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 that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.

A special case is also called the bounded inverse theorem (also called inverse mapping theorem or Banach isomorphism theorem), which states that a bijective bounded linear operator ${\displaystyle T}$ from one Banach space to another has bounded inverse ${\displaystyle T^{-1}}$.

The proof here uses the Baire category theorem, and completeness of both ${\displaystyle E}$ and ${\displaystyle F}$ is essential to the theorem. The statement of the theorem is no longer true if either space is assumed to be only a normed vector space; see § Counterexample.

The proof is based on the following lemmas, which are also somewhat of independent interest. A linear map ${\displaystyle f:E\to F}$ between topological vector spaces is said to be nearly open if, for each neighborhood ${\displaystyle U}$ of zero, the closure ${\displaystyle {\overline {f(U)}}}$ contains a neighborhood of zero. The next lemma may be thought of as a weak version of the open mapping theorem.

Proof: Shrinking ${\displaystyle U}$, we can assume ${\displaystyle U}$ is an open ball centered at zero. We have ${\displaystyle f(E)=f\left(\bigcup _{n\in \mathbb {N} }nU\right)=\bigcup _{n\in \mathbb {N} }f(nU)}$. Thus, some ${\displaystyle {\overline {f(nU)}}}$ contains an interior point ${\displaystyle y}$; that is, for some radius ${\displaystyle r>0}$,

${\displaystyle B(y,r)\subset {\overline {f(nU)}}.}$

Then for any ${\displaystyle v}$ in ${\displaystyle F}$ with ${\displaystyle \|v\|<r}$, by linearity, convexity and ${\displaystyle (-1)U\subset U}$,

${\displaystyle v=v-y+y\in {\overline {f(-nU)}}+{\overline {f(nU)}}\subset {\overline {f(2nU)}}}$,

which proves the lemma by dividing by ${\displaystyle 2n}$.${\displaystyle \square }$ (The same proof works if ${\displaystyle E,F}$ are pre-Fréchet spaces.)

The completeness on the domain then allows to upgrade nearly open to open.

Proof: Let ${\displaystyle y}$ be in ${\displaystyle B(0,\delta )}$ and ${\displaystyle c_{n}>0}$ some sequence. We have: ${\displaystyle {\overline {B(0,\delta )}}\subset {\overline {f(B(0,1))}}}$. Thus, for each ${\displaystyle \epsilon >0}$ and ${\displaystyle z}$ in ${\displaystyle F}$, we can find an ${\displaystyle x}$ with ${\displaystyle \|x\|<\delta ^{-1}\|z\|}$ and ${\displaystyle z}$ in ${\displaystyle B(f(x),\epsilon )}$. Thus, taking ${\displaystyle z=y}$, we find an ${\displaystyle x_{1}}$ such that

${\displaystyle \|y-f(x_{1})\|<c_{1},\,\|x_{1}\|<\delta ^{-1}\|y\|.}$

Applying the same argument with ${\displaystyle z=y-f(x_{1})}$, we then find an ${\displaystyle x_{2}}$ such that

${\displaystyle \|y-f(x_{1})-f(x_{2})\|<c_{2},\,\|x_{2}\|<\delta ^{-1}c_{1}}$

where we observed ${\displaystyle \|x_{2}\|<\delta ^{-1}\|z\|<\delta ^{-1}c_{1}}$. Then so on. Thus, if ${\displaystyle c:=\sum c_{n}<\infty }$, we found a sequence ${\displaystyle x_{n}}$ such that ${\displaystyle x=\sum _{1}^{\infty }x_{n}}$ converges and ${\displaystyle f(x)=y}$. Also,

${\displaystyle \|x\|\leq \sum _{1}^{\infty }\|x_{n}\|\leq \delta ^{-1}\|y\|+\delta ^{-1}c.}$

Since ${\displaystyle \delta ^{-1}\|y\|<1}$, by making ${\displaystyle c}$ small enough, we can achieve ${\displaystyle \|x\|<1}$. ${\displaystyle \square }$ (Again the same proof is valid if ${\displaystyle E,F}$ are pre-Fréchet spaces.)

