Abstract
In this survey, we discuss the classical open problem:
“Does every infinite-dimensional Banach space admit a quotient (by a closed subspace) which is infinite-dimensional and separable?”
We furnish several equivalent formulations of this famous unsolved problem, in terms of certain Baire-type covering and barrel properties of locally convex spaces, and also in terms of the structure theory of (metrizable, normable) (LF)-spaces. We solve the corresponding “Separable Quotient Problem” for the class of (LF)-spaces in the affirmative, by actually constructing the separable quotient. Based on the Baire-type covering properties, all (LF)-spaces are partitioned into three mutually disjoint and sufficiently rich classes, called (LF)1, (LF)2 and (LF)3-spaces. These three classes are then characterized in terms of the sequence space ϕ. We show that every (LF)3-space admits a separable quotient, that is a Fréchet space. Every (LF)2 and (LF)3 space (more generally, all non-strict (LF)-spaces) possesses a defining sequence, each of whose members has a separable quotient. The intimate interaction between the Separable Quotient Problem for Banach spaces, and the existence of metrizable, as well as normable (LF)-spaces will be studied, resulting in a rich supply of metrizable, as well as normable (LF)-spaces. Finally, we discuss “Properly Separable” quotients in the setting of barrelled spaces.
Supported by NSERC Grant No. A-8772. Financial support from the organizers of the International Conference on Functional Analysis and Related Topics in memory of Professor Kôsaku Yosida, is gratefully acknowledged.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
I. AMEMIYA and Y. KŌMURA, Über nicht-vollständige Montelräume, Math. Ann. 177 (1968), 273–277.
J. ARIAS de REYNA, Dense hyperplanes of first category, Math. Ann. 249 (1980), 111–114.
J. ARIAS de REYNA, Normed barely Baire spaces, Israel J. Math. 42 (1982), 33–36.
G. BENNETT and N.J. KALTON, Inclusion theorems for FK-spaces, Canadian J. Math. 25 (1973), 511–524.
P. PÉREZ CARRERAS and J. BONET, Barrelled Locally Convex Spaces, North Holland Mathematics Studies, Vol. 131 (1987).
P. DIEROLF, S. DIEROLF and L. DREWNOWSKI, Remarks and examples concerning unordered Baire-like and ultrabarrelled spaces, Colloq. Math. 39 (1978), 109–116.
J. DIESTEL, S.A. MORRIS and S.A. SAXON, Varieties of Linear Topological Spaces, Trans. Amer. Math. Soc. 172 (1972), 207–230.
J. DIEUDONNÉ and L. SCHWARTZ, La dualité dans les espaces (F) et (LF), Ann. Inst. Fourier (Grenoble) 1 (1950), 61–101.
V. EBERHARDT and W. ROELCKE, Über eine Graphensatz für lineare abbildungen mit metrisierbar Zielräum, Manuscripta Math. 13 (1974), 53–78.
M. EIDELHEIT, Zur Theorie der Systeme linearer Gleichungen, Studia Math. 6 (1936), 139–148.
A. GROTHENDIECK, Produits tensoriels topologiques et espaces nucleaires, Memoirs Amer. Math. Soc. No. 16 (1959)
J. HORVÁTH, Topological Vector Spaces and Distributions, Addison-Wesley 1966.
W. B. JOHNSON and H. P. ROSENTHAL, “On ω* basic sequences, Studia Math. 43 (1972), 77–95.
B. JOSEFSON, Weak sequential convergence in the dual of a Banach space does not imply norm convergence, Bull. Amer. Math. Soc. 81 (1975) 166–168.
B. JOSEFSON, Weak sequential convergence in the dual of a Banach space does not imply norm convergence, Arkiv für Math. 13, (1975), 78–89.
N. KITSUNEZAKI, K. KERA and Y. TERAO, On Unordered Ultra Baire-Like Spaces, TRU Math. 19-1 (1983), 89–92.
G. KÖTHE, Topological Vector Spaces I, Die Grundlehren der Mathematischen Wissenschaften, 159, Springer-Verlag, Berlin-Heidelberg-New York, 1969.
