# Théorème du graphe fermé

Closed graph theorem This article is about closed graph theorems in general topology. For the closed graph theorem in functional analysis, see Closed graph theorem (analyse fonctionnelle). The graph of the cubic function {style d'affichage f(X)=x^{3}-9X} on the interval {style d'affichage [-4,4]} is closed because the function is continuous. The graph of the Heaviside function on {style d'affichage [-2,2]} is not closed, because the function is not continuous.

En mathématiques, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous.

Contenu 1 Graphs and maps with closed graphs 1.1 Examples of continuous maps that do not have a closed graph 2 Closed graph theorem in point-set topology 2.1 For set-valued functions 3 En analyse fonctionnelle 4 Voir également 5 Remarques 6 Références 7 Bibliography Graphs and maps with closed graphs Main article: Closed graph If {style d'affichage f:Xe Y} is a map between topological spaces then the graph of {style d'affichage f} is the set {nom de l'opérateur de style d'affichage {Gr} F:={(X,F(X)):xin X}} ou équivalent, {nom de l'opérateur de style d'affichage {Gr} F:={(X,y)in Xtimes Y_y=f(X)}} It is said that the graph of {style d'affichage f} is closed if {nom de l'opérateur de style d'affichage {Gr} F} is a closed subset of {style d'affichage X fois Y} (with the product topology).

Any continuous function into a Hausdorff space has a closed graph.

Any linear map, {displaystyle L:Xe Y,} between two topological vector spaces whose topologies are (Cauchy) complete with respect to translation invariant metrics, and if in addition (1un) {displaystyle L} is sequentially continuous in the sense of the product topology, then the map {displaystyle L} is continuous and its graph, Gr L, is necessarily closed. inversement, si {displaystyle L} is such a linear map with, in place of (1un), the graph of {displaystyle L} est (1b) known to be closed in the Cartesian product space {style d'affichage X fois Y} , alors {displaystyle L} is continuous and therefore necessarily sequentially continuous.[1] Examples of continuous maps that do not have a closed graph If {style d'affichage X} is any space then the identity map {nom de l'opérateur de style d'affichage {Id} :Xà X} is continuous but its graph, which is the diagonal {nom de l'opérateur de style d'affichage {Gr} nom de l'opérateur {Id} :={(X,X):xin X},} , est fermé dans {displaystyle Xtimes X} si et seulement si {style d'affichage X} is Hausdorff.[2] En particulier, si {style d'affichage X} is not Hausdorff then {nom de l'opérateur de style d'affichage {Id} :Xà X} is continuous but does not have a closed graph.

Laisser {style d'affichage X} denote the real numbers {style d'affichage mathbb {R} } with the usual Euclidean topology and let {style d'affichage Y} dénoter {style d'affichage mathbb {R} } with the indiscrete topology (where note that {style d'affichage Y} is not Hausdorff and that every function valued in {style d'affichage Y} est continue). Laisser {style d'affichage f:Xe Y} be defined by {style d'affichage f(0)=1} et {style d'affichage f(X)=0} pour tous {displaystyle xneq 0} . Alors {style d'affichage f:Xe Y} is continuous but its graph is not closed in {style d'affichage X fois Y} .[3] Closed graph theorem in point-set topology In point-set topology, the closed graph theorem states the following: Théorème du graphe fermé[4] — If {style d'affichage f:Xe Y} is a map from a topological space {style d'affichage X} into a Hausdorff space {style d'affichage Y,} then the graph of {style d'affichage f} is closed if {style d'affichage f:Xe Y} est continue. The converse is true when {style d'affichage Y} is compact. (Note that compactness and Hausdorffness do not imply each other.) Proof First part is essentially by definition.

Second part: For any open {style d'affichage Vsous-ensemble Y} , we check {style d'affichage f^{-1}(V)} is open. So take any {displaystyle xin f^{-1}(V)} , we construct some open neighborhood {style d'affichage U} de {style d'affichage x} , tel que {style d'affichage f(tu)subset V} .

Since the graph of {style d'affichage f} is closed, for every point {style d'affichage (X,y')} on the "vertical line at x", avec {displaystyle y'neq f(X)} , draw an open rectangle {style d'affichage U_{y'}times V_{y'}} disjoint from the graph of {style d'affichage f} . These open rectangles, when projected to the y-axis, cover the y-axis except at {style d'affichage f(X)} , so add one more set {style d'affichage V} .

Naively attempting to take {style d'affichage U:=bigcap _{y'neq f(X)}U_{y'}} would construct a set containing {style d'affichage x} , but it is not guaranteed to be open, so we use compactness here.

Depuis {style d'affichage Y} is compact, we can take a finite open covering of {style d'affichage Y} comme {style d'affichage {V,V_{y'_{1}},...,V_{y'_{n}}}} .

Now take {style d'affichage U:=bigcap _{je=1}^{n}U_{y'_{je}}} . It is an open neighborhood of {style d'affichage x} , since it is merely a finite intersection. We claim this is the open neighborhood of {style d'affichage U} that we want.

