Editing Measure of non-compactness

In [[functional analysis]], two '''measures of non-compactness''' are commonly used; these associate numbers to sets in such a way that [[Compact space|compact]] sets all get the measure 0, and other sets get measures that are bigger according to "how far" they are removed from compactness.

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'') = [[infimum|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:
* ''X'' is bounded if and only if γ(''X'') < ∞. 
* γ(''X'') = γ(''X''<sup>cl</sup>), where ''X''<sup>cl</sup> denotes the [[closure (topology)|closure]] of ''X''. 
* If ''X'' is compact, then γ(''X'') = 0. Conversely, if γ(''X'') = 0 and ''X'' is [[complete space|complete]], then ''X'' is compact. 
* γ(''X'' ∪ ''Y'') = max(γ(''X''), γ(''Y'')) for any two subsets ''X'' and ''Y''.
* γ is continuous with respect to the [[Hausdorff distance]] of sets.

Measures of non-compactness are most commonly used if ''M'' is a [[normed vector space]]. In this case, we have in addition:
* γ(''aX'') = |''a''| γ(''X'') for any [[scalar (mathematics)|scalar]] ''a''
* γ(''X'' + ''Y'') ≤ γ(''X'') + γ(''Y'')
* γ(conv(''X'')) = γ(''X''), where conv(''X'') denotes the [[convex hull]] of ''X''

Note that these measures of non-compactness are useless for subsets of [[Euclidean space]] '''R'''<sup>''n''</sup>: by the [[Heine–Borel theorem]], every bounded closed set is compact there, which means that γ(''X'') = 0 or ∞ according to whether ''X'' is bounded or not.

Measures of non-compactness are however useful in the study of infinite-dimensional [[Banach space]]s, for example. In this context, one can prove that any ball ''B'' of radius ''r'' has α(''B'') = ''r'' and β(''B'') = 2''r''.

== See also ==

* [[Kuratowski's intersection theorem]]

== References ==
# Józef Banaś, [[Kazimierz Goebel]]: ''Measures of noncompactness in Banach spaces'', Institute of Mathematics, Polish Academy of Sciences, Warszawa 1979
# [[Kazimierz Kuratowski]]: ''Topologie Vol I'', PWN. Warszawa 1958
# R.R. Akhmerov, M.I. Kamenskii, A.S. Potapova, A.E. Rodkina and B.N. Sadovskii, ''Measure of Noncompactness and Condensing Operators'', Birkhäuser, Basel 1992

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[[Category:Functional analysis]]