Measures of Noncompactness
In this chapter we consider the basic notions connected with measures of noncompactness (MNCs for brevity) and condensing (or densifying) operators. We define and study in detail the three main and most frequently used MNCs: the Hausdorff MNC χ the Kuratowski MNC α, and the MNC β. We derive a number of formulas that enable us to compute directly the value of the Hausdorff MNC of a set in some concrete spaces. We give the general definition of the notion of an MNC, study the so-called sequential MNCs, and establish their connection with MNCs. We define and study the condensing operators, and we give examples of maps that are condensing with respect to various MNCs. And finally, we bring into consideration the ultimately compact operators and K-operators as natural generalizations of the condensing maps.
KeywordsBanach Space Convex Hull Compact Operator General Definition Ordinal Number
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