Measures of weak noncompactness in Banach spaces

Measures of weak noncompactness are formulae that quantify different characterizations of weak compactness in Banach spaces: we deal here with De Blasi's measure omega and the measure of double limits gamma inspired by Grothendieck's characterization of weak compactness. Moreover for bounded sets H of a Banach space E we consider the worst distance k(H) of the weak*-closure in the bidual (H) over bar of H to E and the worst distance ck(H) of the sets of weak*-cluster points in the bidual of sequences in H to E. We prove the inequalities ck(H) <=((I)) k(H) <=((II)) y (H) <= 2 k(H) <= 2 omega(H) which say that ck, k and y are equivalent. If E has Corson property C then (I) is always an equality but in general constant 2 in (II) is needed: we indeed provide an example for which k(H) = 2ck(H). We obtain quantitative counterparts to Eberlein-Smulyan's and Gantmacher's theorems using gamma. Since it is known that Gantmacher's theorem cannot be quantified using to we therefore have another proof of the fact that gamma and omega are not equivalent. We also offer a quantitative version of the classical Grothendieck's characterization of weak compactness in spaces C(K) using gamma. (c) 2008 Elsevier B.V. All rights reserved.