Geodesic
The Weak Basis Theorem Fails in Non-Locally Convex F-Spaces
Canadian journal of mathematics, Tome 29 (1977) no. 5, pp. 1069-1071

Voir la notice de l'article provenant de la source Cambridge University Press

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). 
Drewnowski, L. The Weak Basis Theorem Fails in Non-Locally Convex F-Spaces. Canadian journal of mathematics, Tome 29 (1977) no. 5, pp. 1069-1071. doi: 10.4153/CJM-1977-104-7
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