 Measure of noncompactness

In functional analysis, two measures of noncompactness are commonly used; these associate numbers to sets in such a way that 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 noncompactness is defined as
 α(X) = inf {r > 0 : there exist finitely many balls of radius r which cover X}
and the Kuratowski measure of noncompactness 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 2r, 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 of sets.
Measures of noncompactness 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 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 noncompactness are useless for subsets of Euclidean space R^{n}: by the HeineBorel 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 noncompactness are however useful in the study of infinitedimensional Banach spaces, for example. In this context, one can prove that any ball B of radius r has α(B) = r and β(B) = 2r.
Categories:
Wikimedia Foundation. 2010.
Look at other dictionaries:
List of mathematics articles (M) — NOTOC M M estimator M group M matrix M separation M set M. C. Escher s legacy M. Riesz extension theorem M/M/1 model Maass wave form Mac Lane s planarity criterion Macaulay brackets Macbeath surface MacCormack method Macdonald polynomial Machin… … Wikipedia
Compact (disambiguation) — Compact may mean: * Compact (newspaper), a broadsheet quality newspaper printed in a tabloid format. * Compact (soap opera), a 1960s British soap opera. * a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an… … Wikipedia
List of general topology topics — This is a list of general topology topics, by Wikipedia page. Contents 1 Basic concepts 2 Limits 3 Topological properties 3.1 Compactness and countability … Wikipedia
List of functional analysis topics — This is a list of functional analysis topics, by Wikipedia page. Contents 1 Hilbert space 2 Functional analysis, classic results 3 Operator theory 4 Banach space examples … Wikipedia
Bounded variation — In mathematical analysis, a function of bounded variation refers to a real valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a… … Wikipedia
Determining the number of clusters in a data set — Determining the number of clusters in a data set, a quantity often labeled k as in the k means algorithm, is a frequent problem in data clustering, and is a distinct issue from the process of actually solving the clustering problem. For a certain … Wikipedia
Alexandra Bellow — (1935 ndash;) is a mathematician who has made substantial contributions to the fields of ergodic theory, probability and analysis. BiographyShe was born in Bucharest, Romania, as Alexandra Bagdasar. Her parents were both physicians. Her mother,… … Wikipedia
Compact operator on Hilbert space — In functional analysis, compact operators on Hilbert spaces are a direct extension of matrices: in the Hilbert spaces, they are precisely the closure of finite rank operators in the uniform operator topology. As such, results from matrix theory… … Wikipedia
Algorithm — Flow chart of an algorithm (Euclid s algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers a and b in locations named A and B. The algorithm proceeds by successive subtractions in two loops: IF the test B ≤ A yields yes… … Wikipedia
Maximum likelihood — In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of a statistical model. When applied to a data set and given a statistical model, maximum likelihood estimation provides estimates for the model s… … Wikipedia