ABSTRACT
In this section we will present the concept of the Kuratowski measure of noncompactness, which will be useful in the sequel sections and chapters. Suppose that E is a real Banach space. Assume that S is a bounded set in E. Let α ( S ) = inf { δ > 0 : S = ⋃ j = 1 n S j : S j ⊂ E , diam ( S j ) < δ , j = 1 , 2 , … , n , n ∈ N } . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003028727/1d426271-33df-4258-80cb-5188d2ed19c2/content/umath1_1.jpg"/> Because S is bounded, we have 0 ≤ α ( S ) < ∞ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003028727/1d426271-33df-4258-80cb-5188d2ed19c2/content/umath1_2.jpg"/>
α(S) is said to be the Kuratowski measure of noncompactness and it is called the noncompactness measure for short.
