| Measure Of Non-compactness |
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| functional analysis | |
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The underlying idea is the following: a bounded set can be covered by a single ball of some radius. Sometimes several balls of a smaller radius can also cover the set. A compact set in fact can be covered by finitely many balls of arbitrary small radius, because it is Totally Bounded . So one could ask: what is the smallest radius that allows to cover the set with finitely many balls? Formally, we start with a Metric Space ''M'' and a subset ''X''. The ball measure of non-compactness is defined as :α(''X'') = Inf {''r'' > 0 : there exist finitely many balls of radius ''r'' which cover ''X''} and the Kuratowski measure of non-compactness is defined as :β(''X'') = inf {''d'' > 0 : there exist finitely many sets of diameter at most ''d'' which cover ''X''} Since a ball of radius ''r'' has diameter at most 2''r'', we have α(''X'') ≤ β(''X'') ≤ 2α(''X''). The two measures α and β share many properties, and we will use γ in the sequel to denote either one of them. Here is a collection of facts:
Measures of non-compactness are most commonly used if ''M'' is a Normed Vector Space . In this case, we have in addition: |
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