Open mapping theorem (functional analysis)

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 which 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 ${\displaystyle X}$  and ${\displaystyle Y}$  are Banach spaces and ${\displaystyle A:X\to Y}$  is a surjective continuous linear operator, then ${\displaystyle A}$  is an open map (that is, if ${\displaystyle U}$  is an open set in ${\displaystyle X,}$  then ${\displaystyle A(U)}$  is open in ${\displaystyle Y}$ ).

This proof uses the Baire category theorem, and completeness of both ${\displaystyle X}$  and ${\displaystyle Y}$  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 ${\displaystyle X}$  and ${\displaystyle Y}$  are taken to be Fréchet spaces.

Proof

Suppose ${\displaystyle A:X\to Y}$  is a surjective continuous linear operator. In order to prove that ${\displaystyle A}$  is an open map, it is sufficient to show that ${\displaystyle A}$  maps the open unit ball in ${\displaystyle X}$  to a neighborhood of the origin of ${\displaystyle Y.}$  Let ${\displaystyle U=B_{1}^{X}(0),V=B_{1}^{Y}(0).}$  Then $${\displaystyle X=\bigcup _{k\in \mathbb {N} }kU.}$$   Since ${\displaystyle A}$  is surjective: $${\displaystyle Y=A(X)=A\left(\bigcup _{k\in \mathbb {N} }kU\right)=\bigcup _{k\in \mathbb {N} }A(kU).}$$   But ${\displaystyle Y}$  is Banach so by Baire's category theorem $${\displaystyle \exists k\in \mathbb {N} :\qquad \left({\overline {A(kU)}}\right)^{\circ }\neq \varnothing .}$$   That is, we have ${\displaystyle c\in Y}$  and ${\displaystyle r>0}$  such that $${\displaystyle B_{r}(c)\subseteq \left({\overline {A(kU)}}\right)^{\circ }\subseteq {\overline {A(kU)}}.}$$   Let ${\displaystyle v\in V,}$  then $${\displaystyle c,c+rv\in B_{r}(c)\subseteq {\overline {A(kU)}}.}$$   By continuity of addition and linearity, the difference ${\displaystyle rv}$  satisfies $${\displaystyle rv\in {\overline {A(kU)}}+{\overline {A(kU)}}\subseteq {\overline {A(kU)+A(kU)}}\subseteq {\overline {A(2kU)}},}$$   and by linearity again, $${\displaystyle V\subseteq {\overline {A(LU)}}}$$   where we have set ${\displaystyle L=2k/r.}$  It follows that for all ${\displaystyle y\in Y}$  and all ${\displaystyle \epsilon >0,}$  there exists some ${\displaystyle x\in X}$  such that $${\displaystyle \qquad \|x\|_{X}\leq L\|y\|_{Y}\quad {\text{and}}\quad \|y-Ax\|_{Y}<\epsilon .\qquad (1)}$$   Our next goal is to show that ${\displaystyle V\subseteq A(2LU).}$  Let ${\displaystyle y\in V.}$  By (1), there is some ${\displaystyle x_{1}}$  with ${\displaystyle \left\|x_{1}\right\|<L}$  and ${\displaystyle \left\|y-Ax_{1}\right\|<1/2.}$  Define a sequence ${\displaystyle \left(x_{n}\right)}$  inductively as follows. Assume: $${\displaystyle \|x_{n}\|<{\frac {L}{2^{n-1}}}\quad {\text{and}}\quad \left\|y-A\left(x_{1}+x_{2}+\cdots +x_{n}\right)\right\|<{\frac {1}{2^{n}}}.\qquad (2)}$$   Then by (1) we can pick ${\displaystyle x_{n+1}}$  so that: $${\displaystyle \|x_{n+1}\|<{\frac {L}{2^{n}}}\quad {\text{and}}\quad \left\|y-A\left(x_{1}+x_{2}+\cdots +x_{n}\right)-A\left(x_{n+1}\right)\right\|<{\frac {1}{2^{n+1}}},}$$   so (2) is satisfied for ${\displaystyle x_{n+1}.}$  Let $${\displaystyle s_{n}=x_{1}+x_{2}+\cdots +x_{n}.}$$   From the first inequality in (2), ${\displaystyle \left(s_{n}\right)}$ is a Cauchy sequence, and since ${\displaystyle X}$  is complete, ${\displaystyle s_{n}}$  converges to some ${\displaystyle x\in X.}$  By (2), the sequence ${\displaystyle As_{n}}$  tends to ${\displaystyle y}$  and so ${\displaystyle Ax=y}$  by continuity of ${\displaystyle A.}$  Also, $${\displaystyle \|x\|=\lim _{n\to \infty }\|s_{n}\|\leq \sum _{n=1}^{\infty }\|x_{n}\|<2L.}$$   This shows that ${\displaystyle y}$  belongs to ${\displaystyle A(2LU),}$  so ${\displaystyle V\subseteq A(2LU)}$  as claimed. Thus the image ${\displaystyle A(U)}$  of the unit ball in ${\displaystyle X}$  contains the open ball ${\displaystyle V/2L}$  of ${\displaystyle Y.}$  Hence, ${\displaystyle A(U)}$  is a neighborhood of the origin in ${\displaystyle Y,}$  and this concludes the proof.

