On weakly compact sets in C(X)

[EN] A subset A of a locally convex space E is called (relatively) sequentially complete if every Cauchy sequence {x(n)}(n=1)(infinity) in E contained in A converges to a point x is an element of A (a point x is an element of E). Asanov and Velichko proved that if X is countably compact, every functionally bounded set in C-p (X) is relatively compact, and Baturov showed that if X is a Lindelof Sigma-space, each countably compact (so functionally bounded) set in C-p (X) is a monolithic compact. We show that if X is a Lindelof Sigma-space, every functionally bounded (relatively) sequentially complete set in C-p (X) or in C-w (X), i. e., in C-k (X) equipped with the weak topology, is (relatively) Gul'ko compact. We get some consequences.This work was supported for the first named author by the Grant PGC2018-094431-B-I00 of Ministry of Science, Innovation and Universities of Spain.Ferrando, JC.; López Alfonso, S. (2021). On weakly compact sets in C(X). Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 115(2):1-8. https://doi.org/10.1007/s13398-020-00987-0181152Arkhangel’skiĭ, A. V.: Topological function spaces. In: Math. Appl. vol. 78, Kluwer Academic Publishers, Dordrecht, Boston, London (1992)Banakh, T., Ka̧kol, J., Śliwa, W.: Josefson-Nissenzweig property for $$C_{p}$$-spaces. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 113, 3015–3030 (2019)Baturov, D.P.: Subspaces of function spaces. Vestnik Moskov. Univ. Ser. I Mat. Mech. 4, 66–69 (1987)Bogachev, V.I., Smolyanov, O.G.: Topological Vector Spaces and Their Applications. Springer, Heidelberg (2017)Buzyakova, R.Z.: In search of Lindelöf $$C_{p}$$ ’s. Comment. Math. Univ. Carolinae 45, 145–151 (2004)Cascales, B., Muñoz, M., Orihuela, J.: The number of $$K$$-determination of topological spaces. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 106, 341–357 (2012)Cembranos, P., Mendoza, J.: Banach Spaces of Vector-Valued Functions. Lecture Notes in Math, vol. 1676. Springer, Berlin, Heidelberg (1997)Ferrando, J.C.: On a Theorem of D.P. Baturov. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 111, 499–505 (2017)Ferrando, J. C.: Descriptive topology for analysts. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 114, Paper No. 107, 34 pp. (2020)Ferrando, J.C., Gabriyelyan, S., Ka̧kol, J., : Metrizable-like locally convex topologies on $$C(X)$$. Topol. Appl. 230, 105–113 (2017)Ferrando, J.C., Ka̧kol, J., Saxon, S. A, : Characterizing $$P$$-spaces in terms of $$C\left( X\right) $$. J. Convex Anal. 22, 905–915 (2015)Ferrando, J.C., López-Pellicer, M.: Covering properties of $$C_{p}\left( X\right) $$ and $$C_{k}\left( X\right) $$ (Filomat, to appear)Floret, K.: Weakly Compact Sets. Lecture Notes in Math, vol. 801. Springer, Berlin, Heidelberg (1980)Gabriyelyan, S.: Ascoli’s theorem for pseudocompact spaces. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 114, Paper No. 174, 10 pp. (2020)Gillman, L., Jerison, M.: Rings of Continuous Functions. Van Nostrand, Princeton (1960)Ka̧kol, J., Kubis, W., López-Pellicer, M., : Descriptive Topology in Selected Topics of Functional Analysis. Springer, Heidelberg (2011)King, D.M., Morris, S.A.: The Stone-Čech compactification and weakly Fréchet spaces. Bull. Austral. Math. Soc. 42, 340–352 (1990)Muñoz, M.: A note on the theorem of Baturov. Bull. Austral. Math. Soc. 76, 219–225 (2007)Orihuela, J.: Pointwise compactness in spaces of continuous functions. J. Lond. Math. Soc. 36, 143–152 (1987)Pełczyński, A., Semadeni, Z.: Spaces of continuous functions (III) (Spaces $$C\left( \Omega \right) $$ for $$\Omega $$ without perfect sets). Studia Math. 18, 211–222 (1959)Robertson, A.P., Robertson, W.: Topological Vector Spaces. Cambridge University Press, Cambridge (1973)Talagrand, M.: Espaces de Banach faiblement $$K$$ -analytiques. Ann. Math. 110, 407–438 (1979)Tkachuk, V.V.: The space $$C_{p}(X)$$: decomposition into a countable union of bounded subspaces and completeness properties. Topol. Appl. 22, 241–253 (1986)Tkachuk, V.V.: A $$C_{p}$$-Theory Problem Book. Topological and Function Spaces. Springer, Heidelberg (2011)Todorcevic, S.: Topics in Topology. Springer, Berlin (1997)Valdivia, M.: Some new results on weak compactness. J. Funct. Anal. 24, 1–10 (1977