Skip to main content
Log in

Weak compactness and metrizability of Mackey*-bounded sets in Fréchet spaces

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

Abstract

Motivated by the density condition in the sense of Heinrich for Fréchet spaces and by some results of Schlüchtermann and Wheeler for Banach spaces, we characterize in terms of certain weakly compact resolutions those Fréchet spaces enjoying the property that each bounded subset of its Mackey* dual is metrizable. We also characterize those Köthe echelon Fréchet spaces \({\lambda _{p}(A)}\) as well as those Fréchet spaces Ck (X) of real-valued continuous functions equipped with the compact-open topology that enjoy this property.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. V. Arkhangel’skiĭ, Topological Function Spaces, Math. Appl. 78, Kluwer Academic Publishers (Dordrecht, 1992)

  2. Avilés, A., Rodríguez, J.: Convex combinations of weak*-convergent sequences and the Mackey topology. Mediterr. J. Math. 13, 4995–5007 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. K. D. Bierstedt and J. Bonet, Some aspects of the modern theory of Fréchet spaces, Rev. R. Acad. Cien. RACSAM, Serie A. Mat., 97 (2003), 159–188

  4. Bierstedt, K.D., Bonet, J.: Density conditions in Fréchet and \((DF)\)-spaces. Rev. Mat. Complut. 2, 59–75 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Bonet and M. Lindstrom, Convergent sequences in duals of Fréchet spaces, in: Functional Analysis (Essen, 1991), Lecture Notes in Pure and Appl. Math. 150, Dekker, (New York, 1993), pp. 391–404

  6. Canela, M.A.: \(K\)-analytic locally convex spaces. Port. Math. 41, 105–117 (1982)

    MathSciNet  MATH  Google Scholar 

  7. Cascales, B., Orihuela, J.: On compactness in locally convex spaces. Math. Z. 195, 365–381 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  8. B. Cascales, J. Ka̧kol and S. A. Saxon, Metrizability vs. Fréchet–Urysohn property, Proc. Amer. Math. Soc., 131 (2003), 3623-3631

  9. J. Diestel, Sequences and Series in Banach Spaces, Graduate Text in Math. 92, Springer (New York, 1984)

  10. Fabian, M., Habala, P., Hájek, P., Montesinos, V., Pelant, J., Zizler, V.: Functional Analysis and Infinite-Dimensional Geometry. Books in Mathematics, Canadian Mathematical Society (2001)

    Book  MATH  Google Scholar 

  11. Ferrando, J.C.: S. S. Gabriyelyan and J. Ka̧kol, Metrizable-like locally convex topologies on \(C (X) \). Topology Appl. 230, 105–113 (2017)

    Article  MathSciNet  Google Scholar 

  12. J. C. Ferrando, S. S. Gabriyelyan and J. Ka̧kol, Functional characterizations of countable Tychonoff spaces (submitted)

  13. J. C. Ferrando and J. Ka̧kol, On precompact sets in spaces \( C_{c} (X) \), Georgian Math. J., 20 (2013), 247–254

  14. K. Floret, Weakly Compact Sets, Lecture Notes in Math. 801, Springer (Berlin, 1980)

  15. S. S. Gabriyelyan, J. Ka̧kol and A. Leiderman, The strong Pytkeev property for topological groups and topological vector spaces, Monatsch. Math., 175 (2014), 519–542

  16. Hagler, J., Odell, E.: A Banach space not containing \(\ell _{1}\) whose dual ball is not weak* sequentially compact. Illinois J. Math. 22, 290–294 (1978)

    MathSciNet  MATH  Google Scholar 

  17. Jarchow, H.: Locally Convex Spaces. B. G, Teubner (Stuttgart (1981)

    Book  MATH  Google Scholar 

  18. J. Ka̧kol, W. Kubiś and M. López-Pellicer, Descriptive Topology in Selected Topics of Functional Analysis, Springer (New York, 2011)

  19. Kōmura, Y.: Some examples on linear topological spaces. Math. Ann. 153, 150–162 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  20. Khurana, S.S.: Weakly compactly generated Fréchet spaces. Internat. J. Math. 2, 721–724 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  21. Köthe, G.: Topological Vector Spaces, vol. I. Springer-Verlag (Berlin, II (1983)

    Book  MATH  Google Scholar 

  22. Mercourakis, S., Stamati, E.: A new class of weakly \(K\)-analytic Banach spaces. Comment. Math. Univ. Carolin. 47, 291–312 (2006)

    MathSciNet  MATH  Google Scholar 

  23. P. Pérez Carreras and J. Bonet, Barrelled Locally Convex Spaces, North-Holland Mathematics Studies 131, North-Holland (Amsterdam, 1987)

  24. Ruess, W.: Locally convex spaces not containing \(\ell _{1}\). Funct. Approx. Comment. Math. 50, 351–358 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Tkachuk, V.V.: A space \(C_{p} (X) \) is dominated by the irrationals if and only if it is \(K\)-analytic. Acta. Math. Hungar. 107, 253–265 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. M. Valdivia, Topics in Locally Convex Spaces, North Holland (Amsterdam, 1982)

  27. Valdivia, M.: Fréchet spaces not containing \(\ell _{1}\). Math. Japon. 38, 397–411 (1993)

    MathSciNet  Google Scholar 

  28. Schlüchtermann, G., Wheeler, R.F.: On Strongly WCG Banach spaces. Math. Z. 199, 387–398 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  29. A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw-Hill (1978)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to J. C. Ferrando.

Additional information

The second named author was supported by GAČR Project 16-34860L and RVO: 67985840.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ferrando, J.C., Ka̧kol, J. Weak compactness and metrizability of Mackey*-bounded sets in Fréchet spaces. Acta Math. Hungar. 157, 254–268 (2019). https://doi.org/10.1007/s10474-018-0866-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10474-018-0866-z

Key words and phrases

Mathematics Subject Classification

Navigation