Related results

Theorem^[2] — Let ${\displaystyle X}$  and ${\displaystyle Y}$  be Banach spaces, let ${\displaystyle B_{X}}$  and ${\displaystyle B_{Y}}$  denote their open unit balls, and let ${\displaystyle T:X\to Y}$  be a bounded linear operator. If ${\displaystyle \delta >0}$  then among the following four statements we have ${\displaystyle (1)\implies (2)\implies (3)\implies (4)}$  (with the same ${\displaystyle \delta }$ )

1. ${\displaystyle \left\|T^{*}y^{*}\right\|\geq \delta \left\|y^{*}\right\|}$  for all ${\displaystyle y^{*}\in Y^{*}}$ ;
2. ${\displaystyle {\overline {T\left(B_{X}\right)}}\supseteq \delta B_{Y}}$ ;
3. ${\displaystyle {T\left(B_{X}\right)}\supseteq \delta B_{Y}}$ ;
4. ${\displaystyle \operatorname {Im} T=Y}$  (that is, ${\displaystyle T}$  is surjective).

Furthermore, if ${\displaystyle T}$  is surjective then (1) holds for some ${\displaystyle \delta >0}$ 

Consequences

The open mapping theorem has several important consequences:

• If ${\displaystyle A:X\to Y}$  is a bijective continuous linear operator between the Banach spaces ${\displaystyle X}$  and ${\displaystyle Y,}$  then the inverse operator ${\displaystyle A^{-1}:Y\to X}$  is continuous as well (this is called the bounded inverse theorem).^[3]
• If ${\displaystyle A:X\to Y}$  is a linear operator between the Banach spaces ${\displaystyle X}$  and ${\displaystyle Y,}$  and if for every sequence ${\displaystyle \left(x_{n}\right)}$  in ${\displaystyle X}$  with ${\displaystyle x_{n}\to 0}$  and ${\displaystyle Ax_{n}\to y}$  it follows that ${\displaystyle y=0,}$  then ${\displaystyle A}$  is continuous (the closed graph theorem).^[4]

Generalizations

Local convexity of ${\displaystyle X}$  or ${\displaystyle Y}$   is not essential to the proof, but completeness is: the theorem remains true in the case when ${\displaystyle X}$  and ${\displaystyle Y}$  are F-spaces. Furthermore, the theorem can be combined with the Baire category theorem in the following manner:

Theorem^[5] — Let ${\displaystyle X}$  be a F-space and ${\displaystyle Y}$  a topological vector space. If ${\displaystyle A:X\to Y}$  is a continuous linear operator, then either ${\displaystyle A(X)}$  is a meager set in ${\displaystyle Y,}$  or ${\displaystyle A(X)=Y.}$  In the latter case, ${\displaystyle A}$  is an open mapping and ${\displaystyle Y}$  is also an F-space.

