The Mackey topology in linearly topologized modules

Subject Areas : Functional analysis

D. P.  Pombo Jr. ¹ (Instituto de Matem\'{a}tica e Estat\'{i}stica, Universidade Federal Fluminense, Niter\'{o}i, RJ Brasil)
P. C. G.  Mauro ² (Centro Interdisciplinar de Ci\^{e}ncias da Natureza, Universidade Federal da Integra\c{c}\~{a}o Latino-Americana, Foz do Igua\c{c}u, PR Brasil)

Received: 2023-04-07 Accepted : 2023-05-07 Published : 2023-06-01

Keywords: linearly topologized modules, Complete discrete valuation rings, Mackey topology,

Abstract :

‎The Mackey topology in the context of Hausdorff linearly topologized modules over a complete discrete valuation ring is introduced and characterizations of this concept are established‎. ‎Moreover the interplay between the concept of Mackey topology and two special classes of linearly topologized modules is discussed‎.

References:

[1] N. Bourbaki, Commutative Algebra, Hermann and Addison-Wesley, Paris and Reading, Massachusetts, 1972.

[2] A. Grothendieck, J. A. Dieudonné, Eléments de Géométrie Algébrique I, Die Grundlehren der mathematischen Wissenschaften 166, Springer-Verlag, 1971.

[3] I. Kaplansky, Dual modules over a valuation ring. I, Proc. Amer. Math. Soc. 4 (1953), 213-219.

[4] P. C. G. Mauro, D. P. Pombo Jr., Barrelled linearly topologized modules over a discrete valuation ring, J. Math. Research. 12 (2020), 77-81.

[5] P. C. G. Mauro, D. P. Pombo Jr., Linearly topologized modules over a discrete valuation ring, Boll. Unione Mat. Ital. 7 (2015), 253-278.

[6] L. Nachbin, Topics on Topological Vector Spaces, Textos de Métodos Matemáticos 31, Instituto de Matemática da UFRJ, 1994.

[7] D. P. Pombo Jr., A note on linear compactness, Sci. Math. Jpn. 63 (2006), 171-173.

[8] D. P. Pombo Jr., Bornological linearly topologized modules over a discrete valuation ring, J. Linear. Topol. Algebra. 11 (2022), 123-128.

[9] P. Schneider, Nonarchimedean Functional Analysis, Springer Monographs in Mathematics, Springer-Verlag, 2002.

[10] J. P. Serre, Corps Locaux, Quatrième édition, Actualités Scientifiques et Industrielles 1296, Hermann, Paris, 1968.

[11] J. P. Serre, A Course in Arithmetic, Third printing, Graduate Texts in Mathematics 7, Springer-Verlag, 1985.

[12] S. Warner, Topological Fields, Notas de Matemática 126, North-Holland, Amsterdam, 1989.