Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-06-01T00:49:49.893Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Stability of Barrelled Topologies II

Published online by Cambridge University Press:  18 May 2009

I. Tweddle  and
F. E. Yeomans
I. Tweddle
Affiliation:
University of Stirling, Stirling, FK9 4LA
F. E. Yeomans
Affiliation:
University of Western Australia, Nedlands, W. Australia6009
Rights & Permissions [Opens in a new window]

Extract

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.

If E is a Hausdorff barrelled space, which does not already have its finest locally convex topology, then the continuous dual E′ may be enlarged within the algebraic dual E*. Robertson and Yeomans [10] have recently investigated whether E can retain the barrelled property under such enlargements. Whereas finite-dimensional enlargements of the dual preserve barrelledness, they have shown that this is not always so for countable-dimensional enlargements E′+M. In fact, if E contains an infinitedimensional bounded set, there always exists a countable-dimensional M for which the Mackey topology τ(E, E′+M) is not barrelled [10, Theorem 2].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 21 , Issue 1 , January 1980 , pp. 91 - 95
Copyright
Copyright © Glasgow Mathematical Journal Trust 1980

References

REFERENCES

1.Bourbaki, N., Espaces vectorieh topologiques, Chap. III–V (Paris, 1967).Google Scholar
2.de Wilde, M., Closed graph theorems and webbed spaces (London, 1978).Google Scholar
3.de Wilde, M. and Houet, C., On increasing sequences of absolutely convex sets in locally convex spaces, Math. Ann. 192 (1971), 257–261.CrossRefGoogle Scholar
4.Dieudonné, J., Sur les propriétés de permanence de certains espaces vectoriels topologiques, Ann. Polon. Math. 25 (1952), 50–55.Google Scholar
5.Drake, F. R., Set theory (Amsterdam, 1974).Google Scholar
6.Grothendieck, A., Topological vector spaces (London, 1973).Google Scholar
7.Mackey, G. W., On infinite-dimensional linear spaces, Trans. Amer. Math. Soc. 57 (1945), 155–207.CrossRefGoogle Scholar
8.Nachbin, L., Topological vector spaces of continuous functions, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 471–474.CrossRefGoogle ScholarPubMed
9.Robertson, A. P. and Robertson, W. J., Topological vector spaces, 2nd edition (Cambridge, 1973).Google Scholar
10.Robertson, W. J. and Yeomans, F. E., On the stability of barrelled topologies I, Bull. Austral. Math. Soc., to appear.Google Scholar
11.Shirota, T., On locally convex vector spaces of continuous functions, Proc. Japan Acad. 30 (1954), 294–298.Google Scholar