About DML-CZ | FAQ | Conditions of Use | Math Archives | Contact Us
  Previous |  Up |  Next

Article

Glöckner, Helge
Tensor products in the category of topological vector spaces are not associative. (English). Commentationes Mathematicae Universitatis Carolinae, vol. 45 (2004), issue 4, pp. 607-614
MSC: 22A05, 46A16, 46A32, 46M05 | MR 2103079 | Zbl 1103.46001
Keywords:
topological tensor product; associative law; non-locally convex space
Summary:
We show by example that the associative law does not hold for tensor products in the category of general (not necessarily locally convex) topological vector spaces. The same pathology occurs for tensor products of Hausdorff abelian topological groups.
References:
[1] Akmalov N.A.: Topologies in the tensor product of abelian groups (in Russian). Dokl. Akad. Nauk UzSSR (1980), 1 3-6. MR 0584817
[2] Bertram W., Glöckner H., Neeb K.-H.: Differential calculus over general base fields and rings. Exp. Math. 22 (2004), 213-282. MR 2069671 | Zbl 1099.58006
[3] Bourbaki N.: Topological Vector Spaces, Chapters 1-5. Springer, Berlin, 1987. MR 0910295 | Zbl 1106.46003
[4] Cohn P.M.: Algebra, Vol. II. 2nd Ed., John Wiley, Chichester, 1989. MR 1006872
[5] Despland J.C.: Produit tensoriels de deux éspaces de Banach. Rev. Roumaine Math. Pures Appl. 27 (1982), 25-32. MR 0658862
[6] Garling D.J.H.: Tensor products of topological Abelian groups. J. Reine Angew. Math. 223 (1966), 164-182. MR 0199305 | Zbl 0145.03401
[7] Iyahen S.O.: On certain classes of linear topological spaces II. J. London Math. Soc. 3 (1971), 609-617. MR 0287272 | Zbl 0215.19402
[8] Grothendieck A.: Produit tensoriels topologiques et éspaces nucléaires. Mem. Amer. Math. Soc. 1955 (1955), no. 16, 140 pp. MR 0075539
[9] Hofmann K.H.: Tensorprodukte lokal kompakter topologischer Gruppen. J. Reine Angew. Math. 216 (1964), 134-149. MR 0167562
[10] Hollstein R.: Über die Metrisierbarkeit von projektiven Tensorprodukten. Manuscripta Math. 9 (1973), 201-209. MR 0328529 | Zbl 0253.46131
[11] Loth P.: Topologically pure extensions. pp.191-201 in: ``Abelian Groups, Rings and Modules'' (Perth, 2000), Contemp. Math. 273, AMS, Providence, 2001. MR 1817162 | Zbl 0980.22007
[12] Sahleh H.: Tensor product of abelian topological groups and $2$nd nilpotent. Int. Math. J. 2 (2002), 11 1081-1087. MR 1930752
[13] Turpin Ph.: Sur certains produits tensoriels topologiques. C.R. Acad. Sci. Paris, Sér. A 291 (1980), 405-407. MR 0596084 | Zbl 0461.46050
[14] Turpin Ph.: Produits tensoriels d'espaces vectoriels topologiques. Bull. Soc. Math. Fr. 110 (1982), 3-13. MR 0662126 | Zbl 0494.46061
[15] Tomášek S.: On tensor products of Abelian groups. Comment. Math. Univ. Carolinae 6 (1965), 73-83. MR 0180839
[16] Tomášek S.: Some remarks on tensor products. ibid. 6 (1965), 85-96. MR 0180840
[17] Waelbroeck L.: The tensor product of a locally pseudo-convex and a nuclear space. Studia Math. 38 (1970), 101-104. MR 0275111 | Zbl 0206.13201
Partner of
EuDML logo