Proof of the theorem: By Baire's category theorem, the first lemma applies. Then the conclusion of the theorem follows from the second lemma. ${\displaystyle \square }$

In general, a continuous bijection between topological spaces is not necessarily a homeomorphism. The open mapping theorem, when it applies, implies the bijectivity is enough:

Even though the above bounded inverse theorem is a special case of the open mapping theorem, the open mapping theorem in turns follows from that. Indeed, a surjective linear operator ${\displaystyle T:E\to F}$ factors as

${\displaystyle T:E{\overset {p}{\to }}E/\operatorname {ker} T{\overset {T_{0}}{\to }}F.}$

Here, ${\displaystyle T_{0}}$ is bijective and thus is a homeomorphism by the bounded inverse theorem; in particular, it is an open mapping. As a quotient map for topological groups is open, ${\displaystyle T}$ is open then.

Because the open mapping theorem and the bounded inverse theorem are essentially the same result, they are often simply called Banach's theorem.

Here is a formulation of the open mapping theorem in terms of the transpose of an operator.

Proof: The idea of 1. ${\displaystyle \Rightarrow }$ 2. is to show: ${\displaystyle y\notin {\overline {T(B_{X})}}\Rightarrow \|y\|>\delta ,}$ and that follows from the Hahn–Banach theorem. 2. ${\displaystyle \Rightarrow }$ 3. is exactly the second lemma in § Statement and proof. Finally, 3. ${\displaystyle \Rightarrow }$ 4. is trivial and 4. ${\displaystyle \Rightarrow }$ 1. easily follows from the open mapping theorem. ${\displaystyle \square }$

Alternatively, 1. implies that ${\displaystyle T^{*}}$ is injective and has closed image and then by the closed range theorem, that implies ${\displaystyle T}$ has dense image and closed image, respectively; i.e., ${\displaystyle T}$ is surjective. Hence, the above result is a variant of a special case of the closed range theorem.

Terence Tao gives the following quantitative formulation of the theorem:^[9]

Proof: 2. ${\displaystyle \Rightarrow }$ 1. is the usual open mapping theorem.

1. ${\displaystyle \Rightarrow }$ 4.: For some ${\displaystyle r>0}$, we have ${\displaystyle B(0,2)\subset T(B(0,r))}$ where ${\displaystyle B}$ means an open ball. Then ${\displaystyle {\frac {f}{\|f\|}}=T\left({\frac {u}{\|f\|}}\right)}$ for some ${\displaystyle {\frac {u}{\|f\|}}}$ in ${\displaystyle B(0,r)}$. That is, ${\displaystyle Tu=f}$ with ${\displaystyle \|u\|<r\|f\|}$.

4. ${\displaystyle \Rightarrow }$ 3.: We can write ${\displaystyle f=\sum _{0}^{\infty }f_{j}}$ with ${\displaystyle f_{j}}$ in the dense subspace and the sum converging in norm. Then, since ${\displaystyle E}$ is complete, ${\displaystyle u=\sum _{0}^{\infty }u_{j}}$ with ${\displaystyle \|u_{j}\|\leq C\|f_{j}\|}$ and ${\displaystyle Tu_{j}=f_{j}}$ is a required solution. Finally, 3. ${\displaystyle \Rightarrow }$ 2. is trivial. ${\displaystyle \square }$

The open mapping theorem may not hold for normed spaces that are not complete. A quickest way to see this is to note that the closed graph theorem, a consequence of the open mapping theorem, fails without completeness. But here is a more concrete counterexample. Consider the space X of sequences x : N → R with only finitely many non-zero terms equipped with the supremum norm. The map T : X → X defined by

${\displaystyle Tx=\left(x_{1},{\frac {x_{2}}{2}},{\frac {x_{3}}{3}},\dots \right)}$

is bounded, linear and invertible, but T⁻¹ is unbounded. This does not contradict the bounded inverse theorem since X is not complete, and thus is not a Banach space. To see that it's not complete, consider the sequence of sequences x⁽ⁿ⁾ ∈ X given by

${\displaystyle x^{(n)}=\left(1,{\frac {1}{2}},\dots ,{\frac {1}{n}},0,0,\dots \right)}$

converges as n → ∞ to the sequence x^(∞) given by

${\displaystyle x^{(\infty )}=\left(1,{\frac {1}{2}},\dots ,{\frac {1}{n}},\dots \right),}$

which has all its terms non-zero, and so does not lie in X.

