Topological tensor products of a Fréchet-Schwartz space and a Banach space

Alfredo Peris

Studia Mathematica (1993)

  • Volume: 106, Issue: 2, page 189-196
  • ISSN: 0039-3223

Abstract

top
We exhibit examples of countable injective inductive limits E of Banach spaces with compact linking maps (i.e. (DFS)-spaces) such that E ⊗ ε X is not an inductive limit of normed spaces for some Banach space X. This solves in the negative open questions of Bierstedt, Meise and Hollstein. As a consequence we obtain Fréchet-Schwartz spaces F and Banach spaces X such that the problem of topologies of Grothendieck has a negative answer for F ⨶ π X . This solves in the negative a question of Taskinen. We also give examples of Fréchet-Schwartz spaces and (DFS)-spaces without the compact approximation property and with the compact approximation property but without the approximation property.

How to cite

top

Peris, Alfredo. "Topological tensor products of a Fréchet-Schwartz space and a Banach space." Studia Mathematica 106.2 (1993): 189-196. <http://eudml.org/doc/216012>.

@article{Peris1993,
abstract = {We exhibit examples of countable injective inductive limits E of Banach spaces with compact linking maps (i.e. (DFS)-spaces) such that $E ⊗_\{ε\} X$ is not an inductive limit of normed spaces for some Banach space X. This solves in the negative open questions of Bierstedt, Meise and Hollstein. As a consequence we obtain Fréchet-Schwartz spaces F and Banach spaces X such that the problem of topologies of Grothendieck has a negative answer for $F ⨶_π X$. This solves in the negative a question of Taskinen. We also give examples of Fréchet-Schwartz spaces and (DFS)-spaces without the compact approximation property and with the compact approximation property but without the approximation property.},
author = {Peris, Alfredo},
journal = {Studia Mathematica},
keywords = {Fréchet-Schwartz spaces; (DFS)-spaces; topological tensor products; approximation property; compact approximation property; countable injective inductive limits; problem of topologies of Grothendieck},
language = {eng},
number = {2},
pages = {189-196},
title = {Topological tensor products of a Fréchet-Schwartz space and a Banach space},
url = {http://eudml.org/doc/216012},
volume = {106},
year = {1993},
}

TY - JOUR
AU - Peris, Alfredo
TI - Topological tensor products of a Fréchet-Schwartz space and a Banach space
JO - Studia Mathematica
PY - 1993
VL - 106
IS - 2
SP - 189
EP - 196
AB - We exhibit examples of countable injective inductive limits E of Banach spaces with compact linking maps (i.e. (DFS)-spaces) such that $E ⊗_{ε} X$ is not an inductive limit of normed spaces for some Banach space X. This solves in the negative open questions of Bierstedt, Meise and Hollstein. As a consequence we obtain Fréchet-Schwartz spaces F and Banach spaces X such that the problem of topologies of Grothendieck has a negative answer for $F ⨶_π X$. This solves in the negative a question of Taskinen. We also give examples of Fréchet-Schwartz spaces and (DFS)-spaces without the compact approximation property and with the compact approximation property but without the approximation property.
LA - eng
KW - Fréchet-Schwartz spaces; (DFS)-spaces; topological tensor products; approximation property; compact approximation property; countable injective inductive limits; problem of topologies of Grothendieck
UR - http://eudml.org/doc/216012
ER -

References

top
  1. [1] K. D. Bierstedt, J. Bonet and A. Galbis, Weighted spaces of holomorphic functions on balanced domains, Michigan Math. J., to appear. Zbl0803.46023
  2. [2] K. D. Bierstedt, J. Bonet and A. Peris, Vector-valued holomorphic germs on Fréchet-Schwartz spaces, Proc. Roy. Irish Acad., to appear. 
  3. [3] K. D. Bierstedt und R. Meise, Induktive Limiten gewichteter Räume stetiger und holomorpher Funktionen, J. Reine Angew. Math. 282 (1976), 186-220. Zbl0318.46034
  4. [4] J. Bonet and J. C. Díaz, The problem of topologies of Grothendieck and the class of Fréchet T-spaces, Math. Nachr. 150 (1991), 109-118. Zbl0754.46043
  5. [5] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), reprint 1966. 
  6. [6] R. Hollstein, Tensor sequences and inductive limits with local partition of unity, Manuscripta Math. 52 (1985), 227-249. Zbl0576.46053
  7. [7] H. Jarchow, Locally Convex Spaces, Math. Leitfäden, B. G. Teubner, Stuttgart 1981. 
  8. [8] W. B. Johnson, Factoring compact operators, Israel J. Math. 9 (1971), 337-345. Zbl0236.47045
  9. [9] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer, Berlin 1979. Zbl0403.46022
  10. [10] P. Pérez Carreras and J. Bonet, Barrelled Locally Convex Spaces, North-Holland Math. Stud. 131, North-Holland, Amsterdam 1987. Zbl0614.46001
  11. [11] A. Peris, Quasinormable spaces and the problem of topologies of Grothendieck, Ann. Acad. Sci. Fenn. Ser. AI Math., to appear. Zbl0789.46006
  12. [12] J. Taskinen, Counterexamples to "Problème des topologies" of Grothendieck, Ann. Acad. Sci. Fenn. Ser. AI Math. Dissertationes 63 (1986). Zbl0612.46069
  13. [13] J. Taskinen, (FBa)- and (FBB)-spaces, Math. Z. 198 (1988), 339-365. Zbl0628.46068
  14. [14] J. Taskinen, The projective tensor product of Fréchet-Montel spaces, Studia Math. 91 (1988), 17-30. Zbl0654.46060
  15. [15] G. Willis, The Compact Approximation Property does not imply the Approximation Property, ibid. 103 (1992), 99-108. Zbl0814.46017

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.