Publication: Completeness properties of locally quasi-convex groups
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2000-04-16
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Elsevier Science
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Abstract
It is natural to extend the Grothendieck theorem on completeness, valid for locally convex topological vector spaces, to Abelian topological groups. The adequate framework to do it seems to be the class of locally quasi-convex groups. However, in this paper we present examples of metrizable locally quasi-convex groups for which the analogue to the Grothendieck theorem does not hold. By means of the continuous convergence structure on the dual of a topological group, we also state some weaker forms of the Grothendieck theorem valid for the class of locally quasi-convex groups. Finally, we prove that for the smaller class of nuclear groups, BB-reflexivity is equivalent to completeness. (C) 2001 Elsevier Science B.V.
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International School of Mathematics G Stampacchia 27th Course: Convergence and Topology.JUN 27-JUL 02, 1998.ERICE, ITALY
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L. Außenhofer, Contributions to the duality theory of Abelian topological groups and to the theory of nuclear groups, Doctoral Dissertation, 1998, Dissertationes Mathematicae, to appear.
W. Banaszczyk, Additive Subgroups of Topological Vector Spaces, Lecture Notes in Math., Vol. 1466, Springer, Berlin, 1991.
E. Binz, Continuous Convergence on C(X), Lecture Notes in Math., Vol. 469, Springer, Berlin, 1975.
E. Binz, H.P. Butzmann, K. Kutzler, Über den c -Dual eines topologischen Vektorraumes, Math. Z. 127 (1972) 70–74.
N. Bourbaki, Topologie Général, Actualités Scientifiques, Hermann, Paris, 1960.
M. Bruguera, Grupos topológicos y grupos de convergencia: Estudio de la dualidad de Pontryagin, Doctoral Dissertation, 1999.
M. Bruguera, M.J. Chasco, Strong reflexivity of Abelian groups, Preprint.
H.-P. Butzmann, Über die c -Reflexivität von Cc (X), Comment. Math. Helv. 127 (1972) 92–101.
H.-P. Butzmann, Pontrjagin-Dualität für topologische Vektorräume, Arch. Math. 28 (1977) 632–637.
H.-P. Butzmann, Duality theory for convergence groups, Topology Appl. 111 (2001) 95–104 (this volume).
M.J. Chasco, Pontryagin duality for metrizable groups, Arch. Math. 70 (1998) 22–28.
M.J. Chasco, E. Martín-Peinador, Binz–Butzmann duality versus Pontryagin duality, Arch. Math. 63 (3) (1994) 264–270.
M.J. Chasco, E. Martín-Peinador, V. Tarieladze, On Mackey topology for groups, Studia Math. 132 (1999) 257–284.
W.W. Comfort, S. Hernández, F.J. Trigos-Arrieta, Relating a locally compact abelian group to its Bohr compactification, Adv. Math. 120 (1996) 322–344.
H.R. Fischer, Limesräume, Math. Ann. 137 (1959) 269–303.
E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, I, Grundl. Math. Wiss., Vol. 115, Springer, Berlin, 1963.
S. Kaplan, Extensions of the Pontryagin duality, I: Infinite products, Duke Math. 15 (1948) 649–658.
J.L. Kelley, Convergence in topology, Math. J. 17 (1950) 277–283.
E. Martín-Peinador, A reflexive admissible topological group must be locally compact, Proc. Amer. Math. Soc. 123 (1995) 3563–3566.
P. Nickolas, Reflexivity of topological groups, Proc. Amer. Math. Soc. 65 (1977) 137–141.
H.H. Schaefer, Topological Vector Spaces, Graduate Texts in Math., Vol. 3, Springer, Berlin, 1970.
M.F. Smith, The Pontryagin duality theorem in linear spaces, Ann. Math. 56 (2) (1952) 248–253.
E.H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.
N.Ya. Vilenkin, The theory of characters of topological Abelian groups with boundedness given, Izv. Akad. Nauk SSSR. Ser. Mat. 15 (1951) 439–462.