Helly's selection theorem
Encyclopedia
In mathematics
, Helly's selection theorem states that a sequence
of functions that is locally of bounded total variation
and uniformly bounded at a point has a convergent subsequence
. In other words, it is a compactness theorem for the space BVloc. It is named for the Austria
n mathematician
Eduard Helly
.
The theorem has applications throughout mathematical analysis
. In probability theory
, the result implies compactness of a tight family of measures
.
and let fn : U → R, n ∈ N, be a sequence of functions. Suppose that
Then there exists a subsequence
fnk, k ∈ N, of fn and a function f : U → R, locally of bounded variation
, such that
s, is due to Barbu and Precupanu:
Let X be a reflexive
, separable Banach space and let E be a closed, convex
subset of X. Let Δ : X → [0, +∞) be positive-definite and homogeneous of degree one
. Suppose that zn is a uniformly bounded sequence in BV([0, T]; X) with zn(t) ∈ E for all n ∈ N and t ∈ [0, T]. Then there exists a subsequence znk and functions δ, z ∈ BV([0, T]; X) such that
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Helly's selection theorem states that a sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
of functions that is locally of bounded total variation
Bounded variation
In mathematical analysis, a function of bounded variation, also known as a BV function, is a real-valued function whose total variation is bounded : the graph of a function having this property is well behaved in a precise sense...
and uniformly bounded at a point has a convergent subsequence
Subsequence
In mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements...
. In other words, it is a compactness theorem for the space BVloc. It is named for the Austria
Austria
Austria , officially the Republic of Austria , is a landlocked country of roughly 8.4 million people in Central Europe. It is bordered by the Czech Republic and Germany to the north, Slovakia and Hungary to the east, Slovenia and Italy to the south, and Switzerland and Liechtenstein to the...
n mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Eduard Helly
Eduard Helly
Eduard Helly was a mathematician and the eponym of Helly's theorem, Helly families, Helly's selection theorem, Helly metric, and the Helly–Bray theorem. In 1912, Helly published a proof of Hahn–Banach theorem, 15 years before Hahn and Banach discovered it independently...
.
The theorem has applications throughout mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
. In probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, the result implies compactness of a tight family of measures
Tightness of measures
In mathematics, tightness is a concept in measure theory. The intuitive idea is that a given collection of measures does not "escape to infinity."-Definitions:...
.
Statement of the theorem
Let U be an open subset of the real lineReal line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
and let fn : U → R, n ∈ N, be a sequence of functions. Suppose that
- (fn) has uniformly boundedBounded variationIn mathematical analysis, a function of bounded variation, also known as a BV function, is a real-valued function whose total variation is bounded : the graph of a function having this property is well behaved in a precise sense...
total variationTotal variationIn mathematics, the total variation identifies several slightly different concepts, related to the structure of the codomain of a function or a measure...
on any W that is compactly embeddedCompactly embeddedIn mathematics, the notion of being compactly embedded expresses the idea that one set or space is "well contained" inside another. There are versions of this concept appropriate to general topology and functional analysis.-Definition :...
in U. That is, for all sets W ⊆ U with compactCompact spaceIn 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...
closureClosure (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...
W̄ ⊆ U,
-
- where the derivative is taken in the sense of tempered distributionsDistribution (mathematics)In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
;- and (fn) is uniformly bounded at a point. That is, for some t ∈ U, { fn(t) | n ∈ N } ⊆ R is a bounded setBounded setIn mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded...
.
- and (fn) is uniformly bounded at a point. That is, for some t ∈ U, { fn(t) | n ∈ N } ⊆ R is a bounded set
Then there exists a subsequence
Subsequence
In mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements...
fnk, k ∈ N, of fn and a function f : U → R, locally of bounded variation
Bounded variation
In mathematical analysis, a function of bounded variation, also known as a BV function, is a real-valued function whose total variation is bounded : the graph of a function having this property is well behaved in a precise sense...
, such that
- fnk converges to f pointwise;
- and fnk converges to f locally in L1 (see locally integrable functionLocally integrable functionIn mathematics, a locally integrable function is a function which is integrable on any compact set of its domain of definition. Their importance lies on the fact that we do not care about their behavior at infinity.- Formal definition :...
), i.e., for all W compactly embedded in U,
-
- and, for W compactly embedded in U,
Generalizations
There are many generalizations and refinements of Helly's theorem. The following theorem, for BV functions taking values in Banach spaceBanach 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, is due to Barbu and Precupanu:
Let X be a reflexive
Reflexive space
In functional analysis, a Banach space is called reflexive if it coincides with the dual of its dual space in the topological and algebraic senses. Reflexive Banach spaces are often characterized by their geometric properties.- Normed spaces :Suppose X is a normed vector space over R or C...
, separable Banach space and let E be a closed, convex
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
subset of X. Let Δ : X → [0, +∞) be positive-definite and homogeneous of degree one
Homogeneous function
In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...
. Suppose that zn is a uniformly bounded sequence in BV([0, T]; X) with zn(t) ∈ E for all n ∈ N and t ∈ [0, T]. Then there exists a subsequence znk and functions δ, z ∈ BV([0, T]; X) such that
- for all t ∈ [0, T],
-
- and, for all t ∈ [0, T],
- and, for all 0 ≤ s < t ≤ T,