H. E. LACEY, Separable Quotients of Banach Spaces, An. Acad. Brasil. Cienc., 44 2 (1972) 185–189.
J. LINDERSTRAUSS, Some Aspects of the Theory of Banach Spaces, Advances in Mathematics 5, (1970) 159–180.
P. P. NARAYANASWAMI, Baire Properties of (LF)-spaces, Note di Mathematica VII, 1–18 (1987)
P. P. NARAYANASWAMI, The Strongest Locally Convex Topology on an ℵ0-dimensional Space, International Conference on Functional Analysis and Related Topics, Hokkaido University, Sept. 1990 (to appear).
P.P. NARAYANASWAMI and S.A. SAXON, (LF)-spaces, Quasi-Baire spaces and the Strongest Locally Convex Topology, Math. Ann. 274 (1986), 627–641.
A. NISSENZWEIG, On ω* sequential convergence, Israel J. Math. 22, (1975), 266–272.
W. ROBERTSON, On Properly Separable Quotients of Strict (LF)-spaces, J. Austral. Math. Soc. (Series A) 47 (1989) 307–312.
W. ROBERTSON and P. P. NARAYANASWAMI, On Properly Separable Quotients and Barrelled Spaces, University of Western Australia Research Reports No. 50 (1988), 1–18.
A.P. ROBERTSON and W. ROBERTSON, On the Closed Graph Theorem, Proc. Glasgow. Math. Assoc. 3 (1956), 9–12.
W.J. ROBERTSON, I. TWEDDLE and F.E. YEOMANS, On the Stability of Barrelled Topologies III, Bull. Austral. Math. Soc. 22 (1980), 99–112.
H. P. ROSENTHAL, On Quasi-Complemented Subspaces of Banach Spaces, with an Appendix on Compactness of Operators from L p(μ) to Lr(v), J. Functional Analysis 4 (1969), 176–214.
S.A. SAXON, Nuclear and Product spaces, Baire-like spaces and the strongest Locally Convex Topology, Math. Ann. 197 (1972), 87–106.
S.A. SAXON, Some Normed Barrelled Spaces which are not Baire, Math. Ann. 209 (1974), 153–160.
S.A. SAXON, Two characterizations of linear Baire spaces, Proc. Amer. Math. Soc. 45 (1974), 204–208.
S. A. SAXON, and M. LEVIN, Every countable-codimensional subspace of a barrelled space is barrelled, Proc. Amer. Math. Soc. 29 No. 1 (1971), 91–96.
S.A. SAXON and P.P. NARAYANASWAMI, Metrizable (LF)-spaces, (db)-spaces and the Separable Quotient Problem, Bull. Austral. Math. Soc. 23 (1981), 65–81.
S.A. SAXON and P.P. NARAYANASWAMI, Metrizable[Normable] (LF)-spaces and two classical problems in Fréchet [Banach] spaces, Studia Math. T. XCIII (1989), 1–16.
S.A. SAXON and A. WILANSKY, The Equivalence of some Banach Space Problems, Colloq. Math. 37 (1977), 217–226.
Y. TERAO, N. KITSUNEZAKI, M. ABE and K. KERA, On (dub)-spaces and Related Topics, TRU Math. 21-1 (1985) 55–60.
A. TODD and S.A. SAXON, A Property of Locally Convex Baire Spaces, Math. Ann. 206 (1973), 23–34.
M. VALDIVIA, Absolutely Convex Sets in Barrelled Spaces, Ann. Inst. Fourier, Grenoble, 21 (1971), 3–13.
M. VALDIVIA, On Weak Compactness, Studia Math. 49 (1973), 35–40.
M. VALDIVIA, On Suprabarrelled Spaces, Functional Analysis, Holomorphy and Approximation Theory, Springer-Verlag, Lecture Notes in Mathematics, 843 (1981), 572–580.
M. VALDIVIA and P. PERÉZ CARRERAS, Sobre espacios (LF) metrizables, Collectanea Math. 33 (1982), 299–303.
A. WILANSKY, Modern Methods in Topological Vector Spaces, McGraw-Hill, Inc. (1978).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1993 Springer-Verlag
About this paper
Cite this paper
Narayanaswami, P.P. (1993). The separable quotient problem for barrelled spaces. In: Komatsu, H. (eds) Functional Analysis and Related Topics, 1991. Lecture Notes in Mathematics, vol 1540. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085487
Download citation
DOI: https://doi.org/10.1007/BFb0085487
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-56471-3
Online ISBN: 978-3-540-47565-1
eBook Packages: Springer Book Archive