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47-452: The notion of equicontinuity was introduced in the late 19th century by the Italian mathematicians Cesare Arzelà and Giulio Ascoli . A weak form of the theorem was proven by Ascoli (1883–1884) , who established the sufficient condition for compactness, and by Arzelà (1895) , who established the necessary condition and gave the first clear presentation of the result. A further generalization of

94-538: A n {\displaystyle a_{0}<a_{1}<\cdots <a_{n}} being a 0 < b 0 < a 1 < a 2 < ⋯ < a n − 1 < b 1 < a n {\displaystyle a_{0}<b_{0}<a_{1}<a_{2}<\cdots <a_{n-1}<b_{1}<a_{n}} ), considering topologies (the standard topology in Euclidean space being

141-419: A compact Hausdorff space X ( Dunford & Schwartz 1958 , §IV.6.7): The Arzelà–Ascoli theorem is thus a fundamental result in the study of the algebra of continuous functions on a compact Hausdorff space . Various generalizations of the above quoted result are possible. For instance, the functions can assume values in a metric space or (Hausdorff) topological vector space with only minimal changes to

188-495: A compact Hausdorff space, and let C ( X ) be the space of real-valued continuous functions on X . A subset F ⊂ C ( X ) is said to be equicontinuous if for every x  ∈ X and every ε > 0 , x has a neighborhood U x such that A set F ⊂ C ( X , R ) is said to be pointwise bounded if for every x  ∈ X , A version of the Theorem holds also in the space C ( X ) of real-valued continuous functions on

235-671: A cover of a set is a subset of the cover that also covers the set. A cover is called an open cover if each of its elements is an open set . Covers are commonly used in the context of topology . If the set X {\displaystyle X} is a topological space , then a cover C {\displaystyle C} of X {\displaystyle X} is a collection of subsets { U α } α ∈ A {\displaystyle \{U_{\alpha }\}_{\alpha \in A}} of X {\displaystyle X} whose union

282-418: A family of functions from a compactly generated Hausdorff space into a uniform space to be compact in the compact-open topology ; see Kelley (1991 , page 234). By definition, a sequence { f n } n ∈ N {\displaystyle \{f_{n}\}_{n\in \mathbb {N} }} of continuous functions on an interval I = [ a , b ] is uniformly bounded if there

329-402: A family of infinitely differentiable functions, whose derivatives of each order are uniformly bounded, has a uniformly convergent subsequence, all of whose derivatives are also uniformly convergent. This is particularly important in the theory of distributions. The argument given above proves slightly more, specifically The limit function is also Lipschitz continuous with the same value K for

376-432: A finite subcover U 1 , ..., U J . There exists an integer K such that each open interval U j , 1 ≤ j ≤ J , contains a rational x k with 1 ≤ k ≤ K . Finally, for any t  ∈  I , there are j and k so that t and x k belong to the same interval U j . For this choice of k , for all n , m > N = max{ N ( ε , x 1 ), ..., N ( ε , x K )}. Consequently,

423-672: A professor in 1880 at the University of Bologna at the Department of analysis. He conducted research in the field of theory of functions . His most famous student was Leonida Tonelli . In 1889 he generalized the Ascoli theorem to Arzelà–Ascoli theorem , an important theorem in the theory of functions. He was a member of the Accademia Nazionale dei Lincei , and of several other academies. This article about an Italian mathematician

470-408: A refinement of the trivial topology ). When subdividing simplicial complexes (the first barycentric subdivision of a simplicial complex is a refinement), the situation is slightly different: every simplex in the finer complex is a face of some simplex in the coarser one, and both have equal underlying polyhedra. Yet another notion of refinement is that of star refinement . A simple way to get

517-803: A subcover is to omit the sets contained in another set in the cover. Consider specifically open covers. Let B {\displaystyle {\mathcal {B}}} be a topological basis of X {\displaystyle X} and O {\displaystyle {\mathcal {O}}} be an open cover of X . {\displaystyle X.} First take A = { A ∈ B :  there exists  U ∈ O  such that  A ⊆ U } . {\displaystyle {\mathcal {A}}=\{A\in {\mathcal {B}}:{\text{ there exists }}U\in {\mathcal {O}}{\text{ such that }}A\subseteq U\}.} Then A {\displaystyle {\mathcal {A}}}

