The density condition in quotients of quasinormable Fréchet spaces

Angela Albanese

Studia Mathematica (1997)

  • Volume: 125, Issue: 2, page 131-141
  • ISSN: 0039-3223

Abstract

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It is proved that a separable Fréchet space is quasinormable if, and only if, every quotient space satisfies the density condition of Heinrich. This answers positively a conjecture of Bonet and Dí az in the class of separable Fréchet spaces.

How to cite

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Albanese, Angela. "The density condition in quotients of quasinormable Fréchet spaces." Studia Mathematica 125.2 (1997): 131-141. <http://eudml.org/doc/216427>.

@article{Albanese1997,
abstract = {It is proved that a separable Fréchet space is quasinormable if, and only if, every quotient space satisfies the density condition of Heinrich. This answers positively a conjecture of Bonet and Dí az in the class of separable Fréchet spaces.},
author = {Albanese, Angela},
journal = {Studia Mathematica},
keywords = {separable Fréchet space; quasinormable; quotient space; density condition of Heinrich},
language = {eng},
number = {2},
pages = {131-141},
title = {The density condition in quotients of quasinormable Fréchet spaces},
url = {http://eudml.org/doc/216427},
volume = {125},
year = {1997},
}

TY - JOUR
AU - Albanese, Angela
TI - The density condition in quotients of quasinormable Fréchet spaces
JO - Studia Mathematica
PY - 1997
VL - 125
IS - 2
SP - 131
EP - 141
AB - It is proved that a separable Fréchet space is quasinormable if, and only if, every quotient space satisfies the density condition of Heinrich. This answers positively a conjecture of Bonet and Dí az in the class of separable Fréchet spaces.
LA - eng
KW - separable Fréchet space; quasinormable; quotient space; density condition of Heinrich
UR - http://eudml.org/doc/216427
ER -

References

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  1. [1] A. Aytuna, P. B. Djakov, A. P. Goncharov, T. Terzioğlu and V. P. Zahariuta, Some open problems in the theory of locally convex spaces, in: Linear Topological Spaces and Locally Complex Analysis I, A. Aytuna (ed.), Metu-Tübitak, Ankara, 1994, 147-164. Zbl0859.46002
  2. [2] S. F. Bellenot, Basic sequences in non-Schwartz Fréchet spaces, Trans. Amer. Math. Soc. 258 (1980), 199-216. Zbl0426.46001
  3. [3] S. F. Bellenot and E. Dubinsky, Fréchet spaces with nuclear Köthe quotients, ibid. 273 (1982), 579-594. Zbl0494.46001
  4. [4] K. D. Bierstedt and J. Bonet, Stefan Heinrich's density condition for Fréchet spaces and the characterization of distinguished Köthe echelon spaces, Math. Nachr. 135 (1988), 149-180. Zbl0688.46001
  5. [5] J. Bonet, A question of Valdivia on quasinormable Fréchet spaces, Canad. Math. Bull. 34 (1991), 301-304. Zbl0698.46002
  6. [6] J. Bonet and J. C. Díaz, Distinguished subspaces and quotients of Köthe echelon spaces, Bull. Polish Acad. Sci. Math. 39 (1991), 177-183. Zbl0777.46007
  7. [7] J. Bonet and J. C. Díaz, The density condition in subspaces and quotients of Fréchet spaces, Monatsh. Math. 117 (1994), 199-212. Zbl0804.46002
  8. [8] J. C. Díaz and C. Fernández, Quotients of Köthe sequence spaces of infinite order, Arch. Math. (Basel) 66 (1996), 207-213. 
  9. [9] A. Grothendieck, Sur les espaces (F) and (DF), Summa Brasil. Math. 3 (1954), 57-123. 
  10. [10] S. Heinrich, Ultrapowers of locally convex spaces and applications I, Math. Nachr. 118 (1984), 211-219. 
  11. [11] H. Jarchow, Locally Convex Spaces, Teubner, Stuttgart, 1981. Zbl0466.46001
  12. [12] G. Köthe, Topological Vector Spaces I, II, Springer, Berlin, 1969 and 1979. Zbl0179.17001
  13. [13] R. Meise and D. Vogt, A characterization of the quasi-normable Fréchet spaces, Math. Nachr. 122 (1985), 141-150. Zbl0583.46002

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