Fréchet spaces with nuclear Köthe quotients
HTML articles powered by AMS MathViewer
- by Steven F. Bellenot and Ed Dubinsky
- Trans. Amer. Math. Soc. 273 (1982), 579-594
- DOI: https://doi.org/10.1090/S0002-9947-1982-0667161-4
- PDF | Request permission
Abstract:
Each separable Fréchet non-Banach space $X$ with a continuous norm is shown to have a quotient $Y$ with a continuous norm and a basis. If, in addition, $Y$ can be chosen to be nuclear, we say that $X$ has a nuclear Köthe quotient. We obtain a (slightly technical) characterization of those separable Fréchet spaces with nuclear Köthe quotients. In particular, separable reflexive Fréchet spaces which are not Banach (and thus Fréchet Montel spaces) have nuclear Köthe quotients.References
- C. Bessaga and Ed Dubinsky, Nuclear Fréchet spaces without bases. III. Every nuclear Fréchet space not isomorphic to $\omega$ admits a subspace and a quotient space without a strong finite-dimensional decomposition, Arch. Math. (Basel) 31 (1978/79), no. 6, 597–604. MR 531575, DOI 10.1007/BF01226497
- C. Bessaga, Bases in certain spaces of continuous functions, Bull. Acad. Polon. Sci. Cl. III. 5 (1957), 11–14. MR 0084734 —, Własności Baz $w$ Przestrzeniach Typu $B$, Prace Mat. 3 (1959), 123-142.
- C. Bessaga, A. Pełczyński, and S. Rolewicz, On diametral approximative dimension and linear homogeneity of $F$-spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 9 (1961), 677–683. MR 132374
- Ed Dubinsky, Perfect Fréchet spaces, Math. Ann. 174 (1967), 186–194. MR 220036, DOI 10.1007/BF01360717
- Ed Dubinsky, Projective and inductive limits of Banach spaces, Studia Math. 42 (1972), 259–263. MR 310578, DOI 10.4064/sm-42-3-259-263
- Ed Dubinsky, The structure of nuclear Fréchet spaces, Lecture Notes in Mathematics, vol. 720, Springer, Berlin, 1979. MR 537039 —, On $(LB)$-spaces and quotients of Fréchet spaces (to appear). M. Eidelheit, Zur Theorie der Systeme linearer Gleichungen, Studia Math. 6 (1936), 139-148.
- W. B. Johnson and H. P. Rosenthal, On $\omega ^{\ast }$-basic sequences and their applications to the study of Banach spaces, Studia Math. 43 (1972), 77–92. MR 310598, DOI 10.4064/sm-43-1-77-92
- J. L. Kelley and Isaac Namioka, Linear topological spaces, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. With the collaboration of W. F. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbe Thue Poulsen, G. Baley Price, Wendy Robertson, W. R. Scott, Kennan T. Smith. MR 0166578
- Gottfried Köthe, Topologische lineare Räume. I, Die Grundlehren der mathematischen Wissenschaften, Band 107, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960 (German). MR 0130551
- V. B. Moscatelli, Fréchet spaces without continuous norms and without bases, Bull. London Math. Soc. 12 (1980), no. 1, 63–66. MR 565487, DOI 10.1112/blms/12.1.63
Bibliographic Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 273 (1982), 579-594
- MSC: Primary 46A12
- DOI: https://doi.org/10.1090/S0002-9947-1982-0667161-4
- MathSciNet review: 667161