SECTION 10

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564-424: Is transitive and reflexive , i.e. a Preorder . It is never asymmetric for X ≠ ∅ {\displaystyle X\neq \emptyset } . Generally speaking, a refinement of a given structure is another that in some sense contains it. Examples are to be found when partitioning an interval (one refinement of a 0 < a 1 < ⋯ <

611-585: Is a stub . You can help Misplaced Pages by expanding it . Open cover In mathematics , and more particularly in set theory , a cover (or covering ) of a set X {\displaystyle X} is a family of subsets of X {\displaystyle X} whose union is all of X {\displaystyle X} . More formally, if C = { U α : α ∈ A } {\displaystyle C=\lbrace U_{\alpha }:\alpha \in A\rbrace }

658-513: Is a topological vector space . Recall that if X {\displaystyle X} is a topological space and Y {\displaystyle Y} is a uniform space (such as any metric space or any topological group , metrisable or not), there is the topology of compact convergence on the set F ( X , Y ) {\displaystyle {\mathfrak {F}}(X,Y)} of functions X → Y {\displaystyle X\rightarrow Y} ; it

705-486: Is a (topological) subspace of X {\displaystyle X} , then a cover of Y {\displaystyle Y} is a collection of subsets C = { U α } α ∈ A {\displaystyle C=\{U_{\alpha }\}_{\alpha \in A}} of X {\displaystyle X} whose union contains Y {\displaystyle Y} , i.e., C {\displaystyle C}

752-408: Is a cover of Y {\displaystyle Y} if That is, we may cover Y {\displaystyle Y} with either sets in Y {\displaystyle Y} itself or sets in the parent space X {\displaystyle X} . Let C be a cover of a topological space X . A subcover of C is a subset of C that still covers X . We say that C

799-492: Is a number M such that for every function   f n   belonging to the sequence, and every x ∈ [ a , b ] . (Here, M must be independent of n and x .) The sequence is said to be uniformly equicontinuous if, for every ε > 0 , there exists a δ > 0 such that whenever | x − y | < δ   for all functions   f n   in the sequence. (Here, δ may depend on ε , but not x , y or n .) One version of

846-613: Is a refinement of O {\displaystyle {\mathcal {O}}} . Next, for each A ∈ A , {\displaystyle A\in {\mathcal {A}},} we select a U A ∈ O {\displaystyle U_{A}\in {\mathcal {O}}} containing A {\displaystyle A} (requiring the axiom of choice). Then C = { U A ∈ O : A ∈ A } {\displaystyle {\mathcal {C}}=\{U_{A}\in {\mathcal {O}}:A\in {\mathcal {A}}\}}

893-459: Is a sequence { f n 1 } of distinct functions in F such that { f n 1 ( x 1 )} converges. Repeating the same argument for the sequence of points { f n 1 ( x 2 )} , there is a subsequence { f n 2 } of { f n 1 } such that { f n 2 ( x 2 )} converges. By induction this process can be continued forever, and so there is a chain of subsequences such that, for each k = 1, 2, 3, ...,

940-438: Is a subcover of O . {\displaystyle {\mathcal {O}}.} Hence the cardinality of a subcover of an open cover can be as small as that of any topological basis. Hence in particular second countability implies a space is Lindelöf . The language of covers is often used to define several topological properties related to compactness . A topological space X is said to be For some more variations see

987-436: Is also possible to extend the statement to functions that are only continuous when restricted to the sets of a covering of X {\displaystyle X} by compact subsets. For details one can consult Bourbaki (1998), Chapter X, § 2, nr 5. Solutions of numerical schemes for parabolic equations are usually piecewise constant, and therefore not continuous, in time. As their jumps nevertheless tend to become small as

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1034-478: Is an open cover if each of its members is an open set (i.e. each U α is contained in T , where T is the topology on X ). A cover of X is said to be locally finite if every point of X has a neighborhood that intersects only finitely many sets in the cover. Formally, C = { U α } is locally finite if for any x ∈ X , {\displaystyle x\in X,} there exists some neighborhood N ( x ) of x such that

1081-560: Is an indexed family of subsets U α ⊂ X {\displaystyle U_{\alpha }\subset X} (indexed by the set A {\displaystyle A} ), then C {\displaystyle C} is a cover of X {\displaystyle X} if ⋃ α ∈ A U α ⊇ X {\displaystyle \bigcup _{\alpha \in A}U_{\alpha }\supseteq X} . Thus

