Vol. 44, No. 2, 1973

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Necessary convexity conditions for the Hahn-Banach theorem in metrizable spaces

Martin G. Ribe

Vol. 44 (1973), No. 2, 715–732

Local convexity appears—by the Hahn-Banach theorem—as a sufficient condition for the (topological) dual of a topological vector space to separate points from closed subspaces. The aim in the present article is to obtain necessary conditions, in terms of local convexity, for the latter statement to hold for a metrizable topological vector space. In particular, certain classes of such spaces are found, for which local convexity is, really, a necessary condition for the dual to separate points from closed subspaces. The course of proof goes via consideration of the more general question how two metrizable vector space topologies on a linear space must be related to each other, given that the class of linear subspaces which are closed in one of them is larger than the class of those closed in the other.

Mathematical Subject Classification
Primary: 46A15
Received: 10 September 1971
Revised: 14 April 1972
Published: 1 February 1973
Martin G. Ribe