
Joel H. Shapiro 

Abstract: It is well known that if E is a locally convex complete metric space then every weak basis for E is a basis [cf., e.g., C. W. McArthur, Colloq. Math. 17 (1967), 7176]. I show here that if E is a complete linear metric space that is not locally convex and has a weak basis and if either there exists a bounded neighbourhood of zero in E or there exists a onetoone continuous linear operator from E into a Banach space, then there exists a weak basis for E that is not a basis. In particular, this phenomenon holds in the Hardy space H^p for 0 < p < 1. 
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