Offener Abbildungssatz (Funktionsanalyse)

Offener Abbildungssatz (Funktionsanalyse) In der Funktionsanalyse, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem[1] (named after Stefan Banach and Juliusz Schauder), is a fundamental result which states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.

Inhalt 1 Klassisch (Banach space) bilden 1.1 Verwandte Ergebnisse 1.2 Konsequenzen 2 Verallgemeinerungen 2.1 Konsequenzen 2.2 Webbed spaces 3 Siehe auch 4 Verweise 5 Bibliography Classical (Banach space) form Open mapping theorem for Banach spaces (Rudin 1973, Satz 2.11) — If {Anzeigestil X} und {Anzeigestil Y} are Banach spaces and {Anzeigestil A:X. Y} is a surjective continuous linear operator, dann {Anzeigestil A} is an open map (das ist, wenn {Anzeigestil U} is an open set in {Anzeigestil X,} dann {Anzeigestil A(U)} is open in {Anzeigestil Y} ).

This proof uses the Baire category theorem, and completeness of both {Anzeigestil X} und {Anzeigestil Y} is essential to the theorem. The statement of the theorem is no longer true if either space is just assumed to be a normed space, but is true if {Anzeigestil X} und {Anzeigestil Y} are taken to be Fréchet spaces.

show Proof Related results Theorem[2] — Let {Anzeigestil X} und {Anzeigestil Y} be Banach spaces, Lassen {Anzeigestil B_{X}} und {Anzeigestil B_{Y}} denote their open unit balls, und lass {Anzeigestil T:X. Y} be a bounded linear operator. Wenn {displaystyle delta >0} then among the following four statements we have {Anzeigestil (1)impliziert (2)impliziert (3)impliziert (4)} (with the same {Anzeigestil-Delta } ) {Anzeigestil links|T^{*}y^{*}Rechts|geq delta left|y^{*}Rechts|} für alle {displaystyle y^{*}in Y^{*}} ; {Anzeigestil {überstreichen {Tleft(B_{X}Rechts)}}supseteq delta B_{Y}} ; {Anzeigestil {Tleft(B_{X}Rechts)}supseteq delta B_{Y}} ; {Anzeigestil Betreibername {Ich bin} T=Y} (das ist, {Anzeigestil T} is surjective).

Außerdem, wenn {Anzeigestil T} is surjective then (1) holds for some {displaystyle delta >0} Consequences The open mapping theorem has several important consequences: Wenn {Anzeigestil A:X. Y} is a bijective continuous linear operator between the Banach spaces {Anzeigestil X} und {Anzeigestil Y,} then the inverse operator {Anzeigestil A^{-1}:Y bis X} is continuous as well (this is called the bounded inverse theorem).[3] Wenn {Anzeigestil A:X. Y} is a linear operator between the Banach spaces {Anzeigestil X} und {Anzeigestil Y,} and if for every sequence {Anzeigestil links(x_{n}Rechts)} in {Anzeigestil X} mit {Anzeigestil x_{n}zu 0} und {displaystyle Ax_{n}to y} es folgt dem {Anzeigestil y=0,} dann {Anzeigestil A} ist kontinuierlich (the closed graph theorem).[4] Generalizations Local convexity of {Anzeigestil X} oder {Anzeigestil Y}  is not essential to the proof, but completeness is: the theorem remains true in the case when {Anzeigestil X} und {Anzeigestil Y} are F-spaces. Außerdem, the theorem can be combined with the Baire category theorem in the following manner: Satz[5] — Let {Anzeigestil X} be a F-space and {Anzeigestil Y} a topological vector space. Wenn {Anzeigestil A:X. Y} is a continuous linear operator, dann entweder {Anzeigestil A(X)} is a meager set in {Anzeigestil Y,} oder {Anzeigestil A(X)=Y.} Im letzteren Fall, {Anzeigestil A} is an open mapping and {Anzeigestil Y} is also an F-space.

Außerdem, in this latter case if {Anzeigestil N} is the kernel of {Anzeigestil A,} then there is a canonical factorization of {Anzeigestil A} in der Form {displaystyle Xto X/N{overset {Alpha }{zu }}Y} wo {displaystyle X/N} is the quotient space (also an F-space) von {Anzeigestil X} by the closed subspace {displaystyle N.} The quotient mapping {displaystyle Xto X/N} is open, and the mapping {Anzeigestil alpha } is an isomorphism of topological vector spaces.[6] Offener Abbildungssatz[7] — Let {Anzeigestil A:X. Y} be a surjective linear map from an complete pseudometrizable TVS {Anzeigestil X} onto a TVS {Anzeigestil Y} and suppose that at least one of the following two conditions is satisfied: {Anzeigestil Y} ist ein Baire-Raum, oder {Anzeigestil X} is locally convex and {Anzeigestil Y} is a barrelled space, Wenn {Anzeigestil A} is a closed linear operator then {Anzeigestil A} is an open mapping. Wenn {Anzeigestil A} is a continuous linear operator and {Anzeigestil Y} is Hausdorff then {Anzeigestil A} ist (a closed linear operator and thus also) an open mapping.

