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

Studia Mathematica (1993)

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

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

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