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

Abstract

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).