functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...

, two measures of non-compactness are commonly used; these associate numbers to sets in such a way that

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

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 In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size� ...

. 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 In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...

''M'' and a subset ''X''. The ball measure of non-compactness is defined as :α(''X'') = inf and the Kuratowski measure of non-compactness is defined as :β(''X'') = inf 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''^cl), where ''X''^cl denotes the closure of ''X''. * If ''X'' is compact, then γ(''X'') = 0. Conversely, if γ(''X'') = 0 and ''X'' is 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 In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a me ...

of sets. Measures of non-compactness are most commonly used if ''M'' is a

normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...

. In this case, we have in addition: * γ(''aX'') = , ''a'', γ(''X'') for any 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 Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

R^''n'': 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 spaces, for example. In this context, one can prove that any ball ''B'' of radius ''r'' has α(''B'') = ''r'' and β(''B'') = 2''r''.

References

# Józef Banaś, Kazimierz Goebel: ''Measures of noncompactness in Banach spaces'', Institute of Mathematics, Polish Academy of Sciences, Warszawa 1979 #

Kazimierz Kuratowski Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics. Biography and studies Kazimierz Kuratowski was born in Warsaw, (t ...

: ''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 {{DEFAULTSORT:Measure Of Non-Compactness Functional analysis

See also

References