位相線形空間における完全完備性について

Abstract

Let E be a separated locally convex topological vector space and E' be its dual space. E is said to be fully complete provided any linear subspace L of E' is weakly closed in E' whenever L ∩ U° is weakly closed for every neighbourhood U of zero in E. A fully complete space is also called B-complete [3]. E is said to be B_Γ-complete provided any weakly dense subspace L of E' is weakly closed in E' whenever L ∩ U°is weakly closed for every neighbourhood U of zero in E [3]. A. Persson [2] introduced the notions of t-polar and weakly t-polar spaces. They are the spaces E which are obtained by replacing the neighbourhood U by a barrel T in the above definitions of a Bcomplete and a B_Γ-complete spaces respectively.
We shall study some generalizations and some relations of these notions. We introduce new spaces, an 〓-polar and a weakly 〓-polar spaces with 〓 a set of barrels in E. These are the spaces obtained by restricting every barrel T of E to that of 〓 in the definitions of t-polar and weakly t-polar spaces. Therefore, when 〓 is the family of all absolutely convex and closed neighbourhoods of zero (resp. all barrels) in E, an 〓-polar space is a fully complete (resp. t-polar) space and a weakly 〓-polar space is a B_Γ-complete (resp. weakly t-polar) space.