Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Hypocontinuity of multiplication on the Clifford algebra of an infinite-dimensional topological vector space


Author: Robert A. Haberstroh
Journal: Proc. Amer. Math. Soc. 50 (1975), 435-442
MSC: Primary 15A66; Secondary 46M99
MathSciNet review: 0379547
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Given a quadratic form on an infinite-dimensional vector space $ E$, useful results have been obtained by imposing on $ E$ the linear topology $ t(V)$ described by Fischer and Gross [4], [5], [6], and investigated by Gross and Miller [9]. It has been shown that, in the induced topology, the Clifford algebra $ C(E)$ is a topological algebra, but that, for topologies strictly finer than $ t(V)$, multiplication need not be continuous. The main result of the present paper asserts that, even for topologies finer than $ t(V)$, desirable conclusions can be drawn if continuity is replaced by hypocontinuity (see [2] for definition).


References [Enhancements On Off] (What's this?)

  • [1] N. Bourbaki, Eléments de mathématique. I: Les structures fondamentales de l'analyse. Fasc. VII, Livre II: Chap. 3: Algèbre multilinéaire, Actualités Sci. Indust., no. 1044, Hermann, Paris, 1948; 2nd ed., 1958. MR 10 #231; 30 #3104.
  • [2] -, Eléments de mathématique. XVIII. Part I. Les structures fondamentales de l'analyse. Livre V: Espaces vectoriels topologiques. Chaps. 3, 4, 5, Actualités Sci. Indust., no. 1229, Hermann, Paris, 1955. MR 17, 1109.
  • [3] -, Eléments de mathematique. XXIV. Part I. Les structures fondamentales de l'analyse. Livre II: Chap. 9: Formes sesquilinêaires et formes quadratiques, Actualités Sci. Indust., no. 1272, Hermann, Paris, 1959. MR 21 #6384.
  • [4] H. R. Fischer and H. Gross, Quadratic forms and linear topologies. I, Math. Ann. 157 (1964), 296–325. MR 0172934
  • [5] Herbert Gross and H. R. Fischer, Quadratic forms and linear topologies. II. Non-real fields 𝑘 and infinite dimensional 𝑘-vector spaces, Math. Ann. 159 (1965), 285–308. MR 0182659
  • [6] Hans R. Fischer and Herbert Gross, Quadratische Formen und lineare Topolgien. III. Tensorprodukte linearer Topologien, Math. Ann. 160 (1965), 1–40 (German). MR 0184062
  • [7] W. H. Greub, Multilinear algebra, Die Grundlehren der Mathematischen Wissenschaften, Band 136, Springer-Verlag New York, Inc., New York, 1967. MR 0224623
  • [8] Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. No. 16 (1955), 140 (French). MR 0075539
  • [9] Herbert Gross and Vinnie H. Miller, Quadratic forms and linear topologies. IV. Continuous forms in infinite dimensional spaces, Comment. Math. Helv. 42 (1967), 132–170. MR 0215874
  • [10] Oma Hamara, Quadratic forms on linearly topologized vector spaces, Portugal. Math. 27 (1968), 15–30. MR 0263849
  • [11] Hans Arwed Keller, Stetigkeitsfragen bei lineartopologischen Cliffordalgebren, Universität Zürich, Zurich, 1971 (German). Inaugural-Dissertation zur Erlangung der philosophischen Doktorwürde vorgelegt der Philosophischen Fakultät II der Universität Zürich. MR 0341097
  • [12] G. Köthe, Topologische linear Räume. I, Die Grundlehren der math. Wissenschaften, Band 107, Springer-Verlag, Berlin 1960; English transl., Topological vector spaces, Die Grundlehren der math. Wissenschaften, Band 159, Springer-Verlag, New York, 1969. MR 40 #1750.
  • [13] A. P. Robertson and W. J. Robertson, Topological vector spaces, Cambridge Tracts in Mathematics and Mathematical Physics, No. 53, Cambridge University Press, New York, 1964. MR 0162118
  • [14] Helmut H. Schaefer, Topological vector spaces, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1966. MR 0193469

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 15A66, 46M99

Retrieve articles in all journals with MSC: 15A66, 46M99


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0379547-8
Keywords: Clifford algebra, linear topologies, hypocontinuity
Article copyright: © Copyright 1975 American Mathematical Society