On weak drop property and quasi-weak drop property

J. H. Qiu¹

¹ Department of Mathematics Suzhou University Suzhou, Jiangsu 215006 People's Republic of China

Studia Mathematica, Tome 156 (2003) no. 2, pp. 189-202

Résumé

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.

Détail

DOI : 10.4064/sm156-2-8

Keywords: every weakly sequentially compact convex set locally convex space has weak drop property every weakly compact convex set has quasi weak drop property example shows quasi weak drop property strictly weaker weak drop property closed bounded convex sets locally convex spaces even spaces quasi complete closed bounded convex subsets quasi complete locally convex spaces quasi weak drop property equivalent weak compactness however closed bounded convex sets sequentially complete locally convex spaces even weak drop property does imply weak compactness quasi complete locally convex space semi reflexive only its closed bounded convex subsets have quasi weak drop property strong duals quasi barrelled spaces semi reflexivity equivalent every closed bounded convex set having quasi weak drop property reflexivity quasi complete quasi barrelled space particular chet space characterized quasi weak drop property space strong dual