en.wikipedia.org

Open mapping theorem (functional analysis) - Wikipedia

️Tue Mar 07 9939

From Wikipedia, the free encyclopedia

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 $T$ from one Banach space to another has bounded inverse $T^{-1}$ .

Statement and proof

[edit]

Open mapping theorem—^[2]^[3] Let $T:E\to F$ be a surjective continuous linear map between Banach spaces (or more generally Fréchet spaces). Then $T$ is an open mapping (that is, if $U\subset E$ is an open subset, then $T(U)$ is open).

The proof here uses the Baire category theorem, and completeness of both $E$ and $F$ 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 $f:E\to F$ between topological vector spaces is said to be nearly open if, for each neighborhood $U$ of zero, the closure ${\overline {f(U)}}$ contains a neighborhood of zero. The next lemma may be thought of as a weak version of the open mapping theorem.

Lemma—^[4]^[5] A linear map $f:E\to F$ between normed spaces is nearly open if the image of $f$ is non-meager in $F$ . (The continuity is not needed.)

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

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

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

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

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

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

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

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

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

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

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

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

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

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:

Corollary (Bounded inverse theorem)—^[8] A continuous bijective linear operator between Banach spaces (or Fréchet spaces) has continuous inverse. That is, the inverse operator is continuous.

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 continuous linear operator $T:E\to F$ factors as

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

Here, $T_{0}$ is continuous and 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, $T$ 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.

Transpose formulation

[edit]

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

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

Alternatively, 1. implies that $T'$ is injective and has closed image and then by the closed range theorem, that implies $T$ has dense image and closed image, respectively; i.e., $T$ is surjective. Hence, the above result is a variant of a special case of the closed range theorem.

Quantative formulation

[edit]

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

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

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

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

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

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

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

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

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

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

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

The completion of X is the space $c_{0}$ 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

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

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

The open mapping theorem has several important consequences:

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

Open mapping theorem for continuous maps^[12]^[15]—Let $A:X\to Y$ be a continuous linear operator from a complete pseudometrizable TVS $X$ onto a Hausdorff TVS $Y.$ If $\operatorname {Im} A$ is nonmeager in $Y$ then $A:X\to Y$ is a (surjective) open map and $Y$ is a complete pseudometrizable TVS. Moreover, if $X$ is assumed to be hausdorff (i.e. a F-space), then $Y$ is also an F-space.

(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 $N$ is the kernel of $A,$ then there is a canonical factorization of $A$ in the form $X\to X/N{\overset {\alpha }{\to }}Y$ where $X/N$ is the quotient space (also an F-space) of $X$ by the closed subspace $N.$ The quotient mapping $X\to X/N$ is open, and the mapping $\alpha$ is an isomorphism of topological vector spaces.^[16]

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

Theorem^[17]—Let $X$ and $Y$ be two F-spaces. Then every continuous linear map of $X$ onto $Y$ is a TVS homomorphism, where a linear map $u:X\to Y$ is a topological vector space (TVS) homomorphism if the induced map ${\hat {u}}:X/\ker(u)\to Y$ is a TVS-isomorphism onto its image.

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

Nearly/Almost open linear maps

A linear map $A:X\to Y$ between two topological vector spaces (TVSs) is called a nearly open map (or sometimes, an almost open map) if for every neighborhood $U$ of the origin in the domain, the closure of its image $\operatorname {cl} A(U)$ is a neighborhood of the origin in $Y.$ ^[18] Many authors use a different definition of "nearly/almost open map" that requires that the closure of $A(U)$ be a neighborhood of the origin in $A(X)$ rather than in $Y,$ ^[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.

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

^ Trèves 2006, p. 166.
^ Rudin 1973, Theorem 2.11.
^ Vogt 2000, Theorem 1.6.
^ Vogt 2000, Lemma 1.4.
^ The first part of the proof of Rudin 1991, Theorem 2.11.
^ ^a ^b Rudin 1991, Theorem 4.13.
^ Vogt 2000, Lemma 1.5.
^ Vogt 2000, Corollary 1.7.
^ ^a ^b Tao, Terence (February 1, 2009). "245B, Notes 9: The Baire category theorem and its Banach space consequences". What's New.
^ Rudin 1973, Corollary 2.12.
^ Rudin 1973, Theorem 2.15.
^ ^a ^b Rudin 1991, Theorem 2.11.
^ Sarnowski, Jarek (October 31, 2020). "Can the inverse operator in Bartle-Graves theorem be linear?". MathOverflow.
^ Borwein, J. M.; Dontchev, A. L. (2003). "On the Bartle–Graves theorem". Proceedings of the American Mathematical Society. 131 (8): 2553–2560. doi:10.1090/S0002-9939-03-07229-0. hdl:1959.13/940334. MR 1974655.
^ ^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.

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.