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This article is cited in 5 scientific papers (total in 5 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.
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
Linking options:
https://www.mathnet.ru/eng/faa3194https://doi.org/10.4213/faa3194 https://www.mathnet.ru/eng/faa/v49/i2/p21
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Abstract page: | 343 | Full-text PDF : | 140 | References: | 54 | First page: | 26 |
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