1128-485: Is bounded in the uniform norm on C ( X ) and in particular is pointwise bounded. Let N ( ε , U ) be the set of all functions in F whose oscillation over an open subset U ⊂ X is less than ε : For a fixed x ∈ X and ε , the sets N ( ε , U ) form an open covering of F as U varies over all open neighborhoods of x . Choosing a finite subcover then gives equicontinuity. This article incorporates material from Ascoli–Arzelà theorem on PlanetMath , which

1175-506: Is contained in some set in C {\displaystyle C} . Formally, In other words, there is a refinement map ϕ : B → A {\displaystyle \phi :B\to A} satisfying V β ⊆ U ϕ ( β ) {\displaystyle V_{\beta }\subseteq U_{\phi (\beta )}} for every β ∈ B . {\displaystyle \beta \in B.} This map

1222-523: Is licensed under the Creative Commons Attribution/Share-Alike License . Cesare Arzel%C3%A0 Cesare Arzelà (6 March 1847–15 March 1912) was an Italian mathematician who taught at the University of Bologna and is recognized for his contributions in the theory of functions , particularly for his characterization of sequences of continuous functions , generalizing the one given earlier by Giulio Ascoli in

1269-489: Is set up so that a sequence (or more generally a filter or net ) of functions converges if and only if it converges uniformly on each compact subset of X {\displaystyle X} . Let C c ( X , Y ) {\displaystyle {\mathcal {C}}_{c}(X,Y)} be the subspace of F ( X , Y ) {\displaystyle {\mathfrak {F}}(X,Y)} consisting of continuous functions, equipped with

1316-508: Is the supremum of the derivatives of functions in the sequence and is independent of n . So, given ε > 0 , let δ = ⁠ ε / 2 K ⁠ to verify the definition of equicontinuity of the sequence. This proves the following corollary: If, in addition, the sequence of second derivatives is also uniformly bounded, then the derivatives also converge uniformly (up to a subsequence), and so on. Another generalization holds for continuously differentiable functions . Suppose that

1363-391: Is the whole space X {\displaystyle X} . In this case we say that C {\displaystyle C} covers X {\displaystyle X} , or that the sets U α {\displaystyle U_{\alpha }} cover X {\displaystyle X} . Also, if Y {\displaystyle Y}

1410-432: Is used, for instance, in the Čech cohomology of X {\displaystyle X} . Every subcover is also a refinement, but the opposite is not always true. A subcover is made from the sets that are in the cover, but omitting some of them; whereas a refinement is made from any sets that are subsets of the sets in the cover. The refinement relation on the set of covers of X {\displaystyle X}

1457-674: The Arzelà–Ascoli theorem . He was a pupil of the Scuola Normale Superiore of Pisa where he graduated in 1869. Arzelà came from a poor household; therefore he could not start his study until 1871, when he studied in Pisa under Enrico Betti and Ulisse Dini . He was working in Florence (from 1875) and in 1878 obtained the Chair of Algebra at the University of Palermo . After that, he became

Arzelà–Ascoli theorem - Misplaced Pages Continue

1504-542: The Arzelà–Ascoli theorem d times to extract a subsequence that converges uniformly in the first coordinate, then a sub-subsequence that converges uniformly in the first two coordinates, and so on. The above examples generalize easily to the case of functions with values in Euclidean space. The definitions of boundedness and equicontinuity can be generalized to the setting of arbitrary compact metric spaces and, more generally still, compact Hausdorff spaces . Let X be

1551-460: The Lipschitz constant. A slight refinement is This holds more generally for scalar functions on a compact metric space X satisfying a Hölder condition with respect to the metric on X . The Arzelà–Ascoli theorem holds, more generally, if the functions   f n   take values in d -dimensional Euclidean space R , and the proof is very simple: just apply the R -valued version of

1598-427: The collection { U α : α ∈ A } {\displaystyle \lbrace U_{\alpha }:\alpha \in A\rbrace } is a cover of X {\displaystyle X} if each element of X {\displaystyle X} belongs to at least one of the subsets U α {\displaystyle U_{\alpha }} . A subcover of

1645-418: The domain. On a compact Hausdorff space X , for instance, the equicontinuity is used to extract, for each ε = 1/ n , a finite open covering of X such that the oscillation of any function in the family is less than ε on each open set in the cover. The role of the rationals can then be played by a set of points drawn from each open set in each of the countably many covers obtained in this way, and

