On topological properties of Fréchet locally convex spaces

Abstract

[EN] We describe the topology of any cosmic space and any N-o-space in terms of special bases defined by partially ordered sets. Using this description we show that a Baire cosmic group is metrizable. Next, we study those locally convex spaces (lcs) E which under the weak topology sigma(E, E') are N-o-spaces. For a metrizable and complete lcs E not containing (an isomorphic copy of) l(1) and satisfying the Heinrich density condition we prove that (E, sigma(E,E')) is an N-o-space if and only if the strong dual of E is separable. In particular, if a Banach space E does not contain l(1), then (E, sigma(E, E')) is an N-o-space if and only if E' is separable. The last part of the paper studies the question: Which spaces (E, sigma(E, E')) are N-o-spaces? We extend, among the others, Michael's results by showing: If E is a metrizable lcs or a (DF)-space whose strong dual E' is separable, then (E, sigma(E, E')) is an N-o-space. Supplementing an old result of Corson we show that, for a Cech-complete Lindelof space X the following are equivalent: (a) X is Polish, (b) C-c(X) is cosmic in the weak topology, (c) the weak*-dual of C-c(X) is an N-o-space.