On weak drop property and quasi-weak drop property
Volume 156 / 2003
Abstract
Every weakly sequentially compact convex set in a locally convex space has the weak drop property and every weakly compact convex set has the quasi-weak drop property. An example shows that the quasi-weak drop property is strictly weaker than the weak drop property for closed bounded convex sets in locally convex spaces (even when the spaces are quasi-complete). For closed bounded convex subsets of quasi-complete locally convex spaces, the quasi-weak drop property is equivalent to weak compactness. However, for closed bounded convex sets in sequentially complete locally convex spaces, even the weak drop property does not imply weak compactness. A quasi-complete locally convex space is semi-reflexive if and only if its closed bounded convex subsets have the quasi-weak drop property. For strong duals of quasi-barrelled spaces, semi-reflexivity is equivalent to every closed bounded convex set having the quasi-weak drop property. From this, reflexivity of a quasi-complete, quasi-barrelled space (in particular, a Fréchet space) is characterized by the quasi-weak drop property of the space and of the strong dual.