Suppose not, then there is some unruly {displaystyle x'in U} tel que {style d'affichage f(X')not in V} , then that would imply {style d'affichage f(X')in V_{y'_{je}}} pour certains {style d'affichage i} by open covering, but then {style d'affichage (X',F(X'))in Utimes V_{y'_{je}}subset U_{y'_{je}}times V_{y'_{je}}} , a contradiction since it is supposed to be disjoint from the graph of {style d'affichage f} .

Non-Hausdorff spaces are rarely seen, but non-compact spaces are common. An example of non-compact {style d'affichage Y} is the real line, which allows the discontinuous function with closed graph {style d'affichage f(X)={commencer{cas}{frac {1}{X}}{texte{ si }}xneq 0,\0{texte{ autre}}fin{cas}}} .

For set-valued functions Closed graph theorem for set-valued functions[5] — For a Hausdorff compact range space {style d'affichage Y} , a set-valued function {style d'affichage F:Xto 2^{Oui}} has a closed graph if and only if it is upper hemicontinuous and F(X) is a closed set for all {style d'affichage xin X} .

In functional analysis Main article: Théorème du graphe fermé (analyse fonctionnelle) Si {style d'affichage T:Xe Y} is a linear operator between topological vector spaces (téléviseurs) then we say that {style d'affichage T} is a closed operator if the graph of {style d'affichage T} est fermé dans {style d'affichage X fois Y} lorsque {style d'affichage X fois Y} is endowed with the product topology.

The closed graph theorem is an important result in functional analysis that guarantees that a closed linear operator is continuous under certain conditions. The original result has been generalized many times. A well known version of the closed graph theorems is the following.

Théorème[6][7] — A linear map between two F-spaces (par exemple. Espaces Banach) is continuous if and only if its graph is closed.

See also Almost open linear map Barrelled space – Topological vector space Closed graph Closed linear operator Discontinuous linear map Kakutani fixed-point theorem – On when a function f: S→Pow(S) on a compact nonempty convex subset S⊂ℝⁿ has a fixed point Open mapping theorem (analyse fonctionnelle) – Condition for a linear operator to be open Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem Webbed space – Space where open mapping and closed graph theorems hold Zariski's main theorem – Theorem of algebraic geometry and commutative algebra Notes References ^ Rudin 1991, p. 51-52. ^ Rudin 1991, p. 50. ^ Narici & Beckenstein 2011, pp. 459–483. ^ Munkres 2000, pp. 163–172. ^ Aliprantis, Charlambos; Kim C. Border (1999). "Chapitre 17". Infinite Dimensional Analysis: A Hitchhiker's Guide (3e éd.). Springer. ^ Schaefer & Wolff 1999, p. 78. ^ Trèves (2006), p. 173 Bibliography Bourbaki, Nicolas (1987) [1981]. Espaces vectoriels topologiques: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer Verlag. ISBN 3-540-13627-4. OCLC 17499190. Folland, Gerald B. (1984), Real Analysis: Modern Techniques and Their Applications (1st ed.), John Wiley & Sons, ISBN 978-0-471-80958-6 Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. Köthe, Gottfried (1983) [1969]. Espaces vectoriels topologiques I. Bases des sciences mathématiques. Volume. 159. Traduit par Garling, DJH. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. M 0248498. OCLC 840293704. Munkres, James R.. (2000). Topologie (Deuxième éd.). Upper Saddle River, New Jersey: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260. Narines, Laurent; Beckenstein, Edouard (2011). Espaces vectoriels topologiques. Mathématiques pures et appliquées (Deuxième éd.). Boca Ratón, Floride: Presse du CRC. ISBN 978-1584888666. OCLC 144216834. Roudine, Walter (1991). Analyse fonctionnelle. Série internationale de mathématiques pures et appliquées. Volume. 8 (Deuxième éd.). New York, New York: McGraw-Hill Sciences/Ingénierie/Maths. ISBN 978-0-07-054236-5. OCLC 21163277. Schäfer, Helmut H.; Wolff, Manfred P.. (1999). Espaces vectoriels topologiques. GTM. Volume. 8 (Deuxième éd.). New York, New York: Springer New York Mentions légales Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. Trèves, François (2006) [1967]. Espaces vectoriels topologiques, Distributions et noyaux. Mineola, NEW YORK.: Publications de Douvres. ISBN 978-0-486-45352-1. OCLC 853623322. Wilanski, Albert (2013). Méthodes modernes dans les espaces vectoriels topologiques. Mineola, New York: Publications de Douvres, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. Zălinescu, Constantin (30 Juillet 2002). Convex Analysis in General Vector Spaces. River Edge, N.J. Londres: World Scientific Publishing. ISBN 978-981-4488-15-0. M 1921556. OCLC 285163112 – via Internet Archive. "Proof of closed graph theorem". PlanèteMath. show vte Analyse fonctionnelle (sujets – glossaire) show vte Espaces vectoriels topologiques (téléviseurs) Catégories: Théorèmes en analyse fonctionnelle

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