Measures of weak noncompactness in Banach spaces

Abstract

Measures 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 inequalities ck ( H ) ⩽ ( I ) k ( H ) ⩽ γ ( H ) ⩽ ( II ) 2 ck ( H ) ⩽ 2 k ( H ) ⩽ 2 ω ( 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 ) = 2 ck ( 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.