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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