Bornological linearly topologized modules over a discrete valuation ring
Subject Areas : Functional analysis
1 - nstituto de Matemática e Estat´ıstica, Universidade Federal Fluminense, Rua Professor Marcos Waldemar de Freitas Reis, s/no, Bloco G, Campus do Gragoatá, 24210-201, Niterói, RJ, Brazil
Keywords: Discrete valuation rings, linearly topologized modules, continuity of linear mappings,
Abstract :
In this work the notion of a bornological linearly topologized mo\-dule over a discrete valuation ring is introduced and it is shown that certain semimetrizable linearly topologized modules are bornological. The main result is a characterization of bornological linearly topologized modules, from which the completeness and the quasi-completeness of certain linearly topologized modules of continuous linear mappings are derived.
[1] N. Bourbaki, Sur certains espaces vectoriels topologiques, Ann. Inst. Fourier. 2 (1950), 5-16.
[2] A. Grothendieck, J. A. Dieudonne, Elements de Geometrie Algebrique I, Die Grundlehren der mathematischen Wissenschaften 166, Springer-Verlag, Berlin-Heidelberg-New York, 1971.
[3] I. Kaplansky, Dual modules over a valuation ring. I, Proc. Amer. Math. Soc. 4 (1953), 213-219.
[4] G. Kothe, Topological Vector Spaces I, Die Grundlehren der mathematischen Wissenschaften 159, Springer-Verlag, Berlin-Heidelberg-New York, 1969.
[5] G. W. Mackey, On convex topological linear spaces, Trans. Amer. Math. Soc. 60 (1946), 520-537.
[6] P. C. G. Mauro, D. P. Pombo Jr., Linearly topologized modules over a discrete valuation ring, Boll. Unione Mat. Ital. 7 (2015), 253-278.
[7] J.-P. Serre, Corps Locaux, Quatrieme edition, Actualites Scientifiques et Industrielles 1296, Hermann, Paris, 1968.
[8] J.-P. Serre, A Course in Arithmetic, Third printing, Graduate Texts in Mathematics 7, Springer-Verlag, New York-Heidelberg-Berlin, 1985.
[9] S. Warner, Topological Fields, Notas de Matemática 126, North-Holland, Amsterdam, 1989.