Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-01-23T18:50:22.075Z Has data issue: false hasContentIssue false
  • English
  • Français

Unconditional convergence and bases

Published online by Cambridge University Press:  20 January 2009

I. Tweddle
I. Tweddle
Affiliation:
University of Stirling
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.

The recent papers (6), (7) of J. T. Marti have revived interest in the concept of extended bases, introduced in (1) by M. G. Arsove and R. E. Edwards. In the present note, two results are established which involve this idea. The first of these, which is given in a more general setting, restricts the behaviour of the coefficients for an extended basis in a certain type of locally convex space. The second result extends the well-known weak basis theorem (1, Theorem 11).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 18 , Issue 4 , December 1973 , pp. 321 - 324
Copyright
Copyright © Edinburgh Mathematical Society 1973

References

REFERENCES

(1) Arsove, M. G. and Edwards, R. E., Generalized bases in topological linear spaces, Studio Math. 19 (1960), 95–113.CrossRefGoogle Scholar
(2) 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
(3) Husain, T., Two new classes of locally convex spaces, Math. Ann. 166 (1966), 289–299.CrossRefGoogle Scholar
(4) Köthe, G., Topologische lineare Räume I (Springer-Verlag, Berlin, 1960).CrossRefGoogle Scholar
(5) Levin, M. and Saxon, S., A note on the inheritance of properties of locally convex spaces by subspaces of countable codimension, Proc. Amer. Math. Soc. 29 (1971), 97–102.CrossRefGoogle Scholar
(6) Marti, J. T., Extended bases for Banach spaces, Illinois J. Math. 15 (1971), 135–143.CrossRefGoogle Scholar
(7) Marti, J. T., A weak basis theorem for non-separable Frechet spaces, J. London Math. Soc. 5 (1972), 8–10.CrossRefGoogle Scholar
(8) Robertson, A. P., Summation and Integration in Linear Topological Spaces (Ph.D. Dissertation, Cambridge, 1954).Google Scholar
(9) Robertson, A. P., On unconditional convergence in topological vector spaces, Proc. Roy. Soc. Edinburgh Sect. A 68 (1969), 145–157.Google Scholar
(10) Robertson, A. P. and Robertson, W. J., Topological Vector Spaces (Cambridge University Press, Cambridge, 1963).Google Scholar