Extending continuous linear functionals in convergence vector spaces

Abstract

Let (E, r) be a convergence vector space, M a subspace of E, and p a linear functional on M continuous in the induced convergence structure. Sufflcient and sometimes necessary conditions are given that (1) (p has a continuous linear extension to the T-adherence M of M; (2) p has a continuous linear extension to E; (3) M is T-closed; (4) every T-closed convex subset of E is a(E, E')-closed. Several examples are included illustrating the extent and limitations of the theory presented. Introduction. Through introduction of an appropriate notion of local convexity, necessary and sufficient conditions are given in order that a subspace M of a convergence vector space (c.v.s.) (E, r) (H. R. Fischer, Limesrdume, Math. Ann. 137 (1959), 269-303) have the Hahn-Banach Property (H.B.P.), namely: Every continuous linear functional p on M has a continuous linear extension to E. This yields an extension of the Hahn-Banach Theorem to a class of c.v.s. satisfying a local convexity condition. Conditions are given insuring that the r-closed and weakly closed subsets of E coincide and, in a c.v.s. where this is the case, that a subspace will have the H.B.P. Prerequisite to this last result is the determination of when every continuous linear functional, p, on M has a continuous linear extension to M, the r-adherence of M. The notion of a nearly closed subspace M of (E, r) is introduced, and it is shown that for nearly closed subspaces, one can always extend p on M continuously to M and that M is r-closed. Subsequently, it is demonstrated that in a strict convergence inductive limit of Frechet spaces, M is nearly closed if and only if every such p on M extends continuously to M if and only if M is r-closed. The final section consists of examples illustrating the extent and limitations of the theory presented. In particular, we (1) provide an example of a locally convex convergence space with a closed subspace which does not have the H.B.P.; (2) provide a characterization of those subspaces M of a strict inductive limit of metrizable spaces in which every continuous linear functional on M has a continuous linear extension to M; and thus (3) characterize those subspaces of a Received by the editors May 15, 1972 and, in revised form, February 14, 1973. AMS (MOS) subject classifications (1970). Primary 46A99. Copyright X 1974, American Mathematical Society