The completion of X is the space ${\displaystyle c_{0}}$ of all sequences that converge to zero, which is a (closed) subspace of the ℓ_p space ℓ_∞(N), which is the space of all bounded sequences. However, in this case, the map T is not onto, and thus not a bijection. To see this, one need simply note that the sequence

${\displaystyle x=\left(1,{\frac {1}{2}},{\frac {1}{3}},\dots \right),}$

is an element of ${\displaystyle c_{0}}$, but is not in the range of ${\displaystyle T:c_{0}\to c_{0}}$. Same reasoning applies to show ${\displaystyle T}$ is also not onto in ${\displaystyle l^{\infty }}$, for example ${\displaystyle x=\left(1,1,1,\dots \right)}$ is not in the range of ${\displaystyle T}$.

The open mapping theorem has several important consequences:

• If ${\displaystyle T:X\to Y}$ is a bijective continuous linear operator between the Banach spaces ${\displaystyle X}$ and ${\displaystyle Y,}$ then the inverse operator ${\displaystyle T^{-1}:Y\to X}$ is continuous as well (this is called the bounded inverse theorem).^[10]
• If ${\displaystyle T:X\to Y}$ 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 ${\displaystyle Tx_{n}\to y}$ it follows that ${\displaystyle y=0,}$ then ${\displaystyle T}$ is continuous (the closed graph theorem).^[11]
• Given a bounded operator ${\displaystyle T:E\to F}$ between normed spaces, if the image of ${\displaystyle T}$ is non-meager and if ${\displaystyle E}$ is complete, then ${\displaystyle T}$ is open and surjective and ${\displaystyle F}$ is complete (to see this, use the two lemmas in the proof of the theorem).^[12]
• An exact sequence of Banach spaces (or more generally Fréchet spaces) is topologically exact.
• The closed range theorem, which says an operator (under some assumption) has closed image if and only if its transpose has closed image (see closed range theorem#Sketch of proof).

The open mapping theorem does not imply that a continuous surjective linear operator admits a continuous linear section. What we have is:^[9]

• A surjective continuous linear operator between Banach spaces admits a continuous linear section if and only if the kernel is topologically complemented.

In particular, the above applies to an operator between Hilbert spaces or an operator with finite-dimensional kernel (by the Hahn–Banach theorem). If one drops the requirement that a section be linear, a surjective continuous linear operator between Banach spaces admits a continuous section; this is the Bartle–Graves theorem.^[13][14]

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:

(The proof is essentially the same as the Banach or Fréchet cases; we modify the proof slightly to avoid the use of convexity,)

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.^[16]

An important special case of this theorem can also be stated as

On the other hand, a more general formulation, which implies the first, can be given:

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 ${\displaystyle Y.}$^[18] 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 ${\displaystyle Y,}$^[18] but for surjective maps these definitions are equivalent. A bijective linear map is nearly open if and only if its inverse is continuous.^[18] Every surjective linear map from locally convex TVS onto a barrelled TVS is nearly open.^[19] The same is true of every surjective linear map from a TVS onto a Baire TVS.^[19]

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

• 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 – Condition for a linear operator to be openPages displaying short descriptions of redirect targets
• 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

• 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.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• Vogt, Dietmar (2000). "Lectures on Fréchet spaces" (PDF). Bergische Universität Wuppertal.
• 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.

• "When is a complex of Banach spaces exact as condensed abelian groups?". MathOverflow. February 6, 2021.