On weak drop property and quasi-weak drop property

J. H. Qiu. "On weak drop property and quasi-weak drop property." Studia Mathematica 156.2 (2003): 189-202. <http://eudml.org/doc/284683>.

@article{J2003,
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.},
author = {J. H. Qiu},
journal = {Studia Mathematica},
keywords = {quasi-complete space; reflexive space; quasi-weak drop property; weak drop property; weakly compact set},
language = {eng},
number = {2},
pages = {189-202},
title = {On weak drop property and quasi-weak drop property},
url = {http://eudml.org/doc/284683},
volume = {156},
year = {2003},
}

TY - JOUR
AU - J. H. Qiu
TI - On weak drop property and quasi-weak drop property
JO - Studia Mathematica
PY - 2003
VL - 156
IS - 2
SP - 189
EP - 202
AB - 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.
LA - eng
KW - quasi-complete space; reflexive space; quasi-weak drop property; weak drop property; weakly compact set
UR - http://eudml.org/doc/284683
ER -