Measures of weak noncompactness in Banach spaces

AbstractMeasures of weak noncompactness are formulae that quantify different characterizations of weak compactness in Banach spaces: we deal here with De Blasi's measure ω and the measure of double limits γ 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¯ 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 inequalitiesck(H)⩽(I)k(H)⩽γ(H)⩽(II)2ck(H)⩽2k(H)⩽2ω(H) which say that ck, k and γ 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 γ. Since it is known that Gantmacher's theorem cannot be quantified using ω we therefore have another proof of the fact that γ and ω are not equivalent. We also offer a quantitative version of the classical Grothendieck's characterization of weak compactness in spaces C(K) using γ