Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T06:25:39.101Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Weak Basis Theorem in F-spaces

Published online by Cambridge University Press:  20 November 2018

Joel H. Shapiro
Joel H. Shapiro*
Affiliation:
Michigan State University, East Lansing, Michigan
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.

It is well-known that every weak basis in a Fréchet space is actually a basis. This result, called the weak basis theorem was first given for Banach spaces in 1932 by Banach [1, p. 238], and extended to Fréchet spaces by Bessaga and Petczynski [3]. McArthur [12] proved an analogue for bases of subspaces in Fréchet spaces, and recently W. J. Stiles [18, Corollary 4.5, p. 413] showed that the theorem fails in the non-locally convex spaces lp (0 < p < 1). The purpose of this paper is to prove the following generalization of Stiles' result.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 6 , December 1974 , pp. 1294 - 1300
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Banach, S., Théorie des opérationes linéaires (Chelsea, New York, 1955).Google Scholar
2. Bessaga, C. and Pelczynski, A., On a class of B0-spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 5 (1957), 379–383.Google Scholar
3. Bessaga, C. and Pelczynski, A., Properties of bases in B0 spaces, Prace. Mat. 3 (1959), 123–142 (Polish).Google Scholar
4. Duren, P. L., Theory of Hp spaces (Academic Press, New York, 1970).Google Scholar
5. Duren, P. L., Romberg, B. W., and Shields, A. L., Linear functionals on Hp spaces with 0 < P < 1. J- Reine Angew. Math. 238 (1969), 32–60.Google Scholar
6. Gregory, D. A. and Shapiro, J. H., Nonconvex linear topologies with the Hahn-Banach extension property, Proc. Amer. Math. Soc. 25 (1970), 902–905.Google Scholar
7. Kalton, N. J., Normalization properties of Schauder bases, Proc. London Math. Soc. 22 (1971), 91–105.Google Scholar
8. Kelley, J., Namioka, I., et. al. Linear topological spaces (Van Nostrand, Princeton, 1963).Google Scholar
9. Krein, M., Milman, D., and Rutman, M., On a property of a basis in a Banach space (Russian), Khark. Zap. Matem. Obsh. 4 (1940), 182.Google Scholar
10. Lindenstrauss, J. and A. Peiczynski, Contributions to the theory of the classical Banach spaces, J. Functional Analysis 8 (1971), 225–248.Google Scholar
11. Marti, J. T., Theory of bases (Springer-Verlag, New York, 1969).Google Scholar
12. McArthur, C. W., The weak basis theorem, Colloq. Math. 17 (1967), 71–76.Google Scholar
13. Shapiro, J. H., Extension of linear functionals on F-spaces with bases, Duke Math. J. 37 (1970), 639–645.Google Scholar
14. Shapiro, J. H., Examples of proper closed, weakly dense subspaces in non-locally convex F-spaces, Israel J. Math. 7 (1969), 369–380.Google Scholar
15. Shapiro, J. H., On compactness and convexity in F-spaces with bases, Indiana Univ. Math. J. 21 (1972), 1073–1090.Google Scholar
16. Shields, A. L. and Williams, D. L., Bounded projections, duality, and multipliers in spaces of analytic functions, Trans. Amer. Math. Soc. 162 (1971), 287–302.Google Scholar
17. Simons, S., The sequence spaces l(pv) and m(pv), Proc. London Math. Soc. 15 (1965), 422–436.Google Scholar
18. Stiles, W. J., On properties of subspaces of lp , 0 < p < 1, Trans. Amer. Math. Soc. 149 (1970), 405–415.Google Scholar