# Satz über geschlossene Graphen 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 (Funktionsanalyse). The graph of the cubic function {Anzeigestil f(x)=x^{3}-9x} on the interval {Anzeigestil [-4,4]} is closed because the function is continuous. The graph of the Heaviside function on {Anzeigestil [-2,2]} is not closed, because the function is not continuous.

In Mathematik, 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.

Inhalt 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 In der Funktionsanalyse 4 Siehe auch 5 Anmerkungen 6 Verweise 7 Bibliography Graphs and maps with closed graphs Main article: Closed graph If {Anzeigestil f:X. Y} is a map between topological spaces then the graph of {Anzeigestil f} is the set {Anzeigestil Betreibername {Gr} f:={(x,f(x)):xin X}} oder gleichwertig, {Anzeigestil Betreibername {Gr} f:={(x,j)in Xtimes Y_y=f(x)}} It is said that the graph of {Anzeigestil f} is closed if {Anzeigestil Betreibername {Gr} f} is a closed subset of {Anzeigestil X mal Y} (with the product topology).

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

Any linear map, {Anzeigestil L:X. Y,} between two topological vector spaces whose topologies are (Cauchy) complete with respect to translation invariant metrics, and if in addition (1a) {Anzeigestil L} is sequentially continuous in the sense of the product topology, then the map {Anzeigestil L} is continuous and its graph, Gr L, is necessarily closed. Umgekehrt, wenn {Anzeigestil L} is such a linear map with, in place of (1a), the graph of {Anzeigestil L} ist (1b) known to be closed in the Cartesian product space {Anzeigestil X mal Y} , dann {Anzeigestil L} is continuous and therefore necessarily sequentially continuous. Examples of continuous maps that do not have a closed graph If {Anzeigestil X} is any space then the identity map {Anzeigestil Betreibername {Id} :X bis X} is continuous but its graph, which is the diagonal {Anzeigestil Betreibername {Gr} Name des Bedieners {Id} :={(x,x):xin X},} , is closed in {displaystyle Xtimes X} dann und nur dann, wenn {Anzeigestil X} is Hausdorff. Im Speziellen, wenn {Anzeigestil X} is not Hausdorff then {Anzeigestil Betreibername {Id} :X bis X} is continuous but does not have a closed graph.

Lassen {Anzeigestil X} denote the real numbers {Anzeigestil mathbb {R} } with the usual Euclidean topology and let {Anzeigestil Y} denote {Anzeigestil mathbb {R} } with the indiscrete topology (where note that {Anzeigestil Y} is not Hausdorff and that every function valued in {Anzeigestil Y} ist kontinuierlich). Lassen {Anzeigestil f:X. Y} be defined by {Anzeigestil f(0)=1} und {Anzeigestil f(x)=0} für alle {displaystyle xneq 0} . Dann {Anzeigestil f:X. Y} is continuous but its graph is not closed in {Anzeigestil X mal Y} . Closed graph theorem in point-set topology In point-set topology, the closed graph theorem states the following: Satz über geschlossene Graphen — If {Anzeigestil f:X. Y} is a map from a topological space {Anzeigestil X} into a Hausdorff space {Anzeigestil Y,} then the graph of {Anzeigestil f} is closed if {Anzeigestil f:X. Y} ist kontinuierlich. The converse is true when {Anzeigestil 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 {displaystyle Vsubset Y} , we check {Anzeigestil f^{-1}(v)} is open. So take any {displaystyle xin f^{-1}(v)} , we construct some open neighborhood {Anzeigestil U} von {Anzeigestil x} , so dass {Anzeigestil f(U)subset V} .

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

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

Seit {Anzeigestil Y} is compact, we can take a finite open covering of {Anzeigestil Y} wie {Anzeigestil {v,V_{y'_{1}},...,V_{y'_{n}}}} .

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

Suppose not, then there is some unruly {displaystyle x'in U} so dass {Anzeigestil f(x')not in V} , then that would imply {Anzeigestil f(x')in V_{y'_{ich}}} für einige {Anzeigestil i} by open covering, but then {Anzeigestil (x',f(x'))in Utimes V_{y'_{ich}}subset U_{y'_{ich}}times V_{y'_{ich}}} , a contradiction since it is supposed to be disjoint from the graph of {Anzeigestil f} .

Non-Hausdorff spaces are rarely seen, but non-compact spaces are common. An example of non-compact {Anzeigestil Y} is the real line, which allows the discontinuous function with closed graph {Anzeigestil f(x)={Start{Fälle}{frac {1}{x}}{Text{ wenn }}xneq 0,\0{Text{ anders}}Ende{Fälle}}} .

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

In functional analysis Main article: Satz über geschlossene Graphen (Funktionsanalyse) Wenn {Anzeigestil T:X. Y} is a linear operator between topological vector spaces (Fernseher) then we say that {Anzeigestil T} is a closed operator if the graph of {Anzeigestil T} is closed in {Anzeigestil X mal Y} Wenn {Anzeigestil X mal 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.

Satz — A linear map between two F-spaces (z.B. 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 (Funktionsanalyse) – 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). "Kapitel 17". Infinite Dimensional Analysis: A Hitchhiker's Guide (3Dr. Ed.). Springer. ^ Schaefer & Wolff 1999, p. 78. ^ Trèves (2006), p. 173 Bibliography Bourbaki, Nikolaus (1987) . Topologische Vektorräume: 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) . Topologische Vektorräume I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Übersetzt von Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. HERR 0248498. OCLC 840293704. Munkres, Jakob R. (2000). Topologie (Zweite Aufl.). Upper Saddle River, NJ: Lehrlingshalle, Inc. ISBN 978-0-13-181629-9. OCLC 42683260. Nasenlöcher, Laurentius; Beckenstein, Eduard (2011). Topologische Vektorräume. Reine und angewandte Mathematik (Zweite Aufl.). Boca Raton, FL: CRC-Presse. ISBN 978-1584888666. OCLC 144216834. Rudin, Walter (1991). Funktionsanalyse. International Series in Pure and Applied Mathematics. Vol. 8 (Zweite Aufl.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. Schäfer, Helmut h.; Wolff, Manfred P. (1999). Topologische Vektorräume. GTM. Vol. 8 (Zweite Aufl.). New York, NY: Springer New York Impressum Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. Trier, Francois (2006) . Topologische Vektorräume, Distributionen und Kernel. Mineola, New York: Dover-Veröffentlichungen. ISBN 978-0-486-45352-1. OCLC 853623322. Wilansky, Albert (2013). Moderne Methoden in topologischen Vektorräumen. Mineola, New York: Dover-Veröffentlichungen, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. Zălinescu, Konstantin (30 Juli 2002). Convex Analysis in General Vector Spaces. River Edge, N.J. London: World Scientific Publishing. ISBN 978-981-4488-15-0. HERR 1921556. OCLC 285163112 – via Internet Archive. "Proof of closed graph theorem". PlanetMath. show vte Functional analysis (Themen – Glossar) show vte Topologische Vektorräume (Fernseher) Kategorien: Sätze in der Funktionalanalysis

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