On weak drop property and quasi-weak drop property

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-reexiv e if and only if its closed bounded convex subsets have the quasi-weak drop property. For strong duals of quasi-barrelled spaces, semi-reexivit y is equivalent to every closed bounded convex set having the quasi-weak drop property. From this, reexivit y of a quasi-complete, quasi-barrelled space (in particular, a Fr echet space) is characterized by the quasi-weak drop property of the space and of the strong dual. 1. Introduction. Let (X;kk) be a Banach space and B(X) the closed unit ball fx 2 X : kxk 1g. Given x0 62 B(X), the convex hull of x0 and B(X) is called the drop generated by x0 and denoted by D(x0;B(X)). Dane s (3) proved that in any Banach space (X;kk), for every closed set A at positive distance from B(X), there exists an x02A such that A =fx0g. Modifying the assumption of the Dane s drop theorem, Role- wicz (23) began the study of the drop property for the closed unit ball. He dened the norm kk to have the drop property if for every closed set A disjoint from B(X) there exists an x0 2 A such that D(x0;B(X)) A = fx0g, and he proved that if the norm kk has the drop property then (X;kk) is reexiv e (see (23, Theorem 5)). Giles, Sims and Yorke (7) dened the following weaker variant: the normkk has the weak drop property if for every weakly sequentially closed set A disjoint from B(X), there exists an x02 A such that D(x0;B(X))\A =fx0g, and they showed that this prop- erty is equivalent to (X;kk) being reexiv e. Moreover Kutzarova (11) and