In functional analysis

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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

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

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 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

Metric space

In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

 M and a subset X. The ball measure of non-compactness is defined as

α(X) = inf

Infimum

In mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used...

 {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 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
  Closure (topology)
  In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...
  
   of X.
• If X is compact, then γ(X) = 0. Conversely, if γ(X) = 0 and X is complete
  Complete space
  In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
  
  , 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
  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 metric space in its own right...
  
   of sets.

Measures of non-compactness are most commonly used if M is a normed vector space

Normed vector space

In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....

. In this case, we have in addition:

• γ(aX) = |a| γ(X) for any scalar
  Scalar (mathematics)
  In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
  
   a
• γ(X + Y) ≤ γ(X) + γ(Y)
• γ(conv(X)) = γ(X), where conv(X) denotes the convex hull
  Convex hull
  In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....
  
   of X

Note that these measures of non-compactness are useless for subsets of Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 Rⁿ: 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

Banach space

In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

s, for example. In this context, one can prove that any ball B of radius r has α(B) = r and β(B) = 2r.