Search results for "Compactness"

MR2502017 (2010c:46055) Angosto, C.; Cascales, B. Measures of weak noncompactness in Banach spaces. Topology Appl. 156 (2009), no. 7, 1412--1421. (Re…

2010

The authors consider for a bounded subset H of a Banach space E the De Blasi measure of weak noncompactness w(H) and the measure of weak noncompactness g(H) based on Grothendieck’s double limit criterion. They also deal with the quantitative characteristics k(H) and ck(H) which represent, respectively, the worst distance to E of the weak*-closure of H in the bidual of E and the worst distance to E of the sets of weak*-cluster points in the bidual of E of sequences in H. The authors prove the following chain of inequalities ck(H) < = k(H) < = g(H) < = 2ck(H) < = 2k(H) < = 2w(H), which, in particular, shows that ck, k and g are equivalent. The authors show that ck = k in the class of Banach s…