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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...

, 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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

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 “si ...

. 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 (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

''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 Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...

, 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 set, non-empty compact space, compact subsets o ...

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

normed vector space The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war ...

. 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 In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...

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, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

R^''n'': by the

Heine–Borel theorem In real analysis, the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset S of Euclidean space \mathbb^n, the following two statements are equivalent: *S is compact, that is, every open cover of S has a finite s ...

, 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 In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...

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''.

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. He worked as a professor at the University of Warsaw and at the Ma ...

: ''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