Abstract

Quite recently W. Ruess [Locally convex spaces not containing ℓ1\ell_{1}, Funct. Approx. Comment. Math. 50 (2014) 351–358] has shown that a wide class of locally convex spaces for which all bounded sets are metrizable enjoy Rosenthal's ℓ1\ell_{1}-dichotomy. Being motivated by this fact we show that for a Tychonoff space XX the bounded sets of Cp(X)C_{p}(X) are metrizable (respectively, the bounded sets of Ck(X)C_{k}(X) are weakly metrizable) if and only if XX is countable. If XX is a PP-space we show that every bounded set in Cp(X)C_{p}(X) is metrizable if and only if XX is countable and discrete. The second part of the paper deals with distinguished Cp(X)C_{p}(X) spaces. Among other things we show that Cp(X)C_{p}(X) is distinguished if and only if the strong topology of the dual coincides with its strongest locally convex topology, and that Cp(X)C_{p}(X) is always distinguished whenever XX is countable.

Author information

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

Juan Carlos Ferrando

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

jc.ferrando@umh.es

Jerzy Kąkol

Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614 Poznań, Poland
and: Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic

kakol@amu.edu.pl

Keywords

  • Countable tightness
  • Fréchet-Urysohn space
  • strong dual
  • strongest locally convex topology
  • distinguished space

Mathematics Subject Classification

54C35, 54E15, 46A03

References

  1. A. V. Arkhangel’skiĭ: Topological Function Spaces, Kluwer, Dordrecht (1992).https://doi.org/10.1007/978-94-011-2598-7Google Scholar
  2. K. D. Bierstedt, J. Bonet: Density conditions in Fréchet and (DF)-spaces, Rev. Mat. Complut. 2 (1989) 59–75.https://doi.org/10.5209/rev_REMA.1989.v2.18067Google Scholar
  3. V. I. Bogachev, O. G. Smolyanov: Topological Vector Spaces and Their Applications, Springer, Cham (2017).https://doi.org/10.1007/978-3-319-57117-1Google Scholar
  4. B. Cascales, J. Orihuela: On compactness in locally convex spaces, Math. Zeitschrift 195 (1987) 365–381.https://doi.org/10.1007/BF01161762Google Scholar
  5. H. G. Dales, F. K. Dashiell Jr., A. T.-M. Lau, D. Strauss: Banach Spaces of Continuous Functions as Dual Spaces, Springer, Cham (2016).https://doi.org/10.1007/978-3-319-32349-7Google Scholar
  6. L. Drewnowski: Another note on Kalton’s theorem, Studia Math. 52 (1975) 233–237.https://doi.org/10.4064/sm-52-3-233-237Google Scholar
  7. J. C. Ferrando, J. Kąkol: Weak compactness and metrizability of Mackey*-bounded sets in Fréchet spaces, Acta Math. Hung. 157 (2019) 254–268.https://doi.org/10.1007/s10474-018-0866-zGoogle Scholar
  8. J. C. Ferrando, J. Kąkol, M. López-Pellicer: Necessary and sufficient conditions for precompact sets to be metrizable, Bull. Austral. Math. Soc. 74 (2006) 7–13.https://doi.org/10.1017/S0004972700035528Google Scholar
  9. J. C. Ferrando, J. Kąkol, S. Saxon: The dual of the locally convex space Cp (X), Funct. Approx. Comment. Math. 50 (2014) 389–399.https://doi.org/10.7169/facm/2014.50.2.11Google Scholar
  10. J. C. Ferrando, J. Kąkol, S. Saxon: Characterizing P -spaces X in terms of Cp (X), J. Convex Analysis 22 (2015) 905–915.https://doi.org/10.68381/jca22048Google Scholar
  11. J. C. Ferrando, S. Moll: Cc (X) spaces with X locally compact, Acta Math Sinica 23 (2007) 1593–1600.https://doi.org/10.1007/s10114-005-0812-1Google Scholar
  12. S. Gabriyelyan: Locally convex spaces and Schur-type properties, Ann. Acad. Sci. Fenn. Math. 44 (2019) 363–378.https://doi.org/10.5186/aasfm.2019.4417Google Scholar
  13. S. Gabriyelyan, J. Kąkol, W. Kubiś, W. Marciszewski: Networks for the weak topology of Banach and Fréchet spaces, J. Math. Anal. Appl. 432 (2015) 1183–1199.https://doi.org/10.1016/j.jmaa.2015.07.037Google Scholar
  14. J. Kąkol, W. Kubiś, M. López-Pellicer: Decriptive Topology in Selected Topics of Functional Analysis, Springer, Cham (2011).https://doi.org/10.1007/978-1-4614-0529-0Google Scholar
  15. G. Köthe: Topological Vector Spaces I, Springer, Berlin et al. (1983).https://doi.org/10.1007/978-3-642-64988-2Google Scholar
  16. P. Pérez Carreras, J. Bonet: Barrelled Locally Convex Spaces, North Holland, Amsterdam et al. (1987).Google Scholar
  17. W. Ruess: Locally convex spaces not containing ℓ1, Funct. Approx. Comment. Math. 50 (2014) 351–358.https://doi.org/10.7169/facm/2014.50.2.9Google Scholar
  18. G. Schlüchtermann, R. F. Wheeler: On strongly WCG Banach spaces, Math. Zeitschrift 199 (1988) 387–398.https://doi.org/10.1007/BF01159786Google Scholar
  19. V. V. Tkachuk: A Cp -Theory Problem Book. Topological and Function Spaces, Springer, New York (2011).https://doi.org/10.1007/978-1-4419-7442-6Google Scholar
  20. A. Wilansky: Topology for Analysis, Krieger, Malabar (1970).Google Scholar

Suggested citation

J. C. Ferrando, J. Kąkol. “Metrizable Bounded Sets in C(X) Spaces and Distinguished C_p(X) Spaces.” Journal of Convex Analysis 26 (2019), No. 4, 1337–1346. https://doi.org/10.68381/jca26070

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.