THE SEPARABLE QUOTIENT PROBLEM FOR (LF)tv-SPACES

Title & Authors
THE SEPARABLE QUOTIENT PROBLEM FOR (LF)tv-SPACES
Sliwa, Wieslaw;

Abstract
In 1981 S. A. Saxon and P. P. Narayanaswami ([10]) showed that every (LF)-space has an infinite dimensional separable quotient. In this note we prove that this fails for (LF)$\small{_{tv}}$-spaces. We construct a wide class of (LF)$\small{_{tv}}$-spaces, which have no infinite dimensional separable quotient.
Keywords
F-spaces;(LF)$\small{_{tv}}$-spaces;separable quotients;
Language
English
Cited by
1.
On isomorphisms of some Köthe function F-spaces, Central European Journal of Mathematics, 2011, 9, 6, 1267
References
1.
N. Adash, B. Ernst, and D. Keim, Topological Vector Spaces, Springer-Verlag, Berlin, 1978.

2.
M. Eidelheit, Zur Theorie der Systeme linear Gleichungen, Studia Math. 6 (1936), 139-148.

3.
H. Jarchow, Locally Convex Spaces, Teubner, Stuttgart, 1981.

4.
J. Kakol and W. Sliwa, Remarks concerning the separable quotient problem, Note Mat. 13 (1993), no. 2, 277-282.

5.
H. E. Lacey, Separable quotients of Banach spaces, An. Acad. Brasil. Ci. 44 (1972), 185-189.

6.
P. Perez Carreras and J. Bonet, Barrelled Locally Convex Spaces, North-Holland Mathematics Studies, 131. Notas de Matematica [Mathematical Notes], 113. North-Holland Publishing Co., Amsterdam, 1987.

7.
A. M. Plichko and M. M. Popov, Symmetric function spaces on atomless probability spaces, Dissertationes Math. (Rozprawy Mat.) 306 (1990), 85 pp.

8.
M. M. Popov, Codimension of subspaces of $L_P(\mu)$ for p < 1, Funktsional. Anal. i Prilozhen. 18 (1984), no. 2, 94-95.

9.
H. P. Rosenthal, On quasi-complemented subspaces of Banach spaces, with an appendix on compactness of operators from $L^P(\mu)$ to $L^{\gamma}(\nu)$, J. Functional Analysis 4 (1969), 176-214.

10.
S. 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.

11.
S. A. Saxon and P. P. Narayanaswami, Metrizable [normable] (LF)-spaces and two classical problems in Frechet [Banach] spaces, Studia Math. 93 (1989), no. 1, 1-16.

12.
W. Sliwa, The separable quotient problem for symmetric function spaces, Bull. Polish Acad. Sci. Math. 48 (2000), no. 1, 13-27.

13.
M. Valdivia and P. P´erez Carreras, Metrizable (LF)-spaces, Collect. Math. 33 (1982), no. 3, 299-303.