Furthermore, in this latter case if ${\displaystyle N}$  is the kernel of ${\displaystyle A,}$  then there is a canonical factorization of ${\displaystyle A}$  in the form

$${\displaystyle X\to X/N{\overset {\alpha }{\to }}Y}$$

 

where ${\displaystyle X/N}$  is the quotient space (also an F-space) of ${\displaystyle X}$  by the closed subspace ${\displaystyle N.}$  The quotient mapping ${\displaystyle X\to X/N}$  is open, and the mapping ${\displaystyle \alpha }$  is an isomorphism of topological vector spaces.^[6]

Open mapping theorem^[7] — Let ${\displaystyle A:X\to Y}$  be a surjective linear map from a complete pseudometrizable TVS ${\displaystyle X}$  onto a TVS ${\displaystyle Y}$  and suppose that at least one of the following two conditions is satisfied:

1. ${\displaystyle Y}$  is a Baire space, or
2. ${\displaystyle X}$  is locally convex and ${\displaystyle Y}$  is a barrelled space,

If ${\displaystyle A}$  is a closed linear operator then ${\displaystyle A}$  is an open mapping. If ${\displaystyle A}$  is a continuous linear operator and ${\displaystyle Y}$  is Hausdorff then ${\displaystyle A}$  is (a closed linear operator and thus also) an open mapping.

Open mapping theorem for continuous maps^[7] — Let ${\displaystyle A:X\to Y}$  be a continuous linear operator from a complete pseudometrizable TVS ${\displaystyle X}$  onto a Hausdorff TVS ${\displaystyle Y.}$  If ${\displaystyle \operatorname {Im} A}$  is nonmeager in ${\displaystyle Y}$  then ${\displaystyle A:X\to Y}$  is a surjective open map and ${\displaystyle Y}$  is a complete pseudometrizable TVS.

The open mapping theorem can also be stated as

Theorem^[8] — Let ${\displaystyle X}$  and ${\displaystyle Y}$  be two F-spaces. Then every continuous linear map of ${\displaystyle X}$  onto ${\displaystyle Y}$  is a TVS homomorphism, where a linear map ${\displaystyle u:X\to Y}$  is a topological vector space (TVS) homomorphism if the induced map ${\displaystyle {\hat {u}}:X/\ker(u)\to Y}$  is a TVS-isomorphism onto its image.

Nearly/Almost open linear maps

A linear map ${\displaystyle A:X\to Y}$  between two topological vector spaces (TVSs) is called a nearly open map (or sometimes, an almost open map) if for every neighborhood ${\displaystyle U}$  of the origin in the domain, the closure of its image ${\displaystyle \operatorname {cl} A(U)}$  is a neighborhood of the origin in ${\displaystyle Y.}$ ^[9] Many authors use a different definition of "nearly/almost open map" that requires that the closure of ${\displaystyle A(U)}$  be a neighborhood of the origin in ${\displaystyle A(X)}$  rather than in ${\displaystyle Y,}$ ^[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]

Open mapping theorem^[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 ${\displaystyle A:X\to Y}$  is a continuous linear bijection from a complete Pseudometrizable topological vector space (TVS) onto a Hausdorff TVS that is a Baire space, then ${\displaystyle A:X\to Y}$  is a homeomorphism (and thus an isomorphism of 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 – Map that satisfies a condition similar to that of being an open map.Pages displaying short descriptions of redirect targets
• Bounded inverse theorem
• Closed graph – Graph of a map closed in the product spacePages displaying short descriptions of redirect targets
• 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 mapsPages displaying wikidata descriptions as a fallback
• 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

1. Trèves 2006, p. 166.
2. Rudin 1991, p. 100.
3. Rudin 1973, Corollary 2.12.
4. Rudin 1973, Theorem 2.15.
5. Rudin 1991, Theorem 2.11.
6. Dieudonné 1970, 12.16.8.
7. Narici & Beckenstein 2011, p. 468.
8. Trèves 2006, p. 170
9. Narici & Beckenstein 2011, pp. 466.
10. Narici & Beckenstein 2011, pp. 467.
11. Narici & Beckenstein 2011, pp. 466−468.
12. 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). [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.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.

This article incorporates material from Proof of open mapping theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.