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.
Similar content being viewed by others
References
A. V. Arkhangel’skiĭ, Topological Function Spaces, Math. Appl. 78, Kluwer Academic Publishers (Dordrecht, 1992)
Avilés, A., Rodríguez, J.: Convex combinations of weak*-convergent sequences and the Mackey topology. Mediterr. J. Math. 13, 4995–5007 (2016)
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
Bierstedt, K.D., Bonet, J.: Density conditions in Fréchet and \((DF)\)-spaces. Rev. Mat. Complut. 2, 59–75 (1989)
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
Canela, M.A.: \(K\)-analytic locally convex spaces. Port. Math. 41, 105–117 (1982)
Cascales, B., Orihuela, J.: On compactness in locally convex spaces. Math. Z. 195, 365–381 (1987)
B. Cascales, J. Ka̧kol and S. A. Saxon, Metrizability vs. Fréchet–Urysohn property, Proc. Amer. Math. Soc., 131 (2003), 3623-3631
J. Diestel, Sequences and Series in Banach Spaces, Graduate Text in Math. 92, Springer (New York, 1984)
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)
Ferrando, J.C.: S. S. Gabriyelyan and J. Ka̧kol, Metrizable-like locally convex topologies on \(C (X) \). Topology Appl. 230, 105–113 (2017)
J. C. Ferrando, S. S. Gabriyelyan and J. Ka̧kol, Functional characterizations of countable Tychonoff spaces (submitted)
J. C. Ferrando and J. Ka̧kol, On precompact sets in spaces \( C_{c} (X) \), Georgian Math. J., 20 (2013), 247–254
K. Floret, Weakly Compact Sets, Lecture Notes in Math. 801, Springer (Berlin, 1980)
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
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)
Jarchow, H.: Locally Convex Spaces. B. G, Teubner (Stuttgart (1981)
J. Ka̧kol, W. Kubiś and M. López-Pellicer, Descriptive Topology in Selected Topics of Functional Analysis, Springer (New York, 2011)
Kōmura, Y.: Some examples on linear topological spaces. Math. Ann. 153, 150–162 (1964)
Khurana, S.S.: Weakly compactly generated Fréchet spaces. Internat. J. Math. 2, 721–724 (1979)
Köthe, G.: Topological Vector Spaces, vol. I. Springer-Verlag (Berlin, II (1983)
Mercourakis, S., Stamati, E.: A new class of weakly \(K\)-analytic Banach spaces. Comment. Math. Univ. Carolin. 47, 291–312 (2006)
P. Pérez Carreras and J. Bonet, Barrelled Locally Convex Spaces, North-Holland Mathematics Studies 131, North-Holland (Amsterdam, 1987)
Ruess, W.: Locally convex spaces not containing \(\ell _{1}\). Funct. Approx. Comment. Math. 50, 351–358 (2014)
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)
M. Valdivia, Topics in Locally Convex Spaces, North Holland (Amsterdam, 1982)
Valdivia, M.: Fréchet spaces not containing \(\ell _{1}\). Math. Japon. 38, 397–411 (1993)
Schlüchtermann, G., Wheeler, R.F.: On Strongly WCG Banach spaces. Math. Z. 199, 387–398 (1988)
A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw-Hill (1978)
Author information
Authors and Affiliations
Corresponding author
Additional information
The second named author was supported by GAČR Project 16-34860L and RVO: 67985840.
Rights and permissions
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-018-0866-z
Key words and phrases
- bounded resolution
- weakly compact resolution
- \({\mathfrak{G}}\)-base of neighborhoods
- K-analytic space
- SWKA space
- SWCG space