Open mapping theorem for continuous maps[7] — Let {Anzeigestil A:X. Y} be a continuous linear operator from an complete pseudometrizable TVS {Anzeigestil X} onto a Hausdorff TVS {Anzeigestil Y.} Wenn {Anzeigestil Betreibername {Ich bin} EIN} is nonmeager in {Anzeigestil Y} dann {Anzeigestil A:X. Y} is a surjective open map and {Anzeigestil Y} is a complete pseudometrizable TVS.

The open mapping theorem can also be stated as Theorem[8] — Let {Anzeigestil X} und {Anzeigestil Y} be two F-spaces. Then every continuous linear map of {Anzeigestil X} onto {Anzeigestil Y} is a TVS homomorphism, where a linear map {Anzeigestil u:X. Y} is a topological vector space (Fernseher) homomorphism if the induced map {Anzeigestil {Hut {u}}:X/ker(u)to Y} is a TVS-isomorphism onto its image.

Nearly/Almost open linear maps A linear map {Anzeigestil A:X. Y} between two topological vector spaces (Fernseher) is called a nearly open map (or sometimes, an almost open map) if for every neighborhood {Anzeigestil U} of the origin in the domain, the closure of its image {Anzeigestil Betreibername {cl} EIN(U)} is a neighborhood of the origin in {Anzeigestil Y.} [9] Many authors use a different definition of "nearly/almost open map" that requires that the closure of {Anzeigestil A(U)} be a neighborhood of the origin in {Anzeigestil A(X)} rather than in {Anzeigestil Y,} [9] but for surjective maps these definitions are equivalent. A bijective linear map is nearly open if and only if its inverse is continuous.[9] Every surjective linear map from locally convex TVS onto a barrelled TVS is nearly open.[10] The same is true of every surjective linear map from a TVS onto a Baire TVS.[10] Offener Abbildungssatz[11] — If a closed surjective linear map from a complete pseudometrizable TVS onto a Hausdorff TVS is nearly open then it is open.

Consequences Theorem[12] — If {Anzeigestil A:X. Y} is a continuous linear bijection from a complete Pseudometrizable topological vector space (Fernseher) auf ein Hausdorff TVS, das ein Baire-Raum ist, dann {Anzeigestil A:X. Y} is a homeomorphism (und damit ein Isomorphismus von TVSs).

Webbed spaces Main article: Webbed space Webbed spaces are a class of topological vector spaces for which the open mapping theorem and the closed graph theorem hold.

See also Almost open linear map Bounded inverse theorem Closed graph Closed graph theorem – Theorem relating continuity to graphs Closed graph theorem (Funktionsanalyse) – Theorems connecting continuity to closure of graphs Open mapping theorem (complex analysis) Surjection of Fréchet spaces – Characterization of surjectivity Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem Webbed space – Space where open mapping and closed graph theorems hold References ^ Trèves 2006, p. 166. ^ Rudin 1991, p. 100. ^ Rudin 1973, Logische Folge 2.12. ^ Rudin 1973, Satz 2.15. ^ Rudin 1991, Satz 2.11. ^ Dieudonné 1970, 12.16.8. ^ Nach oben springen: a b Narici & Beckenstein 2011, p. 468. ^ Trèves 2006, p. 170 ^ Nach oben springen: a b c Narici & Beckenstein 2011, pp. 466. ^ Nach oben springen: a b Narici & Beckenstein 2011, pp. 467. ^ Narici & Beckenstein 2011, pp. 466−468. ^ Narici & Beckenstein 2011, p. 469. Bibliography Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topologische Vektorräume: The Theory Without Convexity Conditions. Vorlesungsunterlagen in Mathematik. Vol. 639. Berlin-New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003. Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (Pdf). Monografie Matematyczne (auf Französisch). Vol. 1. Warschau: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Vom Original archiviert (Pdf) an 2014-01-11. Abgerufen 2020-07-11. Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Abschlusstexte in Mathematik. Vol. 15. New York: Springer. ISBN 978-0-387-90081-0. OCLC 878109401. Bourbaki, Nikolaus (1987) [1981]. 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. Conway, John (1990). A course in functional analysis. Abschlusstexte in Mathematik. Vol. 96 (2und Aufl.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. Dieudonné, Jean (1970), Treatise on Analysis, Band II, Academic Press Edwards, Robert E. (1995). Funktionsanalyse: Theorie und Anwendungen. New York: Dover-Veröffentlichungen. ISBN 978-0-486-68143-6. OCLC 30593138. Grothendieck, Alexander (1973). Topologische Vektorräume. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098. Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. Köthe, Gottfried (1983) [1969]. 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. Nasenlöcher, Laurentius; Beckenstein, Eduard (2011). Topologische Vektorräume. Reine und angewandte Mathematik (Zweite Aufl.). Boca Raton, FL: CRC-Presse. ISBN 978-1584888666. OCLC 144216834. Robertson, Alex P.; Robertson, Wendy J. (1980). Topologische Vektorräume. Cambridge Tracts in Mathematics. Vol. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250. Rudin, Walter (1973). Funktionsanalyse. International Series in Pure and Applied Mathematics. Vol. 25 (First ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 9780070542259. 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. Swartz, Karl (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067. Trier, Francois (2006) [1967]. 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.

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