google Scholar
  7. L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, Princeton (1960).https://doi.org/10.1007/978-1-4615-7819-2Google Scholar
  8. J. Kąkol, W. Kubis, M. López-Pellicer: Descriptive Topology in Selected Topics of Functional Analysis, Springer, Berlin (2011).https://doi.org/10.1007/978-1-4614-0529-0Google Scholar
  9. J. Kąkol, S. A. Saxon, A. R. Todd: Weak barrelledness for C (X) spaces, J. Math. Anal. Appl. 279 (2004) 495–505.https://doi.org/10.1016/j.jmaa.2004.03.015Google Scholar
  10. J. L. Kelley et al.: Linear Topological Spaces, Graduate Texts in Mathematics 36, Springer, New York (1976).Google Scholar
  11. R. A. McCoy: Topological Properties of Spaces of Continuous Functions, Springer, Berlin (2008).Google Scholar
  12. S. A. Saxon: Mackey hyperplanes/enlargements for Tweddle’s space, RACSAM 108 (2014) 1035–1054.https://doi.org/10.1007/s13398-013-0159-xGoogle Scholar
  13. S. A. Saxon: Weak barrelledness vs. P-spaces, in: Descriptive Topology and Functional Analysis, J. C. Ferrando, M. López-Pellicer (eds.), Proceedings in Mathematics & Statistics 80, Springer, Cham (2014) 27–32.https://doi.org/10.1007/978-3-319-05224-3_2Google Scholar
  14. S. A. Saxon, L. M. Sánchez Ruiz: Dual local completeness, Proc. Amer. Math. Soc. 125 (1997) 1063–1070.https://doi.org/10.1090/S0002-9939-97-03864-1Google Scholar
  15. S. A. Saxon, L. M. Sánchez Ruiz: Mackey weak barrelledness, Proc. Amer. Math. Soc. 126 (1998) 3279–3282.https://doi.org/10.1090/S0002-9939-98-04511-0Google Scholar
  16. V. V. Tkachuk: A Cp -Theory Problem Book: Topological and Function Spaces, Springer, Berlin (2011).https://doi.org/10.1007/978-1-4419-7442-6Google Scholar

Suggested citation

J. C. Ferrando, J. Kąkol, S. A. Saxon. “Characterizing P-spaces X in Terms of C_p(X).” Journal of Convex Analysis 22 (2015), No. 4, 905–915. https://doi.org/10.68381/jca22048

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2015. Rights now held by Banach Press LLC.