Separable quotients in ,
, 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
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: The quotient problem has a positive solution for the weak and strong duals of (
an infinite Tichonov space), for Banach spaces
, and even for barrelled
, but not for barrelled spaces in general. The solution is unknown for general
. A locally convex space is properly separable if it has a proper dense
-dimensional subspace. For
quotients, properly separable coincides with infinite-dimensional separable.
has a properly separable algebra quotient if
has a compact denumerable set. Relaxing compact to closed, we obtain the converse as well; likewise for
. And the weak dual of
, which always has an
-dimensional quotient, has no properly separable quotient when
is a P-space of a certain special form
- [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
, 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
- [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
- [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
- [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
- [9]
J. C. Ferrando, Jerzy Kakol, and Stephen A. Saxon, The dual of the locally convex space
, 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 Kakol, and Stephen A. Saxon, Characterizing P-spaces
in terms of
, 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, Mathematische Leitfäden. [Mathematical Textbooks], B. G. Teubner, Stuttgart, 1981. MR 632257
- [13] Jerzy Kakol, 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 Kakol, Stephen A. Saxon, and Aaron R. Todd, Pseudocompact spaces
and
-spaces
, Proc. Amer. Math. Soc. 132 (2004), no. 6, 1703-1712 (electronic). MR 2051131, https://doi.org/10.1090/S0002-9939-04-07279-X
- [15] Jerzy Kakol, 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 Kakol, 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
, Dissertation, Ludwig-Maximilians-Universität, München, 1979.
- [19] Pedro Pérez Carreras and José Bonet, Barrelled locally convex spaces, Notas de Matemática [Mathematical Notes], 113, North-Holland Mathematics Studies, vol. 131, North-Holland Publishing Co., Amsterdam, 1987. MR 880207
- [20]
M. M. Popov, Codimension of subspaces of
for
, 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
to
, 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
), 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,
-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
- [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
- [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
- [33] Russell C. Walker, The Stone-Čech compactification, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 83, Springer-Verlag, New York-Berlin, 1974. MR 0380698
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Additional Information
Jerzy Kakol
Affiliation:
Faculty of Mathematics and Informatics, A. Mickiewicz University, 60-769 Poznań, Matejki 48-49, Poland – and Institute of Mathematics, Czech Academy of Sciences, Zitna 25, Prague, Czech Republic
Email:
kakol@math.amu.edu.pl
Stephen A. Saxon
Affiliation:
Department of Mathematics, University of Florida, P.O. Box 118105, Gainesville, Florida 32611-8105
Email:
stephen_saxon@yahoo.com
DOI:
https://doi.org/10.1090/proc/13360
Keywords:
(Properly) separables quotients,
$C( X) $,
weak barrelledness,
P-spaces
Received by editor(s):
May 3, 2016
Received by editor(s) in revised form:
June 1, 2016, and June 21, 2016
Published electronically:
May 24, 2017
Additional Notes:
Thanks to Professor Aaron R. Todd for vital discussions/encouragement/prequels.
The first author’s research was supported by Generalitat Valenciana, Conselleria d’Educació, Cultura i Esport, Spain, grant PROMETEO/2013/058, and by GACR grant 16-34860L and RVO 67985840 (Czech Republic).
Communicated by:
Thomas Schlumprecht
Article copyright:
© Copyright 2017
American Mathematical Society