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Funktsional. Anal. i Prilozhen., 2015, Volume 49, Issue 2, Pages 21–33 (Mi faa3194)  

This article is cited in 4 scientific papers (total in 4 papers)

Completeness in the Mackey topology

A. J. Guirao, V. Montesinos

Universidad Politécnica de Valencia

Abstract: Bonet and Cascales [Non-complete Mackey topologies on Banach spaces, Bulletin of the Australian Mathematical Society, 81, 3 (2010), 409–413], answering a question of M. Kunze and W. Arendt, gave an example of a norming norm-closed subspace $N$ of the dual of a Banach space $X$ such that $\mu(X,N)$ is not complete, where $\mu(X,N)$ denotes the Mackey topology associated with the dual pair $\langle X,N\rangle$. We prove in this note that we can decide on the completeness or incompleteness of topologies of this form in a quite general context, thus providing large classes of counterexamples to the aforesaid question. Moreover, our examples use subspaces $N$ of $X^*$ that contain a predual $P$ of $X$ (if exists), showing that the phenomenon of noncompleteness that Kunze and Arendt were looking for is not only relatively common but illustrated by “well-located” subspaces of the dual. We discuss also the situation for a typical Banach space without a predual—the space $c_0$—and for the James space $J$.

Keywords: Mackey-star topology, completeness, local completeness, Banach space

Funding Agency Grant Number
Ministerio de Economía y Competitividad de España MTM2008-05396
Ministerio de Ciencia e Innovación de España MTM2011-22417
Federación Española de Enfermedades Raras MTM2008-05396
Fundación Séneca 08848/PI/08
Generalitat Valenciana GV/2010/036
Universitat Politècnica de València PAID-06-09-2829
The first author is supported in part by MICINN and FEDER (project no. MTM2008-05396), by Fundación Séneca (project no. 08848/PI/08), by Generalitat Valenciana (GV/2010/036), and by Universitat Politecnica de Valencia (project no. PAID-06-09-2829). The second author is supported in part by MICINN project no. MTM2011-22417, by Generalitat Valenciana (GV/2010/036), and by Universidad Politecnica de Valencia (project no. PAID-06-09-2829).


DOI: https://doi.org/10.4213/faa3194

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English version:
Functional Analysis and Its Applications, 2015, 49:2, 97–105

Bibliographic databases:

UDC: 517.98+515.1
Received: 13.03.2013

Citation: A. J. Guirao, V. Montesinos, “Completeness in the Mackey topology”, Funktsional. Anal. i Prilozhen., 49:2 (2015), 21–33; Funct. Anal. Appl., 49:2 (2015), 97–105

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Rodriguez J., “On Integration in Banach Spaces and Total Sets”, Quaest. Math.  crossref  isi  scopus
    2. A. J. Guirao, V. Montesinos, V. Zizler, “A note on Mackey topologies on Banach spaces”, J. Math. Anal. Appl., 445:1 (2017), 944–952  crossref  mathscinet  zmath  isi  scopus
    3. V. Bogachev, O. Smolyanov, Topological vector spaces and their applications, Springer Monographs in Mathematics, Springer, Cham, 2017, x + 456 pp.  crossref  mathscinet  zmath  isi
    4. Guirao A.J., Martinez-Cervantes G., Rodriguez J., “Completeness in the Mackey Topology By Norming Subspaces”, J. Math. Anal. Appl., 478:2 (2019), 776–789  crossref  isi
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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