Strongly Generated Banach Spaces and Measures of Noncompactness

Abstract

Abstract To generalize the Hausdorff measure of noncompactness to other classes of bounded sets (like e. g. conditionally weakly compact or Asplund sets), we introduce Grothendieck classes. We deduce integral inequalities for quantities (called Grothendieck measures) related to these classes. As a by‐product, we can answer a question concerning the measure of noncompactness for linear T : X → Y introduced in [14], and generalize a theorem about weak solutions of differential equations in Banach spaces.