Abstract
We introduce the strong Gelfand–Phillips property for locally convex spaces and give several characterizations of this property. We characterize the strong Gelfand–Phillips property among locally convex spaces admitting a stronger Banach space topology. If \(C_{\mathcal {T}}(X)\) is a space of continuous functions on a Tychonoff space X, endowed with a locally convex topology \({\mathcal {T}}\) between the pointwise topology and the compact-open topology, then: (a) the space \(C_{\mathcal {T}}(X)\) has the strong Gelfand–Phillips property iff X contains a compact countable subspace \(K\subseteq X\) of finite scattered height such that for every functionally bounded set \(B\subseteq X\) the complement \(B\setminus K\) is finite, (b) the subspace \(C^b_{\mathcal {T}}(X)\) of \(C_{\mathcal {T}}(X)\) consisting of all bounded functions on X has the strong Gelfand–Phillips property iff X is a compact countable space of finite scattered height.
Similar content being viewed by others
References
Albiac, F., Kalton, N.J.: Topics in Banach Space Theory. Graduate Texts in Mathematics, vol. 233. Springer, New York (2006)
Banakh, T., Gabriyelyan, S.: \(b\)-Feral locally convex spaces (Preprint)
Banakh, T., Gabriyelyan, S.: The Gelfand–Phillips property for locally convex spaces (Under review)
Banakh, T., Gabriyelyan, S.: The Josefson–Nissenzweig property for locally convex spaces. Filomat (accepted)
Banakh, T., Gabriyelyan, S.: Banach spaces with the (strong) Gelfand–Phillips property. Banach J. Math. Anal. 45, Paper No. 24, 14 pp (2022)
Banakh, T., Gabriyelyan, S., Protasov, I.: On uniformly discrete subsets in uniform spaces and topological groups. Mat. Stud. (1) 45, 76–97 (2016)
Bessaga, C., Pełczyński, A.: Spaces of continuous functions, IV. Stud. Math. 19, 53–62 (1960)
Bonet, J., Lindström, M., Valdivia, M.: Two theorems of Josefson–Nissenzweig type for Fréchet spaces. Proc. Am. Math. Soc. 117, 363–364 (1993)
Castillo, J.M.F., González, M., Papini, P.L.: On weak\(^\ast \)-extensible Banach spaces. Nonlinear Anal. 75, 4936–4941 (2012)
Corson, H.: The weak topology of a Banach space. Trans. Am. Math. Soc. 101, 1–15 (1961)
Dales, H.G., Dashiell, F.K., Jr., Lau, A.T.-M., Strauss, D.: Banach Spaces of Continuous Functions as Dual Spaces. Springer, Berlin (2016)
Drewnowski, L.: On Banach spaces with the Gelfand–Phillips property. Math. Z. 193, 405–411 (1986)
Drewnowski, L., Emmanuele, G.: On Banach spaces with the Gelfand–Phillips property, II. Rend. Circ. Mat. Palermo 38, 377–391 (1989)
Edgar, G.A., Wheeler, R.F.: Topological properties of Banach spaces. Pac. J. Math. 115, 317–350 (1984)
Engleking, R.: General Topology. Heldermann Verlag, Berlin (1989)
Fabian, M., Habala, P., Hájek, P., Montesinos, V., Pelant, J., Zizler, V.: Banach Space Theory. The Basis for Linear and Nonlinear Analysis. Springer, New York (2010)
Ghenciu, I., Lewis, P.: The Dunford-Pettis property and the Gelfand-Phillips property, and \(L\)-sets. Colloq. Math. 106, 311–324 (2006)
Jarchow, H.: Locally Convex Spaces. B.G. Teubner, Stuttgart (1981)
Josefson, B.: A Gelfand-Phillips space not containing \(\ell _1\) whose dual ball is not weak\(^\ast \) sequentially compact. Glasg. Math. J. 43, 125–128 (2001)
Ka̧kol, J., Kubiś, W., López-Pellicer, M.: Descriptive Topology in Selected Topics of Functional Analysis. Developments in Mathematics. Springer, Berlin (2011)
Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces. I. Sequence Spaces. Springer, Berlin (1977)
Narici, L., Beckenstein, E.: Topological Vector Spaces, 2nd edn. CRC Press, New York (2011)
Phillips, R.S.: On linear transformations. Trans. Am. math. Soc. 48, 516–541 (1940)
Schlumprecht, T.: Limited sets in Banach spaces. Dissertation, Univ. Munich (1987)
Schlumprecht, T.: Limited sets in \(C(K)\)-spaces and examples concerning the Gelfand–Phillips property. Math. Nachr. 157, 51–64 (1992)
Schmets, J.: Espaces de fonctions Continues. Lecture Notes in Math, vol. 519. Springer, Berlin (1976)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Vesko Valov.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Banakh, T., Gabriyelyan, S. Locally convex spaces with the strong Gelfand–Phillips property. Ann. Funct. Anal. 14, 27 (2023). https://doi.org/10.1007/s43034-023-00255-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s43034-023-00255-3