Separable quotients in 𝐶_{𝑐}(𝑋), 𝐶_{𝑝}(𝑋), and their duals

Kakol, Jerzy; Saxon, Stephen

doi:10.1090/proc/13360

Separable quotients in $C_{c}( X)$ , $C_{p}( X)$ , and their duals

Authors: Jerzy Kakol and Stephen A. Saxon
Journal: Proc. Amer. Math. Soc. 145 (2017), 3829-3841
MSC (2010): Primary 46A08, 46A30, 54C35
DOI: https://doi.org/10.1090/proc/13360
Published electronically: May 24, 2017
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Abstract: The quotient problem has a positive solution for the weak and strong duals of $C_{c}\left ( X\right )$ ( an infinite Tichonov space), for Banach spaces $C_{c}\left ( X\right )$ , and even for barrelled $C_{c}\left ( X\right )$ , but not for barrelled spaces in general. The solution is unknown for general $C_{c}\left ( X\right )$ . A locally convex space is properly separable if it has a proper dense $\aleph _{0}$ -dimensional subspace. For $C_{c}\left ( X\right )$ quotients, properly separable coincides with infinite-dimensional separable. $C_{c}\left ( X\right )$ has a properly separable algebra quotient if has a compact denumerable set. Relaxing compact to closed, we obtain the converse as well; likewise for $C_{p}\left ( X\right )$ . And the weak dual of $C_{p}\left ( X\right )$ , which always has an $\aleph _{0}$ -dimensional quotient, has no properly separable quotient when is a P-space of a certain special form $X=X_\kappa$

