Extension of linear functionals
on F-spaces with bases

Joel H. Shapiro


Duke Math. J. 37 (1970), 639--645
Abstract: I prove that if an F-space (linear topological space with a complete, translation-invariant metric) with a Schauder basis possesses the extension property guaranteed for locally convex spaces by the Hahn-Banach theorem, then the space is actually locally convex. Later Nigel Kalton removed the requirement that the space have a basis. In the other direction James Roberts showed that the requirement of completeness cannot be removed from the translation-invariant metric by constructing a topologically complete metrizable TVS that is not locally convex, but nevertheless does have the Hahn-Banach extension property.
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