Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T14:44:05.396Z Has data issue: false hasContentIssue false
  • English
  • Français

NONCOMPLETE MACKEY TOPOLOGIES ON BANACH SPACES

Part of: Normed linear spaces and Banach spaces; Banach lattices Topological linear spaces and related structures

Published online by Cambridge University Press:  05 March 2010

JOSÉ BONET  and
BERNARDO CASCALES
JOSÉ BONET*
Affiliation:
Instituto Universitario de Matemática Pura y Aplicada IUMPA, Universidad Politécnica de Valencia, E-46071 Valencia, Spain (email: jbonet@mat.upv.es)
BERNARDO CASCALES
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, E-30100 Espinardo (Murcia), Spain (email: beca@um.es)
*
✉For correspondence; e-mail: jbonet@mat.upv.es
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Answering a question of W. Arendt and M. Kunze in the negative, we construct a Banach space X and a norm closed weak* dense subspace Y of the dual X′ of X such that X, endowed with the Mackey topology μ(X,Y ) of the dual pair 〈X,Y 〉, is not complete.

Keywords

Banach spacesMackey topologiesnorming subspacesKrein–Smulyan theorem

MSC classification

Secondary: 46B10: Duality and reflexivity 46A03: General theory of locally convex spaces
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 3 , June 2010 , pp. 409 - 413
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

The research of Bonet was partially supported by FEDER and MEC Project MTM2007-62643 and by GV Prometeo/2008/101. The research of Cascales was supported by FEDER and MEC Project MTM2008-05396 and by Fundación Séneca de la CARM, project 08848/PI/08.

References

[1]Cascales, B., Manjabacas, G. and Vera, G., ‘A Krein–Smulian type result in Banach spaces’, Quart. J. Math. Oxford Ser. (2) 48(190) (1997), 161–167.CrossRefGoogle Scholar
[2]Cascales, B. and Shvydkoy, R., ‘On the Krein–Smulian theorem for weaker topologies’, Illinois J. Math. 47(4) (2003), 957–976.CrossRefGoogle Scholar
[3]Granero, A. S. and Sánchez, M., ‘The class of universally Krein–Smulian spaces’, Bull. London Math. Soc. 39(4) (2007), 529–540.CrossRefGoogle Scholar
[4]Köthe, G., Topological Vector Spaces, Vol. I (Springer, Berlin, 1969).Google Scholar
[5]Kunze, M. C., ‘Semigroups on norming dual pairs and transition operators for Markov processes’, PhD Thesis, Universität Ulm, 2008.Google Scholar
[6]Meise, R. and Vogt, D., Introduction to Functional Analysis (Clarendon, Oxford, 1997).Google Scholar
[7]Wilanski, A., Modern Methods in Topological Vector Spaces (McGraw-Hill, New York, 1978).Google Scholar