1692-494: The family F is equicontinuous, for this fixed ε and for every x in I , there is an open interval U x containing x such that for all f  ∈  F and all s ,  t in I such that s , t ∈ U x . The collection of intervals U x , x  ∈ I , forms an open cover of I . Since I is closed and bounded, by the Heine–Borel theorem I is compact , implying that this covering admits

1739-485: The functions   f n   are continuously differentiable with derivatives f n ′ . Suppose that f n ′ are uniformly equicontinuous and uniformly bounded, and that the sequence {  f n  }, is pointwise bounded (or just bounded at a single point). Then there is a subsequence of the {  f n  } converging uniformly to a continuously differentiable function. The diagonalization argument can also be used to show that

1786-445: The main part of the proof proceeds exactly as above. A similar argument is used as a part of the proof for the general version which does not assume completeness of Y . The Arzela-Ascoli theorem generalises to functions X → Y {\displaystyle X\rightarrow Y} where X {\displaystyle X} is not compact. Particularly important are cases where X {\displaystyle X}

1833-429: The sequence { f n } is uniformly Cauchy , and therefore converges to a continuous function, as claimed. This completes the proof. The hypotheses of the theorem are satisfied by a uniformly bounded sequence {  f n  } of differentiable functions with uniformly bounded derivatives. Indeed, uniform boundedness of the derivatives implies by the mean value theorem that for all x and y , where K

1880-548: The set is finite. A cover of X is said to be point finite if every point of X is contained in only finitely many sets in the cover. A cover is point finite if it is locally finite, though the converse is not necessarily true. A refinement of a cover C {\displaystyle C} of a topological space X {\displaystyle X} is a new cover D {\displaystyle D} of X {\displaystyle X} such that every set in D {\displaystyle D}

1927-415: The statement (see, for instance, Kelley & Namioka (1982 , §8), Kelley (1991 , Chapter 7)): Here pointwise relatively compact means that for each x  ∈  X , the set F x = {  f  ( x ) :   f   ∈ F } is relatively compact in Y . In the case that Y is complete , the proof given above can be generalized in a way that does not rely on the separability of

Arzelà–Ascoli theorem - Misplaced Pages Continue

1974-408: The subsequence { f n k } converges at x 1 , ..., x k . Now form the diagonal subsequence { f } whose m th term f m is the m th term in the m th subsequence { f n m } . By construction, f m converges at every rational point of I . Therefore, given any ε > 0 and rational x k in I , there is an integer N = N ( ε , x k ) such that Since

2021-471: The theorem can be stated as follows: The proof is essentially based on a diagonalization argument . The simplest case is of real-valued functions on a closed and bounded interval: Fix an enumeration { x i } i ∈ N of rational numbers in I . Since F is uniformly bounded, the set of points { f ( x 1 )} f ∈ F is bounded, and hence by the Bolzano–Weierstrass theorem , there

2068-404: The theorem was proven by Fréchet (1906) , to sets of real-valued continuous functions with domain a compact metric space ( Dunford & Schwartz 1958 , p. 382). Modern formulations of the theorem allow for the domain to be compact Hausdorff and for the range to be an arbitrary metric space. More general formulations of the theorem exist that give necessary and sufficient conditions for

2115-501: The time step goes to 0 {\displaystyle 0} , it is possible to establish uniform-in-time convergence properties using a generalisation to non-continuous functions of the classical Arzelà–Ascoli theorem (see e.g. Droniou & Eymard (2016 , Appendix)). Denote by S ( X , Y ) {\displaystyle S(X,Y)} the space of functions from X {\displaystyle X} to Y {\displaystyle Y} endowed with

2162-610: The topology of compact convergence. Then one form of the Arzelà-Ascoli theorem is the following: This theorem immediately gives the more specialised statements above in cases where X {\displaystyle X} is compact and the uniform structure of Y {\displaystyle Y} is given by a metric. There are a few other variants in terms of the topology of precompact convergence or other related topologies on F ( X , Y ) {\displaystyle {\mathfrak {F}}(X,Y)} . It

2209-477: The uniform metric Then we have the following: Whereas most formulations of the Arzelà–Ascoli theorem assert sufficient conditions for a family of functions to be (relatively) compact in some topology, these conditions are typically also necessary. For instance, if a set F is compact in C ( X ), the Banach space of real-valued continuous functions on a compact Hausdorff space with respect to its uniform norm, then it

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