Open mapping theorem (functional analysis): Difference between revisions
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Revision as of 11:16, 17 July 2024
In functional analysis, 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 that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.
Classical (Banach space) form
Open mapping theorem for Banach spaces (Rudin 1973, Theorem 2.11) — If and are Banach spaces and is a surjective continuous linear operator, then is an open map (that is, if is an open set in then is open in ).
This proof uses the Baire category theorem, and completeness of both and is essential to the theorem. The statement of the theorem is no longer true if either space is just assumed to be a normed vector space, but is true if and are taken to be Fréchet spaces.
Proof
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Suppose is a surjective continuous linear operator. In order to prove that is an open map, it is sufficient to show that maps the open unit ball in to a neighborhood of the origin of Let Then Since is surjective: But is Banach so by Baire's category theorem That is, we have and such that Let then By continuity of addition and linearity, the difference satisfies and by linearity again, where we have set It follows that for all and all there exists some such that Our next goal is to show that Let By (1), there is some with and Define a sequence inductively as follows. Assume: Then by (1) we can pick so that: so (2) is satisfied for Let From the first inequality in (2), is a Cauchy sequence, and since is complete, converges to some By (2), the sequence tends to and so by continuity of Also, This shows that belongs to so as claimed. Thus the image of the unit ball in contains the open ball of Hence, is a neighborhood of the origin in and this concludes the proof. |
Related results
Theorem[2] — Let and be Banach spaces, let and denote their open unit balls, and let be a bounded linear operator. If then among the following four statements we have (with the same )
- for all ;
- ;
- ;
- (that is, is surjective).
Furthermore, if is surjective then (1) holds for some
See also: § Quantative version.
Consequences
The open mapping theorem has several important consequences:
- If is a bijective continuous linear operator between the Banach spaces and then the inverse operator is continuous as well (this is called the bounded inverse theorem).[3]
- If is a linear operator between the Banach spaces and and if for every sequence in with and it follows that then is continuous (the closed graph theorem).[4]
Quantative version
T. Tao’s blog post[5] gives the following quantitative formulation of the theorem
Theorem — Let be a bounded operator between Banach spaces. Then the following are equivalent:
- is open.
- is subjective.
- There exists a constant such that, for each in , the equation has a solution with .
- 3. holds for in some dense subspace of .
Proof: 2. 1. is the usual open mapping theorem.
1. 4.: For some , we have where means an open ball. Then for some in . That is, with .
4. 3.: We can write with in the dense subspace and the sum converging in norm. Then, since is complete, with and is a required solution. Finally, 3. 2. is trivial.
Generalizations
Local convexity of or is not essential to the proof, but completeness is: the theorem remains true in the case when and are F-spaces. Furthermore, the theorem can be combined with the Baire category theorem in the following manner:
Open mapping theorem for continuous maps[6][7] — Let be a continuous linear operator from a complete pseudometrizable TVS onto a Hausdorff TVS If is nonmeager in then is a (surjective) open map and is a complete pseudometrizable TVS. Moreover, if is assumed to be hausdorff (i.e. a F-space), then is also an F-space.
Furthermore, in this latter case if is the kernel of then there is a canonical factorization of in the form where is the quotient space (also an F-space) of by the closed subspace The quotient mapping is open, and the mapping is an isomorphism of topological vector spaces.[8]
An important special case of this theorem can also be stated as
Theorem[9] — Let and be two F-spaces. Then every continuous linear map of onto is a TVS homomorphism, where a linear map is a topological vector space (TVS) homomorphism if the induced map is a TVS-isomorphism onto its image.
On the other hand, a more general formulation, which implies the first, can be given:
Open mapping theorem[7] — Let be a surjective linear map from a complete pseudometrizable TVS onto a TVS and suppose that at least one of the following two conditions is satisfied:
- is a Baire space, or
- is locally convex and is a barrelled space,
If is a closed linear operator then is an open mapping. If is a continuous linear operator and is Hausdorff then is (a closed linear operator and thus also) an open mapping.
Nearly/Almost open linear maps
A linear map between two topological vector spaces (TVSs) is called a nearly open map (or sometimes, an almost open map) if for every neighborhood of the origin in the domain, the closure of its image is a neighborhood of the origin in [10] Many authors use a different definition of "nearly/almost open map" that requires that the closure of be a neighborhood of the origin in rather than in [10] but for surjective maps these definitions are equivalent. A bijective linear map is nearly open if and only if its inverse is continuous.[10] Every surjective linear map from locally convex TVS onto a barrelled TVS is nearly open.[11] The same is true of every surjective linear map from a TVS onto a Baire TVS.[11]
Open mapping theorem[12] — If a closed surjective linear map from a complete pseudometrizable TVS onto a Hausdorff TVS is nearly open then it is open.
Consequences
Theorem[13] — If is a continuous linear bijection from a complete Pseudometrizable topological vector space (TVS) onto a Hausdorff TVS that is a Baire space, then is a homeomorphism (and thus an isomorphism of TVSs).
Webbed spaces
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 – Map that satisfies a condition similar to that of being an open map.
- Bounded inverse theorem – Condition for a linear operator to be open
- Closed graph – Graph of a map closed in the product space
- Closed graph theorem – Theorem relating continuity to graphs
- Closed graph theorem (functional analysis) – Theorems connecting continuity to closure of graphs
- Open mapping theorem (complex analysis) – Theorem that holomorphic functions on complex domains are open maps
- 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, Corollary 2.12.
- ^ Rudin 1973, Theorem 2.15.
- ^ https://terrytao.wordpress.com/2009/02/01/245b-notes-9-the-baire-category-theorem-and-its-banach-space-consequences/
- ^ Rudin 1991, Theorem 2.11.
- ^ a b Narici & Beckenstein 2011, p. 468.
- ^ Dieudonné 1970, 12.16.8.
- ^ Trèves 2006, p. 170
- ^ a b c Narici & Beckenstein 2011, pp. 466.
- ^ 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). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. 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 (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
- Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. Vol. 15. New York: Springer. ISBN 978-0-387-90081-0. OCLC 878109401.
- Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: 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. Graduate Texts in Mathematics. Vol. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.
- Dieudonné, Jean (1970), Treatise on Analysis, Volume II, Academic Press
- Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.
- Grothendieck, Alexander (1973). Topological Vector Spaces. 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]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
- Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
- Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
- Rudin, Walter (1973). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 25 (First ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 9780070542259.
- Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
- Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
- Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067.
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