[1] S. A. Argyros, P. Dodos, and V. Kanellopoulos, Unconditional families in Banach spaces, Math. Ann. 341 (2008), no. 1, 15–38. MR 2377468, https://doi.org/10.1007/s00208-007-0179-y
[2] Henri Buchwalter and Jean Schmets, Sur quelques propriétés de l’espace 𝐶_{𝑠} (𝑇), J. Math. Pures Appl. (9) 52 (1973), 337–352 (French). MR 0333687
[3] J. Diestel, Sidney A. Morris, and Stephen A. Saxon, Varieties of linear topological spaces, Trans. Amer. Math. Soc. 172 (1972), 207–230. MR 0316992, https://doi.org/10.1090/S0002-9947-1972-0316992-2
[4] Lech Drewnowski and Robert H. Lohman, On the number of separable locally convex spaces, Proc. Amer. Math. Soc. 58 (1976), 185–188. MR 0417728, https://doi.org/10.1090/S0002-9939-1976-0417728-6
[5] Volker Eberhardt and Walter Roelcke, Über einen Graphensatz für lineare Abbildungen mit metrisierbarem Zielraum, Manuscripta Math. 13 (1974), 53–68 (German, with English summary). MR 0355517, https://doi.org/10.1007/BF01168742
[6] M. Eidelheit, Zur Theorie der Systeme Linearer Gleichungen, Studia Math. 6 (1936), 130-148.
[7] R. E. Edwards, Functional analysis. Theory and applications, Holt, Rinehart and Winston, New York-Toronto-London, 1965. MR 0221256
[8] S. Yu. Favorov, A generalized Kahane-Khinchin inequality, Studia Math. 130 (1998), no. 2, 101–107. MR 1623352, https://doi.org/10.4064/sm-130-2-101-107
[9] J. C. Ferrando, Jerzy Kąkol, and Stephen A. Saxon, The dual of the locally convex space 𝐶_{𝑝}(𝑋), Funct. Approx. Comment. Math. 50 (2014), no. 2, 389–399. MR 3229067, https://doi.org/10.7169/facm/2014.50.2.11
[10] J. C. Ferrando, Jerzy K kol, and Stephen A. Saxon, Characterizing P-spaces 𝑋 in terms of 𝐶_{𝑝}(𝑋), J. Convex Anal. 22 (2015), no. 4, 905–915. MR 3436693
[11] Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199
[12] Hans Jarchow, Locally convex spaces, B. G. Teubner, Stuttgart, 1981. Mathematische Leitfäden. [Mathematical Textbooks]. MR 632257
[13] Jerzy Kąkol, Wiesław Kubiś, and Manuel López-Pellicer, Descriptive topology in selected topics of functional analysis, Developments in Mathematics, vol. 24, Springer, New York, 2011. MR 2953769
[14] Jerzy Kąkol, Stephen A. Saxon, and Aaron R. Todd, Pseudocompact spaces 𝑋 and 𝑑𝑓-spaces 𝐶_{𝑐}(𝑋), Proc. Amer. Math. Soc. 132 (2004), no. 6, 1703–1712. MR 2051131, https://doi.org/10.1090/S0002-9939-04-07279-X
[15] Jerzy Kąkol, Stephen A. Saxon, and Aaron R. Todd, The analysis of Warner boundedness, Proc. Edinb. Math. Soc. (2) 47 (2004), no. 3, 625–631. MR 2097264, https://doi.org/10.1017/S001309150300066X
[16] Jerzy Kąkol, Stephen A. Saxon, and Aaron R. Todd, Barrelled spaces with(out) separable quotients, Bull. Aust. Math. Soc. 90 (2014), no. 2, 295–303. MR 3252012, https://doi.org/10.1017/S0004972714000422
[17] Mark Krein and Selim Krein, On an inner characteristic of the set of all continuous functions defined on a bicompact Hausdorff space, C. R. (Doklady) Acad. Sci. URSS (N.S.) 27 (1940), 427–430. MR 0003453
[18] W. Lehner, Über die Bedeutung gewisser Varianten des Baire'schen Kategorienbegriffs für die Funktionenräume $C_{c}\left ( T\right )$ , Dissertation, Ludwig-Maximilians-Universität, München, 1979.
[19] Pedro Pérez Carreras and José Bonet, Barrelled locally convex spaces, North-Holland Mathematics Studies, vol. 131, North-Holland Publishing Co., Amsterdam, 1987. Notas de Matemática [Mathematical Notes], 113. MR 880207
[20] M. M. Popov, Codimension of subspaces of 𝐿_{𝑝}(𝜇) for 𝑝<1, Funktsional. Anal. i Prilozhen. 18 (1984), no. 2, 94–95 (Russian). MR 745720
[21] W. J. Robertson, On properly separable quotients of strict (LF) spaces, J. Austral. Math. Soc. Ser. A 47 (1989), no. 2, 307–312. MR 1008844
[22] Haskell P. Rosenthal, On quasi-complemented subspaces of Banach spaces, with an appendix on compactness of operators from 𝐿^{𝑝}(𝜇) to 𝐿^{𝑟}(𝜈), J. Functional Analysis 4 (1969), 176–214. MR 0250036
[23] Stephen A. Saxon, Every countable-codimensional subspace of an infinite-dimensional (nonnormable) Fréchet space has an infinite-dimensional Fréchet quotient (isomorphic to 𝜔), Bull. Polish Acad. Sci. Math. 39 (1991), no. 3-4, 161–166. MR 1194484
[24] Stephen A. Saxon, Weak barrelledness versus P-spaces, Descriptive topology and functional analysis, Springer Proc. Math. Stat., vol. 80, Springer, Cham, 2014, pp. 27–32. MR 3238201, https://doi.org/10.1007/978-3-319-05224-3_2
[25] Stephen A. Saxon, Mackey hyperplanes/enlargements for Tweddle’s space, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 108 (2014), no. 2, 1035–1054. MR 3249992, https://doi.org/10.1007/s13398-013-0159-x
[26] Stephen A. Saxon, (𝐿𝐹)-spaces with more-than-separable quotients, J. Math. Anal. Appl. 434 (2016), no. 1, 12–19. MR 3404545, https://doi.org/10.1016/j.jmaa.2015.09.005
[27] Stephen Saxon and Mark Levin, Every countable-codimensional subspace of a barrelled space is barrelled, Proc. Amer. Math. Soc. 29 (1971), 91–96. MR 0280972, https://doi.org/10.1090/S0002-9939-1971-0280972-0
[28] Stephen A. Saxon and P. P. Narayanaswami, Metrizable (LF)-spaces, (db)-spaces, and the separable quotient problem, Bull. Austral. Math. Soc. 23 (1981), no. 1, 65–80. MR 615133, https://doi.org/10.1017/S0004972700006900
[29] S. A. Saxon and L. M. Sánchez Ruiz, Reinventing weak barrelledness, Journal of Convex Analysis 24 (2017), no. 3.
[30] Stephen A. Saxon and Albert Wilansky, The equivalence of some Banach space problems, Colloq. Math. 37 (1977), no. 2, 217–226. MR 0511780, https://doi.org/10.4064/cm-37-2-217-226
[31] Wiesław Śliwa, The separable quotient problem and the strongly normal sequences, J. Math. Soc. Japan 64 (2012), no. 2, 387–397. MR 2916073
[32] Manuel Valdivia, On weak compactness, Studia Math. 49 (1973), 35–40. MR 0333644, https://doi.org/10.4064/sm-49-1-35-40
[33] Russell C. Walker, The Stone-Čech compactification, Springer-Verlag, New York-Berlin, 1974. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 83. MR 0380698