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The Weak Basis Theorem Fails in Non-Locally Convex F-Spaces

Published online by Cambridge University Press:  20 November 2018

L. Drewnowski
L. Drewnowski*
Affiliation:
A. Mickiewicz University, Poznan, Poland
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W. J. Stiles showed in [10, Corollary 4.5] that Banach's weak basis theorem fails in the spaces lp, 0 < p < 1. Then, J. H. Shapiro [9] indicated certain general classes of non-locally convex F-spaces with the same property, and asked whether the weak basis theorem fails in every non-locally convex F-space with a weak basis. Our purpose is to answer this question in the affirmative. In [3] we observed that, essentially, the only case that remained open is that of an F-space with irregular basis (en), i.e. such that snen →0 for any scalar sequence (sn).

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 5 , 01 October 1977 , pp. 1069 - 1071
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Bessaga, C. and Pelczynski, A., An extension of the Krein-Milman-Rutman theorem concerning bases to the case of Bo-spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 5 (1957), 379–383.Google Scholar
2. Drewnowski, L., On minimally subspace-comparable F-spaces, J. Functional Anal, (to appear).Google Scholar
3. Drewnowski, L. F-spaces with a basis which is shrinking but not hyper-shrinking, Studia Math. (to appear).Google Scholar
4. Kalton, N.J., Bases in non-closed subspaces ofco, J. London Math. Soc. (2) 3 (1971), 711–716.Google Scholar
5. Kalton, N.J. Basic sequences in F-spaces and their applications, Proc. Edinburgh Math. Soc. 19 (1974), 151–167.Google Scholar
6. Kalton, N.J. Orlicz sequence spaces without local convexity, Math. Proc. Cambridge Phil. Soc. 51 (1977), 253–277.Google Scholar
7. Kalton, N.J. Quotients of F-spaces, to appear.Google Scholar
8. Przeworska-Rolewicz, D. and Rolewicz, S., Equations in linear spaces (PWN, Warszawa, 1968).Google Scholar
9. Shapiro, J. H., On the weak basis theorem in F-spaces, Can. J. Math. 26 (1974), 1294–1300.Google Scholar
10. Stiles, W. J., On properties of subspaces of lv, 0 < p < 1, Trans. Amer. Math. Soc. 1-9 (1970), 405–415.Google Scholar