# Category: Compactness theorems

Michael selection theorem
In functional analysis, a branch of mathematics, Michael selection theorem is a selection theorem named after Ernest Michael. In its most popular form, it states the following: Let X be a paracompact
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 Rn, the following two statements are equivalent: * S is closed and bounde
Sobolev inequality
In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclu
Eberlein–Šmulian theorem
In the mathematical field of functional analysis, the Eberlein–Šmulian theorem (named after William Frederick Eberlein and Witold Lwowitsch Schmulian) is a result that relates three different kinds of
Gromov's compactness theorem (topology)
In the mathematical field of symplectic topology, Gromov's compactness theorem states that a sequence of pseudoholomorphic curves in an almost complex manifold with a uniform energy bound must have a
Prokhorov's theorem
In measure theory Prokhorov's theorem relates tightness of measures to relative compactness (and hence weak convergence) in the space of probability measures. It is credited to the Soviet mathematicia
Montel's theorem
In complex analysis, an area of mathematics, Montel's theorem refers to one of two theorems about families of holomorphic functions. These are named after French mathematician Paul Montel, and give co
Kuratowski's intersection theorem
In mathematics, Kuratowski's intersection theorem is a result in general topology that gives a sufficient condition for a nested sequence of sets to have a non-empty intersection. Kuratowski's result
Bolzano–Weierstrass theorem
In mathematics, specifically in real analysis, the Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Eucl
Blaschke selection theorem
The Blaschke selection theorem is a result in topology and convex geometry about sequences of convex sets. Specifically, given a sequence of convex sets contained in a bounded set, the theorem guarant
Banach–Alaoglu theorem
In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is
Mumford's compactness theorem
In mathematics, Mumford's compactness theorem states that the space of compact Riemann surfaces of fixed genus g > 1 with no closed geodesics of length less than some fixed ε > 0 in the Poincaré metri
Fréchet–Kolmogorov theorem
In functional analysis, the Fréchet–Kolmogorov theorem (the names of Riesz or Weil are sometimes added as well) gives a necessary and sufficient condition for a set of functions to be relatively compa
Mazur's lemma
In mathematics, Mazur's lemma is a result in the theory of Banach spaces. It shows that any weakly convergent sequence in a Banach space has a sequence of convex combinations of its members that conve
Helly's selection theorem
In mathematics, Helly's selection theorem (also called the Helly selection principle) states that a uniformly bounded sequence of monotone real functions admits a convergent subsequence.In other words
Fraňková–Helly selection theorem
In mathematics, the Fraňková–Helly selection theorem is a generalisation of Helly's selection theorem for functions of bounded variation to the case of regulated functions. It was proved in 1991 by th
Arzelà–Ascoli theorem
The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous funct
Mahler's compactness theorem
In mathematics, Mahler's compactness theorem, proved by Kurt Mahler, is a foundational result on lattices in Euclidean space, characterising sets of lattices that are 'bounded' in a certain definite s
Cantor's intersection theorem
Cantor's intersection theorem refers to two closely related theorems in general topology and real analysis, named after Georg Cantor, about intersections of decreasing nested sequences of non-empty co