Closed graph theorem

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 (análise funcional). The graph of the cubic function {estilo de exibição f(x)=x^{3}-9x} on the interval {estilo de exibição [-4,4]} is closed because the function is continuous. The graph of the Heaviside function on {estilo de exibição [-2,2]} is not closed, because the function is not continuous.

Na matemática, 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.

Conteúdo 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 Na análise funcional 4 Veja também 5 Notas 6 Referências 7 Bibliography Graphs and maps with closed graphs Main article: Closed graph If {estilo de exibição f:Xº Y} is a map between topological spaces then the graph of {estilo de exibição f} is the set {nome do operador de estilo de exibição {Gr} f:={(x,f(x)):xin X}} ou equivalente, {nome do operador de estilo de exibição {Gr} f:={(x,y)in Xtimes Y_y=f(x)}} It is said that the graph of {estilo de exibição f} is closed if {nome do operador de estilo de exibição {Gr} f} is a closed subset of {estilo de exibição Xtimes Y} (with the product topology).

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

Any linear map, {estilo de exibição L:Xº Y,} between two topological vector spaces whose topologies are (Cauchy) complete with respect to translation invariant metrics, and if in addition (1uma) {estilo de exibição L} is sequentially continuous in the sense of the product topology, then the map {estilo de exibição L} is continuous and its graph, Gr L, is necessarily closed. Por outro lado, E se {estilo de exibição L} is such a linear map with, no lugar de (1uma), the graph of {estilo de exibição L} é (1b) known to be closed in the Cartesian product space {estilo de exibição Xtimes Y} , então {estilo de exibição L} is continuous and therefore necessarily sequentially continuous.[1] Examples of continuous maps that do not have a closed graph If {estilo de exibição X} is any space then the identity map {nome do operador de estilo de exibição {Id} :X a X} is continuous but its graph, which is the diagonal {nome do operador de estilo de exibição {Gr} nome do operador {Id} :={(x,x):xin X},} , is closed in {displaystyle Xtimes X} se e apenas se {estilo de exibição X} is Hausdorff.[2] Em particular, E se {estilo de exibição X} is not Hausdorff then {nome do operador de estilo de exibição {Id} :X a X} is continuous but does not have a closed graph.

Deixar {estilo de exibição X} denote the real numbers {estilo de exibição mathbb {R} } with the usual Euclidean topology and let {estilo de exibição Y} denote {estilo de exibição mathbb {R} } with the indiscrete topology (where note that {estilo de exibição Y} is not Hausdorff and that every function valued in {estilo de exibição Y} é contínuo). Deixar {estilo de exibição f:Xº Y} be defined by {estilo de exibição f(0)=1} e {estilo de exibição f(x)=0} para todos {displaystyle xneq 0} . Então {estilo de exibição f:Xº Y} is continuous but its graph is not closed in {estilo de exibição Xtimes Y} .[3] Closed graph theorem in point-set topology In point-set topology, the closed graph theorem states the following: Closed graph theorem[4] — If {estilo de exibição f:Xº Y} is a map from a topological space {estilo de exibição X} into a Hausdorff space {estilo de exibição Y,} then the graph of {estilo de exibição f} is closed if {estilo de exibição f:Xº Y} é contínuo. The converse is true when {estilo de exibição 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 {estilo de exibição Vsubconjunto Y} , we check {estilo de exibição f^{-1}(V)} is open. So take any {displaystyle xin f^{-1}(V)} , we construct some open neighborhood {estilo de exibição U} do {estilo de exibição x} , de tal modo que {estilo de exibição f(você)subset V} .

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

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

Desde {estilo de exibição Y} is compact, we can take a finite open covering of {estilo de exibição Y} Como {estilo de exibição {V,V_{y'_{1}},...,V_{y'_{n}}}} .

Now take {estilo de exibição U:=bigcap _{i=1}^{n}VOCÊ_{y'_{eu}}} . It is an open neighborhood of {estilo de exibição x} , since it is merely a finite intersection. We claim this is the open neighborhood of {estilo de exibição U} that we want.

Suppose not, then there is some unruly {displaystyle x'in U} de tal modo que {estilo de exibição f(x')not in V} , then that would imply {estilo de exibição f(x')in V_{y'_{eu}}} para alguns {estilo de exibição eu} by open covering, but then {estilo de exibição (x',f(x'))in Utimes V_{y'_{eu}}subset U_{y'_{eu}}times V_{y'_{eu}}} , a contradiction since it is supposed to be disjoint from the graph of {estilo de exibição f} .

Non-Hausdorff spaces are rarely seen, but non-compact spaces are common. An example of non-compact {estilo de exibição Y} is the real line, which allows the discontinuous function with closed graph {estilo de exibição f(x)={começar{casos}{fratura {1}{x}}{texto{ E se }}xneq 0,\0{texto{ senão}}fim{casos}}} .

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

In functional analysis Main article: Closed graph theorem (análise funcional) Se {estilo de exibição T:Xº Y} is a linear operator between topological vector spaces (TV) then we say that {estilo de exibição T} is a closed operator if the graph of {estilo de exibição T} is closed in {estilo de exibição Xtimes Y} quando {estilo de exibição Xtimes 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.

Teorema[6][7] — A linear map between two F-spaces (por exemplo. Banach spaces) 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 (análise funcional) – 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). "Capítulo 17". Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer. ^ Schaefer & Wolff 1999, p. 78. ^ Trèves (2006), p. 173 Bibliography Bourbaki, Nicolas (1987) [1981]. Espaços vetoriais topológicos: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlim Nova 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]. Espaços Vetoriais Topológicos I. Noções básicas de ciências matemáticas. Volume. 159. Traduzido por Garling, D.J.H. Nova york: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704. Munkres, James R. (2000). Topologia (Second ed.). Rio Sela Superior, Nova Jersey: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260. Narinas, Lourenço; Beckenstein, Eduardo (2011). Espaços vetoriais topológicos. Matemática pura e aplicada (Second ed.). Boca Raton, FL: Imprensa CRC. ISBN 978-1584888666. OCLC 144216834. Rudin, Walter (1991). Análise funcional. International Series in Pure and Applied Mathematics. Volume. 8 (Second ed.). Nova york, Nova Iorque: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. Schaefer, Helmut H.; Wolff, Manfred P. (1999). Espaços vetoriais topológicos. GTM. Volume. 8 (Second ed.). Nova york, Nova Iorque: Springer Nova York Impressão Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. Trier, Francisco (2006) [1967]. Espaços vetoriais topológicos, Distribuições e Kernels. Mineola, NOVA IORQUE.: Publicações de Dover. ISBN 978-0-486-45352-1. OCLC 853623322. Wilansky, Alberto (2013). Métodos modernos em espaços vetoriais topológicos. Mineola, Nova york: Publicações de Dover, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. Zălinescu, Constantino (30 Julho 2002). Convex Analysis in General Vector Spaces. River Edge, N.J. Londres: World Scientific Publishing. ISBN 978-981-4488-15-0. MR 1921556. OCLC 285163112 – via Internet Archive. "Proof of closed graph theorem". PlanetMath. show vte Functional analysis (tópicos – glossário) mostrar espaços vetoriais topológicos vte (TV) Categorias: Teoremas em análise funcional