Abstract
For a Tychonoff space X, let \(C_k(X)\) and \(C_p(X)\) be the spaces of real-valued continuous functions C(X) on X endowed with the compact-open topology and the pointwise topology, respectively. If X is compact, the classic result of A. Grothendieck states that \(C_k(X)\) has the Dunford–Pettis property and the sequential Dunford–Pettis property. We extend Grothendieck’s result by showing that \(C_k(X)\) has both the Dunford–Pettis property and the sequential Dunford–Pettis property if X satisfies one of the following conditions: (1) X is a hemicompact space, (2) X is a cosmic space (= a continuous image of a separable metrizable space), (3) X is the ordinal space \([0,\kappa )\) for some ordinal \(\kappa \), or (4) X is a locally compact paracompact space. We show that if X is a cosmic space, then \(C_k(X)\) has the Grothendieck property if and only if every functionally bounded subset of X is finite. We prove that \(C_p(X)\) has the Dunford–Pettis property and the sequential Dunford–Pettis property for every Tychonoff space X, and \(C_p(X) \) has the Grothendieck property if and only if every functionally bounded subset of X is finite.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Albanese, A.A., Bonet, J., Ricker, W.J.: Grothendieck spaces with the Dunford–Pettis property. Positivity 14, 145–164 (2010)
Albanese, A.A., Bonet, J., Ricker, W.J.: \(C_0\)-semigroups and mean ergodic operators in a class of Fréchet spaces. J. Math. Anal. Appl. 365, 142–157 (2010)
Albanese, A.A., Bonet, J., Ricker, W.J.: Mean ergodic semigroups of operators. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 106, 299–319 (2012)
Albanese, A.A., Mangino, E.M.: Some permanence results of the Dunford–Pettis and Grothendieck properties in lcHs. Funct. Approx. Comment. Math. 44, 243–258 (2011)
Banakh, T., Gabriyelyan, S., Protasov, I.: On uniformly discrete subsets in uniform spaces and topological groups. Matematychni Studii 1(45), 76–97 (2016)
Bombal, F., Villanueva, I.: On the Dunford–Pettis property of the tensor product of \(C(K)\) spaces. Proc. Am. Math. Soc. 129, 1359–1363 (2001)
Bonet, J., Lindström, M.: Convergent sequences in duals of Fréchet spaces. In: Functional Analysis, Proceedings of the Essen Conference, Marcel Dekker, New York, pp. 391–404 (1993)
Bonet, J., Ricker, W.: Schauder decompositions and the Grothendieck and Dunford-Pettis properties in Köthe echelon spaces of infinite order. Positivity 11, 77–93 (2007)
Borwein, J., Fabian, M., Vanderwerff, J.: Characterizations of Banach spaces via convex and other locally Lipschitz functions. Acta Math. Vietnam. 22, 53–69 (1997)
Bourgain, J.: On the Dunford–Pettis property. Proc. Am. Math. Soc. 81, 265–272 (1981)
Bourgain, J.: The Dunford–Pettis property for the ball-algebras, the polydisc algebras and the Sobolev spaces. Studia Math. 77, 245–253 (1984)
Cima, J.A., Timoney, R.M.: The Dunford–Pettis property for certain planar uniform algebras. Mich. Math. J. 34, 99–104 (1987)
Dales, H.G., Dashiell Jr., F.K., Lau, A.T.-M., Strauss, D.: Banach Spaces of Continuous Functions as Dual Spaces. Springer, Berlin (2016)
Diestel, J.: A survey of results related to the Dunford–Pettis property. Contemp. Math. AMS 2, 15–60 (1980)
Diestel, J.: Sequences and Series in Banach Spaces, Garduate Text in Mathematics 92. Springer, Berlin (1984)
Diestel, J., Uhl Jr., J.J.: Vector Measures, Math. Surveys No. 15. Amer. Math. Soc., Providence (1977)
Edwards, R.E.: Functional Analysis. Reinhart and Winston, New York (1965)
Engelking, R.: General Topology. Heldermann Verlag, Berlin (1989)
Ferrando, J.C., Ka̧kol, J., Saxon, S.: The dual of the locally convex space \(C_p(X)\). Funct. Approx. Comment. Math. 50, 389–399 (2014)
Gabriyelyan, S.: Locally convex spaces and Schur type properties. Ann. Acad. Sci. Fenn. Math. 44, 363–378 (2019)
Grothendieck, A.: Sur les applications linéaires faiblement compactes d’espaces du type \(C(K)\). Can. J. Math. 5, 129–173 (1953)
Jarchow, H.: Locally Convex Spaces. B.G. Teubner, Stuttgart (1981)
Ka̧kol, J., Kubiś, W., Lopez-Pellicer, M.: Descriptive Topology in Selected Topics of Functional Analysis, Developments in Mathematics. Springer, Berlin (2011)
Ka̧kol, J., Saxon, S.A., Tood, A.: Weak barreldness for \(C(X)\) spaces. J. Math. Anal. Appl. 297, 495–505 (2004)
McCoy, R.A., Ntantu, I.: Topological Properties of Spaces of Continuous Functions. Lecture Notes in Math. Springer, Berlin, 1315 (1988)
Michael, E.: \(\aleph _{0}\)-spaces. J. Math. Mech. 15, 983–1002 (1966)
Narici, L., Beckenstein, E.: Topological Vector Spaces, 2nd edn. CRC Press, New York (2011)
Orihuela, J.: Pointwise compactness in spaces of continuous functions. J. Lond. Math. Soc. 36, 143–152 (1987)
Pérez Carreras, P., Bonet, J.: 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)
Acknowledgements
We are deeply indebted to the referees for careful reading and useful suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The second named author gratefully acknowledges the financial support he received from the Center for Advanced Studies in Mathematics of the Ben Gurion University of the Negev during his visit April, 2018.
Rights and permissions
About this article
Cite this article
Gabriyelyan, S., Ka̧kol, J. Dunford–Pettis type properties and the Grothendieck property for function spaces. Rev Mat Complut 33, 871–884 (2020). https://doi.org/10.1007/s13163-019-00336-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13163-019-00336-9