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Banach-Dieudonné Theorem Revisited

Part of: Topological and differentiable algebraic systems Topological linear spaces and related structures

Published online by Cambridge University Press:  09 April 2009

Montserrat Bruguera  and
Elena Martín-Peinador
Montserrat Bruguera
Affiliation:
Dept. de Matemática Aplicada I Universidad Politécnica de CataluñaSpain e-mail: m.montserrat.bruguera@upc.es
Elena Martín-Peinador
Affiliation:
Dept. de Geometría y Topología Universidad Complutense de MadridSpain e-mail: peinador@mat.ucm.es
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Abstract

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We prove that in the character group of an abelian topological group, the topology associated (in a standard way) to the continuous convergence structure is the finest of all those which induce the topology of simple convergence on the corresponding equicontinuous subsets. If the starting group is furthermore metrizable (or even almost metrizable), we obtain that such a topology coincides with the compact-open topology. This result constitutes a generalization of the theorem of Banach-Dieudonné, which is well known in the theory of locally convex spaces.

We also characterize completeness, in the class of locally quasi-convex metrizable groups, by means of a property which we have called the quasi-convex compactness property, or briefly qcp (Section 3).

Keywords

Completemetrizable groupcontinuous convergence structureequicontinuous weak* topologycompact-open topologyBanach-Dieudonné theoremdual grouplocally quasi-convex group.

MSC classification

Secondary: 22A05: Structure of general topological groups 46A04: Locally convex Fréchet spaces and (DF)-spaces 46A19: Other “topological” linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than ${bf R}$, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 75 , Issue 1 , August 2003 , pp. 69 - 83
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Auβenhofer, L., ‘Contributions to the duality theory of abelian topological groups and to the theory of nuclear groups’, Dissertations Math. 384 (1999).Google Scholar
[2]Banaszczyk, W., Additive subgroups of topological vector spaces, Lecture Notes in Math. 1466 (Springer, Berlin, 1991).CrossRefGoogle Scholar
[3]Banaszczyk, W. and Martín-Peinador, E., ‘Weakly pseudocompact subsets of nuclear groups’, J. Pure Appl. Algebra 138 (1999), 99–106.CrossRefGoogle Scholar
[4]Binz, E., Continuous Convergence in C(X), Lecture Notes in Math. 469 (Springer, Heidelberg, 1975).CrossRefGoogle Scholar
[5]Bruguera, M., Grupos topológicos y grupos de convergencia: estudio de la dualidad de Pontryagin (Ph.D. Thesis, University of Barcelona, 1999).Google Scholar
[6]Bruguera, M. and Chasco, M. J., ‘Strong reflexivity of abelian groups’, Czechoslovak Math. J. 51 (2001), 213–224.CrossRefGoogle Scholar
[7]Bruguera, M., Chasco, M. J., Martín-Peinador, E. and Tarieladze, V., ‘Completeness properties of locally quasi-convex groups’, Topology Appl. 111 (2001), 81–93.CrossRefGoogle Scholar
[8]Butzmann, H. P., ‘Pontrjagin-Dualität für topologische Vektorräume’, Arch. Math. 28 (1977), 632–637.CrossRefGoogle Scholar
[9]Chasco, M. J., ‘Pontryagin duality for metrizable groups’, Arch. Math. 70 (1998), 22–28.CrossRefGoogle Scholar
[10]Chasco, M. J., Martín-Peinador, E. and Tarieladze, V., ‘On Mackey topology for groups’, Studia Math. 132 (1999), 257–284.CrossRefGoogle Scholar
[11]Collins, H. S., ‘Completeness and compactness in linear topological spaces’, Trans. Amer. Math. Soc. 79 (1955), 256–280.CrossRefGoogle Scholar
[12]Fischer, H. R., ‘Limesráume’, Math. Ann. 137 (1959), 269–303.CrossRefGoogle Scholar
[13]García, M., Margalef, J., Olano de Lorenzo, C., Outerelo, E. and Pinilla, J. L., Topología I (Ed. Alhambra, Madrid, 1975).Google Scholar
[14]Hernández, S., ‘Pontryagain duality for topological abelian groups’, Math. Z. 238 (2001), 493–503.CrossRefGoogle Scholar
[15]Jarchow, H., Locally convex spaces (B. G. Teubner, Stuttgart, 1981).CrossRefGoogle Scholar
[16]Komura, Y., ‘Some examples on linear topological spaces’, Math. Ann. 153 (1964), 150–162.CrossRefGoogle Scholar
[17]Köthe, G., Topological vector spaces I, Grundlehren Math. Wiss. 159 (Springer, New York, 1969).Google Scholar
[18]Martín-Peinador, E., ‘Constructión de la topología de la convergencia débil en el espacio de Hilbert’, Rev. Mat. Hispano-Americana, 4a Serie 34 (1974), 221–229.Google Scholar
[19]Ostling, E. G. and Wilansky, A., ‘Locally convex topologies and the convex compactness property’, Proc. Cambridge Philos. Soc. 74 (1974), 45–50.CrossRefGoogle Scholar
[20]Schaefer, H. H., Topological vector spaces, Graduate Texts in Math. 3 (Springer, Heidelberg, 1970).Google Scholar
[21]Valdivia, M., ‘On certain topologies on a vector space’, Manuscripta Math. 14 (1974), 241–247.CrossRefGoogle Scholar
[22]Wheeler, R. F., ‘The equicontinuous weak* topology and semi-reflexivity’, Studia Math. 41 (1972), 243–256.CrossRefGoogle Scholar